Internal and free energy in a pair of like-charged colloids: Monte Carlo simulations

The effective interaction between two colloidal particles in a bath of monovalent co- and counterions is studied by means of lattice Monte Carlo simulations with the primitive model. The internal electrostatic energy as a function of the colloid distance is studied fixing the position of the colloids. The free energy of the whole system is obtained introducing a bias parabolic potential, that allows us to sample efficiently small separations between the colloidal particles. For small charges, both the internal and free energy increase when the colloids approach each other, resulting in an effective repulsion driven by the electrostatic repulsion. When the colloidal charge is large enough, on the other hand, the colloid-ion coupling is strong enough to form double layers. The internal energy in this case decreases upon approaching the colloids because more ions enter the double layer. This attractive contribution to the interaction between the colloids is stronger for larger charges and larger ionic concentrations. However, the total free energy increases due to the loss of ionic entropy, and resulting finally in a repulsive interaction potential driven by the entropic contributions. The loss of ionic entropy can be almost quantitatively reproduced with the ideal contribution, the same level of approximation as the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. The overall behavior is captured by the DLVO theory qualitatively, and a comparison is made with the functional form predicted by the theory, showing moderate agreement.     

Gas-liquid phase separation in oppositely charged colloids: stability and interfacial tension

We study the phase behavior and the interfacial tension of the screened Coulomb (Yukawa) restricted primitive model (YRPM) of oppositely charged hard spheres with diameter sigma using Monte Carlo simulations. We determine the gas-liquid and gas-solid phase transitions using free energy calculations and grand-canonical Monte Carlo simulations for varying inverse Debye screening length kappa. We find that the gas-liquid phase separation is stable for kappasigma相似文献   

Electrophoretic properties of highly charged colloids: a hybrid molecular dynamics/lattice Boltzmann simulation study

Using computer simulations, the electrophoretic motion of a positively charged colloid (macroion) in an electrolyte solution is studied in the framework of the primitive model. In this model, the electrolyte is considered as a system of negatively and positively charged microions (counterions and coions, respectively) that are immersed into a structureless medium. Hydrodynamic interactions are fully taken into account by applying a hybrid simulation scheme, where the charged ions (i.e., macroion and electrolyte), propagated via molecular dynamics, are coupled to a lattice Boltzmann (LB) fluid. In a recent electrophoretic experiment by Martin-Molina et al. [J. Phys. Chem. B 106, 6881 (2002)], it was shown that, for multivalent salt ions, the mobility mu initially increases with charge density sigma, reaches a maximum, and then decreases with further increase of sigma. The aim of the present work is to elucidate the behavior of mu at high values of sigma. Even for the case of monovalent microions, a decrease of mu with sigma is found. A dynamic Stern layer is defined that includes all the counterions that move with the macroion while subjected to an external electrical field. The number of counterions in the Stern layer, q(0), is a crucial parameter for the behavior of mu at high values of sigma. In this case, the mobility mu depends primarily on the ratio q(0)/Q (with Q the valency of the macroion). The previous contention that the increase in the distortion of the electric double layer (EDL) with increasing sigma leads to the lowering of mu does not hold for high sigma. In fact, it is shown that the deformation of the EDL decreases with the increase of sigma. The role of hydrodynamic interactions is inferred from direct comparisons to Langevin simulations where the coupling to the LB fluid is switched off. Moreover, systems with divalent counterions are considered. In this case, at high values of sigma the phenomenon of charge inversion is found.     

Simulation of electric double layers undergoing charge inversion: mixtures of mono- and multivalent ions

In this paper, the electric double layer (EDL) of a charged plane in the presence of mixtures of 1:1 and 3:1 electrolytes has been investigated through Monte Carlo (MC) simulations using a nonrestrictive primitive model of EDL. In particular, the charge inversion in colloids (attributable to an accumulation of counterions on the surface) can be better understood by means of the simulations performed here. Moreover, two mechanisms proposed for charge inversion are probed: The formation of a strongly correlated layer (SCL) of multivalent counterions and excluded volume effects (to which we will also refer as ion size correlations). Our results are in agreement with the behavior found experimentally for some model colloids with increasing the concentration of monovalent salt in the presence of trivalent ions, which clearly supports the relevance of ion size correlations. In contrast, certain disagreement with predictions of SCL theories is reported.     

