Sedimentation of a Colloidal Monolayer Down an Inclined Plane
Brennan Sprinkle, Sam Wilken, Shake Karapetyan, Michio Tanaka, Zhe Chen, Joseph R. Cruise, Blaise Delmotte, Michelle M. Driscoll, Paul Chaikin, Aleksandar Donev
We study the driven collective dynamics of a colloidal monolayer sedimentating down an inclined plane. The action of the gravity force parallel to the bottom wall creates a flow around each colloid, and the hydrodynamic interactions among the colloids accelerate the sedimentation as the local density increases. This leads to the creation of a universal "triangular" inhomogeneous density profile, with a traveling density shock at the leading front moving in the downhill direction. Unlike density shocks in a colloidal monolayer driven by applied torques rather than forces [Phys. Rev. Fluids, 2(9):092301, 2017], the density front during sedimentation remains stable over long periods of time even though it develops a roughness on the order of tens of particle diameters. Through experimental measurements and particle-based computer simulations, we find that the Burgers equation can model the density profile along the sedimentation direction as a function of time remarkably well, with a modest improvement if the nonlinear conservation law accounts for the sub-linear dependence of the collective sedimentation velocity on density.
Both polydispersity and the presence of a gravitational field are inherent to essentially any colloidal experiment. While several theoretical works have focused on the effect of polydispersity on the bulk phase behavior of a colloidal system, little is known about the effect of a gravitational field on a polydisperse colloidal suspension. We extend here sedimentation path theory to study sedimentation-diffusion-equilibrium of a mass-polydisperse colloidal system: the particles possess different buoyant masses but they are otherwise identical. The model helps to understand the interplay between…
We investigate the sedimentation equilibrium of a charge stabilized colloidal suspension in the regime of low ionic strength. We analyze the asymptotic behaviour of the density profiles on the basis of a simple Poisson--Boltzmann theory and show that the effective mass we can deduce from the barometric law corresponds to the actual mass of the colloidal particles, contrary to previous studies.
We study self-diffusion and sedimentation in colloidal suspensions of nearly-hard spheres using the multiparticle collision dynamics simulation method for the solvent with a discrete mesh model for the colloidal particles (MD+MPCD). We cover colloid volume fractions from 0.01 to 0.40 and compare the MD+MPCD simulations to Brownian dynamics simulations with free-draining hydrodynamics (BD) as well as pairwise far-field hydrodynamics described using the Rotne--Prager--Yamakawa mobility tensor (BD+RPY). The dynamics in MD+MPCD suggest that the colloidal particles are only partially coupled to the…
We use a mesoscopic simulation technique to study the effect of short-ranged inter-particle attraction on the steady-state sedimentation of colloidal suspensions. Attractions increase the average sedimentation velocity $v_s$ compared to the pure hard-sphere case, and for strong enough attractions, a non-monotonic dependence on the packing fraction $ϕ$ with a maximum velocity at intermediate $ϕ$ is observed. Attractions also strongly enhance hydrodynamic velocity fluctuations, which show a pronounced maximum size as a function of $ϕ$. These results are linked to a complex interplay between hydr…
We report on the differences between colloidal systems left to evolve in the earth's gravitational field and the same systems for which a slow continuous rotation averaged out the effects of particle sedimentation on a distance scale small compared to the particle size. Several systems of micron-sized colloidal particles were studied: a hard sphere fluid, colloids interacting via long-range electrostatic repulsions above the freezing volume fraction, an oppositely charged colloidal system close to either gelation and/or crystallization, colloids with a competing short-range depletion attractio…
Employing a recently developed dynamical density functional theory we study the response of a colloidal sediment above a wall to shear, demonstrating the time dependent changes of the density distribution and its center-of-mass after switching shear either on or off and under oscillatory shear. Following the onset of steady shear we identify two dynamical mechanisms, distinguished by their timescales. Shortly after the onset, a transient enhancement of the packing structure at the wall reflects the self-organization into lanes. On a much longer timescale these effects are transmitted to the bu…
In this work, we use structure and dynamics in sedimentation equilibrium, in the presence of gravity, to examine, $via$ confocal microscopy, a Brownian colloidal system in the presence of an external electric field. The zero field equation of state (EOS) is hard sphere without any re-scaling of particle size, and the hydrodynamic corrections to the long-time self-diffusion coefficient are quantitatively consistent with the expected value for hard spheres. Care is taken to ensure that both the dimensionless gravitational energy, which is equivalent to a Peclet number $Pe_g$, and dipolar strengt…
We study the sedimentation of colloidal gels by using a combination of light scattering, polarimetry and video imaging. The asymptotic concentration profiles $φ(z,t\rightarrow \infty)$ exhibit remarkable scaling properties: profiles for gels prepared at different initial volume fractions and particle interactions can be superimposed onto a single master curve by using suitable reduced variables. We show theoretically that this behavior stems from a power law dependence of the compressive elastic modulus \textit{vs} $φ$, which we directly test experimentally. The sedimentation kinetics comprise…
