Counterion condensation and release in micellar solutions

Counterion condensation and release in micellar solutions are investigated by direct measurement of counterion concentration with ion-selective electrode. Monte Carlo simulations based on the cell model are also performed to analyze the experimental results. The degree of counterion condensation is indicated by the concentration ratio of counterions in the bulk to the total ionic surfactant added, alpha< or =1. The ionic surfactant is completely dissociated below the critical micelle concentration (cmc). However, as cmc is exceeded, the free counterion ratio alpha declines with increasing the surfactant concentration and approaches an asymptotic value owing to counterion condensation to the surface of the highly charged micelles. Micelle formation leads to much stronger electrostatic attraction between the counterion and the highly charged sphere in comparison to the attraction of single surfactant ion with its counterion. A simple model is developed to obtain the true degree of ionization, which agrees with our Monte Carlo results. Upon addition of neutral polymer or monovalent salts, some of the surfactant counterions are released to the bulk. The former is due to the decrease of the intrinsic charge (smaller aggregation number) and the degree of ionization is increased. The latter is attributed to competitive counterion condensation, which follows the Hefmeister series. This consequence indicates that the specific ion effect plays an important role next to the electrostatic attraction.     

Polyelectrolyte solutions: Counterion condensation and intermolecular interactions

This article demonstrates that the neglect of nonlinear effects in the conventional counterion condensation theory for the double layer about a charged cylinder can be significant, especially for phenomena involving intramolecular or intermolecular interactions in dilute solutions. For concentrated solutions the Manning theory derives from a linearized superposition approximation for the potential, in contrast to the cylindrical-cell model, which explicitly treats interactions within an ordered array of parallel cylinders. A new theory which treats interactions explicitly while permitting disorder in two dimensions is presented, and predictions for the osmotic pressure are compared with those from the Manning and cylindrical-cell models.     

Ion condensation on charged patterned surfaces

We study ion condensation on a patterned surface with stripes of alternating charge. The competition between adsorbed ion-ion and adsorbed ion-surface interactions leads to the formation of different strongly correlated structures of condensed ions in the low-temperature limit (LTL). We consider two types of arrangements which have lowest energy in the LTL: (1) ions adsorbed onto the stripe center lines and (2) arrays of dipoles at the interfaces between charged domains. We determine the preferred arrangement as a function of surface charge density, the chemical potential of the ions in the surrounding medium, and the geometric parameters of the system. We determine the conditions for the appearance of more complex ionic patterns by considering simple perturbations of the stripe-centered and dipolar array structures.     

Counterion release from adsorbed highly charged polyelectrolyte: an electrooptical study

The adsorption of sodium poly(4-styrene sulfonate) on oppositely charged beta-FeOOH particles is studied by electrooptics. The focus of this paper is on the release of condensed counterions from adsorbed polyelectrolyte upon surface charge overcompensation. The fraction of condensed Na+ counterions on the adsorbed polyion surface is estimated according to the theory of Sens and Joanny and it is compared with the fraction of condensed counterions on nonadsorbed polyelectrolyte. The relaxation frequency of the electrooptical effect from the polymer-coated particle is found to depend on the polyelectrolyte molecular weight. This is attributed to polarization of the layer from condensed counterions on the polyion surface, being responsible for creation of the effect from particles covered with highly charged polyelectrolyte. The number of the adsorbed chains is calculated also assuming counterion condensation on the adsorbed polyelectrolyte and semiquantative agreement is found with the result obtained from the condensed counterion polarizability of the polymer-coated particle. Our findings are in line with theoretical predictions that the fraction of condensed counterions remains unchanged due to the adsorption of highly charged polyelectrolyte onto weakly charged substrate.     

Charge renormalization of charged spheres based on thermodynamic properties

At strong electrostatic coupling, counterions are accumulated in the vicinity of the surface of the charged particle with intrinsic charge Z. In order to explain the behavior of highly charged particles, effective charge Z(*) is therefore invoked in the models based on Debye-Huckel approximation, such as the Derjaguin-Landau-Verwey-Overbeek potential. For a salt-free colloidal suspension, we perform Monte Carlo simulations to obtain various thermodynamic properties omega in a spherical Wigner-Seitz cell. The effect of dielectric discontinuity is examined. We show that at the same particle volume fraction, counterions around a highly charged sphere with Z may display the same value of omega as those around a weakly charged sphere with Z(*), i.e., omega(Z)=omega(Z(*)). There exists a maximally attainable value of omega at which Z=Z(*). Defining Z(*) as the effective charge, we find that the effective charge passes through a maximum and declines again due to ion-ion correlation as the number of counterions is increased. The effective charge is even smaller if one adopts the Debye-Huckel expression omega(DH). Our results suggest that charge renormalization can be performed by chemical potential, which may be observed in osmotic pressure measurements.     