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.     

Effective electrostatic interactions in solutions of polyelectrolyte stars with rigid rodlike arms

In solutions of star-branched polyelectrolytes, electrostatic interactions between charged arms on neighboring stars can compete with intrastar interactions and rotational entropy to induce anisotropy in the orientational distribution of arms. We explore the influence of arm orientational anisotropy on effective star-star interactions for model stars comprising rigid rodlike arms with evenly spaced charged monomers interacting via an effective screened-Coulomb (Yukawa) potential. Monte Carlo simulation and density-functional theory are used to compute the arm orientational distributions and effective pair potentials between weakly charged stars. For comparison, a torque balance analysis is performed to obtain the configuration and energy of the ground state, in which the torque vanishes on each arm of the two-star system. The degree of anisotropy is found to increase with the strength of electrostatic interactions and proximity of the stars. As two stars begin to overlap, the forward arms are pushed back by interstar arm-arm repulsion, but partially interdigitate due to rotational entropy. At center-center separations approaching complete overlap, the arms relax to an isotropic distribution. For nonoverlapping stars, anisotropy-induced changes in the intra- and interstar arm-arm interactions largely cancel and the effective pair interactions are then well approximated by a simple Yukawa potential, as predicted by linear-response theory for a continuum model of isotropic stars [A. R. Denton, Phys. Rev. E 67, 11804 (2003)]. For overlapping stars, the effective pair interactions in the simple rigid-arm-Yukawa model agree closely with simulations of a molecular model that includes flexible arms and explicit counterions [A. Jusufi et al., Phys. Rev. Lett. 88, 018301 (2002); J. Chem. Phys. 116, 11011 (2002)].     

Molecular dynamics study of charged dendrimers in salt-free solution: effect of counterions

Polyamidoamine dendrimers, being protonated under physiological conditions, represent a promising class of nonviral, nanosized vectors for drug and gene delivery. We performed extensive molecular dynamics simulations of a generic model dendrimer in a salt-free solution with dendrimer's terminal beads positively charged. Solvent molecules as well as counterions were explicitly included as interacting beads. We find that the size of the charged dendrimer depends nonmonotonically on the strength of electrostatic interactions demonstrating a maximum when the Bjerrum length equals the diameter of a bead. Many other structural and dynamic characteristics of charged dendrimers are also found to follow this pattern. We address such a behavior to the interplay between repulsive interactions of the charged terminal beads and their attractive interactions with oppositely charged counterions. The former favors swelling at small Bjerrum lengths and the latter promotes counterion condensation. Thus, counterions can have a dramatic effect on the structure and dynamics of charged dendrimers and, under certain conditions, cannot be treated implicitly.     

The foam analogy: from phases to elasticity

By mapping the interactions of colloidal particles onto the problem of minimizing areas, the physics of foams can be used to understand the phase diagrams of both charged and fuzzy colloids. We extend this analogy to study the elastic properties of such colloidal crystals and consider the face-centered cubic, body-centered cubic and A15 lattices. We discuss two types of soft interparticle potentials corresponding to charged and fuzzy colloids, respectively, and we analyze the dependence of the elastic constants on density as well as on the parameters of the potential. We show that the bulk moduli of the three lattices are generally quite similar, and that the shear moduli of the two non-close-packed lattices are considerably smaller than in the face-centered cubic lattice. We find that in charged colloids, the elastic constants are the largest at a finite screening length, and we discuss a shear instability of the A15 lattice.     

Coarse-graining intermolecular interactions in dispersions of highly charged colloids

Effective pair potentials between charged colloids, obtained from Monte Carlo simulations of two single colloids in a closed cell at the primitive model level, are shown to reproduce accurately the structure of aqueous salt-free colloidal dispersions, as determined from full primitive model simulations by Linse et al. (Linse, P.; Lobaskin, V. Electrostatic Attraction and Phase Separation in Solutions of Like-Charged Colloidal Particles. Phys. Rev. Lett.1999, 83, 4208). Excellent agreement is obtained even when ion-ion correlations are important and is in principle not limited to spherical particles, providing a potential route to coarse-grained colloidal interactions in more complex systems.     