We study theoretically the effect of a gravitational field on the equilibrium behaviour of a colloidal suspension of rods with different length-to-width aspect ratios. The bulk phases of the system are described with analytical equations of state. The gravitational field is then incorporated via sedimentation path theory, which assumes a local equilibrium condition at each altitude of the sample. The bulk phenomenology is significantly enriched by the presence of the gravitational field. In a suspension of elongated rods with five stable phases in bulk, the gravitational field stabilizes up to…
In this paper, we investigate experimentally the non-equilibrium steady state of an active colloidal suspension under gravity field. The active particles are made of chemically powered colloids, showing self propulsion in the presence of an added fuel, here hydrogen peroxide. The active suspension is studied in a dedicated microfluidic device, made of permeable gel microstructures. Both the microdynamics of individual colloids and the global stationary state of the suspension under gravity - density profiles, number fluctuations - are measured with optical microscopy. This allows to connect th…
We give a full account of a recently proposed theory that explicitly relates the bulk phase diagram of a binary colloidal mixture to its phase stacking phenomenology under gravity [Soft Matter 9, 8636 (2013)]. As we demonstrate, the full set of possible phase stacking sequences in sedimentation-diffusion equilibrium originates from straight lines (sedimentation paths) in the chemical potential representation of the bulk phase diagram. From the analysis of various standard topologies of bulk phase diagrams, we conclude that the corresponding sedimentation stacking diagrams can be very rich, eve…
We investigate theoretically the effect of gravity on a plate-sphere colloidal mixture by means of an Onsager-like density functional to describe the bulk, and sedimentation path theory to incorporate gravity. We calculate the stacking diagram of the mixture for two sets of buoyant masses and different values of the sample height. Several stacking sequences appear due to the intricate interplay between gravity, the sample height, and bulk phase separation. These include the experimentally observed floating nematic sequence, which consists of a nematic layer sandwiched between two isotropic lay…
We report on calculations of the reduced sedimentation velocity $U/U_{0}$ in homogenous suspensions of strongly and weakly charged colloidal spheres as a function of particle volume fraction $ϕ$. For dilute suspensions of strongly charged spheres at low salinity, $U/U_{0}$ is well represented by the parametric form $1-pϕ^α$ with a fractional exponent $α=1/3$ and a parameter $p\simeq 1.8$, which is essentially independent from the macroion charge $Z$. This non-linear volume fraction dependence can be quantitatively understood in terms of a model of effective hard spheres with $ϕ$-dependent diam…
Colloidal photonic crystals display peculiar optical properties which make them particularly suitable for application in different fields. However, the low packing fraction of the targeted structures usually poses a real challenge in the fabrication stage. Here, we propose a novel route to colloidal photonic crystals via a binary mixture of hard tetramers and hard spheres. By combining theory and computer simulations, we calculate the phase diagram as well as the stacking diagram of the mixture, and show that a colloidal analogue of the MgCu2 Laves phase -- which can serve as a precursor of a …
We study experimentally and theoretically the sedimentation of gels made of strongly aggregated colloidal particles, focussing on the long time behavior, when mechanical equilibrium is asymptotically reached. The asymptotic gel height is found to vary linearly with the initial height, a finding in stark contrast with a recent study on similar gels [Manley \textit{et al.} 2005 \textit{Phys. Rev. Lett.} \textbf{94} 218302]. We show that the asymptotic compaction results from the balance between gravity pull, network elasticity, and solid friction between the gel and the container walls. Based on…
Gravitational effects in colloidal suspensions can be easily turned off by matching the density of the solid microparticles with the one of the surrounding fluid. By studying the motion of catalytic microswimmers with tunable buoyant weight, we show that this strategy cannot be adopted for active colloidal suspensions. If the average buoyant weight decreases, pronounced accumulation at the top wall of a sample cell is observed due to a counter-alignment of the swimming velocity with the gravitational field. Even when the particles reach a flat wall, gravitational torques still determine the pr…
We present Monte Carlo simulations of colloidal particles pulled into grafted polymer layers by external fields. The insertion free energy of a single colloid into the polymer layer is qualitatively different for surfaces with an ordered and a disordered distribution of grafting points. Moreover, the tendency of colloidal particles to traverse the grafting layer is strongly size dependent. In dense colloidal suspensions, under the influence of sufficiently strong external force, a collective instability allows the colloids to penetrate and form internally ordered, columnar structures spanning …
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 th…
