Computer simulations of charge and steric stabilised colloidal suspensions

Structural correlations in highly asymmetric binary charged colloidal mixtures

We explore structural correlations of strongly asymmetric mixtures of binary charged colloids within the primitive model of electrolytes considering large charge and size ratios of 10 and higher. Using computer simulations with explicit microions, we obtain the partial pair correlation functions between the like-charged colloidal macroions. Interestingly the big-small correlation peak amplitude is smaller than that of the big-big and small-small macroion correlation peaks, which is unfamiliar for additive repulsive interactions. Extracting optimal effective microion-averaged pair interactions between the macroions, we find that on top of non-additive Yukawa-like repulsions an additional shifted Gaussian attractive potential between the small macroions is needed to accurately reproduce their correct pair correlations. For small Coulomb couplings, the behavior is reproduced in a coarse-grained theory with microion-averaged effective interactions between the macroions. However, the accuracy of the theory deteriorates with increasing Coulomb coupling. We emphasize the relevance of entropic interactions exerted by the microions on the macroions. Our results are experimentally verifiable in binary mixtures of micron-sized colloids and like-charge nanoparticles.

Machine learning many-body potentials for colloidal systems

Simulations of colloidal suspensions consisting of mesoscopic particles and smaller species such as ions or depletants are computationally challenging as different length and time scales are involved. Here, we introduce a machine learning (ML) approach in which the degrees of freedom of the microscopic species are integrated out and the mesoscopic particles interact with effective many-body potentials, which we fit as a function of all colloid coordinates with a set of symmetry functions. We apply this approach to a colloid-polymer mixture. Remarkably, the ML potentials can be assumed to be effectively state-independent and can be used in direct-coexistence simulations. We show that our ML method reduces the computational cost by several orders of magnitude compared to a numerical evaluation and accurately describes the phase behavior and structure, even for state points where the effective potential is largely determined by many-body contributions.

From predictive modelling to machine learning and reverse engineering of colloidal self-assembly

An overwhelming diversity of colloidal building blocks with distinct sizes, materials and tunable interaction potentials are now available for colloidal self-assembly. The application space for materials composed of these building blocks is vast. To make progress in the rational design of new self-assembled materials, it is desirable to guide the experimental synthesis efforts by computational modelling. Here, we discuss computer simulation methods and strategies used for the design of soft materials created through bottom-up self-assembly of colloids and nanoparticles. We describe simulation techniques for investigating the self-assembly behaviour of colloidal suspensions, including crystal structure prediction methods, phase diagram calculations and enhanced sampling techniques, as well as their limitations. We also discuss the recent surge of interest in machine learning and reverse-engineering methods. Although their implementation in the colloidal realm is still in its infancy, we anticipate that these data-science tools offer new paradigms in understanding, predicting and (inverse) design of novel colloidal materials.

Self-assembly of charged colloidal cubes

In this work, we show how and why the interactions between charged cubic colloids range from radially isotropic to strongly directionally anisotropic, depending on tuneable factors. Using molecular dynamics simulations, we illustrate the effects of typical solvents to complement experimental investigations of cube assembly. We find that in low-salinity water solutions, where cube self-assembly is observed, the colloidal shape anisotropy leads to the strongest attraction along the corner-to-corner line, followed by edge-to-edge, with a face-to-face configuration of the cubes only becoming energetically favorable after the colloids have collapsed into the van der Waals attraction minimum. Analysing the potential of mean force between colloids with varied cubicity, we identify the origin of the asymmetric microstructures seen in experiment.

