We investigate the assembly of the dipole-like patchy particles confined to a spherical surface by Brownian dynamics simulations. The surface property of the spherical particle is described by the spherical harmonic $Y_{10}$, and the orientation of the particle is defined as the uniaxial axis. On a flat space, we observe a defect-free square lattice with nematic order. On a spherical surface, defects appear due to the topological constraint. As for the director field, four defects of winding number $+1/2$ are observed, satisfying the Euler characteristic. We have found many configurations of the four defects lying near a great circle. Regarding the positional order for the square lattice, eight grain boundary scars proliferate linearly with the sphere size. The positions and orientations of the eight grain boundary scars are strongly related to the four $+1/2$ defect cores.
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…
We study the irreversible aggregation of films of patchy spherical colloids with directional and selective interactions. We report a crossover of the interfacial roughening from the Kardar-Parisi-Zhang (KPZ) to the KPZ with quenched disorder (KPZQ) universality class when the difference between the strong and weak bonds is sufficiently large. We calculate the critical exponents and identify the crossover between the two regimes.
Charge regulation is an important aspect of electrostatics in biological and colloidal systems, where the charges are generally not fixed, but depend on the environmental variables. Here, we analyze the charge regulation mechanism in patchy, inhomogeneously charged spherical particles, such as globular proteins, colloids, or viruses. Together with the multipole expansion of inhomogeneously charged spherical surfaces, the charge regulation mechanism on the level of linear approximation is shown to lead to a mixing between different multipole moments depending on their capacitance -- the respons…
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…
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 imp…
We investigate numerically the irreversible aggregation of patchy spherical colloids on a flat substrate. We consider $n$-patch particles and characterize the dependence of the irreversible aggregation kinetics on $n$. For all values of $n$, considered in this study, the growing interface of the aggregate is in the Kardar-Parisi-Zhang universality class, although the bulk structure exhibits a rich dependence on $n$. In particular, the bulk density varies with $n$ and the network is more ordered for particles with fewer patches. Preferred orientations of the bonds are also observed for networks…
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 report theoretical and numerical evaluations of the phase diagram for patchy colloidal particles of new generation. We show that the reduction of the number of bonded nearest neighbours offers the possibility of generating liquid states (i.e. states with temperature $T$ lower than the liquid-gas critical temperature) with a vanishing occupied packing fraction ($ϕ$), a case which can not be realized with spherically interacting particles. Theoretical results suggest that such reduction is accompanied by an increase of the region of stability of the liquid phase in the ($T$-$ϕ$) plane, possib…
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…
Advances in nanotechnology lead to an increasing interest in how nanoparticles interact with biomembranes. Nanoparticles are wrapped spontaneously by biomembranes if the adhesive interactions between the particles and membranes compensate for the cost of membrane bending. In the last years, the cooperative wrapping of spherical nanoparticles in membrane tubules has been observed in experiments and simulations. For spherical nanoparticles, the stability of the particle-filled membrane tubules strongly depends on the range of the adhesive particle-membrane interactions. In this article, we show …
We study the structural and thermodynamic properties of patchy particle liquids, with a special focus on the role of "color", i.e. specific interactions between individual patches. A possible experimental realization of such "chromatic" interactions is by decorating the particle patches with single-stranded DNA linkers. The complementarity of the linkers can promote selective bond formation between predetermined pairs of patches. By using MD simulations, we compare the local connectivity, the bond orientation order, and other structural properties of the aggregates formed by the "colored" and …
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…
We present a theory to predict the structure and thermodynamics of mixtures of colloids of different diameters, building on our earlier work [J. Chem. Phys. 145, 074904 (2016)] that considered mixtures with all particles constrained to have the same size. The patchy, solvent particles have short-range directional interactions, while the solute particles have short-range isotropic interactions. The hard-sphere mixture without any association site forms the reference fluid. An important ingredient within the multi-body association theory is the description of clustering of the reference solvent …
Colloidal systems (and protein solutions) are often characterized by attractive interactions whose range is much smaller than the particle size. When this is the case and the interaction is spherical, systems obey a generalized law of correspondent states (GLCS), first proposed by Noro and Frenkel [ J.Chem.Phys. 113, 2941 (2000) ]. The thermodynamic properties become insensitive to the details of the potential, depending only on the value of the second virial coefficient B_2 and the density $ρ$. The GLCS does not generically hold for the case of non-spherical potentials. In this Letter we sugg…
A mixture of solvent particles with short-range, directional interactions and solute particles with short-range, isotropic interactions that can bond multiple times is of fundamental interest in understanding liquids and colloidal mixtures. Because of multi-body correlations predicting the structure and thermodynamics of such systems remains a challenge. Earlier Marshall and Chapman developed a theory wherein association effects due to interactions multiply the partition function for clustering of particles in a reference hard-sphere system. The multi-body effects are incorporated in the clust…
