Electrostatic excluded volume

the inverse Debye length k arises only from the counterions and salt ions and not from the charged monomers of the polymer, as given by Equation 4.5. 

It can be shown that for sufficiently large values of k, such that KRg 1 Rg being the radius of gyration of the polymer), the screened Coulomb interaction energy becomes short-ranged. In this limit, the result of Equation 4.12 can be written exactly (Muthukumar 1987) as 

The situation of k = 0 is unphysical, because there are always counterions present in the solution of finite volume to meet the electroneutrality condition. Nevertheless, the behavior of Equation 4.14 is approachable in the asymptotic limit of the Debye length being larger than the radius of gyration of the polymer, KRg 1. Combining the above two limits. 

We shall use these two limits as guides in interpreting the experimental data. For values of KRg intermediate between the limits, the full form of Equation 4.12 is needed. A convenient result of the above argument is that the intersegment electrostatic interaction energy becomes short-ranged as for the 

The first term is due to the uncharged part and the second term is due to the electrostatic part. We call the term AnZp slic as the electrostatic excluded volume parameter, in the limit of high salt concentrations. 

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Where the slope s determined by the plot is the extension coefficient. The I-1/2 dependence viewed in terms of an increased Debye length can be explained as the electrostatic excluded volume contribution. 

In the classical theories of polyelectrolytes, the chain expansion is characterized by the electrostatic excluded volume parameter, zel with ... 

It can be seen that on addition of salt the brush height decreases by a weak power law, ISait oc cs 1/3. This behavior is often called salted brush . Here the surface-linked monolayer behaves like a neutral brush with an enlarged (electrostatic) excluded volume. Note that the grafting density has a strong influence on the amount of free mobile ions per chain inside the brush. Therefore, the height of such polyelectrolyte brushes is expected to become less sensitive to the external salt concentration if chains at high graft density are considered. 

FIG. 14 Dependence of the second virial coefficient on added-salt concentration. NaPSS in NaCl solutions. Results by light scattering from Ref. 41 ( ), Ref. 30 (A) and by osmometry from Ref. 41 (o). The dashed line shows calculation of the second virial coefficient by incorporating electrostatic excluded volume effects after Ref. 30. 

The change in slope as a function of ionomer concentration is seen in more detail in Figure 8 at low concentrations, the value is negative, reflecting the dominant excluded volume term. As the ionomer concentration increases, the electrostatic excluded volume decreases significantly, while (1... 

Reliable theories that accurately correlate or predict polyelectrolyte phase diagrams are lacking. Letting electrostatic excluded volume between chain segments be modeled at the level of the Debye-Hiickel approximation, a modified or effective Flory-Huggins parameter Xeff can be determined via the random phase approximation (141) ... 

Electrostatic Persistence Length (EPL) and Electrostatic Excluded Volume (EEV)... 

One can further imagine that as the neutral polymer becomes charged, the intermolecular repulsive energy will increase, and hence the excluded volume will also increase, due both to the increasing strength of the electric field in the vicinity of the charged polymer, and to the fact that as it stretches out in space it will also occupy and hence exclude more volume. This problem has been formulated in terms of the electrostatic excluded volume (EEV). 

In all these studies, ACM allowed, with the massive data provided, the behavior of the polyelectrolyte conformational and interaction parameters to be followed at a level of detail previously unobtainable by manual gathering of individual data points. Correlations were made between experimental data and of electrostatic persistence length and electrostatic excluded volume theories, with no adjustable parameters. 

An investigation on the effect of added electrolyte valence and species on polyelectrolytes studied using ACM was made in Reference [46], First-order differences in polyelectrolyte conformations, interactions, and hydrodynamics were reported by the authors when electrolytes of different valence and symmetry were added to the polyelectrolyte solution. Each relevant characteristic, however, followed the same scaling relationship to ionic strength but the prefactors were different. Electrostatic persistence length/electrostatic excluded volume calculations without adjustable parameters suggested that the effective linear charge density is considerably lower in the presence of divalent ions than monovalent ions. Consideration of the... 

The expansion of a polyion with decreasing ionic strength is caused by the electrostatic excluded volume only. A local stiffening (concept of the electrostatic persistence length) was not in discussion at the time. 

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. 

However, the prefactor a z lBl is smaller than unity. In the regime of Equation 4.39 being valid, the contribution from the hydrophobic interaction w becomes negligible as the second term of Equation 4.38 is of order 1 jN, whereas the other terms are proportional to N. Similar to the dimensionless excluded volume parameter (Fixman parameter) identified in Section 2.5, the strength of the intrachain electrostatic interaction in salt-free solutions is given by the dimensionless electrostatic excluded volume parameter Zei,<,... 

The limits of high salt and low salt have been addressed in Seetions 4.1.1 and 4.1.2, respeetively. In experiments, the salt coneentrations are not neeessarily in sueh extreme limits. Similarly, the dimensionless exeluded volume parameter wVn and the electrostatic excluded volume parameter defined in Equation 4.40 can assume intermediate values, instead of being either zero or very large. The crossover formula for the free energy that recovers the limits of Equations 4.30 and 4.38 is (Muthukumar 1987)... 

In conclusion, the Fixman-Skolnick theoiy of the electrostatic binary cluster integral for the rodlike segment model considerably reduces the unsatisfartory situation encountered with the conventional bead model, but it still overestimates experimental Bgi- Our imderstanding of the electrostatic excluded-volume effect in charged polymers leaves much to be desired. 

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