We have measured equilibrium sedimentation profiles in a colloidal model system with confocal microscopy. By tuning the interactions, we have determined the gravitational length in the limit of hard-sphere-like interactions, and using the same particles, tested a recent theory [R.van Roij, J. Phys. Cond. Mat. 15, S3569, (2003)], which predicts a significantly extended sedimentation profile in the case of charged colloids with long-ranged repulsions, due to a spontaneously formed macroscopic electric field. For the hard-sphere-like system we find that the gravitational length matches that expec…
We apply a hybrid Molecular Dynamics and mesoscopic simulation technique to study the steady-state sedimentation of hard sphere particles for Peclet numbers (Pe) ranging from 0.08 to 12. Hydrodynamic back-flow 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, non-equilibrium hydrodynamic fluctuations are visible, and their character appears to be independent of the thermal fluctuations.…
Colloidal particles that are confined to an interface effectively form a two-dimensional fluid. We examine the dynamics of such colloids when they are subject to a constant external force, which drives them in a particular direction over the surface. Such a situation occurs, for example, for colloidal particles that have settled to the bottom of their container, when the container is tilted at an angle, so that they `sediment' to the lower edge of the surface. We focus in particular on the case when there are attractive forces between the colloids which causes them to phase separate into regio…
Gravity can affect colloidal suspensions since for micrometer-sized particles gravitational and thermal energies can be comparable over vertical length scales of a few millimeters. In mixtures, each species possesses a different buoyant mass, which can make experimental results counter-intuitive and difficult to interpret. Here, we revisit from a theoretical perspective iconic sedimentation-diffusion-equilibrium experiments on colloidal plate-rod mixtures by van der Kooij and Lekkerkerker. We reproduce their findings, including the observation of five different mesophases in a single cuvette. …
We present a method of organizing incoherent motion of a colloidal suspension to produce synchronized, coherent motion. This method exploits general features of rotational response to time-dependent forcing, and it does not require interaction between the particles. We report two methods of achieving orientational alignment of an ensemble of identical colloids by means of a time-dependent, but spatially uniform forcing: a) a piecewise constant force alternating between two directions and b) a force uniformly rotating about an axis. The physical origin of the forcing may be e.g., sedimentation …
Equilibrium sedimentation density profiles of charged binary colloidal suspensions are calculated by computer simulations and density functional theory. For deionized samples, we predict a colloidal ``brazil nut'' effect: heavy colloidal particles sediment on top of the lighter ones provided that their mass per charge is smaller than that of the lighter ones. This effect is verifiable in settling experiments.
The discrepancy in nucleation rate densities between simulated and experimental hard spheres remains staggering and unexplained. Suggestively, more strongly sedimenting colloidal suspensions of hard spheres nucleate much faster than weakly sedimenting systems. In this work we consider firstly the effect of sedimentation on the structure of colloidal hard spheres, by tuning the density mismatch between solvent and colloidal particles. In particular we investigate the effect on the degree of five fold symmetry present. Secondly we study the size of density fluctuations in these experimental syst…
We investigate sedimentation of model hard sphere-like colloidal dispersions confined in horizontal capillaries using laser scanning confocal microscopy, dynamical density functional theory, and Brownian dynamics computer simulations. For homogenized initial states we obtain quantitative agreement of the results from the respective approaches for the time evolution of the one-body density distribution and the osmotic pressure on the walls. We demonstrate that single particle information can be obtained experimentally in systems that were initialized further out-of-equilibrium such that complex…
The phase behaviour of colloidal dispersions is interesting for fundamental reasons and for technological applications such as photonic crystals and electronic paper. Sedimentation, which in everyday life is relevant from blood analysis to the shelf life of paint, is a means to determine phase boundaries by observing distinct layers in samples that are in sedimentation-diffusion equilibrium. However, disentangling the effects due to interparticle interactions, which generate the bulk phase diagram, from those due to gravity is a complex task. Here we show that a line in the space of chemical p…
We examine sedimentation density profiles of star polymer solutions as an example of colloidal systems in sedimentation equilibrium which exhibit reentrant melting in their bulk phase diagram. Phase transitions between a fluid and a fluid with an intercalated solid are observed below a critical gravitational strength $α^{*}$. Characteristics of the two fluid-solid interfaces in the density profiles occurring in Monte Carlo simulations for $α< α^{*}$ are in agreement with scaling laws put forth in the framework of a phenomenological theory. Furthermore we detect density oscillations at the flui…
We develop a fully continuous model for colloidal suspensions with hydrodynamic interactions. The Navier Stokes Phase Field Crystal (NS-PFC) model combines ideas of dynamic density functional theory with particulate flow approaches and is derived in detail and related to other dynamic density functional theory approaches with hydrodynamic interactions. The derived system is numerically solved using adaptive finite elements and used to analyse colloidal crystallization in flowing environments demonstrating a strong coupling in both directions between the crystal shape and the flow field. We fur…
