Counterion Worm

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. 

The value of Qeg depends on the choice of the cutoff radius One of the convenient choices of is the value ro at which the electrostatic energy of a pair of monovalent ions (IsksTIro) is comparable to the kinetic energy (iksT/l) of an ion. We shall use this choice for Vc in discussing the simulation results for the effective polymer charge below. 

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Figure 4.3 Counterion worm, constructed from the nonoverlapping superposition of spheres of radius Tc centered at each united atom of a skeletal chain. Small filled circles represent counterions.

By following the procedure described in Section 4.1 to construct the counterion worm, the average degree of ionization a as defined by Equation 4.7 is computed. The dependence of a on the electrostatic interaction parameter r is given in Figure 4.10 for several values of N, by fixing the monomer density at p = 8 x and Zc = 1. As is seen clearly, a decreases... 

Figure 4.11 Decrease in the effective charge of the polymer VYith increasing salt concentration for N = 100 at F = 3. (a) Number of counterions and coions inside the counterion worm and (b) net degree of ionization. (From Liu, S. et al., J. Cbem. Pbys., 119, 1813, 2003. With permission.)...

The omnipresence of the counterion worm around a polymer chain controls both the equilibrium and dynamical behavior of the polymer. When a charged... 

Figure 6.1 Net polymer charge as a function of the cutoff radius of the counterion worm, for N = 100, Cp/j = 8 X 10 , and the salt-free case. The vertical dashed line denotes the choice of in this chapter. Circles monovalent counterions squares divalent counterions.

We now consider the effect of salt concentration on a. In the simulation, first a salt-free solution of polyelectrolyte chains and their monovalent counterions are equilibrated. For specifidty, we denote the counterion as X, and there are N counterions (Zc = 1), and take the polymer to be uniformly negatively charged. Next, we add a fixed quantity of salt of either the type XY (Z+ = 1 = Z ) or the type AYz (Z + =2, Z = 1) to the system and again equilibrate the system. We then collect the statistics and compute the average number of various ions (X, Y, and A) in the counterion worm surrounding the polymer defined as above with = 2lo. We illustrate the key results only for the case of N = 100 and Cp = 8 x lO l ... 

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