Spherical Macroions Research Articles

Competition between a Macroion and a Polyelectrolyte in Complexation with an Oppositely Charged Polyelectrolyte

The competition between a spherical macroion and a polyelectrolyte in complexation with an oppositely charged flexible polyelectrolyte has been studied by means of Monte Carlo simulations. A simple model system with focus on the electrostatic interactions has been used to investigate under what conditions the macroion in a polyelectrolyte−macroion complex (excess of polyelectrolyte charges) is exchanged with a polyelectrolyte. It is shown that the macroion is released when the charge of the added polyelectrolyte equals or exceeds that of the polyelectrolyte in the complex. The effect of the polyelectrolyte length, flexibility, and linear charge density on the macroion release has been investigated.

Effects of asymmetric salt and a cylindrical macroion on charge inversion: electrophoresis by molecular dynamics simulations.

The charge inversion phenomenon is studied by molecular dynamics simulations, focusing on size and valence asymmetric salts, and a threshold of surface charge density for charge inversion. The charge inversion criteria by the electrophoretic mobility and the radial distribution functions of ions coincide except around the charge inversion threshold. The reversed electrophoretic mobility increases with the ratio of coion to counterion radii, a(-)/a(+), while it decreases with the ratio of coion to counterion valences, Z(-)/Z(+). The monovalent salt enhances charge inversion of a strongly charged macroion at small salt ionic strength, while it reduces reversed mobility otherwise. A cylindrical macroion is more persistent to monovalent salt than a spherical macroion of the same radius and surface charge density.

Overcharging in Macroions. Effects of Macroion Geometry/Charge Distribution

A random sampling technique is employed to obtain the ground-state configuration energy of an overcharged macroion in that the magnitude of the total counterion charge exceeds the magnitude of the macroion charge. The technique is based on minimizing the total electrostatic energy of the macroion and counterions in salt-free conditions. The algorithm easily reproduces the ground-state energies of an overcharged isolated spherical macroion obtained from Molecular Dynamics simulation in earlier published literature (Messina, R., Holm, C.; Kremer, K. Phys. Rev. E 2001, 64, 021405) for low to intermediate values of the macroion charge, while at higher macroion charge the calculated energies lie below those of the earlier results. The calculations are extended to nonspherical macroion geometries such as spherocylinders and spheroids, and substantial effects on the energetics due to the geometries are noted. A detailed study of counterion distributions is also done for macroions of different charges and geometries.

Depletion and Structural Forces in Confined Polyelectrolyte Solutions

Monte Carlo simulations and density functional calculations have been performed for charged macromolecules confined to planar slits. The force between the confining walls has been evaluated as a function of separation, while keeping the chemical potential of the macromolecules constant. Highly charged spherical particles and flexible polyelectrolyte chains in confinement give rise to depletion and structural oscillatory forces as a function of surface separation. The sign and magnitude of the surface charge of the confining walls have no dramatic effect on the qualitative behavior of the confined liquid. With neutral or oppositely charged surfaces, an accumulation of charged macroions is seen in the slit driven by the repulsive interaction between the macroions, while equally charged surfaces give rise to a pure depletion. The net charge, the range of interaction, and the particle density affect the details of the force curve. For spherical macroions, the period of the oscillations scales approximately as the bulk aggregate concentration, cbulk-1/3. Confined polyelectrolyte chains share some of these properties, but they partly display a different behavior. One clear difference is that the polyelectrolyte net charge, that is, the degree of polymerization, has no effect on the osmotic pressure. This is an indication that polyelectrolyte chains pack not as spheres but rather as cylindrical objects. Another difference is that the effective repulsive interaction between polyelectrolyte chains can be more long ranged and oscillatory forces can appear more readily than for a corresponding solution of equally charged spherical macroions.

Simple Model for the Binding of a Polyelectrolyte to an Oppositely Charged Curved Surface

We present a simple model for the binding of persistence-length segments of a polyelectrolyte to an oppositely charged spherical macroion with radius of the same order as the polymer persistence length. The polyelectrolyte is taken as a rod with elastic bending capacity. The macroion is abstracted as an absolutely rigid circular arc. In the bound state, the polyelectrolyte bends onto the macroion, and some of the opposite charges are mutually neutralized. The curvature of the macroion imposes stability limits on the bound state. The model predicts transitions to bound-state stability in the same conditions (polyelectrolyte charge density and bare persistence length, macroion radius, salt concentration) under which they are observed in experimental studies of the binding of polyelectrolytes to oppositely charged micelles. The calculations also suggest why macroscopic phase separation occurs in experiments below a critical salt concentration but not above.

