Three-body forces between charged colloidal particles

Coarse-Grained Many-Body Potentials of Ligand-Stabilized Nanoparticles from Machine-Learned Mean Forces

Colloidal nanoparticles self-assemble into a variety of superstructures with distinctive optical, structural, and electronic properties. These nanoparticles are usually stabilized by a capping layer of organic ligands to prevent aggregation in the solvent. When the ligands are sufficiently long compared to the dimensions of the nanocrystal cores, the effective coarse-grained forces between pairs of nanoparticles are largely affected by the presence of neighboring particles. In order to efficiently investigate the self-assembly behavior of these complex colloidal systems, we propose a machine-learning approach to construct effective coarse-grained many-body interaction potentials. The multiscale methodology presented in this work constitutes a general bottom-up coarse-graining strategy where the coarse-grained forces acting on coarse-grained sites are extracted from measuring the vectorial mean forces on these sites in reference fine-grained simulations. These effective coarse-grained forces, i.e., gradients of the potential of mean force or of the free-energy surface, are represented by a simple linear model in terms of gradients of structural descriptors, which are scalar functions that are rotationally invariant. In this way, we also directly obtain the free-energy surface of the coarse-grained model as a function of all coarse-grained coordinates. We expect that this simple yet accurate coarse-graining framework for the many-body potential of mean force will enable the characterization, understanding, and prediction of the structure and phase behavior of relevant soft-matter systems by direct simulations. The key advantage of this method is its generality, which allows it to be applicable to a broad range of systems. To demonstrate the generality of our method, we also apply it to a colloid-polymer model system, where coarse-grained many-body interactions are pronounced.

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.

Collective self-optimization of communicating active particles

The quest for how to collectively self-organize in order to maximize the survival chances of the members of a social group requires finding an optimal compromise between maximizing the well-being of an individual and that of the group. Here we develop a minimal model describing active individuals which consume or produce, and respond to a shared resource-such as the oxygen concentration for aerotactic bacteria or the temperature field for penguins-while urging for an optimal resource value. Notably, this model can be approximated by an attraction-repulsion model, but, in general, it features many-body interactions. While the former prevents some individuals from closely approaching the optimal value of the shared "resource field," the collective many-body interactions induce aperiodic patterns, allowing the group to collectively self-optimize. Arguably, the proposed optimal field-based collective interactions represent a generic concept at the interface of active matter physics, collective behavior, and microbiological chemotaxis. This concept might serve as a useful ingredient to optimize ensembles of synthetic active agents or to help unveil aspects of the communication rules which certain social groups use to maximize their survival chances.

Rigorous treatment of pairwise and many-body electrostatic interactions among dielectric spheres at the Debye-Hückel level

Electrostatic interactions among colloidal particles are often described using the venerable (two-particle) Derjaguin-Landau-Verwey-Overbeek (DLVO) approximation and its various modifications. However, until the recent development of a many-body theory exact at the Debye-Hückel level (Yu in Phys Rev E 102:052404, 2020), it was difficult to assess the errors of such approximations and impossible to assess the role of many-body effects. By applying the exact Debye-Hückel level theory, we quantify the errors inherent to DLVO and the additional errors associated with replacing many-particle interactions by the sum of pairwise interactions (even when the latter are calculated exactly). In particular, we show that: (1) the DLVO approximation does not provide sufficient accuracy at shorter distances, especially when there is an asymmetry in charges and/or sizes of interacting dielectric spheres; (2) the pairwise approximation leads to significant errors at shorter distances and at large and moderate Debye lengths and also gets worse with increasing asymmetry in the size of the spheres or magnitude or placement of the charges. We also demonstrate that asymmetric dielectric screening, i.e., the enhanced repulsion between charged dielectric bodies immersed in media with high dielectric constant, is preserved in the presence of free ions in the medium.

