Counterions adsorbability

The polyelectrolyte chain consists of A monovalent charged monomers. The chain is immersed in an aqueous solution. The volume of the system is V. The solution is electroneutral. If the number of counterions adsorbed on the surface of the polyion is M, then the degree of ionization is given by / = 1 — M/N [48]. The salt added to the solution is completely dissociated into n co-ions and n counterions. The salt concentration is, therefore, cs = n /V. 

Ion Exchange (Figure 2-5). Involves replacement of counterions adsorbed onto the substrate from the solution by similarly charged surfactant ions (Wakamatsu, 1968 Rupprecht, 1972 Law, 1966). 

This model, shown in Figure 10.7, divides the doublelayer into two parts, i.e. (i) a fixed layer of strongly adsorbed counterions, adsorbed at specific sites on the surface, and (ii) a diffuse layer of ions similar to that of the Gouy-Chapman model. The fixed layer of ions is known as the Stem layer, and the potential decays rapidly and linearly in this layer. The potential decay is much more gradual in the diffuse layer. In the case of specifically adsorbing ions (multivalent ions, surfactants, etc.) the sign of the Stem potential may be reversed. 

Ion exchange occurs when similarly charged surfactant ions replace counterions adsorbed onto the substrate from the solution. 

The preceding derivation has been for the interfacial excess of a nonionic surfactant. Eor a 1 1 ionic surfactant, it is necessary to account for the fact that both surfactant and counterion adsorb at the interface (creating two molecules of ions per mole of surfactant) and Eq. (4.27) is modified to give... 

Figure 11. Monte Carlo configuration snapshots for counterions adsorbed onto a charged wall (within the slice 0 < x < 0.35nm). The conditions are the same as for results in Fig. 12. The circles representing particles have diameter cr = 0.5nm and are selected arbitrarily for visualization. The first snapshot is for R = O.lnm, essential non-penetrable ions, and the second snapshot is for R = 0.8nm, the fully penetrable ions. The 2D densities of each snapshot are = 2.34nm" and p = 2.68nm", respectively. For comparison, the surface charge density is aje = 2.50nm, indicating overcharging for R = 0.8 nm counterions.

The number of counterions adsorbed on the chain backbone is a difficult quantity to measure experimentally. However, computer simulations have helped to gain an understanding of the counterion cloud around the chain. Since the strongest attractive electric potential for the counterions is generally near the contour of the chain, the ion cloud would dress the chain along its contour. As a result, the chain with its counterions would look like a worm. 

Let us consider a single chain of N monomers in volume V. Each monomer is monovalently charged, and i is the distance between two successive monomers along the polymer. Due to the electroneutrality condition, there are N monovalent counterions. Let M be the number of counterions adsorbed on the polyelectrolyte so that M/N is the degree of counterion adsorption and a = 1 — M/N) is the degree of ionization of the polyelectrolyte. In addition. 

We have seen in Section 8.8 that the EOF velocity generated by electric fields on the counterions adsorbed on immobile charged interfaces can be very significant. In this section, we address the effect of the EOF on the capture rate of the polymer chains in the steady state. As a specific example, consider a positively charged cylindrical pore (Figure 9.6a) with a surface electric potential... 

Htot consists out of 3 parts, namely Eo, which is the Debye electrostatic energy of the PE chain, Ea, the energy of the counterion adsorbed on the PE, and E, , the elastic energy of the stretched, or charged PE [49]. These parameters are defined according to Safranov [49] with the fraction of free counterions, fl, as ... 

The immobile counterions adsorbed to and immediately adjacent to the wall form the compact Stem layer, while the Gouy-Chapman layer comprises the diffuse and mobile counterion layer that is set in motion upon the application of an external electric field. The shear plane separates the Stem and Gouy-Chapman layers and, in simple double-layer models, is the location of the fluid motion s no-slip condition (Figure 7-11). The magnitude of the potential at the wall surface x = 0 decays from the wall, and the bulk fluid far... 

---