Charge renormalization of cylinders and spheres: Ion size effects

The effect of the counterion size on the degree of counterion condensation onto a cylindrical macroion and a spherical one in the absence of salt is studied theoretically within a modified Poisson–Boltzmann approach. We find that excluded volume interactions reduce the degree of condensation. Using a simple variational free energy we show that this reduction can be attributed to an effective increase in the macroion size due to the contribution of the condensed counterions. We also find that for a charged cylinder, the reduction in charge renormalization vanishes at infinite dilution because of the extended nature of the condensed layer. In contrast, excluded volume interactions can reduce the degree of charge renormalization of a sphere even at high dilutions. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 3598–3615, 2004     

Phase behavior of cetyltrimethylammonium surfactants with oligo carboxylate counterions mixed with water and decanol: attraction between charged planes or spheres with oligomeric counterions

Cetyltrimethylammonium surfactants with a range of oligo carboxylate anions bearing 2, 3, or 4 negative charges have been synthesized, and their respective behaviors in binary mixtures with water and in ternary mixtures with added decanol have been investigated. In binary mixtures with water, all surfactants formed nearly spherical micelles at high water contents; however, the interactions between micelles varied strongly with the number of charges in the counterion. Micelles with divalent counterions were generally miscible with water, whereas micelles with tri- or tetravalent counterions demixed in one concentrated and one dilute phase. Addition of decanol resulted in all cases in the appearance of a lamellar phase, and all investigated oligo carboxylate anions (di-, tri-, and tetravalent) gave rise to a strong attraction between the lamellar planes, resulting in a limited swelling (up to 35-40 wt % water) of the lamellar phase in contact with excess water. These experiments confirm the theoretically predicted influence of aggregate geometry (spheres or planes) on the attraction between colloidal aggregates neutralized by multivalent counterions. Further addition of decanol resulted in the appearance of a second birefringent phase in equilibrium with the lamellar phase. SWAXS showed this phase to be lamellar and to display short-range order that disappeared upon heating. This phase is identified as a lamellar gel phase (Lbeta-phase).     

Counterion condensation in short cationic peptides: limiting mobilities beyond the Onsager-Fuoss theory

We investigated the effect of the background electrolyte (BGE) anions on the electrophoretic mobilities of the cationic amino acids arginine and lysine and the polycationic peptides tetraarginine, tetralysine, nonaarginine, and nonalysine. BGEs composed of sodium chloride, sodium propane-1,3-disulfonate, and sodium sulfate were used. For the amino acids, determination of the limiting mobility by extrapolation, using the Onsager-Fuoss (OF) theory expression, yielded consistent estimates. For the peptides, however, the estimates of the limiting mobilities were found to spuriously depend on the BGE salt. This paradox was resolved using molecular modeling. Simulations, on all-atom as well as coarse-grained levels, show that significant counterion condensation, an effect not accounted for in OF theory, occurs for the tetra- and nonapeptides, even for low BGE concentrations. Including this effect in the quantitative estimation of the BGE effect on mobility removed the discrepancy between the estimated limiting mobilities in different salts. The counterion condensation was found to be mainly due to electrostatic interactions, with specific ion effects playing a secondary role. Therefore, the conclusions are likely to be generalizable to other analytes with a similar density of charged groups and OF theory is expected to fail in a predictable way for such analytes.     

Line of charges in electrolyte solution near a half-space I. Counterion condensation

The effect of a half-space on counterion condensation around a line of charges in electrolyte solution is examined in the framework of Debye-Hückel electrostatics. The half-space substrate is allowed to be a conductor, a dielectric, or a semiconductor. Counterions are predicted to be released completely as the line of charges approaches a conducting substrate. When it approaches a dielectric substrate, depending on the ratio of solvent to substrate dielectric constant, there are three possibilities: (1) epsilon(sol)/epsilon(sub) < 1; the counterions are partially (or completely) released; (2) epsilon(sol)/epsilon(sub) = 1; the amount of condensation remains unchanged; and (3) epsilon(sol)/epsilon(sub) > 1; more counterions condense. Depending on the relative magnitude of screening lengths in the semiconductor and in the solution, its effect on condensation follows either that of a metal or that of a dielectric. For the case of a moderately doped silicon substrate, condensation is predicted to be similar to that for a dielectric.     