2D colloids in rotating electric fields: A laboratory of strong tunable three-body interactions

Many-body forces play a prominent role in structure and dynamics of matter, but their role is not well understood in many cases due to experimental challenges. Here, we demonstrate that a novel experimental system based on rotating electric fields can be utilised to deliver unprecedented degree of control over many-body interactions between colloidal silica particles in water. We further show that we can decompose interparticle interactions explicitly into the leading terms and study their specific effects on phase behaviour. We found that three-body interactions exert critical influence over the phase diagram domain boundaries, including liquid-gas binodal, critical and triple points. Phase transitions are shown to be reversible and fully controlled by the magnitude of external rotating electric field governing the tunable interactions. Our results demonstrate that colloidal systems in rotating electric fields are a unique laboratory to study the role of many-body interactions in physics of phase transitions and in applications, such as self-assembly, offering exciting opportunities for studying generic phenomena inherent to liquids and solids, from atomic to protein and colloidal systems.

Core-shell particles in rotating electric and magnetic fields: Designing tunable interactions via particle engineering

Tunable interactions between colloidal particles, governed by external rotating electric or magnetic fields, yield rich capabilities for prospective self-assembly technologies of materials and fundamental particle-resolved studies of phase transitions and transport phenomena in soft matter. However, the role of the internal structure of colloidal particles in the tunable interactions has never been systematically investigated. Here, we study the tunable interactions between composite particles with core-shell structure in a rotating electric field and show that the engineering of their internal structure provides an effective tool for designing the interactions. We generalized an integral theory and studied the tunable interactions between core-shell particles with homogeneous cores (layered particles) and cores with nano-inclusions to reveal the main trends in the interactions influenced by the structure. We found that depending on the materials of the core, shell, and solvent, the interactions with the attractive pairwise part and positive or negative three-body part can be obtained, as well as pairwise repulsion with attractive three-body interactions (for triangular triplets). The latter case is observed for the first time, being unattainable for homogeneous particles but feasible with core-shell particles: Qualitatively similar interactions are inherent to charged colloids (repulsive pairwise and attractive three-body energies), known as a model system of globular proteins. The methods and conclusions of our paper can be generalized for magnetic and 3D colloidal systems. The results make a significant advance in the analysis of tunable interactions in colloidal systems, which are of broad interest in condensed matter, chemical physics, physical chemistry, materials science, and soft matter.

Experimental validation of interpolation method for pair correlations in model crystals

Accurate analysis of pair correlations in condensed matter allows us to establish relations between structures and thermodynamic properties and, thus, is of high importance for a wide range of systems, from solids to colloidal suspensions. Recently, the interpolation method (IM) that describes satisfactorily the shape of pair correlation peaks at short and at long distances has been elaborated theoretically and using molecular dynamics simulations, but it has not been verified experimentally as yet. Here, we test the IM by particle-resolved studies with colloidal suspensions and with complex (dusty) plasmas and demonstrate that, owing to its high accuracy, the IM can be used to experimentally measure parameters that describe interaction between particles in these systems. We used three- and two-dimensional colloidal crystals and monolayer complex (dusty) plasma crystals to explore suitability of the IM in systems with soft to hard-sphere-like repulsion between particles. In addition to the systems with pairwise interactions, if many-body interactions can be mapped to the pairwise ones with some effective (e.g., density-dependent) parameters, the IM could be used to obtain these parameters. The results reliably show that the IM can be effectively used for analysis of pair correlations and interactions in a wide variety of systems and therefore is of broad interest in condensed matter, complex plasma, chemical physics, physical chemistry, materials science, and soft matter.

Long-range electron-electron interaction and charge transfer in protein complexes: a numerical approach

With application to the nitrite reductase hexameric protein complex of Desulfovibrio vulgaris, NrfH2A4, we suggest a strategy to compute the energy landscape of electron transfer in large systems of biochemical interest. For small complexes, the energy of all electronic configurations can be scanned completely on the level of a numerical solution of the Poisson-Boltzmann equation. In contrast, larger systems have to be treated using a pair approximation, which is verified here. Effective Coulomb interactions between neighbouring sites of excess electron localization may become as large as 200 meV, and they depend in a nontrivial manner on the intersite distance. We discuss the implications of strong Coulomb interactions on the thermodynamics and kinetics of charging and decharging a protein complex. Finally, we turn to the effect of embedding the system into a biomembrane.

