Purely entropic systems such as suspensions of hard rods, platelets and spheres show rich phase behavior. Rods and platelets have successfully been used as models to predict the equilibrium properties of liquid crystals for several decades. Over the past years hard particle models have also been studied in the context of non-equilibrium statistical mechanics, in particular regarding the glass transition, jamming, sedimentation and crystallization. Recently suspensions of hard anisotropic particles also moved into the focus of materials scientists who work on conducting soft matter composites. An insulating polymer resin that is mixed with conductive filler particles becomes conductive when the filler percolates. In this context the mathematical topic of connectivity percolation finds an application in modern nano-technology. In this article, we briefly review recent work on the phase behavior, confinement effects, percolation transition and phase transition kinetics in hard particle models. In the first part, we discuss the effects that particle anisotropy and depletion have on the percolation transition. In the second part, we present results on the kinetics of the liquid-to-crystal transition in suspensions of spheres and of ellipsoids.
Using computer simulations, we study the dynamic arrest in a schematic model of colloid-polymer mixtures combining short-ranged attractions with long-ranged repulsions. The arrested gel is a dilute rigid network of colloidal particles bonded due to the strong attractions. Without repulsions, the gel forms at the spinodal through arrested phase separation. In the ergodic suspension at sufficiently high densities, colloidal clusters form temporary networks that percolate space. Recently [Nat. Commun. 7, 11817 (2016)], it has been proposed that the transition of these networks to directed percola…
When strained beyond the linear regime, soft colloidal glasses yield to steady-state plastic flow in a way that is similar to the deformation of conventional amorphous solids. Due to the much larger size of the colloidal particles with respect to the atoms comprising an amorphous solid, colloidal glasses allow to obtain microscopic insight into the nature of the yielding transition, as we illustrate here combining experiments, atomistic simulations, and mesoscopic modeling. Our results unanimously show growing clusters of non-affine deformation percolating at yielding. In agreement with percol…
Rigidity percolation (RP) occurs when mechanical stability emerges in disordered networks as constraints or components are added. Here we discuss RP with structural correlations, an effect ignored in classical theories albeit relevant to many liquid-to-amorphous-solid transitions, such as colloidal gelation, which are due to attractive interactions and aggregation. Using a lattice model, we show that structural correlations shift RP to lower volume fractions. Through molecular dynamics simulations, we show that increasing attraction in colloidal gelation increases structural correlation and th…
We explore structural and dynamical behavior of concentrated colloidal suspensions made up by C-shape particles using Brownian dynamics computer simulations and theory. In particular, we focus on the entanglement process between nearby particles for almost closed C-shapes with a small opening angle. Depending on the opening angle and the particle concentration, there is a percolation transition for the cluster of entangled particles which shows the classical scaling characteristics. In a broad density range below the percolation threshold, we find a stretched exponential function for the dynam…
A discussion on the viscoelastic properties of weakly attractive colloids and the relation to a newly developed percolation theory is presented. The need for taking into account the particle-particle bond lifetime in order to compare rheological and percolation estimates of the sol-gel transition is stressed. Recent molecular dynamics simulations and generalised percolation theory that use a bond lifetime criterion in the definition of aggregates show that the percolation transition can be reconciled with the rheological sol-gel transition. The percolation concept is used here as a "measuring"…
Brownian motion of colloidal particles in the quasi-two-dimensional (qTD) confinement displays distinct kinetic characters from that in bulk. Here we experimentally report a dynamic evolution of Brownian particles in the qTD system. The dynamic system displays a quasi-equilibrium state of colloidal particles performing Brownian motion. In the quasi-equilibrium process, the qTD confinement results in the coupling of particle motions, which slowly dampens the motion and interaction of particles until the final equilibrium state reaches. The theory is developed to explain coupling motions of Brow…
