Infinitely dilute solutions, polyelectrolyte

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... 

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) ... 

The sodium and calcium activity coefficients were determined by specific electrodes, in dilute salt-free solutions (c < 10 equiv/I) of sodium and calcium pectinates. Results were compared to those calculated from Manning s theory (21). This model is proposed for infinitely dilute solutions of rodlike polyelectrolytes and the activity coefficients are directly imposed by the charge parameter ... 

In infinitely dilute solutions, the polyelectrolyte chains are essentially isolated. In Muthukumar s treatment [21] of this limit, the degrees of freedom of counterions... 

Because of the macrcnnolecular character of macroions, the stoidiio-metiically infinite dilution of polyelectrolyte solutions is not a state at which interionic interactions are absent. Such an ideal state cannot be realized in the case of polyelectrolytes unless polymer chains carrying electrically charged ionized groups can be severed in such a way that... 

Considered purely as empirical statements. Equations (97a) and (97b) may be looked at as two-point interpolation formulas. That is, each of these equations obviously is forced to agree with experiment at Xg=0 and JCg= 1. (At x =0, for example. Equation (97a) says that = , the correct value for an infinitely dilute solution of simple salt in the absence of polyelectrolyte for purposes of comparison with the limiting laws, the additivity rules given above have been written as they apply to very dilute solutions.) Equation (97c) for the co-ion activity coefficient may not be given such an empirical rationale Equation (97d) is merely the definition of the mean activity coefficient of a uni-univalent salt combined with Equation (97c). 

In this section, we ignore all interchain interactions and address the situation of polyelectrolyte solutions at infinitely dilute conditions. Since isolated chains are now being considered, the chain label is dropped in this section. 

Therefore we expect Df, identified as the fast diffusion coefficient measured in dynamic light-scattering experiments, in infinitely dilute polyelectrolyte solutions to be very high at low salt concentrations and to decrease to self-diffusion coefficient D KRg 1) as the salt concentration is increased. The above result for KRg 1 limit is analogous to the Nernst-Hartley equation reported in Ref. 33. The theory described here accounts for stmctural correlations inside poly electrolyte chains. 

The viscosity of xanthan solutions is also distinct from that of flexible polyelectrolyte solutions which generally shows a strong Cs dependence [141]. In this connection, we refer to Sho et al. [142] and Liu et al. [143], who measured the intrinsic viscosity and radius of gyration of Na salt xanthan at infinite dilution which were quite insensitive to Cs ( > 0.005 mol/1). Their finding can be attributed to the xanthan double helix which is so stiff that its conformation is hardly perturbed by the intramolecular electrostatic interactions. In fact, it has been shown that the electrostatic persistence length contributes only 10% to the total persistence length even at as low a Cs as 0.005 mol/1 [142]. Therefore, the difference in viscosity behavior between xanthan and flexible polyelectrolyte... 

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... 

Before we will discuss the structure of polyelectrolyte solutions at nonzero densities, we will briefly address the conformational properties of a single chain at infinite dilution in the next section. 

Part V, by Andrey Dobrynin, focuses on simulations of charged polymer systems (polyelectrolytes, polyampholytes). Chains at infinite dilution are examined first, and how electrostatic interactions at various salt concentrations affect conformation is discussed, according to scaling theory and to simulations. Simulation methods for solutions of charged polymers at finite concentration, including explicitly represented ions, are then presented. Summation methods for electrostatic interactions (Ewald, particle-particle particle mesh, fast multipole method) are derived and discussed in detail. Applications of simulations in understanding Manning ion condensation and bundle formation in polyelectrolyte solutions are presented. This chapter puts the recent simulations results, and methods used to obtain them, in the context of the state of the art of the polyelectrolyte theory. 

Experimental study of infinitely-dilute polymers Is difficult and few direct measurements have been reported for configurational properties of isolated polymers in solution. Molecular simulation offers an alternate powerful method for studying the properties of model polymeric systems, including infinitely-dilute polymers in solution. Computer simulations have been performed in several cases for examining the conformational behavior of isolated, uncharged polymers [28-30] and, on a more limited basis, for studying isolated, fu/Zy-zon/zec/polyelectrolytes [31, 32]. Hooper et al [14, 15] recently performed Monte-Carlo computer simulations for a lattice model of an isolated, partially-ionized polyelectrolyte. Here, we present some of the primary results from references 14 and 15, and discuss how these results can improve our understanding of phase behavior in aqueous/polymer systems. 

The molecular-simulation studies discussed here consider the relatively simple case of a single polymer at infinite-dilution on an infinite lattice. Many interesting features of polymer solutions result from interchain interactions (e.g., aggregations in hydrophobic polyelectrolytes) and from interactions of polymers with interfaces (e.g., adsorption and adhesion properties). Computer simulation may also prove useful for studying such polyelectrolyte systems. 

Pessoa and Maurer [110] assume that a polyion might not completely dissociate in an aqueous solution and that the degree of dissociation is independent of the composition of the aqueous solution. They propose the use of experimental data for the osmotic coefficient of an aqueous solution of the single polyelectrolyte at infinite dilution to determine that degree. On the molahty scale the osmotic coefficient is ... 

The osmotic coefficient of an aqueous solution of the polyelectrolyte at infinite dilution... 

The reference state for the chemical potential of the solvent (water) is the pure liquid, whereas for the solute (polyelectrolyte) it is a hypothetical one molal solution of the undissociated polyelectrolyte in water (wp = m° = I mol/ kg water)), where it experiences interactions with water molecules only, i.e., in that reference state the undissociated polyelectrolyte is infinitely diluted in water (mp= 0 in pure water). The difference between the chemical potential of the polyelectrolyte in the real solution p.p T, mp) and in its reference state is calculated in five steps ... 

All expressions given above are absolutely rigorous [3] in the limit of infinite dilution for a system consisting of infinitely long line charges and point ions immersed in a dielectric continuum and subject to the electroneutrality condition (except that interactions between line charges have not been considered). Until this point no indication has been offered that the model should be relevant to real polyelectrolyte solutions except for the excellent agreement with data of the theoretical expressions derived from the model. Indeed, the relevance of the model has been amply discussed [21,64], and only several points will be stressed here. 

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