We propose a minimal model for spherical proteins with aeolotopic pair interactions to describe the equilibrium phase behavior of lysozyme. The repulsive screened Coulomb interactions between the particles are taken into account assuming that the net charges are smeared out homogeneously over the spherical protein surfaces. We incorporate attractive surface patches, with the interactions between patches on different spheres modeled by an attractive Yukawa potential. The parameters entering the attractive Yukawa potential part are determined using information on the experimentally accessed gas-…
We study the critical Casimir interaction between two spherical colloids immersed in a binary liquid mixture close to its critical demixing point. The surface of each colloid prefers one species of the mixture with the exception of a circular patch of arbitrary size, where the other species is preferred. For such objects we calculate, within the Derjaguin approximation, the scaling function describing the critical Casimir potential, and we use it to derive the scaling functions for all components of the forces and torques acting on both colloids. The results are compared with available experim…
Recent simulations have studied the formation of patterns in a binary mixture of immiscible surfactants absorbed onto the surface of a spherical nanoparticle. The resulting patterns (Janus, spots and stripes) were in good agreement with experimental results. We perform dissipative particle dynamics (DPD) simulations to study the patterns obtained by adding a third surfactant to the monolayer as a guide towards increasing the richness and diversity of patchy particles synthesized this way. We predict a variety of new patterns that can be produced through different combinations of simple design …
We report extensive monte-carlo and event-driven molecular dynamics simulations of a liquid composed by particles interacting via hard-sphere interactions complemented by four tetrahedrally coordinated short-range attractive ("sticky") spots, a model introduced several years ago by Kolafa and Nezbeda [J. Kolafa and I. Nezbeda, Mol. Phys. 161, 87 (1987)]. To access the dynamic properties of the model we introduce and implement a new event-driven molecular dynamics algorithm suited to study the evolution of hard bodies interacting, beside the repulsive hard-core, with a short-ranged inter-patch …
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…
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…
Inhomogeneous charge distributions have important repercussions on electrostatic interactions in systems of charged particles but are often difficult to examine theoretically. We investigate how electrostatic interactions are influenced by patchy charge distributions exhibiting certain point group symmetries. We derive a general form of the electrostatic interaction energy of two permeable, arbitrarily charged spherical shells in the Debye-Hückel approximation and apply it to the case of particles with icosahedral, octahedral, and tetrahedral inhomogeneous charge distributions. We analyze in d…
Anisotropic dipole-dipole interaction often plays a key role in biological, soft, and complex matter. For it to induce non-trivial order in the system, there must be additional repulsive interactions or external potentials involved that partially or completely fix the positions of the dipoles. These positions can often be represented as an underlying lattice on which dipole interaction induces orientational ordering of the particles. On lattices in the Euclidean plane, dipoles have been found to assume different ground state configurations depending on the lattice type, with a global ordering …
Behaviour of the model antibody fluid constrained by rigid spherical obstacles: effects of the obstacle-antibody binding
University of Ljubljana, Faculty of Chemistry and Chemical Technology（卢布尔雅那大学化学与化学技术学院）；Yukhnovskii Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine（乌克兰国家科学院尤赫诺夫斯基凝聚态物理研究所）；Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University（利沃夫国立理工大学应用数学与基础科学研究所）
We study the director field around a spherical particle immersed in a uniformly aligned nematic liquid crystal and assume that the molecules prefer a homeotropic orientation at the surface of the particle. Three structures are possible: a dipole, a Saturn-ring, and a surface-ring configuration, which we investigate by numerically minimizing the Frank free energy supplemented by a magnetic-field and a surface term. In the dipole configuration, which is the absolutely stable structure for micron-size particles and sufficiently strong surface anchoring, a twist transition is found and analyzed. W…
We present extensive Molecular Dynamics simulation results for the structure, static and dynamical response of a droplet of 1000 soft spheres carrying extended dipoles and confined to spherical cavities of radii $R=2.5$, 3, and 4 nm embedded in a dielectric continuum of permittivity $ε' \geq 1$. The polarisation of the external medium by the charge distribution inside the cavity is accounted for by appropriate image charges. We focus on the influence of the external permittivity $ε'$ on the static and dynamic properties of the confined fluid. The density profile and local orientational order p…
We consider a model of colloidal spherical particles carrying a permanent dipole moment which is laterally shifted out of the particles' geometrical centres, i.e. the dipole vector is oriented perpendicular to the radius vector of the particles. Varying the shift $δ$ from the centre, we analyze ground state structures for two, three and four hard spheres, using a simulated annealing procedure. We also compare to earlier ground state results. We then consider a bulk system at finite temperatures and different densities. Using Molecular Dynamics simulations, we examine the equilibrium self-assem…
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 …
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…