Percolation and critical phenomena show common features such as scaling and universality. Colloidal particles, immersed in a solvent close to criticality, experience long-range effective forces, named critical Casimir forces. %These originate from the confinement of the solvent critical fluctuations between the colloids. Building on the analogy between critical phenomena and percolation, we explore the possibility of observing long-range forces near a percolation threshold. To this aim we numerically evaluate the effective potential between two colloidal particles dispersed in a chemical sol a…
We investigate percolation in binary and ternary mixtures of patchy colloidal particles theoretically and using Monte Carlo simulations. Each particle has three identical patches, with distinct species having different types of patch. Theoretically we assume tree-like clusters and calculate the bonding probabilities using Wertheim's first-order perturbation theory for association. For ternary mixtures we find up to eight fundamentally different percolated states. The states differ in terms of the species and pairs of species that have percolated. The strongest gel is a trigel or tricontinuous …
We study the non-equilibrium coarsening dynamics of a binary liquid solvent around a colloidal particle in a presence of a time-dependent temperature gradient that emerges after temperature quench of a suitably coated colloid surface. The solvent is maintained at its critical concentration and the colloid is fixed in space. The coarsening patterns near the surface are shown to be strongly dependent on the colloid surface adsorption properties and on the temperature evolution. The temperature gradient alters the morphology of a binary solvent near the surface of a colloid as compared to the coa…
We study the percolation properties for a system of functionalized colloids on patterned substrates via Monte Carlo simulations. The colloidal particles are modeled as hard disks with three equally-distributed attractive patches on their perimeter. We describe the patterns on the substrate as circular potential wells of radius $R_p$ arranged in a regular square or hexagonal lattice. We find a nonmonotonic behavior of the percolation threshold (packing fraction) as a function of $R_p$. For attractive wells, the percolation threshold is higher than the one for clean (non-patterned) substrates if…
Designing and fabricating self-assembled open colloidal crystals have become one major direction in soft matter community because of many promising applications associated with open colloidal crystals. However, most of the self-assembled crystals found in experiments are not open but close-packed. Here by using computer simulation, we systematically investigate the self-assembly of oppositely charged colloidal hard spheres confined between two parallel hard walls, and we find that the confinement can stabilize multi-layer NaCl-like (simple cubic) open crystals. The maximal layers of stable NaC…
We present a new model of patchy particles in which the interaction sites can be activated or deactivated by varying the temperature of the system. We study the thermodynamics of the system by means of Wertheim's first order perturbation theory, and use Flory-Stockmayer theory of polymerization to analyse the percolation threshold. We find a very rich phase behaviour including lower critical points and reentrant percolation.
In a system of colloidal inclusions suspended in a thermalized bath of smaller particles, the bath engenders an attractive force between the inclusions, arising mainly from entropic origins, known as the depletion force. In the case of active bath particles, the nature of the bath-mediated force changes dramatically from an attractive to a repulsive one, as the strength of particle activity is increased. We study such bath-mediated effective interactions between colloidal inclusions in a bath of self-propelled Brownian particles, being confined in a narrow planar channel. Confinement is found …
We study the effective electrostatic interactions between a pair of charged colloidal particles without salt ions while the system is confined in two dimensions. In particular we use a simplified model to elucidate the effects of rotational fluctuations in counterion distribution. The results exhibit effective colloidal attractions under appropriate conditions. Meanwhile, long-range repulsions persist over most of our studied cases. The repulsive forces arise from the fact that in two dimensions the charged colloids cannot be perfectly screened by counterions, as the residual quadrupole moment…
We study the self-assembly of branching-chain networks and crystals in a binary colloidal system with tunable interactions. The particle positions are extracted from microscope images and order parameters are extracted by image processing and statistical analysis. With these, we construct phase diagrams with respect to particle density, ratio and interaction. In order to draw a more complete picture, we complement the experiments with computer simulations. We establish a region in the phase diagram, where bead ratios and interactions are symmetric, promoting percolated structures.
