Counterions, polyelectrolyte dynamics

Although the coupling of counterion dynamics and polyelectrolyte dynamics has been accounted for at the mean field level, the relaxation of counterion cloud needs to be included in comparing with experimental data. 

As discussed extensively in this chapter, most of the surprising properties of polyelectrolyte dynamics are due to the coupling of counterion dynamics with polymer dynamics. But, there is no adequate understanding of how much of the counterions are mobile and how much are effectively condensed on polymer chain backbone. Theoretical attempts [77, 78] on counterion condensation need to be extended to concentrated poly electrolyte solutions. 

In order to resolve these challenges, it is essential to account for chain connectivity, hydrodynamic interactions, electrostatic interactions, and distribution of counterions and their dynamics. It is possible to identify three distinct scenarios (a) polyelectrolyte solutions with high concentrations of added salt, (b) dilute polyelectrolyte solutions without added salt, and (c) polyelectrolyte solutions above overlap concentration and without added salt. If the salt concentration is high and if there is no macrophase separation, the polyelectrolyte solution behaves as a solution of neutral polymers in a good solvent, due to the screening of electrostatic interaction. Therefore for scenario... 

By accounting for the coupling between the dynamics of polyelectrolyte chains and their counterions and salt ions and assuming that small ions relax faster than polyelectrolyte chains, we have derived Df to be... 

In this chapter we describe the use of polyelectrolytes carrying redox-active centers on electrode surfaces with particular emphasis on organized layer-by-layer redox polyelectrolyte multilayers (RPEM). In redox-active polyelectrolyte multilayers the polyion-polyion intrinsic charge compensation can be broken by ion exchange driven by the electrochemical oxidation and reduction forming extrinsic polyion-counterion pairing. In this chapter we describe the structure, dynamics and applications of these systems. 

Recently, the stiff-chain polyelectrolytes termed PPP-1 (Schemel) and PPP-2 (Scheme2) have been the subject of a number of investigations that are reviewed in this chapter. The central question to be discussed here is the correlation of the counterions with the highly charged macroion. These correlations can be detected directly by experiments that probe the activity of the counterions and their spatial distribution around the macroion. Due to the cylindrical symmetry and the well-defined conformation these polyelectrolytes present the most simple system for which the correlation of the counterions to the macroion can be treated by analytical approaches. As a consequence, a comparison of theoretical predictions with experimental results obtained in solution will provide a stringent test of our current model of polyelectrolytes. Moreover, the results obtained on PPP-1 and PPP-2 allow a refined discussion of the concept of counterion condensation introduced more than thirty years ago by Manning and Oosawa [22, 23]. In particular, we can compare the predictions of the Poisson-Boltzmann mean-field theory applied to the cylindrical cell model and the results of Molecular dynamics (MD) simulations of the cell model obtained within the restricted primitive model (RPM) of electrolytes very accurately with experimental data. This allows an estimate when and in which frame this simple theory is applicable, and in which directions the theory needs to be improved. 

First of all, the comparison of the PB-theory and experiment shown in Fig. 8 proceeds virtually without adjustable parameters. The osmotic coefficient (j) is solely determined by the charge parameter polyelectrolyte concentration. The latter parameter determines the cell radius R0 (see the discussion in Sect. 2.1) Figure 8 summarizes the results. It shows the osmotic coefficient of an aqueous PPP-1 solution as a function of counterion concentration as predicted by Poisson-Boltzmann theory, the DHHC correlation-corrected treatment from Sect. 2.2, Molecular Dynamics simulations [29, 59] and experiment [58]. 

Liao, Q., Dobrynin, A.V., and Rubinstein, M. Molecular dynamics simulations of polyelectrolyte solutions Osmotic coefficient and counterion condensation. Macromolecules, 2003, 36, No. 9, p. 3399-3410. 

Since the experimentally determined osmotic coefficient appears to be smaller even than the molecular dynamics results, this indicates effects to be relevant that go beyond the model used for simulation. Most obvious candidates for this are the neglect of additional chemical interactions between the ions and the polyelectrolyte as well as solvation effects, i.e., interactions between the ions or the polyelectrolyte with the water molecules from the solution. It is for instance demonstrated in Ref. 46 that the osmotic coefficient also depends on whether one uses chlorine or iodine counterions. While one could certainly account for the different radii of these ions when computing the distance of closest approach entering the PB equation, the implications of the different hydration energies is much less obvious to incorporate and in principle requires very expensive all-atom simulations. 

