Multivalent Coions Research Articles

Nonmonotonic electrophoretic mobility of rodlike polyelectrolytes by multivalent coions in added salt.

It is well established that when multivalent counterions or salts are added to a solution of highly charged polyelectrolytes (PEs), correlation effects can cause charge inversion of the PE, leading to electrophoretic mobility (EM) reversal. In this work, we use coarse-grained molecular-dynamics simulations to unravel the less understood effect of coion valency on EM reversal for rigid DNA-like PEs. We find that EM reversal induced by multivalent counterions is suppressed with increasing coion valency in the salt added and eventually vanishes. Further, we find that EM is enhanced at fixed low salt concentrations for salts with monovalent counterions when multivalent coions with increasing valency are introduced. However, increasing the salt concentration causes a crossover that leads to EM reversal which is enhanced by increasing coion valency at high salt concentration. Remarkably, this multivalent coion-induced EM reversal persists even for low values of PE linear charge densities where multivalent counterions alone cannot induce EM reversal. These results facilitate tuning PE-PE interactions and self-assembly with both coion and counterion valencies.

Effects of ionic valence on aggregation kinetics of colloidal particles with and without a mixing flow

Hypothesis: The classical Schulze-Hardy rule states that the critical coagulation concentration (CCC) of colloidal particles is inversely proportional to the counter-ionic valence at powers ranging from 2 to 6. However, the inverse Schulze-Hardy rule has recently been proposed, suggesting that the CCC can also be inversely proportional to the co-ionic valence. Previous studies on these rules did not consider the effect of flow on aggregation kinetics and the CCC. This study aims to investigate the effect of multivalent counter-ions and co-ions on aggregation kinetics and the CCCs in systems with and without a mixing flow.Experiments: We measured the aggregation rate coefficients of polystyrene sulfate latex particles as a function of the salt concentration with different ionic species. Furthermore, we analyzed these measurements using theoretical models based on hydrodynamic pair-diffusion in a random flow and trajectory analysis in two steady flows. The analysis was conducted using zeta potentials determined through electrophoretic measurements.Findings: Although the trajectory analysis underestimates the CCCs, the hydrodynamic pair-diffusion model can capture the shift of critical coagulation concentrations in the mixing flow to higher values than those in Brownian aggregation and also shows a better agreement with the experimental results. This result suggests that combining random flow and Brownian diffusion is crucial for developing a consistent framework for predicting both Brownian aggregation and aggregation in a mixing flow.

Schulze-Hardy rule revisited

The classical Schulze-Hardy rule suggests that the critical coagulation concentration (CCC) decreases as the inverse sixth power of the counterion valence. While this dependence can be derived from the theory of Derjaguin, Landau, Verwey, and Overbeek (DLVO), this derivation relies on unrealistic assumptions. In particular, one cannot assume that the electrolytes are symmetric, since one normally works with the better soluble asymmetric electrolytes. For such electrolytes, however, it is essential to distinguish between multivalent counterions and coions. For multivalent counterions, one must consider their strong tendency towards adsorption to the oppositely charged substrates, which leads to low charge densities. In this situation, the CCC increases with the surface charge density, inducing the strong decrease of the CCC with valence. For multivalent coions, the substrates are typically highly charged. In this case, the CCC decreases with increasing ionic valence and is in fact inversely proportional to the valence. This dependence is referred to as the inverse Schulze-Hardy rule.

Expansion and shrinkage of the electrical double layer in charge-asymmetric electrolytes: A non-linear Poisson-Boltzmann description

