Polyelectrolyte molecular theory

G.S. Manning, The molecular theory of polyelectrolyte solution with applications to the properties of polynucleotides. Quart. Rev. Biophys. II, 179—246 (1978). 

While the structure of nonredox polymer and polyelectrolytes thin layers has received much attention in the past [116, 117], only recently has a molecular theory able to treat, from a molecular point of view, redox polyelectrolytes adsorbed on electrodes, been presented [118-120]. The formulation of the theory, its scope, advantages and limitations will be discussed in detail in Section 2.5.2, and therefore we will limit ourselves to show here some predictions that are relevant for the understanding of the structure of polyelectrolyte-modified electrodes. The theory was applied to study the particular system depicted in Figure 2.5, which consists of a single layer of PAH-Os adsorbed on a gold surface thiolated with negatively charged mercapto... 

Figure 2.5 Schematic representation of the Au/MPS/PAH-Os/solution interface modeled in Refs. [118-120] using the molecular theory for modified polyelectrolyte electrodes described in Section 2.5. The red arrows indicate the chemical equilibria considered by the theory. The redox polymer, PAH-Os (see Figure 2.4), is divided into the poly(allyl-amine) backbone (depicted as blue and light blue solid lines) and the pyridine-bipyridine osmium complexes. Each osmium complex is in redox equilibrium with the gold substrate and, dependingon its potential, can be in an oxidized Os(lll) (red spheres) or in a reduced Os(ll) (blue sphere) state. The allyl-amine units can be in a positively charged protonated state (plus signs on the polymer...

In Sections 2.2 and 2.3 we have anticipated some results from the molecular theory for polyelectrolyte-modified electrodes. We will briefly discuss here the formulation... 

Deserno, M., and Holm, C. Theory and simulations of rigid polyelectrolytes. Molecular... 

G. S. Manning, Q. Rei/. Biophys., 11,179 (1978). The Molecular Theory of Polyelectrolyte Solutions with Applications to the Electrostatic Properties of Polynucleotides. 

We have used molecular simulation to examine configurational properties of isolated polyelectrolytes In solution. Studies on hydrophilic polyelectrolytes indicate that existing polyelectrolyte-expansion theories are inconsistent with the physical model on which they are based. Studies on hydrophobic polyelectrolytes are useful for examining the factors which Influence and induce structural transitions in these systems. 

Manning GS (1978) The molecular theory of polyelectrolyte solutions with applications to the electrostatic properties of polynucleotides. Q Rev Biophys 11 179 246... 

Tagliazucchi M, Calvo EJ, Szleifer I (2008) Molecular theory of chemically modified electrodes by redox polyelectrolytes under equilibrium conditions comparison with experiment. J Phys Chem C 112 458 71... 

Analysis of polyelectrolytes in the semi-dilute regime is even more complicated as a result of inter-molecular interactions. It has been established, via dynamic light-scattering and time-dependent electric birefringence measurements, that the behavior of polyelectrolytes is qualitatively different in dilute and semi-dilute regimes. The qualitative behavior of osmotic pressure has been described by a power-law relationship, but no theory approaching quantitative description is available. 

Solvent power parameter entering Flory s theory of dilute solutions, degree of neutralization in polyelectrolyte solutions, free-volume parameter entering Vrentas-Duda theory subscript (1,24) denotes molecular species in solution. 

In this section, we will describe the theory of swelling of polyelectrolyte networks. The simplest problem of this type concerns a network sample swelling freely in an infinite solvent. The solvent may contain some low-molecular-weight salt. This problem will be considered in Sect. 2.1. 

The influence of molar mass, charge density as well as chain branching was also determined in the presence of low molecular mass salt. As seen in Fig. 16, the differences between theory and experiment are more important to low molar masses. In Fig. 16 the concentration dependence of the activity of the low molecular salt has been taken into account when calculating fac=fexp/fo [H4, 126], where fac and fexp are calculated and experimentally determined counterion activity coefficients, respectively f0 is the activity coefficient of the added low molecular salt in aqueous solution without polyelectrolyte. 

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. 

Another viable method to compare experiments and theories are simulations of either the cell model with one or more infinite rods present or to take a solution of finite semi-flexible polyelectrolytes. These will of course capture all correlations and ionic finite size effects on the basis of the RPM, and are therefore a good method to check how far simple potentials will suffice to reproduce experimental results. In Sect. 4.2, we shall in particular compare simulations and results obtained with the DHHC local density functional theory to osmotic pressure data. This comparison will demonstrate to what extent the PB cell model, and furthermore the whole coarse grained RPM approach can be expected to hold, and on which level one starts to see solvation effects and other molecular details present under experimental conditions. 

Up to now, only two sets of data of the osmotic coefficient of rod-like polyelectrolytes in salt-free solution are available 1) Measurements by Auer and Alexandrowicz [68] on aqueous DNA-solutions, and 2) Measurements of polyelectrolyte PPP-1 in aqueous solution [58]. A critical comparison of these data with the PB-cell model and the theories delineated in Sect. 2.2 has been given recently [59]. Here it suffices to discuss the main results of this analysis displayed in Fig. 8. It should be noted that the measurements by the electric birefringence discussed in Sect. 4.1 are the most important prerequisite of this analysis. These data have shown that PPP-1 form a molecularly disperse solution in water and the analysis can therefore assume single rods dispersed in solution [49]. 

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]. 

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