Charge regulation of nonpolar colloids

Individual colloids often carry a charge as a result of the dissociation (or adsorption) of weakly-ionized surface groups. The magnitude depends on the precise chemical environment surrounding a particle, which in a concentrated dispersion is a function of the colloid packing fraction η. Theoretical studies have suggested that the effective charge Z_eff in regulated systems could, in general, decrease with increasing η. We test this hypothesis for nonpolar dispersions by determining Z_eff(η) over a wide range of packing fractions (10^-5 ≤ η ≤ 0.3) using a combination of small-angle X-ray scattering and electrophoretic mobility measurements. All dispersions remain entirely in the fluid phase regime. We find a complex dependence of the particle charge as a function of the packing fraction, with Z_eff initially decreasing at low concentrations before finally increasing at high η. We attribute the non-monotonic density dependence to a crossover from concentration-independent screening at low η, to a high packing fraction regime in which counterions outnumber salt ions and electrostatic screening becomes η-dependent. The efficiency of charge stabilization at high concentrations may explain the unusually high stability of concentrated nanoparticle dispersions which has been reported.

Three-body critical Casimir forces

Within mean-field theory we calculate universal scaling functions associated with critical Casimir forces for a system consisting of three parallel cylindrical colloids immersed in a near-critical binary liquid mixture. For several geometrical arrangements and boundary conditions at the surfaces of the colloids we study the force between two colloidal particles in the direction normal to their axes, analyzing the influence of the presence of a third particle on that force. Upon changing temperature or the relative positions of the particles we observe interesting features such as a change of sign of this force caused by the presence of the third particle. We determine the three-body component of the forces acting on one of the colloids by subtracting the pairwise forces from the total force. The three-body contribution to the total critical Casimir force turns out to be more pronounced for small surface-to-surface distances between the colloids as well as for temperatures close to criticality. Moreover, we compare our results with similar ones for other physical systems such as three atoms interacting via van der Waals forces.

Anisotropic elasticity of experimental colloidal Wigner crystals

Colloidal particles interacting via a long-range repulsion can, in contrast to hard-sphere systems, exhibit crystalline ordering at low volume fraction. Here we experimentally investigate the structure and properties of such "colloidal Wigner crystals." We find a body-centered-cubic crystalline phase at volume fractions of ϕ≳15%, which exhibits large fluctuations of individual particles from their average positions. We determine the three independent crystalline elastic constants and find that these crystals are very compliant and highly anisotropic.

Electrostatics at the nanoscale

Electrostatic forces are amongst the most versatile interactions to mediate the assembly of nanostructured materials. Depending on experimental conditions, these forces can be long- or short-ranged, can be either attractive or repulsive, and their directionality can be controlled by the shapes of the charged nano-objects. This Review is intended to serve as a primer for experimentalists curious about the fundamentals of nanoscale electrostatics and for theorists wishing to learn about recent experimental advances in the field. Accordingly, the first portion introduces the theoretical models of electrostatic double layers and derives electrostatic interaction potentials applicable to particles of different sizes and/or shapes and under different experimental conditions. This discussion is followed by the review of the key experimental systems in which electrostatic interactions are operative. Examples include electroactive and "switchable" nanoparticles, mixtures of charged nanoparticles, nanoparticle chains, sheets, coatings, crystals, and crystals-within-crystals. Applications of these and other structures in chemical sensing and amplification are also illustrated.

Brownian dynamics simulation of the crystallization dynamics of charged colloidal particles

Crystal formation process of charged colloidal particles is investigated using Brownian dynamics (BD) simulations. The particles are assumed to interact with the pair-additive repulsive Yukawa potential. The time evolution of crystallization process and the crystal structure during the simulation are characterized by means of the radial distribution functions (RDF) and mean square displacement (MSD). The simulations show that when the interaction is featured with long-range, particles can spontaneously assemble into body-centered-cubic (BCC) arrays at relatively low particle number density. When the interaction is short-ranged, with increasing the number density particles become trapped into a stagnant disordered configuration before the crystallization could be actualized. The simulations further show that as long as the trapped configurations are bypassed, the face-centered-cubic (FCC) structures can be achieved and are actually more stable than BCC structures.