The spatial extension or thickness of the electrical double layer determines many thermodynamic and electrokinetic properties of charged colloids in solution. In the classical Debye-Hückel formalism of point ions, the thickness of the electrical double layer around a spherical macroion or next to an infinite planar electrode can be characterized by the bulk Debye length of the supporting ionic fluid. As a result, that approach neglects, at least, the influence of the colloidal charge on the spatial extension of the ionic cloud. Given that the Debye-Hückel formalism of point ions is valid only in the limit of very weak colloidal charges, in this work we use the non-linear Poisson-Boltzmann equation to study the thickness of the electrical double layer near a positively charged electrode in spherical and planar geometries, in the presence of several binary charge-asymmetric −1:z+ point-ions electrolytes with monovalent counterions and multivalent coions. The properties of counterions are maintained fixed, whereas the properties of coions (such as their valence and concentration) are varied fulfilling the bulk electroneutrality condition. The thickness of the ionic cloud is quantified here via the recently introduced capacitive compactness idea (Phys. Chem. Chem. Phys. 20 (2018) 262). Physically, this length represents essentially the separation distance between two electrodes associated to the corresponding effective electrical double layer capacitor, in both planar and spherical geometries. Our numerical calculations show that the capacitive compactness obtained via the non-linear Poisson-Boltzmann equation reduces to the Debye length of the supporting bulk electrolyte at the point of zero charge. In the presence of a charge symmetric −1:+1 electrolyte, the capacitive compactness always decreases as a function of the colloidal charge. Contrastingly, the electrical double layer may expand or shrink as a function of the surface charge density, in the presence of multivalent coions, which is confirmed here by primitive model Monte Carlo simulations. This last non-monotonic behaviour of the capacitive compactness for multivalent coions, depending on the colloidal charge, is related to the microscopic behaviour of the local electric field, the mean electrostatic potential, and the net ionic charge per unit volume close to the colloidal surface. At very large colloidal charges, the capacitive compactness of all −1:z+ electrolytes collapses onto a single curve illustrating the dominance of counterions in the non-linear Poisson-Boltzmann theory.

DNA like-charge attraction and overcharging by divalent counterions in the presence of divalent co-ions.

Strongly correlated electrostatics of DNA systems has drawn the interest of many groups, especially the condensation and overcharging of DNA by multivalent counterions. By adding counterions of different valencies and shapes, one can enhance or reduce DNA overcharging. In this paper, we focus on the effect of multivalent co-ions, specifically divalent co-ions such as SO[Formula: see text]. A computational experiment of DNA condensation using Monte Carlo simulation in grand canonical ensemble is carried out where the DNA system is in equilibrium with a bulk solution containing a mixture of salt of different valency of co-ions. Compared to systems with purely monovalent co-ions, the influence of divalent co-ions shows up in multiple aspects. Divalent co-ions lead to an increase of monovalent salt in the DNA condensate. Because monovalent salts mostly participate in linear screening of electrostatic interactions in the system, more monovalent salt molecules enter the condensate leads to screening out of short-range DNA-DNA like charge attraction and weaker DNA condensation free energy. The overcharging of DNA by multivalent counterions is also reduced in the presence of divalent co-ions. Strong repulsions between DNA and divalent co-ions and among divalent co-ions themselves lead to a depletion of negative ions near the DNA surface as compared to the case without divalent co-ions. At large distances, the DNA-DNA repulsive interaction is stronger in the presence of divalent co-ions, suggesting that divalent co-ions' role is not only that of simple stronger linear screening.

Depletion and double layer forces acting between charged particles in solutions of like-charged polyelectrolytes and monovalent salts.

Interaction forces between silica particles were measured in aqueous solutions of the sodium salt of poly(styrene sulphonate) (PSS) and NaCl using the colloidal probe technique based on an atomic force microscope (AFM). The observed forces can be rationalized through a superposition of damped oscillatory forces and double layer forces quantitatively. The double layer forces are modeled using Poisson-Boltzmann (PB) theory for a mixture of a monovalent symmetric electrolyte and a highly asymmetric electrolyte, whereby the multivalent coions represent the polyelectrolyte chains. The effective charge of the polyelectrolyte is found to be smaller than the bare number of charged groups residing on one polyelectrolyte molecule. This effect can be explained by counterion condensation. The interplay between depletion and double layer forces can be further used to predict the phase of the depletion force oscillations. However, this picture holds only at not too elevated concentrations of the polyelectrolyte and salt. At higher salt concentrations, attractive van der Waals forces become important, while at higher polyelectrolyte concentrations, the macromolecules adsorb onto the like-charged silica interface.

Interactions between silica particles in the presence of multivalent coions.

Forces between charged silica particles in solutions of multivalent coions are measured with colloidal probe technique based on atomic force microscopy. The concentration of 1 : z electrolytes is systematically varied to understand the behavior of electrostatic interactions and double-layer properties in these systems. Although the coions are multivalent the Derjaguin, Landau, Verwey, and Overbeek (DLVO) theory perfectly describes the measured force profiles. The diffuse-layer potentials and regulation properties are extracted from the forces profiles by using the DLVO theory. The dependencies of the diffuse-layer potential and regulation parameter shift to lower concentration with increasing coion valence when plotted as a function of concentration of 1 : z salt. Interestingly, these profiles collapse to a master curve if plotted as a function of monovalent counterion concentration.