Model shape transitions of micelles: spheres to cylinders and disks

We present a microscopic analysis of shape transitions of micelles of model linear nonionic surfactants. In particular, symmetric H(4)T(4) and asymmetric H(3)T(6) surfactants have been chosen for the study. In a previous work, it has been observed that symmetric surfactants have a strong tendency to prefer spherical micelles over a wide range of chemical potentials, while asymmetric surfactants undergo shape transitions between a spherical micelle at low concentration to other forms, mainly finite cylindrical micelles. This study combines the application of a two-dimensional single-chain mean-field theory (SCMFT) with Monte Carlo (MC) simulations of exactly the same systems. On the one hand, the characteristics of the SCMFT make this method suitable for free energy calculations, especially for small surfactants, due to the incorporation of relevant microscopic details in the model. On the other hand, MC simulations permit us to obtain a complete picture of the statistical mechanical problem, for the purpose of validation of the mean-field calculations. Our results reveal that the spherical shape for the symmetric surfactant is stable over a large range of surfactant concentrations. However, the asymmetric surfactant undergoes a complex shape transition that we have followed by calculating the standard chemical potential as a function of the aggregation number. The results indicate that the system forms prolate spheroids prior to developing short capped cylinders that gradually grow in length, with some oscillations in the energy of formation. The most important result of our work is the evidence of a bifurcation where, together with the elongated objects, the system can develop oblate aggregates and finally a torus shape similar to a red blood cell.     

Theoretical mass-transfer coefficients for circular cylinders and spheres in cross flow

Based on the mathematical solutions of Levich and Natanson, which describe diffusion to a sphere and a circular cylinder in a fluid stream, theoretical expressions are derived for mass-transfer coefficients in flow-through porous electrodes composed of wires, wire nets and spheres. Those expressions are applicable for theoretical interpretation and estimation of the diffusion-controlled efficiency of electrolysis for flow-through porous electrodes. The numerical solutions of the theoretical expressions derived compare favorably with published experimental results.     

Component dynamics in nanostructured PI-PDMS diblock copolymers with PI segregated in lamellas,cylinders, and spheres

The dynamics of the two components of self-assembled diblock polyisoprene (PI)-polydimethylsiloxane (PDMS) copolymers is investigated by means of broadband dielectric spectroscopy. By varying the size of the PDMS blocks, different PI segregation geometries are obtained, namely lamellas, cylinders, and spheres, with typical sizes in the range 6–16 nm. In this way, we identify the effects of the nanostructure on the dielectric relaxation of the polymer components: the α-relaxation of PI and PDMS blocks as well as the normal mode of PI blocks. Two different situations are explored: (1) the PDMS dynamics is detected in a temperature range where the PI phase is frozen and (2) the PI dynamics is detected in a temperature range where the PDMS phase is highly mobile. Thus, using a single system, the similarities and differences between hard and soft confinement are investigated.     

Phase behaviour of binary mixtures of colloidal charged spheres

As a step towards the modelling of binary metal alloys we here report on the shape of the phase boundary of two deionized charged sphere colloidal suspensions as a function of mixing ratio and particle density. Their size ratios are r = 0.68 and r = 0.56. Both aqueous suspensions of polystyrene copolymer spheres crystallize in a body-centred cubic structure. Interesting differences in the shape of the phase boundary are observed. In the first case a peaked increase of crystal stability was observed for a mixing ratio of p = 0.2–0.3, which gives the fraction of small spheres. Also in the second case the stability of the crystalline phase is larger than expected for an ideal solid solution but over a more extended range of small p. In addition at p = 0.7–0.8 we find a pronounced suppression of crystallization and furthermore some indications of a precipitation of one species at p = 0.9. While the first phase diagram resembles that of a solid solution with possibly the onset of compound formation, the second more resembles a eutectic.     

Diffusiophoretic mobility of charged porous spheres in electrolyte gradients

The diffusiophoretic motion of a polyelectrolyte molecule or charged floc in an unbounded solution of a symmetrically charged electrolyte with a uniform prescribed concentration gradient is analytically studied. The model used for the particle is a porous sphere in which the density of the hydrodynamic frictional segments, and therefore also that of the fixed charges, is constant. The electrokinetic equations which govern the electrostatic potential profile, the ionic concentration distributions (or electrochemical potential energies), and the fluid velocity field inside and outside the porous particle are linearized by assuming that the system is only slightly distorted from equilibrium. Using a regular perturbation method, these linearized equations are solved for a charged porous sphere with the density of the fixed charges as the small perturbation parameter. An analytical expression for the diffusiophoretic mobility of the charged porous sphere in closed form is obtained from a balance between its electrostatic and hydrodynamic forces. This expression, which is correct to the second order of the fixed charge density of the particle, is valid for arbitrary values of kappaa and lambdaa, where kappa is the reciprocal of the Debye screening length, lambda is the reciprocal of the length characterizing the extent of flow penetration inside the particle, and a is the particle radius. Our result to the first order of the fixed charge density agrees with the corresponding solution for the electrophoretic mobility obtained in the literature. In general, the diffusiophoretic mobility of a porous particle becomes greater as the hindrance to the diffusive transport of the solute species inside the particle is more significant.     