Recent theories predict that when a supercooled liquid approaches the glass transition, particle clusters with a special "amorphous order" nucleate within the liquid, which lead to static correlations dictating the dramatic slowdown of liquid relaxation. The prediction, however, has yet to be verified in 3D experiments. Here, we design a colloidal system, where particles are confined inside spherical cavities with an amorphous layer of particles pinned at the boundary. Using this novel system, we capture the amorphous-order particle clusters and demonstrate the development of a static correlat…
We study structural phase transition of polymer-grafted colloidal particles by Monte Carlo simulations on hard spherical particles. The interaction potential, which has a weak repulsive step outside the hard core, was validated with use of the self-consistent field calculations. With this potential, canonical Monte Carlo simulations have been carried out in two and three dimensions using the Metropolis algorithm. At low temperature and high density, we find that the particles start to self-assemble and finally align in strings. By analyzing the cluster size distribution and string length distr…
Under partial confinement, the motion of colloidal particles is restricted to a plane but their dynamics is influenced by hydrodynamic interactions mediated by the unconfined, three--dimensional flow of the embedding fluid. We demonstrate that this dimensionality mismatch induces a characteristic divergence in the two--dimensional collective diffusion coefficient of the colloidal monolayer. This result, independent from the specific interparticle forces in the monolayer, is solely due to the kinematical constraint on the colloidal particles, and it is different from the known divergence of tra…
We combine a heuristic theory of geometric percolation and the Smoluchowski theory of colloid dynamics to predict the impact of shear flow on the percolation threshold of hard spherical colloidal particles, and verify our findings by means of molecular dynamics simulations. It appears that the impact of shear flow is subtle and highly non-trivial, even in the absence of hydrodynamic interactions between the particles. The presence of shear flow can both increase and decrease the percolation threshold, depending on the criterion used for determining whether or not two particles are connected an…
We study a colloidal suspension confined between two quasi-parallel walls as a model system for glass transitions in confined geometries. The suspension is a mixture of two particle sizes to prevent wall-induced crystallization. We use confocal microscopy to directly observe the motion of colloidal particles. This motion is slower in confinement, thus producing glassy behavior in a sample which is a liquid in an unconfined geometry. For higher volume fraction samples (closer to the glass transition), the onset of confinement effects occurs at larger length scales.
Understanding the correlation between structure and rheology in colloidal suspensions is important as these suspensions are crucial in industrial applications. Moreover, colloids exhibit a wide range of structures under confinement that could considerably alter the viscosity. Here, we use a combination of experiments and simulations to elucidate how confinement induced structures alter the relative contributions of hydrodynamic and repulsive forces to produce up to a ten fold change in the viscosity. We use a custom built confocal rheoscope to image the particle configurations of a colloidal s…
Within the framework of a Modified Poisson-Boltzmann theory accounting for steric effects of microions, we prove analytically that the effective pair interactions between like-charge colloids immersed in a confined electrolyte are repulsive. Our approach encompasses and extends previously known results to the case of complete confinement, and further incorporates the finite size of the microions which is absent in the standard Poisson-Boltzmann theory.
Understanding the relationship between colloidal building block shape and self-assembled material structure is important for the development of novel materials by self-assembly. In this regard, colloidal superballs are unique building blocks because their shape can smoothly transition between spherical and cubic. Assembly of colloidal superballs under spherical confinement results in macroscopic clusters with ordered internal structure. By utilizing Small Angle X-Ray Scattering (SAXS), we probe the internal structure of colloidal superball dispersion droplets during confinement. We observe and…
We study the pressure-driven flow of concentrated colloids confined in glass micro-channels at the single particle level using fast confocal microscopy. For channel to particle size ratios $a/\bar{D} \lesssim 30$, the flow rate of the suspended particles shows fluctuations. These turn into regular oscillations for higher confinements ($a/\bar{D} \simeq 20$). We present evidence to link these oscillations with the relative flow of solvent and particles (permeation) and the effect of confinement on shear thickening.
Using video microscopy, we measure local spatial constraints in disordered binary colloidal samples, ranging from dilute fluids to jammed glasses, and probe their spatial and temporal correlations to local dynamics during the glass transition. We observe the emergence of significant correlations between constraints and local dynamics within the Lindemann criterion, which coincides with the onset of glassy dynamics in supercooled liquids. Rigid domains in fluids are identified based on local constraints, and demonstrate a percolation transition near glass transition, accompanied by the emergenc…
In a recent study [Phys. Rev. E 60, 6530 (1999)], Trizac and Raimbault showed that the effective pair interaction between like charged colloids immersed in a cylindrically confined electrolyte remains repulsive even when the size of the micro-ions or the finite longitudinal extension of the confining cylinder are taken into account. Contrary to their claim, we argue that the case of finite longitudinal confinement doesn't always generate repulsive interactions and to illustrate this point we also provide a simple example.