Multilayered crystals of macroions under slit confinement

The crystalline ground state of macroions confined between two neutral parallel plates in the presence of their homogeneously spread counterions is calculated by lattice sum minimization of candidate phases involving up to six layers. For increasing macroion density, a cascade of solid-solid transitions is found involving various multilayered crystals. The cascade includes triangular monolayer and buckled bilayer as well as rhombic, squared and triangular phase structures.

Many-body electrostatic forces between colloidal particles at vanishing ionic strength

Electrostatic forces between small groups of colloidal particles are measured using blinking optical tweezers. When the electrostatic screening length is longer than the interparticle separation, forces are found to be non-pairwise-additive. Both pair and multiparticle forces are well described by the linearized Poisson-Boltzmann equation with constant potential boundary conditions. These findings may play an important role in understanding the structure and stability of a wide variety of systems, from micron-sized particles in oil to aqueous nanocolloids.

Breakdown of the Yukawa model in de-ionized colloidal suspensions

We study effective colloidal interactions in de-ionized colloidal mixtures through sedimentation-diffusion equilibrium. We derive a coarse-grained effective model (EM) and compare its density profiles with those of the computationally much more expensive primitive model (PM) of colloids and counterions in gravity. The EM, which contains not only standard pairwise screened-Coulomb interactions, but also explicit many-body effects by means of a so-called volume term, can quantitatively account for all observed sedimentation phenomena such as lifting of colloids to high altitudes, segregation into layers in mixtures, and floating of heavy colloids on top of lighter ones. Without the volume term there is no quantitative agreement between the PM and EM, even in the present high-temperature limit of interest, showing that de-ionized colloidal suspensions cannot be described by a pairwise Yukawa model.

Sedimentation of charged colloids: the primitive model and the effective one-component approach

Sedimentation-diffusion equilibrium density profiles of suspensions of charge-stabilized colloids are calculated theoretically and by Monte Carlo (MC) simulations, 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, J. Phys.: Condens Matter 17, 2315 (2005)]. Despite the apparently different picture that emerges from the one- and three-component models (repelling colloids pushing each other to high altitude in the former, versus a self-generated electric field that pushes the colloids up in the latter), we find similar colloidal density profiles for both models from theory as well as simulation, thereby suggesting that these pictures represent different viewpoints of the same phenomenon. The sedimentation profiles obtained from an effective one-component model by MC simulations and theory, together with MC simulations of the multicomponent primitive model are consistent among themselves, but differ quantitatively from the results of a theoretical multicomponent description at the Poisson-Boltzmann level. We find that for small and moderate colloid charge the Poisson-Boltzmann theory gives profiles in excellent agreement with the effective one-component theory if a smaller effective charge is used. We attribute this discrepancy to the poor treatment of correlations in the Poisson-Boltzmann theory.

Noncentral forces in crystals of charged colloids

The elastic properties of fcc crystals consisting of charge stabilized colloidal particles are determined from real space imaging experiments using confocal microscopy. The normal modes and the force constants of the crystal are obtained from the fluctuations of the particles around their lattice sites using the equipartition theorem. We show that the Cauchy relation is not fulfilled and that only noncentral many-body forces can account for the elastic properties.

Nonlinear screening and gas-liquid separation in suspensions of charged colloids

We calculate phase diagrams of charged colloidal spheres (valency Z and radius a) in a 1:1 electrolyte from multicentered nonlinear Poisson-Boltzmann theory. Our theory takes into account charge renormalization of the colloidal interactions and volume terms due to many-body effects. For valencies as small as Z = 1 and as large as 10(4) we find a gas-liquid spinodal instability in the colloid-salt phase diagram provided Z lambdaB/a > or similar 24+/-1, where lambdaB is the Bjerrum length.

Effective triplet interactions in nematic colloids

Three-body effective interactions emerging between parallel cylindrical rods immersed in a nematic liquid crystals are calculated within the Landau-de Gennes free-energy description. Collinear, equilateral and midplane configurations of the three colloidal particles are considered. In the last two cases the effective triplet interaction is of the same magnitude and range as the pair one.

