Effective attractions between like-charged colloidal particles

A self-consistent Ornstein-Zernike jellium for highly charged colloids (microgels) in suspensions with added salt

This work discusses a jellium scheme, built within the framework of the multicomponent Ornstein-Zernike (OZ) equation, which is capable of describing the collective structure of suspensions of highly charged colloids with added salt, even in the presence of finite-size multivalent microions. This approach uses a suitable approximation to decouple the microion-microion correlations from the macroion-microion profiles, which in combination with the methodology from the dressed ion theory (DIT) gives a full account of the electrostatic effective potential among the colloids. The main advantages of the present contribution reside in its ability to manage the short-range potentials and non-linear correlations among the microions, as well as its realistic characterization of the ionic clouds surrounding each macroion. The structure factors predicted by this jellium scheme are contrasted with previously reported experimental results for microgel suspensions with monovalent salts (2019Phys. Rev. E100032602), thus validating its high accuracy in these situations. The present theoretical analysis is then extended to microgel suspensions with multivalent salts, which reveals the prominent influence of the counterion valence on the makeup of the effective potentials. Although the induced differences may be difficult to identify through the mesoscopic structure, our results suggest that the microgel collapsing transition may be used to enhance these distinct effects, thus giving a feasible experimental probe for these phenomena.

Volume transition effects on the correlations and effective interactions among highly charged microgels

Recent experimental studies have demonstrated the huge influence that the volume phase transition (VPT) has on the collective structure of highly charged thermo-responsive microgels in aqueous solution with low concentrations of added monovalent salt, thus opening a promising new route for controlling the overall properties of practical colloidal suspensions. We present here an analysis of this structure based on the effective electrostatic potential obtained with the exact methodology of the dressed ion theory (DIT). Starting with a description at the primitive model level, we determine the correlations among the components of our model system (macroions plus monovalent anions and cations) by utilizing the two-density integral equation theory, thus allowing us to consider realistic values for the microgel charges. The resulting microgel structure factors show a good agreement with the reported light scattering measurements, whereas the microscopic pair distributions reveal that in this regime the shrunken states promote an enhanced counterion absorption into the microgels. This packing of counterions inside the microgels induces strongly non-linear correlations among the microions, and in turn provokes a substantial weakening of the microgel-microgel correlations. The ensuing effective interactions are then obtained by contracting the description to the level in which only the macroions are present. We find not only that the magnitude and reach of the corresponding pair potentials are markedly inhibited in the shrunken states, but also that their general form diverges from the conventional screened Coulomb shape. This makes it necessary to rethink the concepts of effective charge and screening length.

Interfacial solvation can explain attraction between like-charged objects in aqueous solution

Over the past few decades, the experimental literature has consistently reported observations of attraction between like-charged colloidal particles and macromolecules in aqueous solution. Examples include nucleic acids and colloidal particles in the bulk solution and under confinement, and biological liquid-liquid phase separation. This observation is at odds with the intuitive expectation of an interparticle repulsion that decays monotonically with distance. Although attraction between like-charged particles can be rationalized theoretically in the strong-coupling regime, e.g., in the presence of multivalent counterions, recurring accounts of long-range attraction in aqueous solution containing monovalent ions at low ionic strength have posed an open conundrum. Here, we show that the behavior of molecular water at an interface-traditionally disregarded in the continuum electrostatics picture-provides a mechanism to explain the attraction between like-charged objects in a broad spectrum of experiments. This basic principle will have important ramifications in the ongoing quest to better understand intermolecular interactions in solution.

Accounting for effective interactions among charged microgels

We introduce a theoretical approach to describe structural correlations among charged permeable spheres at finite particle concentrations. This theory explicitly accounts for correlations among microions and between microions and macroions and allows for the proposal of an effective interaction among macroions that successfully captures structural correlations observed in poly-N-isopropyl acrylamide microgel systems. In our description the bare charge is fixed and independent of the microgel size, the microgel concentration, and the ionic strength, which contrasts with results obtained using linear response approximations, where the bare charge needs to be adapted to properly account for microgel correlations obtained at different conditions.