We study a dense colloidal suspension confined between two quasiparallel glass plates as a model system for a supercooled liquid in confined geometries. We directly observe the three-dimensional Brownian motion of the colloidal particles using laser scanning confocal microscopy. The particles form dense layers along the walls, but crystallization is avoided as we use a mixture of two particle sizes. A normally liquid-like sample, when confined, exhibits slower diffusive motion. Particle rearrangements are spatially heterogeneous, and the shapes of the rearranging regions are strongly influence…
We simulate structural phase behavior of polymer-grafted colloidal particles by molecular Monte Carlo technique. Interparticle potential, which has a finite repulsive square-step outside a rigid core of the colloid, was previously confirmed via numerical self-consistent field calculation. This model potential is purely repulsive. We simulate these model colloids in the canonical ensemble in 2 dimensions and find that these particles containing no interparticle attraction self-assemble and align in a string-like assembly, at low temperature and high density. This string-like colloidal assembly …
The effective dynamics of a colloidal particle immersed in a complex medium is often described in terms of an overdamped linear Langevin equation for its velocity with a memory kernel which determines the effective (time-dependent) friction and the correlations of fluctuations. Recently, it has been shown in experiments and numerical simulations that this memory may depend on the possible optical confinement the particle is subject to, suggesting that this description does not capture faithfully the actual dynamics of the colloid, even at equilibrium. Here, we propose a different approach in w…
The effective interaction between charged colloidal particles confined between two planar like-charged walls is investigated using computer simulations of the primitive model describing asymmetric electrolytes. In detail, we calculate the effective force acting onto a single macroion and onto a macroion pair in the presence of slit-like confinement. For moderate Coulomb coupling, we find that this force is repulsive. Under strong coupling conditions, however, the sign of the force depends on the distance to the plates and on the interparticle distance. In particular, the particle-plate interac…
Colloids held by optical or magnetic tweezers have been used to explore the local rheological properties of a complex medium and to extract work from fluctuations with some appropriate protocols. However, a general theoretical understanding of the interplay between the confinement and the interaction with the environment is still lacking. Here, we explore the statistical properties of the position of a probe confined in a harmonic trap moving at constant velocity and interacting with a bath of colloidal particles maintained at a different temperature. Interactions among particles are accounted…
The effects of confinement on colloidal self-assembly in the case of fixed number of confined particles are studied in the one dimensional lattice model solved exactly in the Grand Canonical Ensemble (GCE) in [J. Pękalski et al. J. Chem. Phys. 142, 014903 (2015)]. The model considers a pair interaction defined by a short-range attraction plus a longer-range repulsion. We consider thermodynamic states corresponding to self-assembly into clusters. Both, fixed and adaptive boundaries are studied. For fixed boundaries, there are particular states in which, for equal average densities, the number o…
We consider the sedimentation of a colloidal gel under confinement in the direction of gravity. The confinement allows us to compare directly experiments and computer simulations, for the same system size in the vertical direction. The confinement also leads to qualitatively different behaviour compared to bulk systems: in large systems gelation suppresses sedimentation, but for small systems sedimentation is enhanced relative to non-gelling suspensions, although the rate of sedimentation is reduced when the strength of the attraction between the colloids is strong. We map interaction paramete…
Using molecular dynamics simulations we study the slow dynamics of a hard sphere fluid confined in a disordered porous matrix. The presence of both discontinuous and continuous glass transitions as well as the complex interplay between single-particle and collective dynamics are well captured by a recent extension of mode-coupling theory for fluids in porous media. The degree of universality of the mode-coupling theory predictions for related models of colloids is studied by introducing size-disparity between fluid and matrix particles, as well as softness in the interactions.
