Nonequilibrium sedimentation of colloids on the particle scale

Settled and unsettled issues in particle settling

Colloid sedimentation has played a seminal role in the development of statistical physics thanks to the celebrated experiments by Perrin, which provided a concrete demonstration of molecular reality and gave strong support to Einstein's theory of Brownian motion. This review, which mostly focuses on settling at low Peclét number, where Brownian fluctuations are dominant, aims to show that a lot more can be learnt both from the sedimentation equilibrium and from the particle settling dynamics of a wide class of systems, ranging from simple colloids to mesogenic suspensions, from soft solids to active particles and living organisms. At the same time, the occurrence of unexpected and surprising effects brings about challenging questions in statistical and fluid mechanics that make sedimentation an exciting field of research.

A multi-axis confocal rheoscope for studying shear flow of structured fluids

We present a new design for a confocal rheoscope that enables uniform uniaxial or biaxial shear. The design consists of two precisely positioned parallel plates with a gap that can be adjusted down to 2 ±0.1 μm, allowing for the exploration of confinement effects. By using our shear cell in conjunction with a biaxial force measurement device and a high-speed confocal microscope, we are able to measure the real-time biaxial stress while simultaneously imaging the material three-dimensional structure. We illustrate the importance of the instrument capabilities by discussing the applications of this instrument in current and future research topics in colloidal suspensions.

Sedimentation of a two-dimensional colloidal mixture exhibiting liquid-liquid and gas-liquid phase separation: a dynamical density functional theory study

We present dynamical density functional theory results for the time evolution of the density distribution of a sedimenting model two-dimensional binary mixture of colloids. The interplay between the bulk phase behaviour of the mixture, its interfacial properties at the confining walls, and the gravitational field gives rise to a rich variety of equilibrium and non-equilibrium morphologies. In the fluid state, the system exhibits both liquid-liquid and gas-liquid phase separation. As the system sediments, the phase separation significantly affects the dynamics and we explore situations where the final state is a coexistence of up to three different phases. Solving the dynamical equations in two-dimensions, we find that in certain situations the final density profiles of the two species have a symmetry that is different from that of the external potentials, which is perhaps surprising, given the statistical mechanics origin of the theory. The paper concludes with a discussion on this.

Speeding up of sedimentation under confinement

We show an increase of the sedimentation velocity as small particles are confined in circular capillaries. In general, confinement slows down sedimentation. But, we show that at low Reynolds numbers and in 1D confinement this is not the case. Particle sedimentation velocity is not homogeneous, which can lead to the formation of structures. These structures are enhanced and stabilized in the presence of walls and in the absence of other dissipative mechanisms. As a consequence, it is possible to achieve sedimentation velocities that even exceed the Stokes velocity. The segregation at critical capillary diameters has been directly observed using a large scale model. These simple experiments offer a new insight into the old problem of sedimentation under confinement.

Tunable soft structure in charged fluids confined by dielectric interfaces

Fluids of charged particles act as the supporting medium for chemical reactions and physical, dynamical, and biological processes. The local structure in an electrolytic background is deformed by micro- and nanoscopic polarizable objects. Vice versa, the forces between the objects are regulated by the cohesive properties of the background. We study here the range and strength of these forces and the microscopic origin from which they emerge. We find the forces to be sensitively dependent on the material properties of the charged fluid and the immersed solutes. The induced interactions can be varied over decades, offering high tunability and aided by accurate theory, control in experiments and applications. To distinguish correlational effects from simple ionic screening, we describe electrolyte-induced forces between neutral objects. The interplay of thermal motion, short-range repulsions, and electrostatic forces is responsible for a soft structure in the fluid. This structure changes near polarizable interfaces and causes diverse attractions between confining walls that seem well-exploited by microbiological systems. For parameters that correspond to monovalent electrolytes in biologically and technologically relevant aqueous environments, we find induced forces between nanoscopic areas of the order of piconewtons over a few nanometers.

Density functional theory for hard polyhedra

Using the framework of geometry-based fundamental-measure theory, we develop a classical density functional for hard polyhedra and their mixtures and apply it to inhomogeneous fluids of Platonic solids near a hard wall. As revealed by Monte Carlo simulations, the faceted shape of the polyhedra leads to complex layering and orientational ordering near the wall, which is excellently reproduced by our theory. These effects can be verified in real-space experiments on polyhedral colloids.