We study crystal nucleation under the influence of sedimentation in a model of colloidal hard spheres via Brownian Dynamics simulations. We introduce two external fields acting on the colloidal fluid: a uniform gravitational field (body force), and a surface field imposed by pinning a layer of equilibrium particles (rough wall). We show that crystal nucleation is suppressed in proximity of the wall due to the slowing down of the dynamics, and that the spatial range of this effect is governed by the static length scale of bond orientational order. For distances from the wall larger than this le…
Sedimentation-diffusion equilibrium density profiles of suspensions of charge-stabilized colloids are calculated theoretically and by Monte Carlo simulation, both for a one-component model of colloidal particles interacting through pairwise screened-Coulomb repulsions and for a three-component model of colloids, cations, and anions with unscreened-Coulomb interactions. We focus on a state point for which experimental measurements are available [C.P. Royall et al., J. Phys.: Cond. Matt. {\bf 17}, 2315 (2005)]. Despite the apparently different picture that emerges from the one- and three-compone…
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…
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…
We theoretically study sedimentation-diffusion equilibrium of dilute binary, ternary, and polydisperse mixtures of colloidal particles with different buoyant masses and/or charges. We focus on the low-salt regime, where the entropy of the screening ions drives spontaneous charge separation and the formation of an inhomogeneous macroscopic electric field. The resulting electric force lifts the colloids against gravity, yielding highly nonbarometric and even nonmonotonic colloidal density profiles. The most profound effect is the phenomenon of segregation into layers of colloids with equal mass-…
The influence of hydrodynamic interactions on lane formation of oppositely charged driven colloidal suspensions is investigated using Brownian dynamics computer simulations performed on the Rotne-Prager level of the mobility tensor. Two cases are considered, namely sedimentation and electrophoresis. In the latter case the Oseen contribution to the mobility tensor is screened due to the opposite motion of counterions. The simulation results are compared to that resulting from simple Brownian dynamics where hydrodynamic interactions are neglected. For sedimentation, we find that hydrodynamic int…
By using the critical Casimir force, we study the attractive strength dependent aggregation of colloids with and without gravity by means of Near Field scattering. Significant differences were seen between microgravity and ground experiments, both in the structure of the formed fractal aggregates as well as the kinetics of growth. Ground measurements are severely affected by sedimentation resulting in reaction limited behavior. In microgravity, a purely diffusive behavior is seen reflected both in the measured fractal dimensions for the aggregates as well as the power law behavior in the rate …
We introduce a model gel system in which colloidal forces, structure, and rheology are measured by balancing the requirements of rheological and microscopy techniques with those of optical tweezers. Sterically stabilized poly(methyl methacrylate) (PMMA) colloids are suspended in cyclohexane (CH) and cyclohexyl bromide (CHB) with dilute polystyrene serving as a depletion agent. A solvent comprising of 37% weight fraction CH provides sufficient refractive index contrast to enable optical trapping, while maintaining good confocal imaging quality and minimal sedimentation effects on the bulk rheol…
Dynamic density functional theory (DDFT) is a promising approach for predicting the structural evolution of a drying suspension containing one or more types of colloidal particles. The assumed free-energy functional is a key component of DDFT that dictates the thermodynamics of the model and, in turn, the density flux due to a concentration gradient. In this work, we compare several commonly used free-energy functionals for drying hard-sphere suspensions including local-density approximations based on the ideal-gas, virial, and Boublík-Mansoori-Carnahan-Starling-Leland (BMCSL) equations of sta…
Active colloids constitute a novel class of materials composed of colloidal-scale particles locally converting chemical energy into motility, mimicking micro-organisms. Evolving far from equilibrium, these systems display structural organizations and dynamical properties distinct from thermalized colloidal assemblies. Harvesting the potential of this new class of systems requires the development of a conceptual framework to describe these intrinsically nonequilibrium systems. We use sedimentation experiments to probe the nonequilibrium equation of state of a bidimensional assembly of active Ja…
We present a complete reciprocal description of particle motion inside multi-component fluids that extends the conventional Onsager formulation of non-equilibrium transport to systems where the thermodynamic forces are non-uniform on the colloidal scale. Based on the dynamic length and time scale separation in suspensions, the particle flux is shown to be related to the volume-averaged coupling between the Stokes flow tensor and the thermodynamic force density acting on the fluid. The flux is then expressed in terms of thermodynamic quantities that can be computed from the interfacial properti…
