Theories polyelectrolyte model

The previously described measurements have been performed on lipids in aqueous solutions, but lipid bilayers also swell in some other solvents (12) and the results of such measurements compare quite well with the aqueous case. In addition, hydration (solvation) forces act between DNA polyelectrolytes (13) and polysaccharides (14). These facts make the interpretation of the forces even more complicated and it is no wonder that different approaches to explain the nature of this solvation force exist. So far no truly ab initio theory has been proposed. The existing theories include models based on the electrostatic approach, the free energy approach, and an approach based on the entropic or protrusion model. 

Ionic polymer gels Polyelectrolyte gels Modeling Numerical simulation Statistical theory Transport model Continuum model Coupled multi-field formulation... 

SC/PRISM theory has also been applied to solutions of polyelectrolytes modeled as rigid rods [138,139]. Here, SC/PRISM accurately reproduces the long-range... 

D. Soumpasis, J. Chem. Phys., 69, 3190 (1978). Debye-HUckel Theory of Model Polyelectrolytes. 

Stigter, D, A Comparison of Manning s Polyelectrolyte Theory with the Cylindrical Gouy Model, Journal of Physical Chemistry 82, 1603, 1978. 

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...

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. 

The cell model is a commonly used way of reducing the complicated many-body problem of a polyelectrolyte solution to an effective one-particle theory [24-30]. The idea depicted in Fig. 1 is to partition the solution into subvolumes, each containing only a single macroion together with its counterions. Since each sub-volume is electrically neutral, the electric field will on average vanish on the cell surface. By virtue of this construction different sub-volumes are electrostatically decoupled to a first approximation. Hence, the partition function is factorized and the problem is reduced to a singleparticle problem, namely the treatment of one sub-volume, called cell . Its shape should reflect the symmetry of the polyelectrolyte. Reviews of the basic concepts can be found in [24-26]. 

Recently, Nyquist et al. [39] tried to develop a theory for rods of finite size. These authors used a two-state model for the counterions and employed a random phase approximation in order to calculate the osmotic coefficient (j) of rod-like polyelectrolytes [39]. An important goal of this work was to reproduce in the zero density limit the correct osmotic coefficient of 1 instead of the Manning limiting value which is due to the unphysical infinite rod assumption employed. The model presented in ref. [39], however, seems to overestimate considerably the osmotic coefficient when compared to experimental data (see below Sect. 4.2). 

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. 

As already indicated in Sect. 2, the osmotic coefficient 0 provides a sensitive test for the various models describing the electrostatic interaction of the counterions with the rod-like macroion. It is therefore interesting to first compare the PB theory to simulations of the RPM cell model [26, 29] in order to gain a qualitative understanding of the possible failures of the PB theory. In a second step we compare the first experimental values 0 obtained on polyelectrolyte PPP-1 [58] quantitatively to PB theory and simulations [59]. 

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]. 

The fact that Poisson-Boltzmann theory overestimates the osmotic coefficient is well-known in literature. Careful studies of typical flexible polyelectrolytes in solution ([2, 23] and further references given there) indicated that agreement of the Poisson-Boltzmann cell model and experimental data could only be achieved if the charge parameter was renormalized to a higher value. To justify this procedure it was assumed that the flexible polyelectrolytes adopt a locally helical or wiggly main chain in solution. Hence,... 

An important effect not taken into account by the various models discussed in Sect. 2 is the specific interactions of the counterions with the macroion. It is well-known that counterions may even be complexated by macroions and these effects have been discussed abundantly in the early literature in the field [24]. From the above discussion it now becomes clear that these effects must be traced back to specific effects which are not related to the electrostatic interaction of counterions and macroions. Hence, hydrophobic interactions related to subtle changes in the hydratation shell of the counterions could be responsible for this small but significant discrepancy of the electrostatic theory and experiment. Further studies using the PPP-polyelectrolytes will serve for a quantitative understanding of these effects which are outside of the scope of the present review. 

The osmotic coefficient obtained experimentally from polyelectrolyte PPP-1 having monovalent counterions compares favorably with the prediction of the PB cell model [58]. The residual differences can be explained only partially by the shortcomings of the PB-theory but must back also to specific interactions between the macroions and the counterions [59]. SAXS and ASAXS applied to PPP-2 demonstrate that the radial distribution n(r) of the cell model provides a sufficiently good description of experimental data. 

Abstract In this chapter we review recent advances which have been achieved in the theoretical description and understanding of polyelectrolyte solutions. We will discuss an improved density functional approach to go beyond mean-field theory for the cell model and an integral equation approach to describe stiff and flexible polyelectrolytes in good solvents and compare some of the results to computer simulations. Then we review some recent theoretical and numerical advances in the theory of poor solvent polyelectrolytes. At the end we show how to describe annealed polyelectrolytes in the bulk and discuss their adsorption properties. 

The PRISM (Polymer-Reference-Interaction-Site model) theory is an extension of the Ornstein-Zernike equation to molecular systems [20-22]. It connects the total correlation function h(r)=g(r) 1, where g(r) is the pair correlation function, with the direct correlation function c(r) and intramolecular correlation functions (co r)). For a primitive model of a polyelectrolyte solution with polymer chains and counterions only, there are three different relevant correlation functions the monomer-monomer, the counterion-counterion, and the monomer-counterion correlation function [23, 24]. Neglecting chain end effects and considering all monomers as equivalent, we obtain the following three PRISM equations for a homogeneous and isotropic system in Fourier space ... 

Another kind of self-consistent theory used a continuum diffusion representation to describe the distribution of segments.21 23 The segments were assumed to be subjected to an external potential of a mean field. An analytical approximation of the latter self-consistent theory was suggested by Milner et al. (the MWC model)24 on the basis of the observation that at high stretching the partition function of the brush is dominated by the classical path as the most probable distribution. Under this assumption, it was found that the self-consistent field is parabolic and leads to a parabolic distribution of the monomer density. Similar theories for polyelectrolyte brushes25 27 also adopted the parabolic distribution approximation. 

In polyelectrolyte systems, the theory is adjusted either to the point charge model assuming a distribution of point charges on the polymer chain or to the dipole-ion theory considering an ion pair as a dipole. Their potential energies are expressed as... 

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