Unification of dynamic density functional theory for colloidal fluids to include inertia and hydrodynamic interactions: derivation and numerical experiments

Starting from the Kramers equation for the phase-space dynamics of the N-body probability distribution, we derive a dynamical density functional theory (DDFT) for colloidal fluids including the effects of inertia and hydrodynamic interactions (HI). We compare the resulting theory to extensive Langevin dynamics simulations for both hard rod systems and three-dimensional hard sphere systems with radially symmetric external potentials. As well as demonstrating the accuracy of the new DDFT, by comparing with previous DDFTs which neglect inertia, HI, or both, we also scrutinize the significance of including these effects. Close to local equilibrium we derive a continuum equation from the microscopic dynamics which is a generalized Navier-Stokes-like equation with additional non-local terms governing the effects of HI. For the overdamped limit we recover analogues of existing configuration-space DDFTs but with a novel diffusion tensor.

The unbearable heaviness of colloids: facts, surprises, and puzzles in sedimentation

Sedimentation has played a key role in the development of colloid science. In fact, it is because of the celebrated experiments by Perrin, yielding a concrete demonstration of molecular reality and giving strong support to Einstein's theory of Brownian motion, that colloids enter the realm of basic physics. Subsequent investigations have shown that a lot more can be learnt both from sedimentation equilibrium and from particle settling dynamics. These advances, together with new experimental approaches, will be reviewed in this paper. Yet, we shall also show that inquiring about gravity settling is far from being a closed matter: for instance, the concept of buoyancy for a settling colloidal mixture is far from being obvious. Moreover, sedimentation holds novel surprises, such as colloidal inflations and settling disasters, showing that a simple external field like gravity may induce mind-boggling, and theoretically challenging effects.

Synthesis of monodisperse, highly cross-linked, fluorescent PMMA particles by dispersion polymerization

We describe a facile method to synthesize sterically stabilized monodisperse fluorescent poly(methyl methacrylate) (PMMA) colloids in the polar solvent mixture water/methanol with either a core-shell or a homogeneously cross-linked structure by dispersion polymerization. The particles were sterically stabilized by the polymer poly(vinylpyrrolidone) (PVP). The morphology of the particles was controlled by varying the moment at which the gradual addition of cross-linker and dye was started. The absence of these extra agents at a time when the particle nuclei formed reduced the negative effects on this important process to a minimum and produced a core-shell structure, whereas an essentially homogeneously cross-linked fluorescent polymer colloid structure could be obtained by reducing the starting time of the addition of dye and cross-linker to zero. Three different dyes were chemically incorporated into the polymer network. Such dyes are important for the use of the particles in confocal scanning laser microscopy studies aimed at characterizing concentrated dispersions quantitatively in real space. A series of PMMA particles with different sizes were obtained through the variation of the weight ratio of solvents and the content of cross-linker. Furthermore, the swelling properties of the cross-linked PMMA particles in a good solvent (tetrahydrofuran) were investigated. The particles were stable in polar solvents (water and formamide) but could also successfully be transferred to apolar solvents such as decahydronaphthalene (decalin). The PVP stabilizer also allowed the particles to be permanently bonded in flexible strings by the application of an external electric field.

Modelling the evaporation of thin films of colloidal suspensions using dynamical density functional theory

Recent experiments have shown that various structures may be formed during the evaporative dewetting of thin films of colloidal suspensions. Nanoparticle deposits of strongly branched 'flower-like', labyrinthine and network structures are observed. They are caused by the different transport processes and the rich phase behaviour of the system. We develop a model for the system, based on a dynamical density functional theory, which reproduces these structures. The model is employed to determine the influences of the solvent evaporation and of the diffusion of the colloidal particles and of the liquid over the surface. Finally, we investigate the conditions needed for 'liquid-particle' phase separation to occur and discuss its effect on the self-organized nanostructures.

