Polyelectrolyte dilute solution

While the condition of stoichiometric neutrality invariably must hold for a macroscopic system such as a space-network polyelectrolyte gel, its application to the poly electrolyte molecule in an infinitely dilute solution may justifiably be questioned. In a polyelectrolyte gel of macroscopic size the minute excess charge is considered to occur in the surface layer (the gel being conductive), which is consistent with the assumption that the potential changes abruptly at the surface. This change is never truly abrupt, for it must take place throughout a layer extending to a depth which is of the order of magnitude of the... 

In this article I review some of the simulation work addressed specifically to branched polymers. The brushes will be described here in terms of their common characteristics with those of individual branched chains. Therefore, other aspects that do not correlate easily with these characteristics will be omitted. Explicitly, there will be no mention of adsorption kinetics, absorbing or laterally inhomogeneous surfaces, polyelectrolyte brushes, or brushes under the effect of a shear. With the purpose of giving a comprehensive description of these applications, Sect. 2 includes a summary of the theoretical background, including the approximations employed to treat the equifibrium structure of the chains as well as their hydrodynamic behavior in dilute solution and their dynamics. In Sect. 3, the different numerical simulation methods that are appHcable to branched polymer systems are specified, in relation to the problems sketched in Sect. 2. Finally, in Sect. 4, the appHcations of these methods to the different types of branched structures are given in detail. 

The electrophoretic mobility p of a polyelectrolyte chain in an infinitely dilute solution containing an added salt at concentration c under a constant external electric field E, as defined through... 

In this section we consider the motion of a uniformly charged flexible polyelectrolyte in an infinitely dilute solution under an externally imposed uniform electric field E. The objective is to calculate the electrophoretic mobility p defined by... 

Summarizing, the electrophoretic mobility of a flexible polyelectrolyte chain in infinitely dilute solutions is given by Eq. (156) ... 

As seen in the preceding section, the counterions play a crucial role in the mobility of the polyelectrolyte molecules. Even in the absence of an external electric field, the counterions exert an induced electric field in the immediate environment of a charged segment which in turn significantly modifies the collective diffusion coefficient of the polymer. This additional contribution is absent for uncharged polymers, where the cooperative diffusion coefficient Dc is given by the Stokes-Einstein law in dilute solutions. 

A polyelectrolyte solution contains the salt of a polyion, a polymer comprised of repeating ionized units. In dilute solutions, a substantial fraction of sodium ions are bound to polyacrylate at concentrations where sodium acetate exhibits only dissoci-atedions. Thus counterion binding plays a central role in polyelectrolyte solutions [1], Close approach of counterions to polyions results in mutual perturbation of the hydration layers and the description of the electrical potential around polyions is different to both the Debye-Huckel treatment for soluble ions and the Gouy-Chapman model for a surface charge distribution, with Manning condensation of ions around the polyelectrolyte. 

The analysis described above is useful for modelling colligative properties but does not address polyelectrolyte conformations. Polyelectrolyte conformations in dilute solution have been calculated using the worm-like chain model [103,104], Here, the polymer conformation is characterized by a persistence length (a measure of the local chain stiffness) [96]. One consequence of the... 

In concluding this section, we should touch upon phase boundary concentration data for poly(p-benzamide) dimethylacetamide + 4% LiCl [89], poly(p-phenylene terephthalamide) (PPTA Kevlar)-sulfuric acid [90], and (hydroxy-propyl)cellulose-dichloroacetic acid solutions [91]. Although not included in Figs. 7 and 8, they show appreciable downward deviations from the prediction by the scaled particle theory for the wormlike hard spherocylinder. Arpin and Strazielle [30] found a negative concentration dependence of the reduced viscosity for PPTA in dilute Solution of sulfuric acid, as often reported on polyelectrolyte systems. Therefore, the deviation of the Ci data for PPTA in sulfuric acid from the scaled particle theory may be attributed to the electrostatic interaction. For the other two systems too, the low C] values may be due to the protonation of the polymer, because the solvents of these systems are very polar. 

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. 

If we take into consideration that the lowest experimentally possible polyelectrolyte concentration cp is approximately 10 6 monomol L 1, it follows from Table 8 that the diluted solution state, cp 2000, i.e. if Mn >320,000 g-mol The theoretical treatment and the experimental studies of the concentration dependent behavior of polyelectrolytes in solution is usually restricted to the case with or without an excess of a low molecular electrolyte. A relatively limited amount of data exist for similar concentrations of polyelectrolytes and low molecular mass salt [97]. 

The molecular characterization of polyelectrolytes in general, and of DADMAC polymers in particular is complicated for several reasons. First, in aqueous solution the individual properties of the macromolecules are dominated by Coulom-bic interactions. Therefore, the resulting polyelectrolyte effects have to be suppressed through the addition of low molecular electrolyte, such as NaCl. The increase of the ionic strength results in a decrease of the chain stiffness of the polyelectrolyte molecules (see Sect. 5). The chains then revert to the coil dimensions of neutral macromolecules in dilute solutions. However, problems may still arise, particularly since the mode of action of these effects is quite different in various characterization methods [27]. 

The understanding of flexible polyelectrolytes in dilute solutions of low ionic strength still presents a considerable challenge in macromolecular science despite of many decades of research [1-5]. This is due to the long-range nature of the Coulombic forces between the charged macromolecules. In the case of flexible polyelectrolytes, a decrease of the ionic strength may lead to... 

The investigation of the electric birefringence is an excellent tool for the study of the PPP polyelectrolytes because this method is highly sensitive and therefore particularly suited for very dilute solutions [40-42]. At low field strength, the birefringence observed in solutions or suspensions of non-interacting molecules or particles rises with the square of field strength (Kerr s law) and in proportion with concentration [43] ... 

Polyelectrolyte solutions exhibit liquid-like order in dilute solutions which diminishes at high concentrations (cf. Figure 8 [23, 24, 26-28]. At infinite dilution gmm(r) has a value close to zero at small separations and... 

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