Derivation of the inverse Schulze-Hardy rule.

The inverse Schulze-Hardy rule was recently proposed based on experimental observations. This rule describes an interesting situation of the aggregation of charged colloidal particles in the presence of the multivalent coions. Specifically, it can be shown that the critical coagulation concentration is inversely proportional to the coion valence. Here the derivation of the inverse Schulze-Hardy rule based on purely theoretical grounds is presented. This derivation complements the classical Schulze-Hardy rule, which describes the multivalent counterion systems.

Effect of Ionic Correlations on the Surface Forces in Thin Liquid Films: Influence of Multivalent Coions and Extended Theory.

Experimental data for the disjoining pressure of foam films stabilized by anionic surfactant in the presence of 1:1, 1:2, 1:3, and 2:2 electrolytes: NaCl, Na2SO4, Na3Citrate, and MgSO4 are reported. The disjoining pressure predicted by the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory coincides with the experimental data in the case of a 1:1 electrolyte, but it is considerably greater than the measured pressure in all other cases. The theory is extended to account for the effects of ionic correlations and finite ionic radii. Original analytical expressions are derived for the local activity coefficient, electrostatic disjoining pressure, and asymptotic screening parameter. With the same parameter of counterion binding as for a 1:1 electrolyte, the curves predicted by the extended theory are in perfect agreement with the experimental data for 1:2 and 1:3 electrolytes. In comparison with the DLVO theory, the effect of ionic correlations leads to more effective screening of electrostatic interactions, and lower electric potential and counterion concentrations in the film’s midplane, resulting in lower disjoining pressure, as experimentally observed. The developed theory is applicable to both multivalent coions and multivalent counterions. Its application could remove some discrepancies between theory and experiment observed in studies with liquid films from electrolyte solutions.

The non-dominance of counterions in charge-asymmetric electrolytes: non-monotonic precedence of electrostatic screening and local inversion of the electric field by multivalent coions.

The asymptotic convergence of the thermodynamic and structural properties of unequally-sized charge-symmetric ions in strong electric fields was postulated more than thirty years ago by Valleau and Torrie as the dominance of counterions via the non-linear Poisson-Boltzmann theory [Valleau and Torrie, J. Chem. Phys., 1982, 76, 4623]. According to this mean field prescription, the properties of the electrical double layer near a highly charged electrode immersed in a size-asymmetric binary electrolyte converge to those of a size-symmetric electrolyte if the properties of counterions are the same in both instances. On the other hand, some of the present authors have shown that, in fact, counterions do not dominate the electrical properties of a spherical macroion in the presence of unequally-sized ions, symmetric in valence, if ion correlations and ionic excluded volume effects are taken into account consistently. These ingredients are neglected in the classical Poisson-Boltzmann picture. In the present work, we show the occurrence of the non-dominance of counterions in the opposite scenario, that is, when ions are equally-sized but asymmetric in valence. This is performed in the presence of highly charged colloidal surfaces of spherical and planar geometries for different ionic volume fractions. In addition to the phenomenon of non-dominance of counterions, our simulations and theoretical data also exhibit a non-monotonic order or precedence in the mean electrostatic potential, or electrostatic screening, at the Helmholtz plane of a charged colloid. This interesting behaviour is analyzed as a function of the coion's valence, the ionic volume fraction, and the charge and size of the colloidal particle. All these phenomena are explained in terms of the decay of the electric field near the colloidal surface, and by the appearance of a local inversion of both the electric field and the integrated surface charge density of the colloidal particle in the presence of monovalent counterions and multivalent coions.

Suppression and promotion of charge inversion in the presence of multivalent coions.

We report charge inversion using Monte Carlo calculations for a negatively charged surface in aqueous solutions involving coions of different charges and monovalent counterions. It is shown that a rise in the valence of coions at moderate concentrations can substantially promote charge inversion for the surface charge values of biological relevance, regardless of the representation of surface charges but dependent in a nontrivial way on polarization effects resulting from dielectric discontinuity. These obtained characteristics challenge the traditional belief that the coions are generally considered to suppress charge inversion and expose the important role of coions of higher valence in tailoring the effective interactions of biomolecules with the cell membrane.