We have studied the effective response of composites of spherical particles each having a dielectric profile which varies along the radius of the particles. We developed a first-principles approach to compute the dipole moment of the individual spherical particle and hence the effective dielectric response of a dilute suspension. The approach has been applied to two model dielectric profiles, for which exact solutions are available. Moreover, we used the exact results to validate the results from the differential effective dipole approximation, recently developed to treat graded spherical part…
In this work we present a study on the characterization of ordered and disordered hyperuniform point distributions on spherical surfaces. In spite of the extensive literature on disordered hyperuniform systems in Euclidean geometries, to date few works have dealt with the problem of hyperuniformity in curved spaces. As a matter of fact, some systems that display disordered hyperuniformity, like the space distribution of photoreceptors in avian retina, actually occur on curved surfaces. Here we will focus on the local particle number variance and its dependence on the size of the sampling windo…
We have employed the first-principles approach to compute the effective response of composites of graded spherical particles of arbitrary conductivity profiles. We solve the boundary-value problem for the polarizability of the graded particles and obtain the dipole moment as well as the multipole moments. We provide a rigorous proof of an {\em ad hoc} approximate method based on the differential effective multipole moment approximation (DEMMA) in which the differential effective dipole approximation (DEDA) is a special case. The method will be applied to an exactly solvable graded profile. We …
We have studied the effective response of composites of spherical particles with a dielectric profile which varies along the radius of the particles. We developed a differential effective dipole theory to compute the dipole moment of individual spherical particle and hence the effective dielectric response of a dilute suspension. The theory has been applied to two model dielectric profiles. Moreover, we compare the approximate results with the exact results of the power-law profile and the agreement is excellent.
The importance of electrostatic interactions in soft matter and biological systems can often be traced to non-uniform charge effects, which are commonly described using a multipole expansion of the corresponding charge distribution. The standard approach when extracting the charge distribution of a given system is to treat the constituent charges as points. This can, however, lead to an overestimation of multipole moments of high order, such as dipole, quadrupole, and higher moments. Focusing on distributions of charges located on a spherical surface -- characteristic of numerous biological ma…
We introduce a simple spherical model whose structural properties are similar to the ones generated by models with directional interactions, by employing a binary mixture of large and small hard spheres, with a square-well attraction acting only between particles of different size. The small particles provide the bonds between the large ones. With a proper choice of the interaction parameters, as well as of the relative concentration of the two species, it is possible to control the effective valence. Here we focus on a specific choice of the parameters which favors tetrahedral ordering and st…
Microbullet particles, cylinders with one blunt and one spherical end, offer a novel platform to study the effects of anisotropy and curvature on colloidal assembly in complex fluids. Here, we disperse microbullets in 4-cyano-4'-pentylbiphenyl (5CB) nematic liquid crystal (NLC) cells and form oriented elastic dipoles with a nematic point defect located near the curved end. This feature allows us to study particle interactions as a function of dipole alignment. By careful control of the surface anchoring at the particle surface and the confining boundaries, we study the interactions and assembl…
We analyze a minimal model for a rigid spherical microswimmer and explore the consequences of its extended surface on the interplay between its self-propulsion and flow properties. The model is the first order representation of microswimmers, such as bacteria and algae, with rigid bodies and flexible propelling appendages. The flow field of such a microswimmer at finite distances significantly differs from that of a point-force (Stokeslet) dipole. For a suspension of microswimmers, we derive the grand mobility matrix that connects the motion of an individual swimmer to the active and passive f…
The self-assembly of anisotropic patchy particles with triangular shape was studied by experiments and computer simulations. The colloidal particles were synthesized in a two-step seeded emulsion polymerization process, and consist of a central smooth lobe connected to two rough lobes at an angle of $\sim$90$^{\circ}$, resembling the shape of a "Mickey Mouse" head. Due to the difference in overlap volume, adding an appropriate depletant induces an attractive interaction between the smooth lobes of the colloids only, while the two rough lobes act as steric constraints. The essentially planar ge…
We run numerical simulations of strongly confined suspensions of model spherical swimmers called "squirmers". Because of the confinement, the Stokeslet dipoles generated by the particles are quickly screened and the far-field flow is dominated by the source dipole for all the different kinds of squirmers. However, we show that the collective behavior of the suspension still depends on the self-propelling mechanism of the swimmers as polar states can only be observed for neutral squirmers. We demonstrate that the near-field hydrodynamic interactions play a crucial role in the alignment of the o…
We present a theoretical study of electrorotation (ER) of two spherical particles under the action of a rotating electric field. When the two particles approach and finally touch, the mutual polarization interaction between the particles leads to a change in the dipole moment of the individual particle and hence the ER spectrum, as compared to that of the well-separated particles. The mutual polarization effects are captured by the method of multiple images. From the theoretical analysis, we find that the mutual polarization effects can change the characteristic frequency at which the maximum …
