Charge-stabilized Colloidal Suspensions Research Articles

Effective interactions, structure, and pressure in charge-stabilized colloidal suspensions: Critical assessment of charge renormalization methods.

Charge-stabilized colloidal suspensions display a rich variety of microstructural and thermodynamic properties, which are determined by electro-steric interactions between all ionic species. The large size asymmetry between molecular-scale microions and colloidal macroions allows the microion degrees of freedom to be integrated out, leading to an effective one-component model of microion-dressed colloidal quasi-particles. For highly charged colloids with strong macroion-microion correlations, nonlinear effects can be incorporated into effective interactions by means of charge renormalization methods. Here, we compare and partially extend several practical mean-field methods of calculating renormalized colloidal interaction parameters, including effective charges and screening constants, as functions of concentration and ionic strength. Within the one-component description, we compute structural and thermodynamic properties from the effective interactions and assess the accuracy of the different methods by comparing predictions with elaborate primitive-model simulations [P. Linse, J. Chem. Phys. 113, 4359 (2000)]. We also compare various prescriptions for the osmotic pressure of suspensions in Donnan equilibrium with a salt ion reservoir and analyze instances where the macroion effective charge becomes larger than the bare one. The methods assessed include single-center cell, jellium, and multi-center mean-field theories. The strengths and weaknesses of the various methods are critically assessed, with the aim of guiding optimal and accurate implementations.

Electrostatic interactions between charge regulated spherical macroions.

We study the interaction between two charge regulating spherical macroions with dielectric interior and dissociable surface groups immersed in a monovalent electrolyte solution. The charge dissociation is modelled via the Frumkin-Fowler-Guggenheim isotherm, which allows for multiple adsorption equilibrium states. The interactions are derived from the solutions of the mean-field Poisson-Boltzmann type theory with charge regulation boundary conditions. For a range of conditions we find symmetry breaking transitions from symmetric to asymmetric charge distribution exhibiting annealed charge patchiness, which results in like-charge attraction even in a univalent electrolyte-thus fundamentally modifying the nature of electrostatic interactions in charge-stabilized colloidal suspensions.

Structure and stability of charged colloid-nanoparticle mixtures.

Physical properties of colloidal materials can be modified by addition of nanoparticles. Within a model of like-charged mixtures of particles governed by effective electrostatic interactions, we explore the influence of charged nanoparticles on the structure and thermodynamic phase stability of charge-stabilized colloidal suspensions. Focusing on salt-free mixtures of particles of high size and charge asymmetry, interacting via repulsive Yukawa effective pair potentials, we perform molecular dynamics simulations and compute radial distribution functions and static structure factors. Analysis of these structural properties indicates that increasing the charge and concentration of nanoparticles progressively weakens correlations between charged colloids. We show that addition of charged nanoparticles to a suspension of like-charged colloids can induce a colloidal crystal to melt and can facilitate aggregation of a fluid suspension due to attractive van der Waals interactions. We attribute the destabilizing influence of charged nanoparticles to enhanced screening of electrostatic interactions, which weakens repulsion between charged colloids. This interpretation is consistent with recent predictions of an effective interaction theory of charged colloid-nanoparticle mixtures.

The continuous modeling of charge-stabilized colloidal suspensions in shear flows

Flows of concentrated colloidal suspensions may exhibit a rich set of behaviors due to both hydrodynamic and colloidal interactions between the particles. Colloidal flows are generally modeled with an effective Navier–Stokes equation and a mass balance for the solid phase involving a diffusion coefficient given by the generalized Stokes–Einstein relation. This picture corresponds to a near equilibrium regime in which entropic and colloidal forces dominate over hydrodynamic interactions, the latter being totally ignored. On the other hand, suspension flows far from equilibrium require the modeling of significant hydrodynamic stresses responsible in particular for shear-induced migration, a phenomenon known to occur in some industrial processes involving colloids, such as ultrafiltration. The choice of the proper model ingredients requires a knowledge of the domains in parameter space in which colloidal or hydrodynamic effects are dominant. In this article, such a phase diagram is established for a channel flow of charge-stabilized colloids with a version of the suspension balance model including both colloidal and hydrodynamic effects at the continuous level. It is shown that the classical Peclet number is not sufficient to characterize the flow regime. The phase boundary between the colloidal and hydrodynamic regimes exhibits an original shape explained by the dependence of electrostatic interactions with the colloidal surface charge, and in particular by the phenomenon of ionic condensation. We also show that the phase diagram can be predicted based on the knowledge of a rescaled Peclet number comparing the hydrodynamic stress scale to the bulk modulus of the suspension. The criterion determined here provides important guidelines for the efficient modeling of colloidal flows.