Monte Carlo Simulations of Polyion−Macroion Complexes. 2. Polyion Length and Charge Density Dependence

The complexation between a polyion and an oppositely charged spherical macroion in the framework of the primitive model has been studied by the use of Monte Carlo simulations. The polyion length, linear charge density, and bare persistence length are varied systematically, while the properties of the macroion are kept constant. The polyion charge to macroion charge ratio is varied between 1/4 and 4. The structure of the complex is investigated by direct visualization; polyion bead complexation probability; loop, tail, and train characteristics; degree of overcharging; and tail joint probability functions. The strongest complexes are observed for flexible chains, where the polyion is folded around the macroion. In the case of fully flexible chains, a transition from a collapsed state to a fluctuating two-tail state and eventually to a one-tail state are observed as the chain length is increased. As the stiffness is increased, several complex structures, such as multiloop, single-loop, and solenoid arrangements, and finally a structure involving only a single contact between the polyion and the macroion occur. In particular, for long and highly charged polyions, a transition from the one-tail state to a two-tail state appears as the chain stiffness is increased. A discussion with recent theories and other simulation studies is also provided.

Effective charge versus bare charge: an analytical estimate for colloids in the infinite dilution limit

We propose an analytical approximation for the dependence of the effective charge on the bare charge for spherical and cylindrical macro-ions as a function of the size of the colloid and salt content, for the situation of a unique colloid immersed in a sea of electrolyte (where the definition of an effective charge is non ambiguous). Our approach is based on the Poisson-Boltzmann (PB) mean-field theory. Mathematically speaking, our estimate is asymptotically exact in the limit $\kappa a\gg 1$, where $a$ is the radius of the colloid and $\kappa$ the inverse screening length. In practice, a careful comparison with effective charges parameters obtained by numerically solving the full non-linear PB theory proves that it is good down to $\kappa a\sim 1$. This is precisely the limit appropriate to treat colloidal suspensions. A particular emphasis is put on the range of parameters suitable to describe both single and double strand DNA molecules under physiological conditions.

Counterion penetration and effective electrostatic interactions in solutions of polyelectrolyte stars and microgels.

Counterion distributions and effective electrostatic interactions between spherical macroions in polyelectrolyte solutions are calculated via second-order perturbation (linear response) theory. By modeling the macroions as continuous charge distributions that are permeable to counterions, analytical expressions are obtained for counterion profiles and effective pair interactions in solutions of star-branched and microgel macroions. The counterions are found to penetrate stars more easily than microgels, with important implications for screening of bare macroion interactions. The effective pair interactions are Yukawa in form for separated macroions, but are softly repulsive and bounded for overlapping macroions. A one-body volume energy, which depends on the average macroion concentration, emerges naturally in the theory and contributes to the total free energy.

Comparison of the Thermal and Concentration Criteria for Determining the Effective Charge of a Spherical Colloidal Particle in a Salt-Free Suspension

The techniques of canonical Monte Carlo simulations are employed in a comparative study of two recently proposed criteria for the determination of the renormalized or effective charge of a spherical macroion in a charge-stabilized colloidal suspension. The effective charge is determined using (i) the thermal criterion, where the effective charge consists of the intrinsic macroion charge reduced by the integrated thermodynamically bound counterion charge, the bound counterions being defined as those counterions that are within and up to the radial distance (from the macroion center) at which the reduced mean electrostatic potential energy equals the thermal energy, and (ii) the concentration criterion, where the bound counterions are those that lie within the distance at which the local concentration of the counterions equals its average value. It is found that the former criterion applies to the entire range of bare macroion charge/macroion radius ratios investigated, whereas the latter criterion is more suited to large values of this ratio. The observation that the effective charge/bare charge ratio (Z e f f /Z m ) depends only on the bare charge/macroion radius ratio (Z m /r m ) suggests that all macroions with the same numerical value of the Z m /r m ratio are equivalent systems and that one can therefore study highly charged colloids by using a more manageable set of parameters in a simulation. The calculations are extended to divalent counterion systems, and comparisons are also made with other criteria for determining the effective macroion charge in the literature.