We report an extensive molecular dynamics study on the translational dynamics of a hybrid system composed of dipolar soft spheres (DSS), representing ferromagnetic particles, suspended in a liquid crystal (LC) matrix. We observe that the LC matrix strongly modifies the dynamics of the DSS. In the isotropic regime, the DSS show a crossover from subdiffusive to normal diffusive behavior at long times, with an increase of the subdiffusive regime as the dipolar coupling strength is increased. In the nematic regime, the LC matrix, due to collective reorientation of LC particles, imposes a cylindric…
It is general wisdom that likely charged colloidal particles repel each other when suspended in liquids. This is in perfect agreement with mean field theories being developed more than 60 years ago. Accordingly, it was a big surprise when several groups independently reported long-ranged attractive components in the pair potential U(r) of equally charged colloids. This so-called like-charge attraction (LCA) was only observed in thin sample cells while the pair-interaction in unconfined suspensions has been experimentally confirmed to be entirely repulsive. Despite considerable experimental and…
We theoretically investigate general properties of driven (sheared) colloidal suspensions in confinement, based on methods of classical density functional theory. In the absence of an exact closed (Smoluchowski-) equation for the one-particle density under shear, we formulate a set of general conditions for approximations, and show that a simple closure fulfills them. The exact microscopic stress tensor is identified. Exemplifying the situation near a wall (oriented parallel to the direction of shear), we note that the microscopic shear stress is not necessarily homogeneous. Formulating a seco…
Using grand canonical Monte Carlo simulations, we investigate the percolation behavior of a square well fluid with an ultra-short range of attraction in three dimension (3D) and in confined geometry. The latter is defined through two parallel and structureless walls (slit-pore). We focus on temperatures above the critical temperature of the (metastable) condensation transition of the 3D system. Investigating a broad range of systems sizes, we first determine the percolation thresholds, i. e., the critical packing fraction for percolation $η_{c}$. For the slit-pore systems, $η_{c}$ is found to …
We study a two-dimensional binary mixture of patchy colloids in sedimentation-diffusion-equilibrium using Monte Carlo simulation and Wertheim's theory. By tuning the buoyant masses of the colloids we can control the gravity-induced sequence of fluid stacks of differing density and percolation properties. We find complex stacking sequences with up to four layers and reentrant network formation, consistently in simulations and theoretically using only the bulk phase diagram as input. Our theory applies to general patchy colloidal mixtures and is relevant to understanding experiments under gravit…
In his Comment (cond-mat/0104060) to [Phys. Rev. E 60, 6530 (1999)], Mateescu shows that while the effective interactions remain repulsive when the specific size of the micro-ions is taken into account via a Modified Poisson-Boltzmann equation, a similar conclusion cannot be reached for the situation of complete lateral confinement. This point is correct but has already been considered in a more general study [Phys. Rev. E 62, R1465 (2000), where repulsion is generically obtained]; moreover, we argue that it illustrates the irrelevancy of the notion of pair potential in completely confined con…
The mechanical properties of crystalline materials can be substantially modified under confinement. Such modified macroscopic properties are usually governed by the altered microstructures and internal stress fields. Here, we use a parallel plate geometry to apply a quasi-static squeeze flow crushing a colloidal polycrystal while simultaneously imaging it with confocal microscopy. The confocal images are used to quantify the local structure order and, in conjunction with Stress Assessment from Local Structural Anisotropy (SALSA), determine the stress at the single-particle scale. We find that …
When captured by a flat nematic-isotropic interface, colloidal particles can be dragged by it. As a result spatially periodic structures may appear, with the period depending on a particle mass, size, and interface velocity~\cite{west.jl:2002}. If liquid crystal is sandwiched between two substrates, the interface takes a wedge-like shape, accommodating the interface-substrate contact angle and minimizing the director distortions on its nematic side. Correspondingly, particles move along complex trajectories: they are first captured by the interface and then `glide' towards its vertex point. Ou…