Doubled heterogeneous crystal nucleation in sediments of hard sphere binary-mass mixtures

Crystallization during the sedimentation process of a binary colloidal hard spheres mixture is explored by Brownian dynamics computer simulations. The two species are different in buoyant mass but have the same interaction diameter. Starting from a completely mixed system in a finite container, gravity is suddenly turned on, and the crystallization process in the sample is monitored. If the Peclet numbers of the two species are both not too large, crystalline layers are formed at the bottom of the cell. The composition of lighter particles in the sedimented crystal is non-monotonic in the altitude: it is first increasing, then decreasing, and then increasing again. If one Peclet number is large and the other is small, we observe the occurrence of a doubled heterogeneous crystal nucleation process. First, crystalline layers are formed at the bottom container wall which are separated from an amorphous sediment. At the amorphous-fluid interface, a secondary crystal nucleation of layers is identified. This doubled heterogeneous nucleation can be verified in real-space experiments on colloidal mixtures.

Effects of hydrodynamic interactions in binary colloidal mixtures driven oppositely by oscillatory external fields

The collective dynamics in a binary mixture of colloidal particles which are driven in opposite directions by an external oscillatory field is examined by computer simulations in two spatial dimensions. Both Brownian dynamics (BD) computer simulations, which ignore solvent-mediated hydrodynamic interactions between the colloidal particles, and multi-particle collision dynamics (MPCD) simulations, which include hydrodynamic interactions, are employed. We first review recent results obtained by BD. Depending on the driving frequency and amplitude, lane formation parallel to the drive and band formation perpendicular to the drive occur. Band formation is stable only in a finite window of oscillation frequencies and driving strengths and is taken over by lane formation if the driving force is increased or the oscillation frequency is decreased. MPCD simulations, on the other hand, reveal that band formation is blurred by hydrodynamic interactions. During the front collisions of oppositely driven particles there is a strong vortical movement of the solvent which tends to mix particles and broaden the interface of the bands. This can either lead to a novel intermittent dynamical behaviour or to band rupture into local clusters. These effects, which are absent for BD, are characterized by the strengths of the enstrophy and its spectrum. We finally discuss possible experimental realizations of the models employed.

Band formation in mixtures of oppositely charged colloids driven by an ac electric field

We present experiments on pattern formation in a Brownian system of oppositely charged colloids driven by an ac electric field. Using confocal laser scanning microscopy we observe complete segregation of the two particle species into bands perpendicular to a field of sufficient strength when the frequency is in a well-defined range. Because of its Brownian nature the system spontaneously returns to the equilibrium mixture after the field is turned off. We show that band formation is linked to the time scale associated with collisions between particles moving in opposite directions.

Micro-rheology on (polymer-grafted) colloids using optical tweezers

Optical tweezers are experimental tools with extraordinary resolution in positioning (± 1 nm) a micron-sized colloid and in the measurement of forces (± 50 fN) acting on it-without any mechanical contact. This enables one to carry out a multitude of novel experiments in nano- and microfluidics, of which the following will be presented in this review: (i) forces within single pairs of colloids in media of varying concentration and valency of the surrounding ionic solution, (ii) measurements of the electrophoretic mobility of single colloids in different solvents (concentration, valency of the ionic solution and pH), (iii) similar experiments as in (i) with DNA-grafted colloids, (iv) the nonlinear response of single DNA-grafted colloids in shear flow and (v) the drag force on single colloids pulled through a polymer solution. The experiments will be described in detail and their analysis discussed.

Dynamics of colloids in confined geometries

We discuss the Brownian dynamics of colloids in confinement with special emphasis on the influence of the solvent dynamics. We review the derivation of a dynamic density functional theory (DDFT) including some aspects of hydrodynamic interactions and its application to the micro-rheology of suspensions. In particular we discuss the failure of Stokes' law in suspensions and non-equilibrium solvent structure mediated interactions. With regard to hydrodynamic chromatography we also discuss the stationary transport of particles in narrow channels, and the reasons for the failure of DDFT in this situation.