Capillary Structured Suspensions from In Situ Hydrophobized Calcium Carbonate Particles Suspended in a Polar Liquid Media

We demonstrate that capillary suspensions can be formed from hydrophilic calcium carbonate particles suspended in a polar continuous media and connected by capillary bridges formed of minute amounts of an immiscible secondary liquid phase. This was achieved in two different polar continuous phases, water and glycerol, and three different oils, oleic acid, isopropyl myristate, and peppermint oil as a secondary liquid phase. The capillary structuring of the suspension was made possible through local in situ hydrophobization of the calcium carbonate particles dispersed in the polar media by adding very small amounts of oleic acid to the secondary liquid phase. We observed a strong increase in the viscosity of the calcium carbonate suspension by several orders of magnitude upon addition of the secondary oil phase compared with the same suspension without secondary liquid phase or without oleic acid. The stability and the rheological properties of the obtained capillary structured materials were studied in relation to the physical properties of the system such as the particle size, interfacial tension between the primary and secondary liquid phases, as well as the particle contact angle at this liquid-liquid interface. We also determined the minimal concentrations of the secondary liquid phase at fixed particle concentration as well as the minimal particle concentration at fixed secondary phase concentration needed to form a capillary suspension. Capillary suspensions formed by this method can find application in structuring pharmaceutical and food formulations as well as a variety of home and personal care products.

Specific ion effects on the electrokinetic properties of iron oxide nanoparticles: experiments and simulations

We report experimental and simulation studies on ion specificity in aqueous colloidal suspensions of positively charged, bare magnetite nanoparticles. Magnetite has the largest saturation magnetization among iron oxides and relatively low toxicity, which explain why it has been used in multiple biomedical applications. Bare magnetite is hydrophilic and the sign of the surface charge can be changed by adjusting the pH, its isoelectric point being in the vicinity of pH = 7. Electrophoretic mobility of our nanoparticles in the presence of increasing concentrations of different anions showed that anions regarded as kosmotropic are more efficient in decreasing, and even reversing, the mobility of the particles. If the anions were ordered according to the extent to which they reduced the particle mobility, a classical Hofmeister series was obtained with the exception of thiocyanate, whose position was altered. Monte Carlo simulations were used to predict the diffuse potential of magnetite in the presence of the same anions. The simulations took into account the ion volume, and the electrostatic and dispersion forces among the ions and between the ions and the solid surface. Even though no fitting parameters were introduced and all input data were estimated using Lifshitz theory of van der Waals forces or obtained from the literature, the predicted diffusion potentials of different anions followed the same order as the mobility curves. The results suggest that ionic polarizabilities and ion sizes are to a great extent responsible for the specific ion effects on the electrokinetic potential of iron oxide particles.

Nonadditivity of nanoparticle interactions

Understanding interactions between inorganic nanoparticles (NPs) is central to comprehension of self-organization processes and a wide spectrum of physical, chemical, and biological phenomena. However, quantitative description of the interparticle forces is complicated by many obstacles that are not present, or not as severe, for microsize particles (μPs). Here we analyze the sources of these difficulties and chart a course for future research. Such difficulties can be traced to the increased importance of discreteness and fluctuations around NPs (relative to μPs) and to multiscale collective effects. Although these problems can be partially overcome by modifying classical theories for colloidal interactions, such an approach fails to manage the nonadditivity of electrostatic, van der Waals, hydrophobic, and other interactions at the nanoscale. Several heuristic rules identified here can be helpful for discriminating between additive and nonadditive nanoscale systems. Further work on NP interactions would benefit from embracing NPs as strongly correlated reconfigurable systems with diverse physical elements and multiscale coupling processes, which will require new experimental and theoretical tools. Meanwhile, the similarity between the size of medium constituents and NPs makes atomic simulations of their interactions increasingly practical. Evolving experimental tools can stimulate improvement of existing force fields. New scientific opportunities for a better understanding of the electronic origin of classical interactions are converging at the scale of NPs.

Capillary forces in suspension rheology

The rheology of suspensions (solid particles dispersed in a fluid) is controlled primarily through the volume fraction of solids. We show that the addition of small amounts of a secondary fluid, immiscible with the continuous phase of the suspension, causes agglomeration due to capillary forces and creates particle networks, dramatically altering the bulk rheological behavior from predominantly viscous or weakly elastic to highly elastic or gel-like. This universal phenomenon is observed for a rich variety of particle/liquid systems, independent of whether the second liquid wets the particles better or worse than the primary liquid. These admixtures form stable suspensions where settling would otherwise occur and may serve as a precursor for microporous polymer foams, or lightweight ceramics.