Aggregation of Colloidal Particles in the Presence of Multivalent Co-Ions: The Inverse Schulze-Hardy Rule.

Shifts of the critical coagulation concentration (CCC) in particle suspensions in salt solutions containing multivalent co-ions and monovalent counterions are rationalized. One observes that the CCC is inversely proportional to the valence, and this behavior is referred to as the inverse Schulze-Hardy rule. This dependence is established by means of measurements of the stability ratio for positively and negatively charged latex particles with time-resolved light scattering. The same dependence is equally suggested by calculations of the CCC with the Derjaguin, Landau, Verwey, and Overbeek (DLVO) theory, whereby the full Poisson-Boltzmann equation for the asymmetric electrolytes has to be used. The latter aspect is essential, since in the case of multivalent co-ions the surface charge is principally neutralized by monovalent counterions. This rule complements the classical Schulze-Hardy rule, which applies in the case of multivalent counterions, and states that the CCC is inversely proportional to the sixth power of the valence.

Long-ranged and soft interactions between charged colloidal particles induced by multivalent coions.

Forces between charged particles in aqueous solutions containing multivalent coions and monovalent counterions are studied by the colloidal probe technique. Here, the multivalent ions have the same charge as the particles, which must be contrasted to the frequently studied case where multivalent ions have the opposite sign as the substrate. In the present case, the forces remain repulsive and are dominated by the interactions of the double layers. The valence of the multivalent coion is found to have a profound influence on the shape of the force curve. While for monovalent coions the force profile is exponential down to separations of a few nanometers, the interaction is much softer and longer-ranged in the presence of multivalent coions. The force profiles in the presence of multivalent coions and in the mixtures of monovalent and multivalent coions can be accurately described by Poisson-Boltzmann theory. These results are accurate for different surfaces and even in the case of highly charged particles. This behavior can be explained by the fact that the force profile follows the near-field limit to much larger distances for multivalent coions than for monovalent ones. This limit corresponds to the conditions with no salt, where the coions are expelled between the two surfaces.

Structure of Colloidal Solution in Presence of Mixed Electrolytes: A Solvent Restricted Primitive Model Study

The structure of colloidal solution in presence of mixed electrolytes is studied using Monte Carlo simulation and density functional theory, based on a four-component model of the spherical double layer. In this model the ions and solvent molecules are treated as charged and neutral hard spheres, respectively, having equal diameter, and in addition the mixture of mono- and multivalent co-ions are also considered. The macroion is considered as a uniformly charged hard sphere surrounded by the electrolyte and the solvent. The density functional theory is based on a partially perturbative scheme, where the electrical part is calculated through perturbation with respect to uniform density and the hard sphere contribution is approximated using a weighted density approach. The theory is found to be in quantitative agreement with the Monte Carlo simulation results, for singlet density as well as the mean electrostatic potential profiles. The system is studied over a wide range of parametric conditions, viz. with different ionic valences as well as size, at varying electrolyte concentration ratio of mono- and multivalent co-ions of mixed electrolytes, at different surface charge densities, and radius of the macroion. The present work reflects that even a simple primitive model for the solvent is able to manipulate the hard-sphere and electrostatic correlations of the diffuse double layer in the ionic density as well as mean electrostatic potential profiles.

Polyelectrolyte Brush Interaction with Multivalent Ions

We present a scaling theory to describe the equilibrium structure of an annealed polyelectrolyte brush interacting with multivalent salt ions. We demonstrate that at low ionic strengths of the solution the polyelectrolyte brush swells with increases in the bulk concentration of salt, Φ s . When multivalent counterions are added to the solution, the scaling exponents describing the dependences of the brush thickness H on Φ s become a function of the valency q of the counterion. We also find that, when such a brush is compressed, the shape of the force versus deformation profile depends on q. In the case of multivalent coions, the characteristics of the brush depend on the net concentration of charge, qΦ s , of the added ions. As a result, the force versus deformation profile obeys a single scaling law as a function of qΦ s . At high concentrations of salt, the features of the salt ions affect the properties of the annealed brush through the net ionic strength of the solution.