Computer simulations of colloidal transport on a patterned magnetic substrate

We study the transport of paramagnetic colloidal particles on a patterned magnetic substrate with kinetic Monte Carlo and Brownian dynamics computer simulations. The planar substrate is decorated with point dipoles in either parallel or zigzag stripe arrangements and exposed to an additional external magnetic field that oscillates in time. For the case of parallel stripes we find that the magnitude and direction of the particle current is controlled by the tilt angle of the external magnetic field. The effect is reliably obtained in a wide range of ratios between temperature and magnetic permeability. Particle transport is achieved only when the period of oscillation of the external field is greater than a critical value. For the case of zigzag stripes a current is obtained using an oscillating external field normal to the substrate. In this case, transport is possible only in the vertex of the zigzag, giving rise to a narrow stream of particles. The magnitude and direction of the particle current are found to be controlled by a combination of the zigzag angle and the distance of the colloids from the substrate. Metropolis Monte Carlo and Brownian dynamics simulations predict results that are in good agreement with each other. Using kinetic Monte Carlo we find that at high density the particle transport is hindered by jamming.

General nonequilibrium theory of colloid dynamics

A nonequilibrium extension of Onsager's canonical theory of thermal fluctuations is employed to derive a self-consistent theory for the description of the statistical properties of the instantaneous local concentration profile n(r,t) of a colloidal liquid in terms of the coupled time-evolution equations of its mean value n(r,t) and of the covariance [Formula in text] of its fluctuations δn(r,t)=n(r,t)-n(r,t). These two coarse-grained equations involve a local mobility function b(r,t) which, in its turn, is written in terms of the memory function of the two-time correlation function [Formula in text]. For given effective interactions between colloidal particles and applied external fields, the resulting self-consistent theory is aimed at describing the evolution of a strongly correlated colloidal liquid from an initial state with arbitrary mean and covariance n(0)(r) and σ(0)(r,r') toward its equilibrium state characterized by the equilibrium local concentration profile n(eq)(r) and equilibrium covariance σ(eq)(r,r'). This theory also provides a general theoretical framework to describe irreversible processes associated with dynamic arrest transitions, such as aging, and the effects of spatial heterogeneities.

DDFT for Brownian particles and hydrodynamics

Classical dynamic density functional theory (DDFT) has developed into a versatile tool for describing the dynamics of overdamped Brownian particles. The underlying equilibrium density functional is a quantitative model for colloidal suspensions. Non-equilibrium properties of these systems are strongly influenced by the solvent hydrodynamics. In this paper we discuss ways to introduce certain aspects of solvent hydrodynamics into the DDFT, in particular advection by a flowing solvent and hydrodynamic interactions among suspended particles as well as with container walls.

Experimental study of random-close-packed colloidal particles

A collection of spherical particles can be packed tightly together into an amorphous packing known as "random close packing" (RCP). This structure is of interest as a model for the arrangement of molecules in simple liquids and glasses, as well as the arrangement of particles in sand piles. We use confocal microscopy to study the arrangement of colloidal particles in an experimentally realized RCP state. We image a large volume containing more than 450,000 particles with a resolution of each particle position to better than 0.02 particle diameters. While the arrangement of the particles satisfies multiple criteria for being random, we also observe a small fraction (less than 3%) of tiny crystallites (4 particles or fewer). These regions pack slightly better and are thus associated with locally higher densities. The structure factor of our sample at long length scales is nonzero, S(0)=0.049±0.008, suggesting that there are long wavelength density fluctuations in our sample. These may be due to polydispersity or tiny crystallites. Our results suggest that experimentally realizable RCP systems may be different from simulated RCP systems, in particular, with the presence of these long wavelength density fluctuations.

Non-equilibrium relaxation and near-arrest dynamics in colloidal suspensions

In this work we propose a theory to describe the irreversible diffusive relaxation of the local concentration of a colloidal dispersion that proceeds toward its stable thermodynamic equilibrium state, but which may in the process be trapped in metastable or dynamically arrested states. The central assumption of this theory is that the irreversible relaxation of the macroscopically observed mean value [Formula: see text] of the local concentration of colloidal particles is described by a diffusion equation involving a local mobility b(*)(r,t) that depends not only on the mean value [Formula: see text] but also on the covariance [Formula: see text] of the fluctuations [Formula: see text]. This diffusion equation must hence be solved simultaneously with the relaxation equation for the covariance σ(r,r';t), and here we also derive the corresponding relaxation equation. The dependence of the local mobility b(*)(r,t) on the mean value and the covariance is determined by a self-consistent set of equations involving now the spatially and temporally non-local time-dependent correlation functions, which in a uniform system in equilibrium reduces to the self-consistent generalized Langevin equation (SCGLE) theory of colloid dynamics. The resulting general theory considers the possibility that these relaxation processes occur under the influence of external fields, such as gravitational forces acting in the process of sedimentation. In this paper, however, we describe a simpler application, in which the system remains spatially uniform during the irreversible relaxation process, and discuss the general features of the glass transition scenario predicted by this non-equilibrium theory.