A common neighbor analysis of crystallization kinetics and excess entropy of charged spherical colloids.

The topological analysis tool known as the common neighbor analysis (CNA) is used for the first time in this work to analyze crystallization kinetics and excess entropy of charge-stabilized colloidal suspensions. For this purpose, Brownian dynamics computer simulations are implemented to investigate the crystallization kinetics of homogeneously melted colloidal crystals that are composed of hard-core-screened-Coulomb interacting particles. The results are in agreement with recent static structure factor measurements that could indicate the presence of icosahedral units in the metastable melt, and with the fact that weakly screened charged colloids crystallize into body-centered-cubic (bcc) ordering. A two-step crystallization pathway is found, in which the population of bcc-subunit CNA-pairs satisfactorily obeys a Verhulst model. Moreover, the CNA helped to unveil that the excess entropy obeys a quasi-universal functional form, relating the behavior of colloidal, molecular, and metallic liquid systems. The work contributes to the scientific understanding of the crystallization pathway of charged colloids, and to the development of new ways to assess the degree of crystalline order, starting from the excess entropy.

Population-balance description of shear-induced clustering, gelation and suspension viscosity in sheared DLVO colloids.

Application of shear flow to charge-stabilized aqueous colloidal suspensions is ubiquitous in industrial applications and as a means to achieve controlled field-induced assembly of nanoparticles. Yet, applying shear flow to a charge-stabilized colloidal suspension, which is initially monodisperse and in quasi-equilibrium leads to non-trivial clustering phenomena (and sometimes to a gelation transition), dominated by the complex interplay between DLVO interactions and shear flow. The quantitative understanding of these strongly nonequilibrium phenomena is still far from being complete. By taking advantage of a recent shear-induced aggregation rate theory developed in our group, we present here a systematic numerical study, based on the governing master kinetic equation (population-balance) for the shear-induced clustering and breakup of colloids exposed to shear flow. In the presence of sufficiently stable particles, the clustering kinetics is characterized by an initial very slow growth, controlled by repulsion. During this regime, particles are slowly aggregating to form clusters, the reactivity of which increases along with their size growth. When their size reaches a critical threshold, a very rapid, explosive-like growth follows, where shear forces are able to overcome the energy barrier between particles. This stage terminates when a dynamic balance between shear-induced aggregation and cluster breakage is reached. It is also observed that these systems are characterized by a cluster mass distribution that for a long time presents a well-defined bimodality. The model predictions are quantitatively in excellent agreement with available experimental data, showing how the theoretical picture is able to quantitatively account for the underlying nonequilibrum physics.

Cluster formation in binary charge-stabilized colloidal suspensions confined to a two-dimensional plane

Hypernetted chain (HNC) integral equation theory has been used to study the structural features of binary charged stabilized colloidal suspensions confined to a two-dimensional plane. The particles interact via purely repulsive Yukawa intermolecular potential, the inverse screening length scaled by the average distance between strongly interacting components of the mixture (dimensionless screening parameter) being 1, 3 and 5. Results of HNC theory for one-component systems are found to be in very good agreement with that of simulation, in the parameter range of our study. Binary Yukawa systems with dimensionless screening parameters 1 and 3 are found to exhibit diffuse clusters of the weakly interacting particles, marked by the emergence of a cluster peak in the corresponding partial structure factor curves. No cluster peak is found in the system with the screening parameter 5. For the entire range of mixture parameters, the strongly interacting particles remain homogeneously distributed.

Modeling Fine-Particle (Dusty) Plasmas and Charge-Stabilized Colloidal Suspensions as Inhomogeneous Yukawa Systems

In order to give a basis for the structure and correlation analysis of fine particle (dusty) plasma and colloidal suspensions, thermodynamic treatment of mixtures of macroscopic and microscopic charged particles within the adiabatic response of the latter is extended to include the case where the system is finite and weakly inhomogeneous. It is shown that the effective potential for macroscopic particles is composed of two elements: mutual Yukawa repulsion and a confining (attractive) Yukawa potential from their `shadow' (the average charge density of macroscopic particles multiplied by the minus sign). The result clarifies the relation between two approaches hitherto taken where either a parabolic one-body potential is assumed or the average distribution is assumed to be flat with finite extension. Since the satisfaction of the charge neutrality is largely enhanced by the existence of macroscopic particles, the assumption of the flat electrostatic potential and therefore flat average distribution of macroscopic particles in the domain of their existence is expected to be closer to reality than the assumption of the parabolic potential in that domain.