Theory of planar polyelectrolyte brush immersed in solution of asymmetric salt

The analytical self-consistent field theory of polymer brushes and the Scheutjens–Fleer numerical theory are applied in a study of the equilibrium behaviour of a planar polyelectrolyte brush immersed in a solution of an asymmetric salt. A scaling-type diagram comprising various regimes of brush behaviour is constructed and verified by numerical calculations. It is shown that for a highly asymmetric salt with multivalent coions there are two different salt-controlled regimes of brush behaviour. At large salt contents, highly charged coions penetrate into the brush and dominate the screening of the electrostatic repulsion between grafted polyions (salted brush; SB). In this regime the brush thickness H scales as H ∼ I − 1 3 , where I is the ionic strength of the outer solution (made up of coion and counterion contributions). At relatively small salt contents, highly charged coions are excluded from the brush whereas counterions equal out their concentrations in the brush and outer solution (counterion brush; CB). In this regime the screening of electrostatic repulsion in the brush is performed by mobile counterions and the brush thickness scales as H ∼ φ −1 s, where φ s is the concentration of salt counterions only in the outer solution. Possible experiments to verify these theoretical predictions are discussed.

Double-layer forces in nonadsorbing ionic micellar solutions and polyelectrolyte solutions

A report is given of a series of surface force measurements between molecularly smooth mica surfaces in solutions of sodium dodecyl sulfate (SDS) and sodium polystyrene sulfonate (NaPSS). The micelles and polyions act as coions to the mica surface. This allows an investigation of the effect of large multivalent coions on the range and nature of the double-layer forces. Below the critical micelle concentration (CMC) of SDS, it is found that the forces are well described by assuming the SDS to be completely dissociated. Above the CMC, the micelles together with their effectively “bound” counterions do not contribute to the electrostatic screening. In NaPSS solutions, at constant monomolar concentration (i.e., concentration of polymer segments), the distance range of the double-layer forces increases with increasing chain length. Given that the bulk counterion activity at constant monomolar segment concentration is independent of the chain length, this observation strongly suggests that a polyelectrolyte depletion (together with a concurrent counterion depletion) occurs between the surfaces, which becomes enhanced with increasing molecular weight. Double-layer forces in the presence of divalent counterions increase in magnitude on addition of NaPSS. This is attributed to preferential counterion condensation on the polyelectrolyte backbone: depleted NaPSS effectively scavenges divalent ions from the ionic double-layers between the surfaces and releases a corresponding concentration of previously depleted monovalent counterions.

Sorption of copper from dilute aqueous solutions onto poly(vinyl mercaptan)

A hydrophilic poly(vinyl mercaptan) (PVM) has been prepared from poly(vinyl alcohol) by partial conversion of the OH groups to SH groups. The resin with a free mercaptan content of 2.23 meq/g has a copper sorption capacity of 70 mg/g. The polymercaptan used as a sodium salt, PVM(Na), has several-fold higher sorption capacity for copper than the same resin used in thiol from, PVM(H). From a solution containing Cu(II), copper is sorbed mostly as Cu(II) by PVM(Na) and as Cu(I) by PVM(H). Significantly, there is a large increase in the copper sorption capacity of both PVM(H) and PVM(Na) in the presence of sodium chloride in solution. In the absence of added NaCl, the copper sorption capacities of PVM(H) and PVM(Na) are influenced in different ways by the type of coion in solution. Thus, PVM(H) takes up more copper from copper(II) chloride solution than from copper(II) sulfate solution under comparable conditions, while the opposite is true for PVM(Na). These features can be explained by the formation of copper(II) chloro complexes in the presence of added chloride, more efficient Donnan exclusion of multivalent coions, and greater degree of hydrolysis of copper(II) chloride than copper(II) sulfate. The copper sorption on PVM(Na) is highly sensitive to pH, decreasing rapidly both at lower and higher pH from a maximum at pH ≈ 5. In contrast, the sorption on PVM(H) is fairly constant in the mildly acidic pH range and increases rapidly above pH ≈ 9, approaching the sorption capacity of PVM(Na).

Interactions of Multivalent Coions with Sodium Iota Carrageenan and Sodium Alginate by Self-Diffusion Measurements

Interactions of Multivalent Coions with Sodium Iota Carrageenan and Sodium Alginate by Self-Diffusion Measurements

Interactions of multivalent coions and sodium ions with polyelectrolytes by diffusion studies

Interactions of multivalent coions and sodium ions with polyelectrolytes by diffusion studies