Electrostatics and aggregation: how charge can turn a crystal into a gel

The crystallization of proteins or colloids is often hindered by the appearance of aggregates of low fractal dimension called gels. Here we study the effect of electrostatics upon crystal and gel formation using an analytic model of hard spheres bearing point charges and short range attractive interactions. We find that the chief electrostatic free energy cost of forming assemblies comes from the entropic loss of counterions that render assemblies charge-neutral. Because there exists more accessible volume for these counterions around an open gel than a dense crystal, there exists an electrostatic entropic driving force favoring the gel over the crystal. This driving force increases with increasing sphere charge, but can be counteracted by increasing counterion concentration. We show that these effects cannot be fully captured by pairwise-additive macroion interactions of the kind often used in simulations, and we show where on the phase diagram to go in order to suppress gel formation.     

Effect of quenched size polydispersity on the fluid-solid transition in charged colloidal suspensions

We study the effect of quenched size polydispersity on the phase behavior of charged colloidal suspensions using free-energy calculations in Monte Carlo simulations. The colloids are assumed to interact with a hard-core repulsive Yukawa (screened-Coulomb) interaction with constant surface potential, so that the particles are polydisperse both in size and charge. In addition, we take the size distribution to be fixed in both the fluid and crystal phase (no size fractionation is allowed). We study the fluid-solid transition for various screening lengths and surface potentials, finding that upon increasing the size polydispersity the freezing transition shifts toward higher packing fractions and the density discontinuity between the two coexisting phases diminishes. Our results provide support for a terminal polydispersity above which the freezing transition disappears.     

Effect of polymer size and chain length on depletion interactions between two colloids

A density functional theory based on the weighted density has been developed to investigate the depletion interactions between two colloids immersed in a bath of the binary polymer mixtures, where the colloids are modeled as hard spheres and the polymers as freely jointed tangent hard-sphere chain mixtures. The theoretical calculations for the depletion forces between two colloids induced by the polymer are in good agreement with the computer simulations. The effects of polymer packing fraction, degree of polymerization, polymer/polymer size ratio, colloid/polymer size ratio on the depletion interactions, and colloid-colloid second virial coefficient B2 due to polymer-mediated interactions have been studied. With increasing the polymer packing fraction, the depletion interaction becomes more long ranged and the attractive interaction near the colloid becomes deeper. The effect of degree polymerization shows that the long chain gives a more stable dispersion for colloids rather than the short chain. The strong effective colloid-colloid attraction appears for the large colloid/polymer and polymer/polymer size ratio. The location of maximum repulsion Rmax is found to appear Rmax approximately sigmac+Rg2 for the low polymer packing fraction and this is shifted to smaller separation Rmax approximately sigmac+sigmap2 with increasing the polymer packing fraction, where sigmap2 and Rg2 are the small-particle diameter and the radius of gyration of the polymer with the small-particle diameter, respectively.     

Density functional theory for Yukawa fluids

We develop an approximate field theory for particles interacting with a generalized Yukawa potential. This theory improves and extends a previous splitting field theory, originally developed for counterions around a fixed charge distribution. The resulting theory bridges between the second virial approximation, which is accurate at low particle densities, and the mean-field approximation, accurate at high densities. We apply this theory to charged, screened ions in bulk solution, modeled to interact with a Yukawa potential; the theory is able to accurately reproduce the thermodynamic properties of the system over a broad range of conditions. The theory is also applied to "dressed counterions," interacting with a screened electrostatic potential, contained between charged plates. It is found to work well from the weak coupling to the strong coupling limits. The theory is able to reproduce the counterion profiles and force curves for closed and open systems obtained from Monte Carlo simulations.     