Drying of a Charge-Stabilized Colloidal Suspension in Situ Monitored by Vertical Small-Angle X-ray Scattering

We report a first application of vertical small-angle X-ray scattering to investigate the drying process of a colloidal suspension by overcoming gravity related restrictions. From the observation of the drying behavior of charge-stabilized colloidal silica in situ, we find the solidification of the colloidal particles exhibits an initial ordering, followed by a sudden aggregation when they overcome an electrostatic energy barrier. The aggregation can be driven not only by capillary pressure but also by thermal motion of the particles. The dominating contribution is determined by the magnitude of the energy barrier at the transition, which significantly decreases during drying due to an increased ionic strength.

Colloidal electroconvection in a thin horizontal cell. III. Interfacial and transient patterns on electrodes

Previously we have reported a family of convective patterns formed by charge-stabilized aqueous colloidal suspensions under constant (dc) vertical electric fields [Y. Han and D. G. Grier, J. Chem. Phys. 122, 164701 (2005); and ibid. 125, 144707 (2006)]. These patterns form in the bulk when electrokinetic forces act in the opposite direction to gravity. Here, we report on cellular patterns that silica colloidal spheres form on a horizontal electrode when electrokinetic forces act in the same direction as gravity. We suggest that these cellular patterns form as a result of bulk electroconvection mediated by charge injection into the supporting aqueous electrolyte. This charge-injection mechanism also accounts for some aspects of electroconvective pattern formation in our earlier reports. Cellular patterns reorganize themselves into distinct transient patterns after the driving voltage is turned off. These transients cast new light on the complex interplay between the motions of charged colloidal spheres and the ionic relaxation of water undergoing electrolysis.

Decay of interfacial fluid ordering probed by X-ray reflectivity

To what extent is the ordering of fluids at interfaces governed by the bulk structure of the fluid? In order to address this question, we have studied a dense, charge-stabilized colloidal suspension at the fluid–air interface and at two different solid–fluid interfaces using specular X-ray reflectivity. The experimental data are well described by a simple model of a stratified fluid, with the wall–particle potential mainly affecting the position of the first particle layer with respect to the interface. The decay of the fluid's density profile can for all the studied interfaces be described by a characteristic wavelength and a decay length, both of which are independently determined from the bulk phase using small-angle X-ray scattering. This latter finding is consistent with theoretical predictions and recent surface-force experiments.

Renormalized jellium mean-field approximation for binary mixtures of charged colloids

In this work the renormalized jellium model of colloidal suspensions, originally proposed by Trizac and Levin [Phys. Rev. E 69, 031403 (2004)], is extended to study mechanisms of charge renormalization in binary mixtures of charged colloids. We here apply our recent reformulation that introduces the requirement of self-consistency directly into the Poisson-Boltzmann equation, i.e., the background charge is explicitly replaced by the effective one, thus facilitating the whole charge renormalization scheme. We briefly discuss the reformulated model for monodisperse charged suspensions composed of either spheres or rods. In particular, we put emphasis on the effects of the surface charge variation, mixture composition, and particle size on the charge regulation of charge-stabilized colloidal suspensions.