The study of protein mass transport via atomistic simulation requires time and length scales beyond the computational capabilities of modern computer systems. The raspberry model for colloidal particles in combination with the mesoscopic hydrodynamic method of lattice Boltzmann facilitates coarse-grained simulations that are on the order of microseconds and hundreds of nanometers for the study of diffusive transport of protein-like colloid particles. The raspberry model reproduces linearity in resistance to motion versus particle size and correct enhanced drag within cylindrical pores at off-c…
Equilibrium gels of colloidal particles can be realized through the introduction of a second species, a linker that mediates the bonds between the colloids. A gel forming binary mixture whose linkers can self-assemble into linear chains while still promoting the aggregation of particles is considered in this work. The particles are patchy particles with $f_C$ patches of type $C$ and the linkers are patchy particles with $2$ patches of type $A$ and $f_B$ patches of type B. The bonds between patches of type $A$ ($AA$ bonds) promote the formation of linear chains of linkers. Two different ways (m…
Confined samples of liquid crystals are characterized by a variety of topological defects and can be exposed to external constraints such as extreme confinements with nontrivial topology. Here we explore the intrinsic structure of smectic colloidal layers dictated by the interplay between entropy and an imposed external topology. Considering an annular confinement as a basic example, a plethora of competing states is found with nontrivial defect structures ranging from laminar states to multiple smectic domains and arrays of edge dislocations which we refer to as Shubnikov states in formal ana…
We review recent progress in the theoretical description of anisotropic hard colloidal particles. The shapes considered range from rods and dumbbells to rounded cubes, polyhedra and to biaxial particles with arbitrary shape. Our focus is on both static and dynamical density functional theory and on computer simulations. We describe recent results for the structure, dynamics and phase behaviour in the bulk and in various confining geometries, e.g. established by two parallel walls which reduce the dimensionality of the system to two dimensions. We also include recent theoretical modelling for a…
We study the Brownian motion of a charged colloid, confined between two charged walls, for small separation between the colloid and the walls. The system is embedded in an ionic solution. The combined effect of electrostatic repulsion and reduced diffusion due to hydrodynamic forces results in a specific motion in the direction perpendicular to the confining walls. The apparent diffusion coefficient at short times as well as the diffusion characteristic time are shown to follow a sigmoid curve as function of a dimensionless parameter. This parameter depends on the electrostatic properties and …
We analyse a binary mixture of colloidal parallel hard cylindrical particles with identical diameters but dissimilar lengths $L_1$ and $L_2$, with $s=L_2/L_1=3$, confined by two parallel hard walls in a planar slit-pore geometry, using a fundamental--measure density functional theory. This model presents nematic (N) and two types of smectic (S) phases, with first- and second-order N-S bulk transitions and S-S demixing, and surface behaviour at a single hard wall which includes complete wetting by the S phase mediated (or not) by an infinite number of surface-induced layering (SIL) transitions.…
We study the phase behavior of colloidal suspensions the solvents of which are considered to be binary liquid mixtures undergoing phase segregation. We focus on the thermodynamic region close to the critical point of the accompanying miscibility gap. There, due to the colloidal particles acting as cavities in the critical medium, the spatial confinements of the critical fluctuations of the corresponding order parameter result in the effective, so-called critical Casimir forces between the colloids. Employing an approach in terms of effective, one-component colloidal systems, we explore the pos…
Geometric confinement strongly influences the behavior of microparticles in liquid environments. However, to date, nonspherical particle behaviors close to confining boundaries, even as simple as planar walls, remain largely unexplored. Here, we measure the height distribution and orientation of colloidal dumbbells above walls by means of digital in-line holographic microscopy. We find that while larger dumbbells are oriented almost parallel to the wall, smaller dumbbells of the same material are surprisingly oriented at preferred angles. We determine the total height-dependent force acting on…
The dynamical arrest of attractive colloidal particles into out-of-equilibrium structures, known as gelation, is central to biophysics, materials science, nanotechnology, and food and cosmetic applications, but a complete understanding is lacking. In particular, for intermediate particle density and attraction, the structure formation process remains unclear. Here, we show that the gelation of short-range attractive particles is governed by a nonequilibrium percolation process. We combine experiments on critical Casimir colloidal suspensions, numerical simulations, and analytical modeling with…
