Motivated by observations of heterogeneous domain structure on the surface of cells, we consider a minimal model to describe the dynamics of phase separation on the surface of a spherical particle. Finite-size effects on the curved particle surface lead to the formation of long-lived, metastable states for which the density is distributed in patches over the particle surface. We study the time evolution and stability of these states as a function of both the particle size and the thermodynamic parameters. Finally, by connecting our findings with studies of patchy particles, we consider the implications for self-assembly in many-particle systems.
Review paper. Chapter 6 in Janus Particles Synthesis, Self-Assembly, and Applications pages 108-137, Edited by Steve Granick and Shan Jiang (RCS Publishing 2012) http://www.rsc.org/Shop/books/2012/9781849734233.asp
We explore the use of templated self-assembly to facilitate the formation of complex target structures made from patchy particles. First, we consider the templating of high-symmetry shell structures around a spherical core particle. We find that nucleation around the core particle can inhibit aggregate formation, a process which often hinders self-assembly. In particular, this new assembly pathway allows dodecahedral shells to form readily, whereas these structures never form in the absence of the template. Secondly, we consider the self-assembly of multi-shell structures, where the central ic…
Self-assembly is the mechanism that controls the formation of well defined structures from disordered pre-existing parts. Despite the importance of self-assembly as a manufacturing method and the increasingly large number of experimental realizations of complex self-assembled nano aggregates, theoretical predictions are lagging behind. Here we show that for a non-trivial self-assembly phenomenon, originating branched loop-less clusters, it is possible to derive a fully predictive parameter-free theory of equilibrium self-assembly by combining the Wertheim theory for associating liquids with th…
The rapid progress in precisely designing the surface decoration of patchy colloidal particles offers a new, yet unexperienced freedom to create building entities for larger, more complex structures in soft matter systems. However, it is extremely difficult to predict the large variety of ordered equilibrium structures that these particles are able to undergo under the variation of external parameters, such as temperature or pressure. Here we show that, by a novel combination of two theoretical tools, it is indeed possible to predict the self-assembly scenario of patchy colloidal particles: on…
Patchy colloids are promising candidates for building blocks in directed self-assembly. To be successful the surface patterns need to both be simple enough to be synthesized, while feature-rich enough to cause the colloids to self-assemble into desired structures. Achieving this is a challenge for traditional synthesis methods. Recently it has been suggested that the surface pattern themselves can be made to self-assemble. In this paper we show that a wide range of functional structures can be made to self-assemble using this approach. More generally we present a design path for hierarchical t…
We propose a new strategy for robust high-quality self-assembly of non-trivial periodic structures out of patchy particles, and investigate it with Brownian Dynamics (BD) simulations. Its first element is the use of specific patch-patch and shell-shell interactions between the particles, that can be implemented through differential functionalization of patched and shell regions with specific DNA strands. The other key element of our approach is the use of layer-by-layer protocol that allows to avoid a formations of undesired random aggregates. As an example, we design and self-assemble in sili…
A new theory is developed to describe the equilibrium adsorption and self-assembly of patchy colloids in microchannels. The adsorption theory is developed in classical density functional theory, with the adsorbed phase and fluid phase modeled using thermodynamic perturbation theory. In this work we propose that the introduction of patches on the colloids greatly enhances the temperature dependent and reversible adsorption of colloids in microchannels. It is shown how bulk fluid density, patch size, temperature and channel diameter can be manipulated to achieve the adsorption and self-assembly …
Effective colloid-colloid interactions can be tailored through the addition of a complex cosolute. Here we investigate the case of a cosolute made by self-assembling patchy particles. Depending on the valence, these particles can form either polymer chains or branched structures. We numerically calculate the effective potential $V_{eff}$ between two colloids immersed in a suspension of reversible patchy particles, exploring a wide region of the cosolute phase diagram and the role of valence. In addition to well-known excluded volume and depletion effects, we find that, under appropriate condit…
In this paper, we develop a thermodynamic perturbation theory to describe the self-assembly of patchy colloids which exhibit both patch-patch attractions as well as patch / non-patch attractions. That is, patches attract other patches as well as the no patch region. In general these attractions operate on different energy scales, which allows for controlled self-assembly as well as anomalous phase behavior. As an application we apply the model to the study of liquid water.
