Counterions localization

Abstract Aqueous solutions of star-like polyelectrolytes (PEs) exhibit distinctive features that originate from the topological complexity of branched macromolecules. In a salt-free solution of branched PEs, mobile counterions preferentially localize in the intramolecular volume of branched macroions. Counterion localization manifests itself in a dramatic reduction of the osmotic coefficient in solutions of branched polyions as compared with those of linear PEs. The intramolecular osmotic pressure, created by entrapped counterions, imposes stretched conformations of branches and this leads to dramatic intramolecular conformational transitions upon variations in environmental conditions. In this chapter, we overview the theory of conformations and stimuli-induced conformational transitions in star-like PEs in aqueous solutions and compare these to the data from experiments and Monte Carlo and molecular dynamics simulations. 

Due to the counterion localization, conformations of branched macroions that comprise strongly dissociating groups (charge is quenched) are almost insensitive to the addition of salt, up to relatively high salt concentrations. The ability of a branched polyion to maintain a virtually constant ionic strength in its interior is of special interest for potential applications, where a controlled (buffered) microenvironment is essential (e.g., colloidal bionanoreactors, smart nanocontainers for biologically active molecules, etc.). 

The objective of this chapter is to present an overview of the existing theories on eonformations of star-branched PEs and to compare these to experimental data and the results of computer simulations. Some emphasis is made on the effect of counterion localization and its consequences for the conformations of branched PEs. 

We start with a brief reminder on the scaling theory of nonionic star-branched polymers (Sect. 2), and proceed with the scaling model of a PE star polymer in a salt-free dilute solution (Sect. 3). We then discuss the physical basis of counterion localization and its manifestation in branched PEs of different topologies (Sect. 4).)... 

A quantitative analysis of counterion localization in a salt-free solution of star-like PEs is carried out on the basis of an exact numerical solution of the corresponding Poisson-Boltzmann (PB) problem (Sect. 5). Here, the conformational degrees of freedom of the flexible branches are accounted for within the Scheutjens-Fleer self-consistent field (SF-SCF) framework. The latter is used to prove and to quantify the applicability of the concept of colloidal charge renormalization to PE stars, that exemplify soft charged colloidal objects. The predictions of analytical and numerical SCF-PB theories are complemented by results of Monte Carlo (MC) and molecular dynamics (MD) simulations. The available experimental data on solution properties of PE star polymers are discussed in the light of theoretical predictions (Sect. 6). 

A theoretical analysis of the effect of counterion localization in a dilute solution of weakly charged branched polyions of different topologies [31-33] and ionic microgels [34, 35], was performed on the basis of a cell model, similar to that used here for a star-like PE. The elastic term in the free energy that accounts for the conformational entropy of a uniformly swollen branched macromolecule, has to be specified depending on the polyion topology. The shape of the cell might also be modified. For example, in the case of a molecular PE brush, a cylindrical instead of spherical cell should be used. 

Hence, counterion localization occurs when the number of generations in the star-burst polymer, g log2(Af/n), reaches some characteristic value, which is controlled by the same combination of the parameters, as for a PE star. 

The characteristic branching parameter (grafting density), n/m = specifies the onset of counterion localization inside the molecular brush. Note that in the osmotic regime, the spacers get fully extended, /t m. It is therefore not surprising, that the counterion localization in a cylindrical molecular brush coincides (in scaling terms) with the Manning condensation threshold [25] for a charged cylinder, qh = 1. 

A quantitative analysis of counterion localization in a salt-free solution of star-like PEs is described in [29, 37]. Radial distributions for both the electrostatic potential and the density of counterions were obtained by a numerical solution of the corresponding PB problem within a cell model. The conformational degrees of freedom of the branches of a central star were accounted for within the SF-SCF method [120]. Due to the computational efficiency, the SF-SCF framework allows for a systematic study of a many-armed star with sufficiently long arms in a large cell. The range of the parameters that could be covered by the SF-SCF method exceeds that of contemporary MD and MC simulations. 

Fig. 5 Fraction of counterions localized inside the star (at r

Experiments on PE stars show extremely low values of the osmotic coefficient in salt-free star solution, thus proving the concept of counterion localization. Poten-tiometric titration experiments confirm theoretical predictions concerning a shift of the effective pATa upon an increase in the number of arms. 

It was taken into account in the models developed in Refs 75, 78 and 80 that some portion of the counter-ions is bound to surface-active ions within the Stern-Helmholtz (S-H) layer, while another (unbound) portion is located within the diffuse region of the DEL. The equivalent relations of Eqs (56)-(58) in this case contain the difference F — y"X instead of Fj, where F x is the adsorption of counterions localized within the monolayer. It follows from the model described by Eqs (56)-(58) that if all counterions are lo-... 

What are the limits of the approximated expression Eq. 4 Mainly those due to the mean-field nature of PB. For, say, 99 % of the studied systems, the ions are monovalent, ion-ion correlations in water can be safely ignored, and the standard expression is valid. This is no more the case in presence of multivalent counterions (or monovalent ions in solvent of low e). That opens to the fascinating concept of electrostatic attraction between hke-charged colloids, subject of numerous false analyses, debates, and controversies in the literature for 30 years. Figure 1 presents Monte Carlo (MC) simulations data for the force vs. separation law within the primitive model (two latex colloids and ions in continuous solvent) in presence of counterions of increasing valence. While the PB/DLVO prediction remains everywhere repulsive, the exact MC behavior deviates at intermediate separation and develops an attractive well deeper and deeper as the valence increases above 3. This non mean-field effect is due to the repulsions and correlations among the counterions localized in the intersticial region (discreteness of the condensed layer). The same type of colloidal attraction is responsible for a liquid-gas (concentrated solution-dilute solution) phase separation, observed... 

---