Nonlinear effects in charge stabilized colloidal suspensions

Molecular-dynamics simulations are used to study the effective interactions in charged stabilized colloidal suspensions. We focus on highly charged macroions in the limit of low salt concentrations. Within this regime, nonlinear corrections to Debye-Hückel (DH) theory have to be considered. For non-bulk-like systems, such as isolated pairs or triples of macroions, we show that nonlinear effects can become relevant, which cannot be described by the charge renormalization concept [S. Alexander, J. Chem. Phys. 80, 5776 (1984)]. For an isolated pair of macroions, we find an almost perfect qualitative agreement between our simulation data and DH theory. However, on a quantitative level, neither DH theory nor the charge renormalization concept can be confirmed in detail. This seems mainly to be related to the fact that for small ion concentrations, microionic layers can strongly overlap, whereas, simultaneously, excluded volume effects are less important. In the case of isolated triples, where we compare between coaxial and triangular geometries, we find attractive corrections to pairwise additivity in the limit of small macroion separations and salt concentrations. These triplet interactions arise if all three microionic layers around the macroions exhibit a significant overlap. In contrast to the case of two isolated colloids, the charge distribution around a macroion in a triple is found to be anisotropic.

Renormalization in charged colloids: non-monotonic behaviour with the surface charge

The static structure factor S(q) is measured for a set of deionized latex dispersions with different numbers of ionizable surface groups per particle and similar diameters. For a given volume fraction, the height of the main peak of S(q), which is a direct measure of the spatial ordering of latex particles, does not increase monotonically with the number of ionizable groups. This behaviour cannot be described using the classical renormalization scheme based on the cell model. We analyse our experimental data using a renormalization model based on the jellium approximation, which predicts the weakening of the spatial order for moderate and large particle charges.

Re-entrant melting and freezing in a model system of charged colloids

We studied the phase behavior of charged and sterically stabilized colloids using confocal microscopy in a low polarity solvent (dielectric constant 5.4). Upon increasing the colloid volume fraction we found a transition from a fluid to a body centered cubic crystal at 0.0415+/-0.0005, followed by reentrant melting at 0.1165+/-0.0015. A second crystal of different symmetry, random hexagonal close packed, was formed at a volume fraction around 0.5, similar to that of hard spheres. We attribute the intriguing phase behavior to the particle interactions that depend strongly on volume fraction, mainly due to the changes in the colloid charge. In this low polarity system the colloids acquire charge through ion adsorption. The low ionic strength leads to fewer ions per colloid at elevated volume fractions and consequently a density-dependent colloid charge.

How does column packing microstructure affect column efficiency in liquid chromatography?

Full three-dimensional computer simulations of the fluid flow and dispersion characteristics of model nonporous chromatographic packings are reported. Interstitial porosity and packing defects are varied in an attempt to understand the chromatographic consequences of the packing microstructure. The tracer zone dispersion is calculated in the form of plate height as a function of fluid velocity for seven model particle packs where particles are selectively removed from the packs in clusters of varying size and topology. In an attempt to examine the consequences of loose but random packs, the velocities and zone dispersion of seven defect-free packs are simulated over the range 0.36< or =epsilon< or =0.50, where epsilon is the interstitial porosity. The results indicate that defect-free loose packings can give good chromatographic efficiency but the efficiency can vary depending on subtle details of the pack. When the defect population increases, the zone dispersion increases accordingly. For a particle pack where 6% of the particles are removed from an epsilon=0.36 pack, approximately 33% of the column efficiency is lost. These results show that it is far more important in column packing to prevent defect sites leading to inhomogeneous packing rather than obtaining the highest density pack with the smallest interstitial void volume.

Phase separation in charge-stabilized colloidal suspensions: influence of nonlinear screening