Effective electrostatic interactions in colloid-nanoparticle mixtures

Interparticle interactions and bulk properties of colloidal suspensions can be substantially modified by the addition of nanoparticles. Extreme asymmetries in size and charge between colloidal particles and nanoparticles present severe computational challenges to molecular-scale modeling of such complex systems. We present a statistical mechanical theory of effective electrostatic interactions that can greatly ease large-scale modeling of charged colloid-nanoparticle mixtures. By applying a sequential coarse-graining procedure, we show that a multicomponent mixture of charged colloids, nanoparticles, counterions, and coions can be mapped first onto a binary mixture of colloids and nanoparticles and then onto a one-component model of colloids alone. In a linear-response approximation, the one-component model is governed by a single effective pair potential and a one-body volume energy, whose parameters depend nontrivially on nanoparticle size, charge, and concentration. To test the theory, we perform molecular dynamics simulations of the two-component and one-component models and compute structural properties. For moderate electrostatic couplings, colloid-colloid radial distribution functions and static structure factors agree closely between the two models, validating the sequential coarse-graining approach. Nanoparticles of sufficient charge and concentration enhance screening of electrostatic interactions, weakening correlations between charged colloids and destabilizing suspensions, consistent with experiments.

A cell-model study on counterion fluctuations in macroionic systems: effect of non-extensiveness in entropy

Rigorously speaking, entropy is slightly non-extensive, and this non-extensiveness, which characterizes the degree of fluctuations, can contribute to effective interactions between mesoscopic objects. In this paper, we consider a pair of macroions, each accompanied by 1000 counterions, and with a cell-model description we demonstrate that the slow variation of non-extensiveness in counterion entropy over macroionic distance leads to an effective long-range attraction between the macroions. With the aid of Monte Carlo simulation and a Bragg-Williams theory including counterion number fluctuations, we find the depth of attraction in free energy to be approximately 0.2k(B)T. The observation in our cell-model study provides an insight for further understanding of effective interactions in real macroionic systems.

Large counterions boost the solubility and renormalized charge of suspended nanoparticles

Colloidal particles are ubiquitous in biology and in everyday products such as milk, cosmetics, lubricants, paints, or drugs. The stability and aggregation of colloidal suspensions are of paramount importance in nature and in diverse nanotechnological applications, including the fabrication of photonic materials and scaffolds for biological assemblies, gene therapy, diagnostics, targeted drug delivery, and molecular labeling. Electrolyte solutions have been extensively used to stabilize and direct the assembly of colloidal particles. In electrolytes, the effective electrostatic interactions among the suspended colloids can be changed over various length scales by tuning the ionic concentration. However, a major limitation is gelation or flocculation at high salt concentrations. This is explained by classical theories, which show that the electrostatic repulsion among charged colloids is significantly reduced at high electrolyte concentrations. As a result, these screened colloidal particles are expected to aggregate due to short-range attractive interactions or dispersion forces as the salt concentration increases. We discuss here a robust, tunable mechanism for colloidal stability by which large counterions prevent highly charged nanoparticles from aggregating in salt solutions with concentrations up to 1 M. Large counterions are shown to generate a thicker ionic cloud in the proximity of each charged colloid, which strengthens short-range repulsions among colloidal particles and also increases the corresponding renormalized colloidal charge perceived at larger separation distances. These effects thus provide a reliable stabilization mechanism in a broad range of biological and synthetic colloidal suspensions.

Effective electrostatic interactions in mixtures of charged colloids

We present a theory of effective electrostatic interactions in polydisperse suspensions of charged macroions, generalizing to mixtures a theory previously developed for monodisperse suspensions. Combining linear response theory with a random phase approximation for microion correlations, we coarse grain the microion degrees of freedom to derive general expressions for effective macroion-macroion pair potentials and a one-body volume energy. For model mixtures of charged hard-sphere colloids, we give explicit analytical expressions. The resulting effective pair potentials have the same general form as predicted by linearized Poisson-Boltzmann theory, but consistently incorporate dependence on macroion density and excluded volume via the Debye screening constant. The volume energy, which depends on the average macroion density, contributes to the free energy and so can influence thermodynamic properties of deionized suspensions. To validate the theory, we compute radial distribution functions of binary mixtures of oppositely charged colloidal macroions from molecular dynamics simulations of the coarse-grained model (with implicit microions), taking effective pair potentials as input. Our results agree closely with corresponding results from more computationally intensive Monte Carlo simulations of the primitive model (with explicit microions). Simulations of a mixture with large size and charge asymmetries indicate that charged nanoparticles can enhance electrostatic screening of charged colloids. The theory presented here lays a foundation for future large-scale modeling of complex mixtures of charged colloids, nanoparticles, and polyelectrolytes.