Adsorption of polymer chains onto charged spheres: Experiment and theory

A self-consistent field (SCF) theory for the adsorption of polyelectrolyte chains onto oppositely charged spheres is considered. It is demonstrated that the criterion for critical adsorption shows a different behavior for small and large curvature of the surface. Experiments give indeed evidence for the power-law behavior as theoretically predicted for large curvature.     

Strong and weak adsorptions of polyelectrolyte chains onto oppositely charged spheres

We investigate the complexation of long thin polyelectrolyte (PE) chains with oppositely charged spheres. In the limit of strong adsorption, when strongly charged PE chains adapt a definite wrapped conformation on the sphere surface, we analytically solve the linear Poisson-Boltzmann equation and calculate the electrostatic potential and the energy of the complex. We discuss some biological applications of the obtained results. For weak adsorption, when a flexible weakly charged PE chain is localized next to the sphere in solution, we solve the Edwards equation for PE conformations in the Hulthen potential, which is used as an approximation for the screened Debye-Huckel potential of the sphere. We predict the critical conditions for PE adsorption. We find that the critical sphere charge density exhibits a distinctively different dependence on the Debye screening length than for PE adsorption onto a flat surface. We compare our findings with experimental measurements on complexation of various PEs with oppositely charged colloidal particles. We also present some numerical results of the coupled Poisson-Boltzmann and self-consistent field equation for PE adsorption in an assembly of oppositely charged spheres.     

Sedimentation velocity and potential in concentrated suspensions of charged porous spheres

The body-force-driven migration in a homogeneous suspension of polyelectrolyte molecules or charged flocs in an electrolyte solution is analyzed. The model used for the particle is a porous sphere in which the density of the hydrodynamic frictional segments, and therefore also that of the fixed charges, is constant. The effects of particle interactions are taken into account by employing a unit cell model. The overlap of the electric double layers of adjacent particles is allowed and the relaxation effect in the double layer surrounding each particle is considered. The electrokinetic equations which govern the electrostatic potential profile, the ionic concentration (or electrochemical potential energy) distributions, and the fluid velocity field inside and outside the porous particle in a unit cell are linearized by assuming that the system is only slightly distorted from equilibrium. Using a regular perturbation method, these linearized equations are solved for a symmetrically charged electrolyte with the density of the fixed charges as the small perturbation parameter. An analytical expression for the settling velocity of the charged porous sphere is obtained from a balance among its gravitational, electrostatic, and hydrodynamic forces. A closed-form formula for the sedimentation potential in a suspension of identical charged porous spheres is also derived by using the requirement of zero net electric current. The dependence of the sedimentation velocity and potential of the suspension on the particle volume fraction and other properties of the particle-solution system is found to be quite complicated.     

Primary electroviscous effect in a suspension of charged porous spheres

Primary electroviscous effect for a dilute suspension of porous spheres with fixed volumetric charge density is investigated theoretically. In the absence of flow, the electrical potential and solution charge density are assumed to satisfy the linearized Poisson-Boltzmann equation. With incorporation of the electrical body force, the Brinkman equation and the Stokes equation are used to govern the fluid flow inside and outside a sphere. The theory is formulated by assuming weak deviation of the charge cloud from its equilibrium state. However, the electrical body force is not restricted to be small compared to the viscous force in the fluid momentum equation. The results show that the double layer distortion is increased with increasing particle permeability, thereby enhancing the relative importance of its stress contribution. Nonetheless, the intrinsic viscosity remains a decreasing function of permeability, similar to the case of uncharged particles.     

Mean spherical model for a mixture of charged spheres and hard dipoles

A solution to the mean spherical model for a mixture of charged hard spheres of equal concentration and opposite charge, and hard dipoles, all of equal size, is given. The excess thermodynamic properties and the structure of the system depend on only 3 parameters, which are the solution of a set of 3 nonlinear, algebraic equations.