Note: Renormalized jellium model for charged colloids revisited

Recently, it has been demonstrated that electrostatic interactions between charged colloids in contact with a symmetric electrolyte can accurately be described by using the so-called renormalized jellium (RJ) model proposed by Trizac and Levin.1 This mean-field approximation, based on the Poisson–Boltzmann equation (PBE), requires that the charge of the background around a tagged colloid be renormalized self-consistently to coincide with the colloid effective charge; such a requirement leads to precise thermodynamic properties even in highly charged colloidal suspensions.2–5 In this note, we revisit the renormalized jellium approximation and, in particular, provide a simple explicit recipe that facilitates the renormalization procedure being easy of implementing in different situations. We here apply such recipe in the case of charged spheres and discuss its extension to the case of charged rods. The system under consideration is a charged colloidal suspension of volume fraction η composed of, obviously, charged colloids of radius a, and counter ions in contact with a symmetric (1:1) salt reservoir of concentration 2cs , with cs the density of positive or negative salt ions; the solvent is included through the dielectric constant e. The RJ model assumes that the charge of Nc − 1 colloidal particles around a tagged macroion is smeared out in the whole suspension to form a homogeneous background with charge Zbacke, e being the elementary charge. This background charge is Zback = Zbare, with Zbare the colloidal bare charge, and not known a priori, but it is determined self-consistently to be equal to the effective charge, Zeff, of the tagged macroion.1 An extraordinary advantage of the RJ model is that Zeff is directly associated to the system osmotic pressure, the screening parameter, and the effective pair interaction between colloids (when it is explicitly considered at the level of the Yukawa approximation).1, 2 Nevertheless, since it is an explicit function of the system state, i.e., Zeff = Zeff(η, Zbare, cs), its evaluation depends on the specific conditions of the system and, currently, the entire charge renormalization procedure is carried out through an iterative process.6 This iterative procedure demands the construction of a function of the form Zeff = Zeff(Zback, Zbare). The selfconsistently condition is reached in the intersection of the function with the straight line Zeff = Zback. However, it is still possible to reformulate this scheme and avoid the full iterative procedure to gain clarity in the way in which the RJ can be applied and extended to study the physical properties of charge-stabilized colloidal suspensions. The complete original procedure can then be reformulated as follows. We start with the requirement of selfconsistency from the beginning. This means that the condition Zback = Zeff must be explicitly incorporated into the PBE. Our main assumption is that there exists a unique Zeff for a given Zbare; this avoids completely the inclusion of Zback and facilitates drastically the renormalization scheme. Additionally, one should also rephrase the typical boundary conditions (BCs) to solve the PBE with simple BCs at one point [see Eqs. (3) and (4)]. To achieve this, we use the fact that the farfield solution of the PBE has a Yukawa-like form that depends on Zeff. Therefore, in the case of charged spherical colloids, the PBE now reads d2φ dr2 + 2 r dφ dr = −3η ZeffλB a − ρ+(∞)e−φ + ρ−(∞)e, (1)

Poisson–Boltzmann theory of charged colloids: limits of the cell model for saltysuspensions

Thermodynamic properties of charge-stabilized colloidal suspensions and polyelectrolytesolutions are commonly modelled by implementing the mean-field Poisson–Boltzmann(PB) theory within a cell model. This approach models a bulk system by a singlemacroion, together with counterions and salt ions, confined to a symmetricallyshaped, electroneutral cell. While easing numerical solution of the nonlinear PBequation, the cell model neglects microion-induced interactions and correlationsbetween macroions, precluding modelling of macroion ordering phenomena. Analternative approach, which avoids the artificial constraints of cell geometry, exploitsthe mapping of a macroion–microion mixture onto a one-component model ofpseudo-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 incorporatingan effective (renormalized) macroion charge. Combining charge renormalizationand linearized PB theories, in both the cell model and an effective-interaction(cell-free) model, we compute osmotic pressures of highly charged colloids andmonovalent microions, in Donnan equilibrium with a salt reservoir, over a range ofconcentrations. By comparing predictions with primitive model simulation data for salt-freesuspensions, and with predictions from nonlinear PB theory for salty suspensions, wechart the limits of both the cell model and linear-screening approximations inmodelling bulk thermodynamic properties. Up to moderately strong electrostaticcouplings, the cell model proves accurate for predicting osmotic pressures of deionized(counterion-dominated) suspensions. With increasing salt concentration, however, therelative contribution of macroion interactions to the osmotic pressure grows, leadingpredictions from the cell and effective-interaction models to deviate. No evidence isfound for a liquid–vapour phase instability driven by monovalent microions. Theseresults may guide applications of PB theory to colloidal suspensions and other softmaterials.

Charge Renormalization, Thermodynamics, and Structure of Deionized Colloidal Suspensions

In charge-stabilized colloidal suspensions, highly charged macroions, dressed by strongly correlated counterions, carry an effective charge that can be substantially reduced (renormalized) from the bare charge. Interactions between dressed macroions are screened by weakly correlated counterions and salt ions. Thermodynamic and structural properties of colloidal suspensions depend sensitively on the magnitudes of both the effective charge and the effective screening constant. Combining a charge renormalization theory of effective electrostatic interactions with Monte Carlo simulations of a one-component model, we compute osmotic pressures and pair distribution functions of deionized colloidal suspensions. This computationally practical approach yields close agreement with corresponding results from large-scale simulations of the primitive model up to modest electrostatic coupling strengths.