Poisson-Boltzmann theory of charged colloids: limits of the cell model for salty suspensions

Thermodynamic properties of charge-stabilized colloidal suspensions and polyelectrolyte solutions are commonly modelled by implementing the mean-field Poisson-Boltzmann (PB) theory within a cell model. This approach models a bulk system by a single macroion, together with counterions and salt ions, confined to a symmetrically shaped, electroneutral cell. While easing numerical solution of the nonlinear PB equation, the cell model neglects microion-induced interactions and correlations between macroions, precluding modelling of macroion ordering phenomena. An alternative approach, which avoids the artificial constraints of cell geometry, exploits the mapping of a macroion-microion mixture onto a one-component model of pseudo-macroions governed by effective interparticle interactions. In practice, effective-interaction models are usually based on linear-screening approximations, which can accurately describe strong nonlinear screening only by incorporating an effective (renormalized) macroion charge. Combining charge renormalization and linearized PB theories, in both the cell model and an effective-interaction (cell-free) model, we compute osmotic pressures of highly charged colloids and monovalent microions, in Donnan equilibrium with a salt reservoir, over a range of concentrations. By comparing predictions with primitive model simulation data for salt-free suspensions, and with predictions from nonlinear PB theory for salty suspensions, we chart the limits of both the cell model and linear-screening approximations in modelling bulk thermodynamic properties. Up to moderately strong electrostatic couplings, the cell model proves accurate for predicting osmotic pressures of deionized (counterion-dominated) suspensions. With increasing salt concentration, however, the relative contribution of macroion interactions to the osmotic pressure grows, leading predictions from the cell and effective-interaction models to deviate. No evidence is found for a liquid-vapour phase instability driven by monovalent microions. These results may guide applications of PB theory to colloidal suspensions and other soft materials.

Studies on electrostatic interactions of colloidal particles in two dimensions: a modeling approach

We study the effective electrostatic interactions between a pair of charged colloidal particles without salt ions while the system is confined in two dimensions. In particular, we use a simplified model to elucidate the effects of rotational fluctuations in counterion distribution. The results exhibit effective colloidal attractions under appropriate conditions. Meanwhile, long-range repulsions persist over most of our studied cases. The repulsive forces arise from the fact that in two dimensions, the charged colloids cannot be perfectly screened by counterions, as the residual quadrupole moments contribute to the repulsions at longer range. By applying multiple expansions, we find that the attractive forces observed at short range are mainly contributed by electrostatic interactions among higher-order electric moments. We argue that the scenario for attractive interactions discussed in this work is applicable to systems of charged nanoparticles or colloidal solutions with macroions.

Colloidal attraction induced by a temperature gradient

Colloidal crystals are of extreme importance for applied research and for fundamental studies in statistical mechanics. Long-range attractive interactions, such as capillary forces, can drive the spontaneous assembly of such mesoscopic ordered structures. However, long-range attractive forces are very rare in the colloidal realm. Here we report a novel strong, long-ranged attraction induced by a thermal gradient in the presence of a wall. By switching the thermal gradient on and off, we can rapidly and reversibly form stable hexagonal 2D crystals. We show that the observed attraction is hydrodynamic in nature and arises from thermally induced slip flow on particle surfaces. We used optical tweezers to measure the force law directly and compare it to an analytical prediction based on Stokes flow driven by Marangoni-like forces.

Closure-based density functional theory applied to interfacial colloidal fluids

The behavior of dense colloidal fluids near surfaces can now be probed in great detail with experimental techniques like confocal microscopy. In fact, we are approaching a point where quantitative comparisons of experiment with particle-level theory, such as classical density functional theory (DFT), are appropriate. In a forward sense, we may use a known surface potential to predict a particle density distribution function from DFT; in an inverse sense, we may use an experimentally measured particle density distribution function to predict the underlying surface potential from DFT. In this paper, we tested the ability of the closure-based DFT of Zhou and Ruckenstein (J. Chem. Phys. 2000, 112, 8079-8082) to perform forward and inverse calculations on potential models commonly employed for colloidal particles and surfaces. To reduce sources of uncertainty in this initial study, Monte Carlo simulation results played the role of experimental data. The combination of Rogers-Young and modified-Verlet closures consistently performed well across the different potential models. For a reasonable range of choices of the density, temperature, and potential parameters, the inversion procedure yielded particle-surface potentials to an accuracy on the order of 0.1kT.

