High Salt Limit

As discussed in Section 4.1.6, the intrachain electrostatic interaction becomes short ranged for high salt concentrations and its strength can be simply added to the excluded volume parameter without charges. It follows If om Equations 4.12 and 4.16 that an effective excluded volume parameter Veg can be defined as 

By defining v as (Equation 2.65), the effective excluded volume parameter Weff = is given by 

Therefore, it is only the copying of the results derived in Section 2.5, with the substitution of Weff for w that needs to be done to derive the dependence of polymer size on Bjerrum length, Debye length, and chain length. By adding the free energy contributions from the chain connectivity and the sum of excluded volume effects from hydrophobicity and electrostatic repulsion, we rewrite Equation 2.71 as 

Analogous to the derivation of Equations 2.73 and 2.74 for an uncharged polymer, a crossover formula for the radius of gyration of a flexible polyelectrolyte in a solution with high enough salt is obtained as follows. Substituting Equation 2.45 for the free energy of chain connectivity and Equation 2.68 (with w replaced by Weg) for the excluded volume effect in Equation 2.67, we get 

Similar approximate formulas have been derived (Ghosh et al. 2001) for semifiexible chains by treating the electrostatic interaction as electrostatic excluded volume. The validity and relevance of such formulas for experimental systems remain to be tested. 

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As shown in Ref. 48, l is intimately related to the static correlation length In infinitely dilute solutions is proportional to Rg. In semidilute solutions, is proportional to and respectively, in low and high salt limits. In... 

Elere, as in the formulae that follow next, there is a double screening in this high salt limit the interaction across the distance Z with a screening factor e Kml/l and then another interaction across Z again with e Kml/l screening to effect the correlation in mutual perturbation. 

When we include salt in the solution and look at the high salt limit, the situation is more complex. The only length characterizing the exponential decay of close to the surface is the DH screening length. Hence, using di/f/ irh ere yields i/fj aeK, and therefore from Eq. (65),... 

The estimation of the PE layer charge can be obtained by using the expression for D and Cm in this high salt limit (Eqs. 63-65), yielding... 

In the high salt limit, UbCp -C Oion, the contribution of the translational entropy of mobile ions, disproportionated between the interior and the exterior of the corona, is equivalent to a renormalization of the second virial coefficient of monomer-monomer interactions, as va Veff = Va + [see (84)]. 

In the high salt limit, icRg 1, it can be shown (Muthukumar 1996a) from Equations 7.99 and 7.100 that... 

VL stoichiometric reaction, mild, simple high cost limited to NH2 salt heating to ca small specialties, sulfations ... 

While these model predictions have been confirmed experimentally [11] in a number of cases, significant deviations have been observed [73] in both limits of very low and high salt concentrations in systems of wormlike micelles. 

High salt concentrations up to 15-20% can be found in wastewater from dyestuff industries. Moreover, textile manufacturers located on coastal areas can cause pollution of seawater. The biological treatability of wastewater with a high saline concentration is limited because most of the microorganisms that are able to degrade azo dyes are not active in these conditions, in which the selection of halophilic or halotolerant bacteria capable to degrade azo dyes is necessary [79]. 

In infinitely dilute silutions, l is proportional to N and and consequently the radius of gyration Rg is proportional to N and N, respectively, in low-salt (k 0) and high salt KRg > 1) limits as already pointed out. In semidilute solutions, l is proportional to and and consequently Rg is... 

The scattering function g k) is a function of static correlation length as given by Eqs. (225)-(227). For semidilute solutions at high salt concentrations, Dc follows from Eqs. (226) and (282) in the —> 0 limit. 

It must be pointed out that, in the state-of-the-art electrolytes, the actual high-temperature limits for application in cells are usually not set by the upper boundary of the liquid range because other factors might push the limits far lower. For example, the upper boundary of the liquid range is 90 °C for DMC-, 110 °C for EMC-, and 120 °C for DEC-based electrolytes, all of which are far above the high-temperature limit set by the salt LiPFe (70 °C). ... 

Even if LiPFe is replaced by more thermally stable salts, the thermal stability of passivation films on both the anode and the cathode would still keep the high-temperature limits lower than 90 °C, as do the thermal stability of the separator (<90 °C for polypropylene), the chemical stability of the insulating coatings/sealants used in the cell packaging, and the polymeric binder agents used in both cathode and anode composites. 

Membrane Properties. The reverse osmosis performance of the bentonite-doped membrane under brackish water conditions is compared to that of the reference membrane in Figure 5 (I, reference membrane II, with organophilic bentonite). At low salt rejection the bentonite membrane again shows a higher initial flux than the reference membrane, the performance of the two becoming identical at the high rejection limit. 

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