Fabrication of large binary colloidal crystals with a NaCl structure

Binary colloidal crystals offer great potential for tuning material properties for applications in, for example, photonics, semiconductors and spintronics, because they allow the positioning of particles with quite different characteristics on one lattice. For micrometer-sized colloids, it is believed that gravity and slow crystallization rates hinder the formation of high-quality binary crystals. Here, we present methods for growing binary colloidal crystals with a NaCl structure from relatively heavy, hard-sphere-like, micrometer-sized silica particles by exploring the following external fields: electric, gravitational, and dielectrophoretic fields and a structured surface (colloidal epitaxy). Our simulations show that the free-energy difference between the NaCl and NiAs structures, which differ in their stacking of the hexagonal planes of the larger spheres, is very small (approximately 0.002 k(B)T). However, we demonstrate that the fcc stacking of the large spheres, which is crucial for obtaining the pure NaCl structure, can be favored by using a combination of the above-mentioned external fields. In this way, we have successfully fabricated large, 3D, oriented single crystals having a NaCl structure without stacking disorder.

Selectivity in binary fluid mixtures: static and dynamical properties

Selectivity of particles in a region of space can be achieved by applying external potentials to influence the particles in that region. We investigate static and dynamical properties of size selectivity in binary fluid mixtures of two particles sizes. We find that by applying an external potential that is attractive to both kinds of particles, due to crowding effects, this can lead to one species of particles being expelled from that region, while the other species is attracted into the region where the potential is applied. This selectivity of one species of particle over the other in a localized region of space depends on the density and composition of the fluid mixture. Applying an external potential that repels both kinds of particles leads to selectivity of the opposite species of particles to the selectivity with attractive potentials. We use equilibrium and dynamical density-functional theory to describe and understand the static and dynamical properties of this striking phenomenon. Selectivity by some ion channels is believed to be due to this effect.

Formation of bubbles, blobs, and surface cusps in complex plasmas

Investigations of the dynamical evolution of a complex plasma, in which a vertical temperature gradient compensates gravity, were carried out. At low power the formation of microparticle bubbles, blobs, and spraying cusps was observed. This activity can be turned on and off by changing control parameters, such as the rf power and the gas pressure. Several observational effects indicate the presence of surface tension, even at small "nanoscales" of a few 100's of particles. By tracing the individual microparticle motion the detailed (atomistic) dynamics can be studied as well as the pressure dependence of the forces. A possible mechanism that could drive the observed phenomena is analogous to the Rayleigh-Taylor instability.

Derivation of the phase-field-crystal model for colloidal solidification

The phase-field-crystal model is by now widely used in order to predict crystal nucleation and growth. For colloidal solidification with completely overdamped individual particle motion, we show that the phase-field-crystal dynamics can be derived from the microscopic Smoluchowski equation via dynamical density-functional theory. The different underlying approximations are discussed. In particular, a variant of the phase-field-crystal model is proposed which involves less approximations than the standard phase-field-crystal model. We finally test the validity of these phase-field-crystal models against dynamical density-functional theory. In particular, the velocities of a linear crystal front from the undercooled melt are compared as a function of the undercooling for a two-dimensional colloidal suspension of parallel dipoles. Good agreement is only obtained by a drastic scaling of the free energies in the phase-field-crystal model in order to match the bulk freezing transition point.

Oscillatory driven colloidal binary mixtures: axial segregation versus laning

Using Brownian dynamics computer simulations we show that binary mixtures of colloids driven in opposite directions by an oscillating external field exhibit axial segregation in sheets perpendicular to the drive direction. The segregation effect is stable only in a finite window of oscillation frequencies and driving strengths and is taken over by lane formation in the direction of the driving field if the driving force is increased or the oscillation frequency is decreased. In the crossover regime, bands tilted relative to the drive direction are observed. Possible experiments to verify the axial segregation are discussed.