Colloidal electrostatic interactions near a conducting surface

Like-charged colloidal spheres dispersed in de-ionized water are supposed to repel each other. Instead, artifact-corrected video microscopy measurements reveal an anomalous long-ranged like-charge attraction in the interparticle pair potential when the spheres are confined to a layer by even a single-charged glass surface. These attractions can be masked by electrostatic repulsions at low ionic strengths. Coating the bounding surfaces with a conducting gold layer suppresses the attraction. These observations suggest a possible mechanism for the anomalous confinement-induced attractions.

Long-time self-diffusion of charged colloidal particles: electrokinetic and hydrodynamic interaction effects

The authors analyze the long-time self-diffusion of charge-stabilized colloidal macroions in nondilute suspensions using a mode-coupling scheme developed for multicomponent suspensions of interacting Brownian spheres. In this scheme, all ionic species, including counterions and electrolyte ions, are treated on an equal footing as charged hard spheres undergoing overdamped Brownian motion. Hydrodynamic interactions between all ions are accounted for on the far-field level. We show that the influence on the colloidal long-time self-diffusion coefficient arising from the relaxation of the microionic atmosphere surrounding the colloids, the so-called electrolyte friction effect, is usually insignificant in comparison with the friction contributions arising from direct and hydrodynamic interactions between the colloidal particles. This finding is true even for small colloid concentrations unless the mobility difference between colloidal particles and microions is not large. Furthermore, we observe an interesting nonmonotonic density dependence of the colloidal long-time self-diffusion coefficient in suspensions with low amount of added salt. We show that this unusual density dependence is due to colloid-colloid hydrodynamic interactions.

On attractive interaction of a colloid pair of like charge at infinite dilution

Numerical data on the potential of mean force W(r) at infinite dilution of a highly charged colloid pair embedded in a 1:1 electrolyte are reported. The authors obtain attractive minima (W<0) at short interparticle distance in these potential functions in hypernetted chain (HNC) approximation, as salt concentration is increased. These minima, however, disappear in all system sets studied when a self-consistent Zerah-Hansen (ZH) closure is used. The authors infer that the attractive minima obtained in a HNC closure are spurious and result from the neglect of bridge diagrams in HNC approximation. An expression of bridge function, which the ZH closure in effect incorporates in W(r) to remove attractive minima, is derived in terms of modification of correlation functions. Features of repulsive pair potentials obtained using the ZH closure, their dependence on particle charge and salt concentration, and their agreement with those of the Derajguin-Landau-Verwey-Overbeek theory are investigated.

A simple phenomenological fix for the dielectric constant within the reference interaction site model approach

The effective interactions among ions immersed in water are studied by means of the effective pair potentials (EPPs) [J. Chem. Phys. 2002, 117, 6133] obtained after contracting (integrating out) the degrees of freedom of the solvent molecules. The dressed interaction site theory (DIST) leads to a simple way of adjusting the effective dielectric constant of the model solvent to its experimental value at standard conditions. The molecular structure of the solvent is mirrored in the structure of the short-ranged component of the induced EPPs, with noticeable differences between the cases with trivial (ideal gas) and nontrivial (experimental) values of the dielectric constant. The shape of these EPPs remains almost invariant over the whole range of salt concentrations considered here. The asymptotic behavior of the EPP between two macroions obtained after contracting the supporting electrolyte (water molecules plus small ions) is also briefly discussed.

Polypeptide foldings obtained with effective pair potentials

We present a model of protein folding which is based on a potential function that describes the effective interaction between two amino acids (alanines, in this case). Our model is consistent with the formation of two important secondary structures, namely, an alpha-helix and a beta-ladder. In each case, we estimate the density of states using a random walk in energy space. This function allows the direct calculation of certain thermodynamic properties. By means of the configurational temperature, we also verify that the obtained polypeptides are in their native state.

Influence of the diffuse layer overcharging or undercharging on the stability of charged interfaces: a restricted grand canonical Monte Carlo study

We have applied a restricted grand canonical Monte Carlo procedure to describe, in the framework of the primitive model, the counterion exchange mechanism between diffuse layers of counterions surrounding segregated charged lamellae. The net charge transfer between the dense and dilute domains is shown to vary as a function of the valence of the neutralizing counterions: undercharging of the dense interlayer is detected in the presence of monovalent counterions and overcharging with divalent counterions. Furthermore, no net reduction of the swelling pressure is detected for monovalent counterions, while a large enhancement of the net interlamellar attraction is found for charged lamellae neutralized by divalent counterions.