The dynamic behavior of linear charged polyelectrolytes in aqueous solution is not yet understood. The interpretation of dynamic light scattering (DLS) of aqueous solutions of sodium poly(styrene sulfonate) (NaPSS) is particularly complicated. The intensity correlation function shows a bimodal shape with two characteristic decay rates, differing sometimes by two or three orders of magnitude, termed fast and slow modes. The hrst observations in low salt concentration or salt free solution were reported by Lin et al. [31] for aqueous solutions of poly(L-lysine). Their results are described in terms of an extraordinary-ordinary phase transition. An identical behavior was hrst observed by M. Drifford et al. in NaPSS [32], Extensive studies on this bimodal decay on NaPSS in salt-free solution, or solutions where the salt concentration is increased slowly, have been reported [33-36]. The fast mode has been attributed to different origins such as the coupled diffusion of polyions and counterions [34,37,38] or to cooperative fluctuations of polyelectrolyte network [33,39] in the semidilute solutions. 

Winkler RG, Gold M, Reineker P. Collapse of polyelectrolyte macromolecules by counterion condensation and ion pair formation a molecular dynamics simulation study. Phys. Rev. Lett. 1998 80 3731-3734. 

Irrespective of these models for ion binding, the two-state approach is inherent in the dynamic frictional treatment of conductivity in polyelectrolyte solutions, since strongly interacting counterions have to be classified as... 

According to the fluctuation-dissipation theorem [1], the electrical polarizability of polyelectrolytes is related to the fluctuations of the dipole moment generated in the counterion atmosphere around the polyions in the absence of an applied electric field [2-4], Here we calculate the fluctuations by computer simulation to determine anisotropy of the electrical polarizability Aa of model DNA fragments in salt-free aqueous solutions [5-7]. The Metropolis Monte Carlo (MC) Brownian dynamics method [8-12] is applied to calculate counterion distributions, electric potentials, and fluctuations of counterion polarization. 

Ookubo N, Hirai Y, Ito K, Hayakawa R. Anisotropic counterion polarization and their dynamics in aqueous polyelectrolytes as studied by frequency-do-main electric birefringence relaxation spectroscopy. Macromolecules 1989 22 1359-1366. 

As far as deactivation by T1+ is concerned, a fluorescent label attached to PMAA is considered [95,96] to undergo a mixture of static and dynamic quenching. (Static quenching [1] can be defined as a process that occurs too fast to resolve within the timescale of the experiment. In other words, a ground-state interaction or complex forms between the quencher and the fluorophore before excitation. Such a situation would perhaps be not unexpected when counterions condense in high concentrations to a polyelectrolyte backbone in close proximity to a fluorescent label.)... 

Figure 21. Biomesogen regulations between structure and phase (left to right and top to bottom) visualization of biomesogen polyelectrolyte regulations exemplified by DNA/water-shell/(hydrated) counterion-cloud pattern statics and dynamics arbitrary DNA/ counterion-cloud/water-domain arrangements symbolizing operative biomesogen nucleations between structure and phase [7a, 29, 33 a, c, f, p, q].

NMR spectroscopy is a most effective and significant method for observing the structure and dynamics of polymer complexes both in solution and in the solid state. The widest application of NMR spectroscopy is in the field of structure determination. The identification of certain atoms or groups in a molecule as well as their position relative to each other can be obtained by one-, two-, and three-dimensional NMR. Information about the local dynamics, molecular and supramolecular organization, and the counterion condensation can be obtained. The counterion condensation and polyelectrolyte complexation between TEAC in poly(acrylic acid) was studied by Ni using C NMR. 

Fig. 5 Exchange broadening and slow down of rotational diffusion by dynamic electrostatic attachment of counterion spin probes to a polyelectrolyte. Left panel Effect of exchange broadening on the spin probe EPR spectrum. Right panel The broad wings of the center line indicate strong enrichment of counterions close to the polyelectrolyte. Broadening of the low-field and high-field lines compared to the center line stems from a slow down of the reorientation of the probe and is caused by formation of contact ion pairs with a subnanosecond lifetime (data provided by D. Hinderberger).

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. 

More complex systems such as solutions containing macroions and short flexible coimterions have recently been simulated using the primitive model of electrolytes [112]. Solutions of macroions with simple coimterions at different amounts of oppositely charged polyelectrolyte have also been investigated, and the sequence complexation phase separation redissolution was observed [113]. Similar simulations where the macroion represented lysozyme have also been performed [114]. Finally, by using a related soft-sphere model, the dynamics and, in particular, the self-diffusion of the macroions and the counterions have been investigated by employing Brownian dynamics simulation [115]. 

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