Monte Carlo simulation and molecular theory of tethered polyelectrolytes

We investigate the structure of end-tethered polyelectrolytes using Monte Carlo simulations and molecular theory. In the Monte Carlo calculations we explicitly take into account counterions and polymer configurations and calculate electrostatic interaction using Ewald summation. Rosenbluth biasing, distance biasing, and the use of a lattice are all used to speed up Monte Carlo calculation, enabling the efficient simulation of the polyelectrolyte layer. The molecular theory explicitly incorporates the chain conformations and the possibility of counterion condensation. Using both Monte Carlo simulation and theory, we examine the effect of grafting density, surface charge density, charge strength, and polymer chain length on the distribution of the polyelectrolyte monomers and counterions. For all grafting densities examined, a sharp decrease in brush height is observed in the strongly charged regime using both Monte Carlo simulation and theory. The decrease in layer thickness is due to counterion condensation within the layer. The height of the polymer layer increases slightly upon charging the grafting surface. The molecular theory describes the structure of the polyelectrolyte layer well in all the different regimes that we have studied.     

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.     

Entropic effects in the electric double layer of model colloids with size-asymmetric monovalent ions

The structure of the electric double layer of charged nanoparticles and colloids in monovalent salts is crucial to determine their thermodynamics, solubility, and polyion adsorption. In this work, we explore the double layer structure and the possibility of charge reversal in relation to the size of both counterions and coions. We examine systems with various size-ratios between counterions and coions (ion size asymmetries) as well as different total ion volume fractions. Using Monte Carlo simulations and integral equations of a primitive-model electric double layer, we determine the highest charge neutralization and electrostatic screening near the electrified surface. Specifically, for two binary monovalent electrolytes with the same counterion properties but differing only in the coion's size surrounding a charged nanoparticle, the one with largest coion size is found to have the largest charge neutralization and screening. That is, in size-asymmetric double layers with a given counterion's size the excluded volume of the coions dictates the adsorption of the ionic charge close to the colloidal surface for monovalent salts. Furthermore, we demonstrate that charge reversal can occur at low surface charge densities, given a large enough total ion concentration, for systems of monovalent salts in a wide range of ion size asymmetries. In addition, we find a non-monotonic behavior for the corresponding maximum charge reversal, as a function of the colloidal bare charge. We also find that the reversal effect disappears for binary salts with large-size counterions and small-size coions at high surface charge densities. Lastly, we observe a good agreement between results from both Monte Carlo simulations and the integral equation theory across different colloidal charge densities and 1:1-electrolytes with different ion sizes.     

Particle Monte Carlo simulation of string-like colloidal assembly in two and three dimensions

We simulate structural phase behavior of polymer-grafted colloidal particles by molecular Monte Carlo technique. The interparticle potential, which has a finite repulsive square-step outside a rigid core of the colloid, was previously confirmed via numerical self-consistent field calculation. This model potential is purely repulsive. We simulate these model colloids in the canonical ensemble in two and three dimensions and find that these particles containing no interparticle attraction self-assemble and align in a string-like assembly, at low temperature and high density. This string-like colloidal assembly is related to percolation phenomena. Analyzing the cluster size distribution and the average string length, we build phase diagrams and discover that the average string length diverges around the region where the melting transition line and the percolation transition line cross. This result is similar to Ising spin systems, in which the percolation transition line and the order-disorder line meet at a critical point.     

Theoretical aspects and computer simulations of flexible charged oligomers in salt-free solutions

The structural and thermodynamic properties of a model solution containing flexible charged oligomers and an equivalent number of counterions were studied by means of the canonical Monte Carlo simulation and integral equation theory. The oligomers were represented as freely jointed chains of charged hard spheres. In accordance with the primitive model of electrolyte solutions, the counterions were modeled as charged hard spheres and the solvent as a dielectric continuum. Simulations were performed for a set of model parameters, independently varying the chain length and concentration of the oligomers. Structural properties in the form of pair distribution functions were calculated as functions of model parameters. In addition, thermodynamic properties such as the excess energy of solution and the excess chemical potential of counterions were obtained. These properties were correlated with the conformational averages of oligomers as reflected in the end-to-end distances and radii of gyration obtained from the simulations. The relation with the experimental data for heats of dilution and for the activity coefficient is discussed. Finally, theories based on Wertheim's integral equation approach (product reactant Ornstein-Zernike approach) [J. Stat. Phys. 42, 477 (1986)] in the so-called polymer mean spherical and polymer hypernetted chain approximations were tested against the new and existing computer simulations. For the values of parameters examined in this study, the integral equation theory yields semiquantitative agreement with computer simulations.