Effective interactions in polydisperse colloidal suspensions investigated using Ornstein-Zernike integral equations

We present a mean of calculating the effective interactions in polydisperse colloidal suspension from liquid state integral equation theory. The method is based on Lado's expansion of correlation functions in a suitable set of orthogonal polynomials. The outlined approach is subsequently used to investigate the effects of polydispersity on the effective potentials for model systems with attractive and repulsive bare interactions. The dominant effect of polydispersity of the smaller species is to weaken the effective potentials between big colloidal particles. This can be exploited as another way of tuning the interactions in colloidal suspensions to match the desired properties.

Coarse-graining in suspensions of charged nanoparticles

A coarse-grain description of nanocolloidal suspensions in the presence of an added salt is presented here. It enables us to simulate trajectories of the nanoparticles from effective functions that depend on average densities of salt ions. In practice, the ion-averaged effective potential is used as input of a Brownian dynamics (BD) simulation. This potential may be derived by various methods, ranging from purely analytical to fully numerical ones. For the description of dynamical properties, this simulation also requires an effective diffusion coefficient that must be calculated or experimentally determined, and that accounts for the effects of microions on the mobility of the nanoparticles. The different versions of our coarse-graining procedure are applied to the case of a maghemite suspension, for which an explicit description of all ions would be very time-consuming.

Toward the description of electrostatic interactions between globular proteins: Potential of mean force in the primitive model

Monte Carlo simulations are used to calculate the exact potential of mean force between charged globular proteins in aqueous solution. The aim of the present paper is to study the influence of the ions of the added salt on the effective interaction between these nanoparticles. The charges of the model proteins, either identical or opposite, are either central or distributed on a discrete pattern. Contrarily to Poisson-Boltzmann predictions, attractive, and repulsive direct forces between proteins are not screened similarly. Moreover, it has been shown that the relative orientations of the charge patterns strongly influence salt-mediated interactions. More precisely, for short distances between the proteins, ions enhance the difference of the effective forces between (i) like-charged and oppositely charged proteins, (ii) attractive and repulsive relative orientations of the proteins, which may affect the selectivity of protein/protein recognition. Finally, such results observed with the simplest models are applied to a more elaborate one to demonstrate their generality.

New coarse-graining procedure for the dynamics of charged spherical nanoparticles in solution

A multiscale strategy based on the Brownian dynamics (BD) simulation method is presented here. It leads to an approximate but realistic reproduction of the dynamics of charged nanoparticles in suspension. This method is particularly suited to systems containing highly dissymmetric electrolytes with added salts, such as micellar suspensions or protein solutions. The coarse-graining procedure leads to a description where only the translational degrees of freedom of the nanoparticles are left, all the degrees of freedom related to the smallest solutes being rigorously averaged out. The authors' contribution aims at quantitatively evaluating the influence of the eliminated forces on the dynamics of the nanoparticles. For this purpose, an effective diffusion coefficient has to be calculated. In practice, this effective diffusion coefficient is taken as an input of a coarse-grained simulation that uses the potential of mean force between nanoparticles. The procedure has been validated by the quantitative comparison between the coarse-grained calculations and BD simulations at the "microscopic" level of description (which explicitly include microions). For a model of aqueous solutions of 10-1 electrolyte with a 1-1 added salt, the agreement is found to be excellent. This new method allows us to compute the diffusion coefficients of nanoparticles with a computation time at least one order of magnitude lower than with explicit BD.