We report on the self-assembly of inverse patchy colloids (IPC) using Monte Carlo simulations in two-dimensions. The IPC model considered in this work corresponds to either bipolar colloids or colloids decorated with complementary DNA on their surfaces, where only patch and non-patch parts attract. The patch coverage is found to be a dominant factor in deciding equilibrium self-assembled structures. In particular, both regular square and triangular crystals are found to be stable at 0.5 patch coverage. Upon decreasing the patch coverage to 0.33, the regular square crystal is destablized; inste…
Concentrated solutions of short blunt-ended DNA duplexes, down to 6 base pairs, are known to order into the nematic liquid crystal phase. This self-assembly is due to the stacking interactions between the duplex terminals that promotes their aggregation into poly-disperse chains with a significant persistence length. Experiments show that liquid crystals phases form above a critical volume fraction depending on the duplex length. We introduce and investigate via numerical simulations, a coarse-grained model of DNA double-helical duplexes. Each duplex is represented as an hard quasi-cylinder wh…
A coarse-grained computational model is used to investigate the effect of a fluid membrane on patchy-particle assembly into biologically-relevant structures motivated by viral cores and clathrin. For cores, we demonstrate a non-monotonic dependence of the promotion of assembly on membrane stiffness. If the membrane is significantly deformable, cores are enveloped in buds, although this effect is suppressed for very flexible membranes. In the less deformable regime, we observe no marked enhancement for cores, even for strong adhesion to the surface. For clarthrin-like particles, we again observ…
Open lattices are characterized by low volume-fraction arrangements of building blocks, low coordination number, and open spaces between building blocks. The self-assembly of these lattices faces the challenge of mechanical instability due to their open structures. We theoretically investigate the stabilizing effects of entropy in the self-assembly of open lattices from patchy particles. A preliminary account of these findings and their comparison to experiment was presented recently [X. Mao, Q. Chen, S. Granick, \textit{Nat. Mater.}, \textbf{12}, 217 (2013)]. We found that rotational entropy …
Open structures can display a number of unusual properties, including a negative Poisson's ratio, negative thermal expansion, and holographic elasticity, and have many interesting applications in engineering. However, it is a grand challenge to self-assemble open structures at the colloidal scale, where short-range interactions and low coordination number can leave them mechanically unstable. In this paper we discuss the self-assembly of open structures using triblock Janus particles, which have two large attractive patches that can form multiple bonds, separated by a band with purely hard-sph…
In this work we develop the first theory to model self assembling mixtures of single patch colloids and colloids with spherically symmetric attractions. In the development of the theory we restrict the interactions such that there are short ranged attractions between patchy and spherically symmetric colloids, but patchy colloids do not attract patchy colloids and spherically symmetric colloids do not attract spherically symmetric colloids. This results in the temperature, density and composition dependent reversible self assembly of the mixture into colloidal star molecules. This type of mixtu…
We report on self-assemblies formed from spherical patchy particles interacting by a long-range attraction through a patch region in a two-dimensional system. We performed Monte Carlo simulations to find stable structures in a system with constant number of particles under constant temperature and constant pressure (NPT system), in which particles interact via the Kern--Frenkel potential. We also performed Brownian dynamics simulations employing an interaction potential similar to the Kern--Frenkel potential to study the formation of those structures. For long-range attractive potentials, we d…
Nanoparticles with "sticky patches" have long been proposed as building blocks for the self-assembly of complex structures. The synthetic realizability of such patchy particles, however, greatly lags behind predictions of patterns they could form. Using computer simulations, we show that structures of the same genre can be obtained from a solution of simple isotropic spheres, provided control only over their sizes and a small number of binding affinities. In a first step, finite clusters of well-defined structure and composition emerge from natural dynamics with high yield. In effect a kind of…