Charge renormalization, effective interactions, and thermodynamics of deionized colloidalsuspensions

Thermodynamic properties of charge-stabilized colloidal suspensions depend sensitively onthe effective charge of the macroions, which can be substantially lower than the bare chargein the case of strong counterion–macroion association. A theory of charge renormalizationis proposed, combining an effective one-component model of charged colloids with athermal criterion for distinguishing between free and associated counterions. The theorypredicts, with minimal computational effort, osmotic pressures of deionized suspensions ofhighly charged colloids in close agreement with large-scale simulations of the primitivemodel.

Density Dependent Interactions and Structure of Charged Colloidal Dispersions in the Weak Screening Regime

We determine the structure of charge-stabilized colloidal suspensions at low ionic strength over an extended range of particle volume fractions using a combination of light and small angle neutron scattering experiments. The variation of the structure factor with concentration is analyzed within a one-component model of a colloidal suspension. We show that the observed structural behavior corresponds to a nonmonotonic density dependence of the colloid effective charge and the mean interparticle interaction energy. Our findings are corroborated by similar observations from primitive model computer simulations of salt-free colloidal suspensions.

Phase separation of charge-stabilized colloids: A Gibbs ensemble Monte Carlo simulation study

The fluid phase behavior of charge-stabilized colloidal suspensions is explored by applying a variant of the Gibbs ensemble Monte Carlo simulation method to a coarse-grained one-component model with implicit microions and solvent. The simulations take as input linear-response approximations for the effective electrostatic interactions--a hard-sphere-Yukawa pair potential and a one-body volume energy. The conventional Gibbs ensemble trial moves are supplemented by exchange of (implicit) salt between coexisting phases, with acceptance probabilities influenced by the state dependence of the effective interactions. Compared with large-scale simulations of the primitive model, with explicit microions, our computationally practical simulations of the one-component model closely match the pressures and pair distribution functions at moderate electrostatic couplings. For macroion valences and couplings within the linear-response regime, deionized aqueous suspensions with monovalent microions exhibit separation into macroion-rich and macroion-poor fluid phases below a critical salt concentration. The resulting pressures and phase diagrams are in excellent agreement with predictions of a variational free energy theory based on the same model.

Polymorph selection during the crystallization of Yukawa systems

Using molecular-dynamics simulations, we study the crystallization of supercooled liquids of charge-stabilized colloidal suspensions, modeled by the Yukawa (screened-Coulomb) potential. By modifying the value of the screening parameter lambda, we are able to invert the stability of the body-centered cubic (bcc) and face-centered cubic (fcc) polymorphs and study the crystal nucleation and growth in the domain of stability of each polymorph. We show that the crystallization mechanism strongly depends on the value of lambda. When bcc is the stable polymorph (lambda=3), the crystallization mechanism is straightforward. Both kinetics and thermodynamics favor the formation of the bcc particles and polymorph selection takes place early during the nucleation step. When fcc is the stable polymorph (lambda=10), the molecular mechanism is much more complex. First, kinetics favor the formation of bcc particles during the nucleation step. The growth of the post-critical nucleus proceeds through the successive cross-nucleation of the stable fcc polymorph on the metastable hcp polymorph as well as of the hcp polymorph on the fcc polymorph. As a result, polymorph selection occurs much later, i.e., during the growth step, than for lambda=3. We then extend our findings established in the case of homogeneous crystal nucleation to a situation of practical interest, i.e., when a seed of the stable polymorph is used. We demonstrate that the growth from the (111) face of a perfect fcc crystal into the melt proceeds through the same mechanisms.

Macroion virial contribution to the osmotic pressure in charge-stabilized colloidal suspensions

Our interest goes to the different virial contributions to the equation of state of charged colloidal suspensions. Neglect of surface effects in the computation of the colloidal virial term leads to spurious and paradoxical results. This pitfall is one of the several facets of the danger of a naive implementation of the so called one component model, where the microionic degrees of freedom are integrated out to only keep in the description the mesoscopic (colloidal) degrees of freedom. On the other hand, due incorporation of wall induced forces dissolves the paradox brought forth in the naive approach, provides a consistent description, and confirms that for salt-free systems, the colloidal contribution to the pressure is dominated by the microionic one. Much emphasis is put on the no salt case but the situation with added electrolyte is also discussed.