Phase behavior of a suspension of colloidal hard rods and nonadsorbing polymer

We study the phase behavior of a mixture of colloidal hard rods with a length-to-diameter ratio of L/sigma(c)=5 and nonadsorbing ideal polymer. We map our binary mixture onto an effective one-component system by integrating out the degrees of freedom of the polymer coils. We derive a formal expression for the exact effective Hamiltonian of the colloidal rods, i.e., it includes all effective many-body interactions and it is related to the exact free volume available for the polymer. We determine numerically on a grid the free volume available for the ideal polymer coils "on the fly" for each colloidal rod configuration during our Monte Carlo simulations. This allows us to go beyond first-order perturbation theory, which employs the pure hard-rod system as reference state. We perform free energy calculations for the isotropic, nematic, smectic, and crystal phase using thermodynamic integration and common tangent constructions are used at fixed polymer fugacities to map out the phase diagram. The phase behavior is determined for size ratios q=sigma(p)/sigma(c)=0.15, 0.5, and 1, where sigma(p) is the diameter of the polymer coils. The phase diagrams based on the full effective Hamiltonian are compared with those obtained from first-order perturbation theory, from simulations using the effective pair potential approximation to the effective Hamiltonian, and with those based on an empiric effective depletion potential for the rods. We find that the many-body character of the effective interactions stabilizes the nematic and smectic phases for large q, while the effective pair potential description overestimates the attractive interactions and favors, hence, a broad isotropic-crystal coexistence.

Anomalous small-angle X-ray scattering: analyzing correlations and fluctuations in polyelectrolytes

We review recent structural investigations done by anomalous small-angle X-ray scattering (ASAXS). ASAXS uses the dependence of the scattering length of a given element if the energy of the incident X-ray beam is near the absorption edge of this element. The analysis of the ASAXS data leads to three partial intensities. We show that the comparison of these three partial intensities leads to valuable information in fluctuating systems. This has been demonstrated from data derived from recent molecular dynamics simulations of charged colloidal spheres. Moreover, it is shown that the three partial intensities can be obtained from experimental ASAXS data indeed. As an example for this analysis, we discuss recent ASAXS data referring to rod-like polyelectrolytes. These polyelectrolytes consist of a stiff poly(p-phenylene) backbone with attached charged groups that are balanced by bromine counterions. The three partial intensities can be determined experimentally and compared to the prediction of the Poisson-Boltzmann cell model. Quantitative agreement is found demonstrating the strong correlation of the counterions to the rod-like macroion. ASAXS is thus shown to furnish information not available by the conventional small-angle scattering experiment.

Phase behavior and structure of model colloid-polymer mixtures confined between two parallel planar walls

Using Gibbs ensemble Monte Carlo simulations and density functional theory we investigate the fluid-fluid demixing transition in inhomogeneous colloid-polymer mixtures confined between two parallel plates with separation distances between one and ten colloid diameters covering the complete range from quasi-two-dimensional to bulklike behavior. We use the Asakura-Oosawa-Vrij model in which colloid-colloid and colloid-polymer interactions are hard-sphere like, while the pair potential between polymers vanishes. Two different types of confinement induced by a pair of parallel walls are considered--namely, either through two hard walls or through two semipermeable walls that repel colloids but allow polymers to freely penetrate. For hard (semipermeable) walls we find that the capillary binodal is shifted towards higher (lower) polymer fugacities and lower (higher) colloid fugacities as compared to the bulk binodal; this implies capillary condensation (evaporation) of the colloidal liquid phase in the slit. A macroscopic treatment is provided by a symmetric Kelvin equation for general binary mixtures based on the proximity in chemical potentials of statepoints at capillary coexistence and the reference bulk coexistence. Results for capillary binodals compare well with those obtained from the classic version of the Kelvin equation due to [Evans and Marini Bettolo Marconi, J. Chem. Phys. 86, 7138 (1987)] and are quantitatively accurate away from the fluid-fluid critical point, even at small wall separations. However, the significant shift of the critical polymer fugacity towards higher values upon increasing confinement, as found in simulations, is not reproduced. For hard walls the density profiles of polymers and colloids inside the slit display oscillations due to packing effects for all statepoints. For semipermeable walls either similar structuring or flat profiles are found, depending on the statepoint considered.