Charge inversion in colloidal systems

We investigate spherical macroions in the strong Coulomb coupling regime within the primitive model in salt-free environment. Molecular dynamics (MD) simulations are used to elucidate the effect of discrete macroion charge distribution on charge inversion. A systematic comparison is made with the charge inversion obtained in the conventional continuous charge distribution. Furthermore the effect of multivalent counterions is reported.

Complexation of DNA with positive spheres: Phase diagram of charge inversion and reentrant condensation

The phase diagram of a water solution of DNA and oppositely charged spherical macroions is studied. DNA winds around spheres to form beads-on-a-string complexes resembling the chromatin 10 nm fiber. At small enough concentration of spheres these “artificial chromatin” complexes are negative, while at large enough concentrations of spheres the charge of DNA is inverted by the adsorbed spheres. Charges of complexes stabilize their solutions. In the plane of concentrations of DNA and spheres the phases with positive and negative complexes are separated by another phase, which contains the condensate of neutral DNA–spheres complexes. Thus, when the concentration of spheres grows, DNA–spheres complexes experience condensation and resolubilization (or reentrant condensation). Phenomenological theory of the phase diagram of reentrant condensation and charge inversion is suggested. Parameters of this theory are calculated by microscopic theory. It is shown that an important part of the effect of a monovalent salt on the phase diagram can be described by the nontrivial renormalization of the effective linear charge density of DNA wound around a sphere, due to the Onsager–Manning condensation. We argue that our phenomenological phase diagram or reentrant condensation is generic to a large class of strongly asymmetric electrolytes. Possible implications of these results for the natural chromatin are discussed.

Strong electrostatic interactions in spherical colloidal systems.

We investigate spherical macroions in the strong Coulomb coupling regime within the primitive model in salt-free environment. We first show that the ground state of an isolated colloid is naturally overcharged by simple electrostatic arguments illustrated by the Gillespie rule. We furthermore demonstrate that in the strong Coulomb coupling this mechanism leads to ionized states and thus to long range attractions between like-charged spheres. We use molecular dynamics simulations to study in detail the counterion distribution for one and two highly charged colloids for the ground state as well as for finite temperatures. We compare our results in terms of a simple version of a Wigner crystal theory and find excellent qualitative and quantitative agreement.

Primitive Model Electrophoresis

The electrophoresis of a spherical macroion is calculated with the new primitive model electrophoresis (PME) theory, which incorporates the ionic size. The results are compared with the classical theory of Wiersema, O'Brien, and White. The PME mobility, as a function of the macroion's surface charge (σ) or zeta potential (ζ), is found to depend on the ionic valence, radius and concentration, and macroion hydrodynamical radius (A); i.e., it is not universal with κA, as predicted by the classical theory. The mobility is very nonlinear as a function of ζ. This behavior is related to the nonlinear dependence of ζ, as a function of σ, when ionic size is included in the theory. In the classical theory ζ is a monotonic function of σ. Important quantitative and/or qualitative differences between PME and the classical theory are found. However, in the limit of zero ionic diameter, and/or low salt concentration, and low macroion charge, the new theory reduces to the classical theory. The agreement of the PME with experimental data is very good. In particular, the prediction of reversed mobility is corroborated by recent experimental data.

Gas–liquid phase separation in charged colloidal systems

A model system of charged spherical macroions and oppositely charged point counterions with a macroion-charge to counterion-charge ratio Zr=10 was investigated by employing a temperature and density scaling Monte Carlo simulation method. This approach allowed for a determination of the relative free energies among arbitrary number of thermodynamic states over a substantial region of the state space. Here, relative free energies were determined for the macroion volume fraction interval 0.04<φM<0.24 at different values of the electrostatic coupling parameter ΓII≡ZI2LB/RM, where ZI denotes the counterion charge, LB the Bjerrum length, and RM the macroion radius. The binodal curve separating the one-phase system and a gas-liquid two-phase region was determined near the critical point which was determined to φMcr=0.15 and ΓIIcr=2.6.