The phase behavior of charge-stabilized colloidal suspensions is modeled by a combination of response theory for electrostatic interparticle interactions and variational theory for free energies. Integrating out degrees of freedom of the microions (counterions, salt ions), the macroion-microion mixture is mapped onto a one-component system governed by effective macroion interactions. Linear response of microions to the electrostatic potential of the macroions results in a screened-Coulomb (Yukawa) effective pair potential and a one-body volume energy, while nonlinear response modifies the effective interactions [A. R. Denton, Phys. Rev. E 70, 031404 (2004)]. The volume energy and effective pair potential are taken as input to a variational free energy, based on thermodynamic perturbation theory. For both linear and first-order nonlinear effective interactions, a coexistence analysis applied to aqueous suspensions of highly charged macroions and monovalent microions yields bulk separation of macroion-rich and macroion-poor phases below a critical salt concentration, in qualitative agreement with predictions of related linearized theories [R. van Roij, M. Dijkstra, and J.-P. Hansen, Phys. Rev. E 59, 2010 (1999); P. B. Warren, J. Chem. Phys. 112, 4683 (2000)]. It is concluded that nonlinear screening can modify phase behavior but does not necessarily suppress bulk phase separation of de-ionized suspensions.

Volume terms for charged colloids: a grand-canonical treatment

We present a study of thermodynamic properties of suspensions of charged colloids on the basis of linear Poisson-Boltzmann theory. We calculate the effective Hamiltonian of the colloids by integrating out the ionic degrees of freedom grand canonically. This procedure not only yields the well-known pairwise screened-Coulomb interaction between the colloids, but also additional volume terms that affect the phase behavior and the thermodynamic properties, such as the osmotic pressure. These calculations are greatly facilitated by the grand-canonical character of our treatment of the ions and allow for relatively fast computations compared to earlier studies in the canonical ensemble. Moreover, the present derivation of the volume terms are relatively simple, make a direct connection with Donnan equilibrium, yield an explicit expression for the effective screening constant, and allow for extensions to include, for instance, nonlinear effects.

Melting line of charged colloids from primitive model simulations

We develop an efficient simulation method to study suspensions of charged spherical colloids using the primitive model. In this model, the colloids and the co- and counterions are represented by charged hard spheres, whereas the solvent is treated as a dielectric continuum. In order to speed up the simulations, we restrict the positions of the particles to a cubic lattice, which allows precalculation of the Coulombic interactions at the beginning of the simulation. Moreover, we use multiparticle cluster moves that make the Monte Carlo sampling more efficient. The simulations are performed in the semigrand canonical ensemble, where the chemical potential of the salt is fixed. Employing our method, we study a system consisting of colloids carrying a charge of 80 elementary charges and monovalent co- and counterions. At the colloid densities of our interest, we show that lattice effects are negligible for sufficiently fine lattices. We determine the fluid-solid melting line in a packing fraction eta-inverse screening length kappa plane and compare it with the melting line of charged colloids predicted by the Yukawa potential of the Derjaguin-Landau-Verwey-Overbeek theory. We find qualitative agreement with the Yukawa results, and we do not find any effects of many-body interactions. We discuss the difficulties involved in the mapping between the primitive model and the Yukawa model at high colloid packing fractions (eta>0.2).

Structure of highly asymmetric hard-sphere mixtures: An efficient closure of the Ornstein-Zernike equations

A simple modification of the reference hypernetted chain (RHNC) closure of the multicomponent Ornstein-Zernike equations with bridge functions taken from Rosenfeld's hard-sphere bridge functional is proposed. Its main effect is to remedy the major limitation of the RHNC closure in the case of highly asymmetric mixtures--the wide domain of packing fractions in which it has no solution. The modified closure is also much faster, while being of similar complexity. This is achieved with a limited loss of accuracy, mainly for the contact value of the big sphere correlation functions. Comparison with simulation shows that inside the RHNC no-solution domain, it provides a good description of the structure, while being clearly superior to all the other closures used so far to study highly asymmetric mixtures. The generic nature of this closure and its good accuracy combined with a reduced no-solution domain open up the possibility to study the phase diagram of complex fluids beyond the hard-sphere model.