Assembly of protein complexes like virus shells, the centriole, the nuclear pore complex or the actin cytoskeleton is strongly determined by their spatial structure. Moreover it is becoming increasingly clear that the reversible nature of protein assembly is also an essential element for their biological function. Here we introduce a computational approach for the Brownian dynamics of patchy particles with anisotropic assemblies and fully reversible reactions. Different particles stochastically associate and dissociate with microscopic reaction rates depending on their relative spatial positio…
Confinement or geometric frustration is known to alter the structure of soft matter, including copolymeric melts, and can consequently be used to tune structure and properties. Here we investigate the self-assembly of ABC and ABB 3-miktoarm star copolymers confined to a shell using coarse-grained Dissipative Particle Dynamics simulations. In bulk and flat geometries the ABC stars form hexagonal tilings, but this is topologically prohibited in a spherical geometry which normally is alleviated by forming pentagonal tiles. However, the molecular architecture of the ABC stars implies an additional…
We systematically study the design of simple patchy sphere models that reversibly self-assemble into monodisperse icosahedral clusters. We find that the optimal patch width is a compromise between structural specificity (the patches must be narrow enough to energetically select the desired clusters) and kinetic accessibility (they must be sufficiently wide to avoid kinetic traps). Similarly, for good yields the temperature must be low enough for the clusters to be thermodynamically stable, but the clusters must also have enough thermal energy to allow incorrectly formed bonds to be broken. Ord…
Patchy particles is the name given to a large class of systems of mesoscopic particles characterized by a repulsive core and a discrete number of short-range and highly directional interaction sites. Numerical simulations have contributed significantly to our understanding of the behaviour of patchy particles, but, although simple in principle, advanced simulation techniques are often required to sample the low temperatures and long time scales associated with their self-assembly behaviour. In this work we review the most popular simulation techniques that have been used to study patchy partic…
We numerically study a simple fluid composed of particles having a hard-core repulsion, complemented by two short-ranged attractive (sticky) spots at the particle poles, which provides a simple model for equilibrium polymerization of linear chains. The simplicity of the model allows for a close comparison, with no fitting parameters, between simulations and theoretical predictions based on the Wertheim perturbation theory, a unique framework for the analytic prediction of the properties of self-assembling particle systems in terms of molecular parameter and liquid state correlation functions. …
We employ Monte Carlo simulations to investigate the self-assembly of patchy colloidal dumbbells interacting via a modified Kern-Frenkel potential by probing the system concentration and dumbbell shape. We consider dumbbells consisting of one attractive sphere with diameter $σ_1$ and one repulsive sphere with diameter $σ_2$ and center-to-center distance $d$ between the spheres. For three different size ratios, we study the self-assembled structures for different separations $l = 2d/(σ_1+σ_2)$ between the two spheres. In particular, we focus on structures that can be assembled from the homogene…
Inverse patchy colloids are nano- to micro-scale particles with a surface divided into differently charged regions. This class of colloids combines directional, selective bonding with a relatively simple particle design: owing to the competitive interplay between the orientation-dependent attraction and repulsion -- induced by the interactions between like/oppositely charged areas -- experimentally accessible surface patterns are complex enough to favor the stabilization of specific structures of interest. Most important, the behavior of heterogeneously charged units can be ideally controlled …
Triblock terpolymers exhibit a rich self-organization behavior including the formation of fascinating cylindrical core-shell structures with a phase separated corona. After crystallization-induced self-assembly of polystryrene-(block)-polyethylene-(block)-poly(methyl methacrylate) triblock terpolymers (abbreviated as SEMs = Styrene-Ethylene-Methacrylates) from solution, worm-like core-shell micelles with a patchy corona of polystryrene and poly(methyl methacrylate) were observed by transmission electron microscopy. However, the solution structure is still a matter of debate. Here, we present a…
The addition of co-solutes to colloidal suspensions is often employed to induce tunable depletion interactions. In this work we investigate effective colloidal interactions arising from binary co-solute mixtures of hard spheres and patchy particles. By changing the relative concentration of the two species, we show that the resulting effective potential $V_\text{eff}$ continuously changes from the one obtained for a single-component hard sphere co-solute to that mediated by the single-component patchy particle co-solute. Interestingly, we find that, independently of the relative concentration …