Dynamical density functional theory for colloidal dispersions including hydrodynamic interactions

A dynamical density functional theory (DDFT) for translational Brownian dynamics is derived which includes hydrodynamic interactions. The theory reduces to the simple Brownian DDFT proposed by Marconi and Tarazona (U. Marini Bettolo Marconi and P. Tarazona, J. Chem. Phys. 110, 8032 (1999); J. Phys.: Condens. Matter 12, A413 (2000)) when hydrodynamic interactions are neglected. The derivation is based on Smoluchowski's equation for the time evolution of the probability density with pairwise hydrodynamic interactions. The theory is applied to hard-sphere colloids in an oscillating spherical optical trap which switches periodically in time from a stable confining to an unstable potential. Rosenfeld's fundamental measure theory for the equilibrium density functional is used and hydrodynamics are incorporated on the Rotne-Prager level. The results for the time-dependent density profiles are compared to extensive Brownian dynamics simulations which are performed on the same Rotne-Prager level and excellent agreement is obtained. It is further found that hydrodynamic interactions damp and slow the dynamics of the confined colloid cluster in comparison to the same situation with neglected hydrodynamic interactions.

Colloids dragged through a polymer solution: Experiment, theory, and simulation

We present microrheological measurements of the drag force on colloids pulled through a solution of lambda-DNA (used here as a monodisperse model polymer) with an optical tweezer. The experiments show a drag force that is larger than expected from the Stokes formula and the independently measured viscosity of the DNA solution. We attribute this to the accumulation of DNA in front of the colloid and the reduced DNA density behind the colloid. This hypothesis is corroborated by a simple drift-diffusion model for the DNA molecules, which reproduces the experimental data surprisingly well, as well as by corresponding Brownian dynamics simulations.

Dynamical density functional theory with hydrodynamic interactions and colloids in unstable traps

A density functional theory for colloidal dynamics is presented which includes hydrodynamic interactions between the colloidal particles. The theory is applied to the dynamics of colloidal particles in an optical trap which switches periodically in time from a stable to an unstable confining potential. In the absence of hydrodynamic interactions, the resulting density breathing mode exhibits huge single peaked oscillations in the trap center which become double peaked and damped by hydrodynamic interactions. The predicted dynamical density fields are in good agreement with Brownian dynamics computer simulations.

Interplay between hydrodynamic and Brownian fluctuations in sedimenting colloidal suspensions

We apply a hybrid molecular dynamics and mesoscopic simulation technique to study the steady-state sedimentation of hard sphere particles for Peclet number (Pe) ranging from 0.08 to 12. Hydrodynamic backflow causes a reduction of the average sedimentation velocity relative to the Stokes velocity. We find that this effect is independent of Pe number. Velocity fluctuations show the expected effects of thermal fluctuations at short correlation times. At longer times, nonequilibrium hydrodynamic fluctuations are visible, and their character appears to be independent of the thermal fluctuations. The hydrodynamic fluctuations dominate the diffusive behavior even for modest Pe number, while conversely the short-time fluctuations are dominated by thermal effects for surprisingly large Pe numbers. Inspired by recent experiments, we also study finite sedimentation in a horizontal planar slit. In our simulations distinct lateral patterns emerge, in agreement with observations in the experiments.

Dynamical density functional theory for anisotropic colloidal particles

We generalize the formalism of dynamical density functional theory for translational Brownian dynamics toward that of anisotropic colloidal particles which perform both translational and rotational Brownian motion. Using a mean-field approximation for the density functional and a Gaussian-segment model for the rod interaction, the dynamical density functional theory is then applied to a concentrated rod suspension in a confined slab geometry made by two parallel soft walls. The walls are either expanded or compressed and the relaxation behavior is investigated for an equilibrated starting configuration. We find distinctly different orientational ordering during expansion and compression. During expansion we observe preferential parallel ordering of the rods relative to the wall while during compression there is homeotropic ordering perpendicular to the wall. We find a nonexponential relaxation behavior in time. Furthermore, an external field which aligns the rods perpendicular to the walls is turned on or switched off and similar differences in the relaxational dynamics are found. Comparing the theoretical predictions to Brownian dynamics computer simulation data, we find good agreement.