Effect of counterion species on colloidal crystal

The effect of counterion species on the colloidal crystal structure in a dispersion was carefully investigated as a function of the degree of neutralization (alpha) by the ultra-small-angle X-ray scattering technique. The nearest neighbor interparticle distance (2D(exp)) first increased with decreasing alpha, and then decreased after passing through the maximum. This behavior was confirmed for K(+), Li(+), Ca(2+), TMA(+) (tetramethylammonium) as a counterion, and Na(+) in our previous report (Harada, T.; Matsuoka, H.; Ikeda, T.; Yamaoka, H. Langmuir 2000, 16, 1612). However, the alpha value of the maximum position (alpha(max)) largely depended on the counterion species, and it was in the order K(+) < Na(+) < TMA(+) approximately Li(+). This behavior was well characterized by the specific features of each ion: the alpha(max) map could be well superimposed in the Stokes radius-crystal ion radius relationship of counterions. The alpha(max) dependence on Stokes radius was very similar to that of the B coefficient by Jones and Dole except in the case of Ca(2+). In principle, the smaller the value for B, the smaller alpha(max), indicating that a water structure breaker such as K(+) can more easily destroy the colloidal crystal structure. In other words, the effect of the counterion species on colloidal crystal stability follows the Hofmeister series. Including Ca(2+), the relationship was linear for the alpha(max) values plotted as a function of the limiting equivalent conductivity of small ions. A counterion with larger conductivity would be a stronger breaker for the colloidal crystal structure.

Influence of Confinement on the Electrostatic Interaction between Charged Colloids: a (N,V,T) Monte Carlo Study within Hyperspherical Geometry

Monte Carlo simulations within closed hyperspherical geometry are used to analyze the ionic distribution around two confined charged colloids to determine the origin of the net attraction recently reported in the literature. A scaling procedure is used to compare our numerical results obtained with small ideal colloids with the conclusion of the measurements performed with large silica colloids. Although no electrostatic attraction is detected under confinement, our simulations exhibit a significant reduction of the electrostatic repulsion between charged colloids confined between two weakly charged walls. After rescaling to reproduce the electrostatic repulsion between large confined colloids, our numerical results are qualitatively consistent with the reported attraction because we reasonably expect a reduction of the electrostatic force between such confined colloids below the order of magnitude of their van der Waals attraction.

Nonlinear screening and effective electrostatic interactions in charge-stabilized colloidal suspensions

A nonlinear response theory is developed and applied to electrostatic interactions between spherical macroions, screened by surrounding microions, in charge-stabilized colloidal suspensions. The theory describes leading-order nonlinear response of the microions (counterions, salt ions) to the electrostatic potential of the macroions and predicts microion-induced effective many-body interactions between macroions. A linear response approximation [A.R. Denton, Phys. Rev. E 62, 3855 (2000)] yields an effective pair potential of screened-Coulomb (Yukawa) form, as well as a one-body volume energy, which contributes to the free energy. Nonlinear response generates effective many-body interactions and essential corrections to both the effective pair potential and the volume energy. By adopting a random-phase approximation (RPA) for the response functions, and thus neglecting microion correlations, practical expressions are derived for the effective pair and triplet potentials and for the volume energy. Nonlinear screening is found to weaken repulsive pair interactions, induce attractive triplet interactions, and modify the volume energy. Numerical results for monovalent microions are in good agreement with available ab initio simulation data and demonstrate that nonlinear effects grow with increasing macroion charge and concentration and with decreasing salt concentration. In the dilute limit of zero macroion concentration, leading-order nonlinear corrections vanish. Finally, it is shown that nonlinear response theory, when combined with the RPA, is formally equivalent to the mean-field Poisson-Boltzmann theory and that the linear response approximation corresponds, within integral-equation theory, to a linearized hypernetted-chain closure.

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.

Three-body interactions in colloidal systems

We present a direct measurement of three-body interactions in a colloidal system comprised of three charged colloidal particles. Two of the particles have been confined by means of a scanned laser tweezers to a line-shaped optical trap where they diffused due to thermal fluctuations. Upon the approach of a third particle, attractive three-body interactions have been observed. The results are in qualitative agreement with additionally performed nonlinear Poisson-Boltzmann calculations, which also allow us to investigate the microionic density distributions in the neighborhood of the interacting colloidal particles.

Direct measurement of three-body interactions amongst charged colloids

Three-body interactions amongst three charged colloidal particles are measured in a deionized aqueous solution. Two of the particles are confined to an optical line trap while the third one is approached by means of a focused laser beam. From the observed particle configurations we extract the three-body potential which is found to be attractive and roughly of the same magnitude and range as the pair interactions. In addition, numerical calculations are performed which show qualitative agreement with the experimental results.