Effect of many-body interactions on the bulk and interfacial phase behavior of a model colloid-polymer mixture

We study a model suspension of sterically stabilized colloidal particles and nonadsorbing ideal polymer coils, both in bulk and adsorbed against a planar hard wall. By integrating out the degrees of freedom of the polymer coils, we derive a formal expression for the effective one-component Hamiltonian of the colloids. We employ an efficient Monte Carlo simulation scheme for this mixture based on the exact effective colloid Hamiltonian; i.e., it incorporates all many-body interactions. The many-body character of the polymer-mediated effective interactions between the colloids yields bulk phase behavior and adsorption phenomena that differ substantially from those found for pairwise simple fluids. We determine the phase behavior for size ratios q=sigma(p)/sigma(c)=1, 0.6, and 0.1, where sigma(c) and sigma(p) denote the diameters of the colloids and polymer coils, respectively. For q=1 and 0.6, we find both a fluid-solid and a stable colloidal gas-liquid transition with an anomalously large bulk liquid regime caused by the many-body interactions. We compare the phase diagrams obtained from simulations with the results of the free-volume approach and with direct simulations of the true binary mixture. Although we did not simulate the polymer coils explicitly, we are able to obtain the three partial structure factors and radial distribution functions. We compare our results with those obtained from density functional theory and the Percus-Yevick approximation. We find good agreement between all results for the structure. We also study the mixture in contact with a single hard wall for q=1. Upon approach of the gas-liquid binodal, we find far from the triple point, three layering transitions in the partial wetting regime.

Colloidal crystal assembly on topologically patterned templates

A variety of methods have been successfully used to produce crystals of colloidal particles made of polymeric (or silica) microspheres. Achieving highly accurate growth and control of the packing symmetry, packing efficiency and packing quality of these crystals is of paramount importance for many applications, for example in photonics. If colloidal crystals are formed in self-assembly processes, it is usually the most densely packed (111) set of planes that terminates the crystal-air interface. However, often exposure of (given) different packing facets is required at the crystal surface. In addition, there is ( in photonics) need for crystals exhibiting lower than the tightest packing, and possessing also lower degrees of packing symmetry. These requirements demand development of various engineering approaches for controlled particle assembly in regular structures. The synthesis of polymer colloidal particles with different sizes, shapes and surface charge density is first briefly outlined. The various interactions and forces that control growth for a broad range of colloidal crystals are subsequently discussed. In the main section of this review we give an account of various template-assisted, graphoepitaxial assembly approaches to produce colloidal crystals with tailored packing structures and controlled crystal orientation with respect to the topologically patterned substrates used to direct the assembly process. In the outlook we also describe various selected emerging approaches, which have the potential to produce crystals with low degree of packing symmetries, for example using direct one-to-one colloidal particle assembly.

Modeling of drying in films of colloidal particles

The process of film formation on a solid substrate from polymer colloid dispersion during solvent evaporation has been investigated by means of the Monte Carlo simulation method. Colloid particles are modeled as hard spheres. Time evolution of the colloid density distribution and coverage of the solid substrate are studied. Both density and structure of colloid film is shown to depend strongly on the evaporation rate. At a low evaporation rate, the coexistence of hexagonal and tetragonal domains of dried colloid monolayer has been observed. The results of monolayer structure are in good agreement with the confocal scanning laser microscopy observations of Dullens et al. (2004).

Capillary waves in a colloid-polymer interface

The structure and the statistical fluctuations of interfaces between coexisting phases in the Asakura-Oosawa model [J. Chem. Phys. 22, 1255 (1954)] for a colloid-polymer mixture are analyzed by extensive Monte Carlo simulations. We make use of a recently developed grand canonical cluster move with an additional constraint stabilizing the existence of two interfaces in the (rectangular) box that is simulated. Choosing very large systems, of size L x L x D with L=60 and D=120, measured in units of the colloid radius, the spectrum of capillary wave-type interfacial excitations is analyzed in detail. The local position of the interface is defined in terms of a (local) Gibbs surface concept. For small wave vectors capillary wave theory is verified quantitatively, while for larger wave vectors pronounced deviations show up. When one analyzes the data in terms of the concept of a wave vector-dependent interfacial tension, a monotonous decrease of this quantity with increasing wave vector is found. Limitations of our analysis are critically discussed.