We report theoretical and numerical evaluations of the phase diagram for a model of patchy particles. Specifically we study hard-spheres whose surface is decorated by a small number f of identical sites ("sticky spots'') interacting via a short-range square-well attraction. We theoretically evaluate, solving the Wertheim theory, the location of the critical point and the gas-liquid coexistence line for several values of f and compare them to results of Gibbs and Grand Canonical Monte Carlo simulations. We study both ordered and disordered arrangements of the sites on the hard-sphere surface an…
We introduce a scheme to design patchy particles so that a given target crystal is the global free-energy minimum at sufficiently low temperature. A key feature is a torsional component to the potential that only allows binding when particles have the correct relative orientations. In all examples studied, the target crystal structures readily assembled on annealing from a low-density fluid phase, albeit with the simpler target structures assembling more rapidly. The most complex example was a clathrate with 46 particles in its primitive unit cell. We also explored whether the structural infor…
Enskog theory is extended for the description of the self-diffusion coefficient of patchy colloidal fluid in disordered porous media. The theory includes the contact values of fluid-fluid and fluid-matrix pair distribution functions that are modified to include the dependence from the so-called probe particle porosity, $ϕ$, in order to correctly describe the effects of trapping the fluid particles by a matrix. The proposed expressions for the modified contact values of fluid-fluid and fluid-matrix pair distribution functions include three terms. Namely, a hard sphere contribution obtained by u…
Motivated by recent experimental findings in chemical synthesis of colloidal particles, we draw an analogy between self-assembly processes occurring in biological systems (e.g. protein folding) and a new exciting possibility in the field of material science. We consider a self-assembly process whose elementary building blocks are decorated patchy colloids of various types, that spontaneously drive the system toward a unique and predetermined targeted macroscopic structure. To this aim, we discuss a simple theoretical model -- the Kern-Frenkel model -- describing a fluid of colloidal spherical …
The assembly of filamentous bundles with controlled diameters is common in biological systems and desirable for the development of nanomaterials. We discuss dynamical simulations and free energy calculations on patchy spheres with chiral pair interactions that spontaneously assemble into filamentous bundles. The chirality frustrates long-range crystal order by introducing twist between interacting subunits. For some ranges of system parameters this constraint leads to bundles with a finite diameter as the equilibrium state, and in other cases frustration is relieved by the formation of defects…
We consider a fluid of hard spheres bearing one or two uniform circular adhesive patches, distributed so as not to overlap. Two spheres interact via a ``sticky'' Baxter potential if the line joining the centers of the two spheres intersects a patch on each sphere, and via a hard sphere potential otherwise. We analyze the location of the fluid-fluid transition and of the percolation line as a function of the size of the patch (the fractional coverage of the sphere's surface) and of the number of patches within a virial expansion up to third order and within the first two terms (C0 and C1) of a …
We present a generic model to describe the fractal self-assembly of proteins in terms of electrostatic and hydrophobic interactions. The predictions of the model were correlated with the simulated fractals obtained using patchy diffusion-limited aggregation, and the experimentally observed Amyloid-\b{eta} (A\b{eta}) fractal self-assembly using confocal microscopy. The molecular docking was used to determine the properties of the patches on A\b{eta} oligomers. Similar patch properties were used to design the particles for simulation. In agreement with the model predictions, the free energy of f…
Self-assembly of Janus (or `patchy') particles is dependent on the precise interaction between neighbouring particles. Here, the orientations of two amphiphilic Janus spheres within a dimer in an explicit fluid are studied with high geometric resolution. Molecular dynamics simulations and first-principles energy calculations are used with hard- and soft-sphere Lennard-Jones potentials, and temperature and hydrophobicity are varied. The most probable centre-centre-pole angles are in the range 40° to 55°, with pole-to-pole alignment not observed due to orientational entropy. Angles near 90° are …