Structure and phase separation in solutions of like–charged colloidal particles

Structure and phase stability of model systems of charged spherical macroions and point counterions interacting solely through hard–sphere and Coulomb interactions, as predicted by extensive Monte Carlo simulations, will be presented. The part of the reduced parameter space investigated includes nearly all experimental colloidal solutions (in the salt–free limit) in which the colloids are carrying up to 80 (monovalent counterions) or 160 (divalent counterions) elementary charges. The effects on the structure of varying the counterion charge, the macroion charge, the macroion size, the temperature, the dielectric permittivity and the electrolyte density are presented. At a high electrostatic coupling, the system displays a thermodynamic instability and separates into two phases of widely differing electrolyte concentration. The location of the binodal surface and critical points will be discussed.

Overcharging of a macroion by an oppositely charged polyelectrolyte

Complexation of a polyelectrolyte with an oppositely charged spherical macroion is studied for both salt-free and salty solutions. When a polyelectrolyte winds around the macroion, its turns repel each other and form an almost equidistant solenoid. It is shown that this repulsive correlations of turns lead to the charge inversion: more polyelectrolyte winds around the macroion than it is necessary to neutralize it. The charge inversion becomes stronger with increasing concentration of salt and can exceed 100%. Monte-Carlo simulation results agree with our analytical theory.

Complexation of a polyelectrolyte with oppositely charged spherical macroions: Giant inversion of charge

Complexation of a long flexible polyelectrolyte (PE) molecule with oppositely charged spherical particles such as colloids, micelles, or globular proteins in a salty water solution is studied. PE binds spheres winding around them, while spheres repel each other and form almost periodic necklace. If the total charge of PE in the solution is larger than total charge of spheres, repulsive correlations of PE turns on a sphere lead to inversion of the net charge of each sphere. In the opposite case when the total charge of spheres is larger, we predict another correlation effect; spheres bind to the PE in such a great number that they invert the charge of the PE. The inverted charge by absolute value can be larger than the bare charge of PE even when screening by monovalent salt is weak. At larger concentrations of monovalent salt, the inverted charge can reach giant proportions. Near the isoelectric point where total charges of spheres and PE are equal, necklaces condense into macroscopic bundles. Our theory is in qualitative agreement with recent experiments on micelles–PE systems.

Phase coexistence in colloidal suspensions: an analytic Poisson-Boltzmann treatment.

We solve the linearized Poisson-Boltzmann equation analytically, subject to justifiable approximations, for a suspension containing a large number of identical spherical macroions under conditions of constant surface charge and zero added salt, in order to investigate the phase behavior of charge-stabilized colloidal suspensions. Our results for the electrostatic part of the Helmholtz free energy lead to an interaction which resembles the intermolecular interaction in the theory of molecular fluids. When combined with the ideal gas free energy of the counterions, this produces a van der Waals loop in the pV diagram, indicating coexistence between phases with different densities, for certain values of the macroion radius and charge. We also derive an expression for the surface potential of the macroions, and clarify the interpretation of the Poisson-Boltzmann equation.

Effect of colloidal charge discretization in the primitive model

The effect of fixed discrete colloidal charges in the primitive model is investigated for spherical macroions. Instead of considering a central bare charge, as it is traditionally done, we distribute \textit{discrete} charges randomly on the sphere. We use molecular dynamics simulations to study this effect on various properties such as overcharging, counterion distribution and diffusion. In the vicinity of the colloid surface the electrostatic potential may considerably differ from the one obtained with a central charge. In the strong Coulomb coupling, we showed that the colloidal charge discretization qualitatively influences the counterion distribution and leads to a strong colloidal charge-counterion pair association. However, we found that \textit{charge inversion} still persists even if strong pair association is observed.

Interaction between oppositely charged micelles or globular proteins.

Monte Carlo simulations and the hypernetted chain theory are used to study the interaction between spherical macroions of opposite charge immersed in a solution of monovalent or divalent simple electrolyte. These calculations represent the first step toward studying phase behavior and precipitation kinetics in solutions containing a mixture of macroions with positive and negative net charges. The potential of mean force between colloidal particles is determined as a function of colloid-colloid separation. In addition to having an opposite sign, the calculated potential of mean force is found to be stronger and longer-ranged than observed in the case of equally charged macroparticles. The difference is more pronounced in the presence of divalent counterions and is especially noticeable when we compare distinct Coulombic and hard-core collision contributions to the interaction between equally and oppositely charged colloids. The present observations suggest the dominance of attractive forces between globally neutral but electrostatically heterogeneous macroparticles. While our numerical results cannot be successfully analyzed by existing theories, they provide useful guidance and benchmark data for the development of advanced analytic descriptions.