When mixtures of hard-sphere-like colloids do not behave as mixtures of hard spheres

The validity of the concept of "hard-sphere-like" particles for mixtures of colloids is questioned from a theoretical point of view. This concerns the class of pseudobinary mixtures in which the nonsteric interactions between the colloids are "residual" (with very small range and moderate strength). It is shown that contrary to common expectation, such interactions may have unexpected consequences on the theoretical phase diagram. The distinction between this situation and true solute-solvent mixtures is emphasized.

Grand canonical Monte Carlo simulation of a model colloid–polymer mixture: Coexistence line, critical behavior, and interfacial tension

Grand canonical Monte Carlo simulations are used to study phase separation in a simple colloid-polymer model, the so-called Asakura-Oosawa model. To overcome the problem of small acceptance rates of the grand-canonical moves, cluster moves are introduced. Successive umbrella sampling, recently introduced by Virnau and Muller [J. Chem. Phys. 120, 10925 (2004)], is used to access the phase-separated regime. The unmixing binodal and the interfacial tension are measured and compared to theoretical predictions. By means of finite-size scaling, the behavior close to the critical point is also investigated. Close to criticality, we observe substantial deviations from mean-field behavior.

Effect of three-body interactions on the phase behavior of charge-stabilized colloidal suspensions

We study numerically the effect of attractive triplet interactions on the phase behavior of suspensions of highly charged colloidal particles at low salinity. In our computer simulations, we employ the pair and triplet potentials that were obtained from a numerical Poisson-Boltzmann study [Phys. Rev. E 66, 011402 (2002)]]. On the basis of free energy calculations, we determine the phase diagram of an aqueous suspension of identical spheres of diameter sigma=32 nm and charge Z=80 as a function of colloid concentration and salinity, both for the purely pairwise additive system and for the system with pair and triplet interactions. The main effect of including the triplet interactions is a destabilization of the body-centered-cubic (bcc) crystal phase in favor of the face-centered-cubic (fcc) crystal phase. As a consequence the phase diagram features the coexistence of a rather dilute fluid with an almost-close-packed fcc phase at low salinity and bcc-fcc coexistence with a big density jump at intermediate salinity. The triplet attractions do not affect the phase behavior at sufficiently high salinity; under these conditions the system is well described by the pairwise potential.

Overcharging in colloids: beyond the Poisson-Boltzmann approach

A broad range of manufactured products and biological fluids are colliods. The ability to understand and control the processes (of scientific, technological and industrial interest) in which such colloids are involved relies upon a precise knowledge of the electrical double layer. The traditional approach to describing this ion cloud around colloidal particles has been the Gouy-Chapman model developed on the basis of the Poisson-Boltzmann equation. Since the early 1980s, however, more sophisticated theoretical treatments have revealed both quantitative and qualitative deficiencies in the Poisson-Boltzmann theory, particularly at high ionic strengths and/or high surface charge densities. This review deals with these novel approaches, which are mostly computer simulations and approximate integral equation theories based on the so-called primitive model. Special attention is paid to phenomena that cannot be accounted for by the classic theory as a result of neglecting ion size correlations, such as overcharging, namely, the counterion concentration in the immediate neighborhood of the surface is so large that the particle surface is overcompensated. Other illustrative examples are the nonmonotonic behavior of the electrostatic potential and attractive interactions between equally charged surfaces. These predictions are certainly remarkable and, on paper, they can have an effect on experimentally measurable quantities (for instance, electrophoretic mobility). Even so, these new approaches have scarcely been applied in practice. Thus a critical survey on the relevance of ion size correlation in real systems is also included. Overcharging of macroions can also be brought about by adsorption of oppositely charged polyelectrolytes. Noteworthy examples and theoretical approaches for them are also briefly reviewed.

Three-body forces between charged colloidal particles

Within nonlinear Poisson-Boltzmann theory we calculate the pair and triplet interactions between charged colloidal spheres, specifically in the nonlinear regime of low salt concentrations and high charges. We find repulsive pair interactions and attractive triplet interactions. Within a van der Waals-like mean-field theory we estimate in which parameter regime a gas-liquid coexistence is to be expected.