Brownian Dynamics is used to study self-assembly in a hybrid system of istotropic particles (IPs), combined with anisotropic building blocks that represent special "designer particles". Those are modeled as spherical patchy particles (PPs) with binding only allowed between their patches and IPs. In this study, two types of PPs are considered: Octahedral PPs (Oh-PPs) and Square PPs (Sq-PPs), with octahedral and square arrangements of patches, respectively. The self-assembly is additionally facilitated by the simulated annealing procedure. The resultant structures are characterized by a combinat…
We simulate the assembly of DNA copolymers from two types of short duplexes (short double strands with a single-stranded overhang at each end), as described by the oxDNA model. We find that the statistics of chain lengths can be well reproduced by a simple theory that treats the association of particles into ideal (i.e., non-interacting) clusters as a reversible chemical reaction. The reaction constants can be predicted either from Santalucia's theory or from Wertheim's thermodynamic perturbation theory of association for spherical patchy particles. Our results suggest that theories incorporat…
A matrix formulation is derived for the calculation of the swimming speed and the power required for swimming of an assembly of rigid spheres immersed in a viscous fluid of infinite extent. The spheres may have arbitrary radii and may interact with elastic forces. The analysis is based on the Stokes mobility matrix of the set of spheres, defined in low Reynolds number hydrodynamics. For small amplitude swimming optimization of the swimming speed at given power leads to an eigenvalue problem. The method allows straightforward calculation of the swimming performance of structures modeled as asse…
Patchy particles are a popular paradigm for the design and synthesis of nanoparticles and colloids for self-assembly. In "traditional" patchy particles, anisotropic interactions arising from patterned coatings, functionalized molecules, DNA, and other enthalpic means create the possibility for directional binding of particles into higher-ordered structures. Although the anisotropic geometry of non-spherical particles contributes to the interaction patchiness through van der Waals, electrostatic, and other interactions, how particle shape contributes entropically to self-assembly is only now be…
We use Monte Carlo simulations and free-energy techniques to show that binary solutions of penta- and hexavalent two-dimensional patchy particles can form thermodynamically stable quasicrystals even at very narrow patch widths, provided their patch interactions are chosen in an appropriate way. Such patchy particles can be thought of as a coarse-grained representation of DNA multi-arm `star' motifs, which can be chosen to bond with one another very specifically by tuning the DNA sequences of the protruding arms. We explore several possible design strategies and conclude that DNA star tiles tha…
We realise an antibacterial nanomaterial based on the self-limited assembly of patchy plasmonic colloids, obtained by adsorption of lysozyme to gold nanoparticles. The possibility of selecting the size of the assemblies within several hundred nanometres allows for tuning their optical response in a wide range of frequencies from visible to near infrared. We also demonstrate an aggregation-dependent modulation of the catalytic activity, which results in an enhancement of the antibacterial performances for assemblies of the proper size. The gained overall control on structure, optical properties…
Quasicrystals are fascinating structures, characterized by strong positional order but lacking the periodicity of a crystal. In colloidal systems, quasicrystals are typically predicted for particles with complex or highly specific interactions, which makes experimental realization difficult. Here, we propose an ideal colloidal model system for quasicrystal formation: binary mixtures of hard spheres sedimented onto a flat substrate. Using computer simulations, we explore both the close-packing and spontaneous self-assembly of these systems over a wide range of size ratios and compositions. Surp…
Hard spheres are an important benchmark of our understanding of natural and synthetic systems. In this work, colloidal experiments and Monte Carlo simulations examine the equilibrium and out-of-equilibrium assembly of hard spheres of diameter $σ$ within cylinders of diameter $σ\leq D\leq 2.82σ$. Although in such a system phase transitions formally do not exist, marked structural crossovers are observed. In simulations, we find that the resulting pressure-diameter structural diagram echoes the densest packing sequence obtained at infinite pressure in this range of $D$. We also observe that the …
We apply a simple model system of patchy particles to study monodisperse self-assembly, using the Platonic solids as target structures. We find marked differences between the assembly behaviours of the different systems. Tetrahedra, octahedra and icosahedra assemble easily, while cubes are more challenging and dodecahedra do not assemble. We relate these differences to the kinetics and thermodynamics of assembly, with the formation of large disordered aggregates a particular important competitor to correct assembly. In particular, the free energy landscapes of those targets that are easy to as…