Molecular multivalent electrolytes: microstructure and screening lengths

We study small rod-like molecular electrolytes solutions with their corresponding atomic counterions. The asymptotic length scales (decay length and wavelength) of the structural correlations are analyzed using the formalism of the dressed interaction site theory (DIST). The correlation functions are determined using the reference interaction site model equation complemented with a mixed approach in which the hypernetted-chain closure is used for the repulsive interactions, and the mean spherical approximation is used for the attractive interactions. The results from this scheme are in good agreement with the Monte Carlo computer simulations reported here. The asymptotic properties of the correlation functions of this molecular system are compared against those corresponding to two related simple (atomic) electrolyte models. The main conclusion is that the molecular structure of the ions lowers by two orders of magnitude the concentration at which the transition from monotonic to oscillatory decay occurs.

Cluster formation in two-Yukawa fluids

We present a different and efficient method for implementing the analytical solution of Ornstein-Zernike equation for two-Yukawa fluids in the mean spherical approximation. We investigate, in particular, the conditions for the formation of an extra low-Q peak in the structure factor, which we interpret as due to cluster formation in the two-Yukawa fluid when the interparticle potential is composed of a short-range attraction and a long-range repulsion. We then apply this model to interpret the small angle neutron scattering data for protein solutions at moderate concentrations and find out that the presence of a peak centered at Q=0 (zero-Q peak) besides the regular interaction peak due to charged proteins implies an existence of long-range attractive interactions besides the charge repulsion.

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.

Dipolar effective interaction in a fluid of charged spheres near a dielectric plate

Static correlations in a classical fluid of charged spheres at equilibrium are studied in the vicinity of an insulating wall characterized by its dielectric constant. It is well known that the deformations of screening clouds induced by the presence of the wall result into an effective f(alphaalpha('))(x,x('))/y(3) interaction in the pair distribution function between two charges e(alpha) and e(alpha(')) located at distances x and x(') from the wall and separated by a large distance y along the wall. We investigate the structure of f(alphaalpha('))(x,x(')). The method is based on systematic resummations in the Mayer diagrammatics, which are valid both in the bulk and in an inhomogeneous situation. The screened potential phi arising in the formalism happens to coincide with the linearized mean-field approximation for the immersion free energy of two external unit charges. Phi is shown to decay as a repulsive f(phi)(x,x('))/y(3) interaction, whatever the density profiles may be. f(phi)(x,x(')) takes a factorized dipolar structure f(phi)(x,x('))=D(phi)(x)D(phi)(x(')) for distances x and x(') larger than the maximum of the closest approach distances b(alpha)'s to the wall for every species alpha. Moreover, we devise a reorganization of resummed diagrammatics, which is adequate for the determination of the large-distance behavior of correlations, and we prove that, when all species have the same approach distance b to the wall, f(alphaalpha('))(x,x(');b)=D(alpha)(x)D(alpha('))(x(')). In this case, the leading tail of the effective electrostatic interaction between two like charges at the same distance x from a single wall is repulsive. Results are independent of charge magnitudes, of excluded-volume sphere sizes, and of the existence of a surface charge on the wall. It holds whether charges are concentrated at sphere centers or uniformly spread over their surfaces. Comparison is made with an experiment about dilute colloids where the linearized mean-field approximation proves to be relevant. At equilibrium attraction between like charges in confined geometry might arise from purely electrostatic charge-charge interactions only through correlation effects not taken into account in the latter approximation.

Confinement-induced colloidal attractions in equilibrium

The Poisson-Boltzmann theory for colloidal electrostatic interactions predicts that charged colloidal spheres dispersed in water should repel each other, even when confined by charged surfaces. Direct measurements on highly charged polystyrene spheres, however, reveal strong, long-ranged confinement-induced attractions that have yet to be explained. We demonstrate that anomalous attractions also characterize the equilibrium pair potential for more weakly charged colloidal silica spheres sedimented into a monolayer above a glass surface. This observation substantially expands the range of conditions for which mean-field theory incorrectly predicts the sign of macroions' interactions, and provides new insights into how confinement induces long-ranged like-charge attractions.

Effective pair potentials between protein amino acids

We present an effective potential describing the interaction between pairs of residues (alanines, in this case) that belong to a protein. The effective potential is extracted from an experimental correlation function, by means of the Ornstein-Zernike equation together with a closure approximation. It is found that the most relevant features of the effective potential are consistent with the formation of two different secondary structures of proteins.