Polyion counterion pairing

In this chapter we describe the use of polyelectrolytes carrying redox-active centers on electrode surfaces with particular emphasis on organized layer-by-layer redox polyelectrolyte multilayers (RPEM). In redox-active polyelectrolyte multilayers the polyion-polyion intrinsic charge compensation can be broken by ion exchange driven by the electrochemical oxidation and reduction forming extrinsic polyion-counterion pairing. In this chapter we describe the structure, dynamics and applications of these systems. 

The organization of the chapter is straightforward. Section II reviews counterion condensation theory for an isolated polyion in the framework of the second stage as indicated above. Then in Sec. Ill the radial distribution function for the counterions and coions is discussed with emphasis on an inverted region at intermediate distances from the polyion. The pair potential for two identical polyions is also discussed in this section, and an inverted attraction is highlighted (an inverted repulsion is found for two polyions identical but for opposite charge). Finally, we review our work on polyion clustering in Sec. IV. 

Not all ions are mobile within the ionic atmosphere of the polyion. A proportion are localized and site-bound-a concept apparently first suggested by Harris Rice (1954). Localized ion binding is equivalent to the formation of an ion-pair in simple electrolytes. Experimental evidence comes mainly from studies on monovalent counterions. 

Polyelectrolyte complexes formed by polyion pairing are of special interest, including protein-polyelectrolyte interactions such as protein-DNA complexes. A special case of polyelectrolyte complexes are polyelectrolyte multilayers (PEM) on surfaces formed by ion pairing, van der Waals interactions and counterion release of oppositely charged polyelectrolytes [2, 3]. 

A very versatile approach to the formation of multilayer films has been developed by Decher, based on polyelectrolytes. If a solid substrate with ionic groups at the surface is dipped into a solution of a complementary polyelectrolyte, an ultrathin, essentially monomolecular film of the polyion is adsorbed [340]. The adsorption is based on pairing of surface bound ionic sites with oppositely charged ions, bound to the macromolecule. The polymers adsorb in an irregular flattened coil structure and only part of the polymer ions can be paired with the surface ions (Figure 29a). Ionic sites which remain with small counterions provide anchor points for a next layer formed by a complementary polyelectrolyte [342,343]. This way multilayer polyelectrolyte films can be prepared layer-by-layer just by dipping a suitable substrate alternately in an aqueous solution of polyanions and polycations. The technique can be employed with nearly all soluble charged polymers and results in films with a... 

Ray J, Manning GS. Effect of counterion valence and polymer charge density on the pair potential of two polyions. Macromolecules 1997 30 5739-5744. 

A recent attempt to extend the scope of counterion condensation theory to the calculation of counterion-polyion, coion-polyion, and polyion-polyion pair potentials retains structural idealization but may nonetheless be capable of generating useful new information about the real molecular structure of polyelectrolyte solutions [57-59], The validity of the first and second stages of the theory, as discussed above, has been heavily documented, including the physical reality of the condensed layer and the onset of condensation at a critical charge density [55,56,60,61], In contrast, the inverted forces predicted by our extended theory have yet to be confirmed by experiment or simulation. We will argue, however, that their presence is at least suggested by current experimental knowledge. 

In this section, we discuss the interaction of a counterion and a rodlike segment of a polyion as a function of separation distance r between the two [59], We consider as well the coion-polyion [59] and polyion-polyion pair potentials [57,58]. In the latter case, the two polyions may be identically charged or oppositely charged. For each type of pair, there is a polyion selfenergy of the form of Eq. 1 (for the polyion-polyion pairs the factor P is replaced by 2P). The essential difference from Eq. 1 is that the number of condensed counterions 9 is now a function of the pair separation distance, 9 = 9(r). Similarly, in the transfer free energy Eq. 2, both 9 and the condensed layer partition function Q depend on r. In addition to the self-assem-... 

Similar expressions apply to the other types of pairs, and all of them contain the zeroth-order modified Bessel function of the second kind K0(xr), or, simply, the Bessel K0 function. In Figure 1 we show a graph of the smoothly decreasing nonoscillatory Bessel K0 function, which can subsequently be contrasted with the pair potentials w(r) that emerge only after the three free energy terms (polyion self-energy, counterion transfer, and direct pair interaction) are added and minimized, a procedure that involves determination of the functions 0(r) and Q(r). 

We discuss some of the results, beginning with the number of condensed counterions in the case of the counterion-polyion pair. When a counterion is brought from infinity to a distance r in the far Debye-Hiickel-like region, the number of counterions condensed on the polyion remains at the same constant value as on the isolated polyion, 9 r) = 1 i When the counterion continues its approach and enters the near region, however, the number of condensed counterions per polyion charge is changed to a different constant value, 9(r) = 1 t (1 IP). Since P9 is the total number of... 

Plots of the pair potentials w(r) exhibiting inverted forces may be seen in our other publications. The counterion-polyion potential is attractive in the near and far regions but inverted (repulsive) in the intermediate region. For like-charged polyions, the polyion-polyion potential is repulsive in the near and far regions but attractive for intermediate distances. Here we show graphs of the radial distribution functions g(r) = exp[—w(r)]. In Figure 5... 

FIG. 4 The partition function of the counterion layer condensed on a pair of identical rodlike polyions in parallel orientation with separation distance r. Debye screening length 30 A (0.01 M NaCl) polyion charge spacing 1.7 A. 

FIG. 8 A calculated cross-section of the cylindrically symmetric distribution of condensed counterions held in common by a pair of identical parallel rodlike polyions when the partition function for the condensed layer is interpreted as a free volume. The numerical scale is in A, and the polyions pierce the page at 15 A. The Debye length equals 30 A, so the theoretically calculated condensed layer lies inside the Debye atmosphere, as required on physical grounds. Polymer charge spacing 1.7 A. 

There is ample experimental evidence that identically charged polymers in the presence of ordinary univalent counterions have a tendency to form loose clusters in solution [65-70], and we have asked whether the attractive polyion-polyion potential discussed in Sec. Ill can stabilize a finite-sized cluster of parallel rodlike polyions without leading to precipitation [71,72]. The theoretical problem is complicated by a failure of pairwise additivity the work of assembling N polyions is not equal to the work of assembling the N(N — l)/2 polyion pairs, each in isolation from the other N 2 polyions. To be sure, the Debye-Htickel interaction term for a cluster (the generalization of Eq. 5 above) takes the form of a pairwise sum over polyions,... 

Figures 1, 2, and 3 show three different views of the counterion distribution for a 64 base-pair DNA at a polymer concentration of 1.2 mM nucleotide residues. They are drawn by superposing a number (3,968) of uncorrelated configurations of counterions collected during the simulation. Two kinds of counterions, one of them deserving identification with Manning s condensed counterions [44,45], at least qualitatively, and the other forming a diffuse ion cloud, are clearly recognizable in distinct spatial distributions. Further in Figure 1, the prolate spheroidal symmetry of the distribution, and in Figures 2 and 3, depletion of the diffuse ion cloud at the ends of the polyion, may be noticed.

To express counterion distributions more quantitatively, counterion concentration c+ profiles for a 64 base-pair DNA at various polymer concentrations are plotted in Figure 4 as functions of the radial coordinate r measured from the axis of the DNA cylinder at its center and in Figure 5 as functions of the z coordinate along the surface of the cylinder. The very high counterion concentration ( 3 M) on the surface of the polyion rapidly decreases in both radial and longitudinal directions, and dilution of the polymer concentration has the slightest effect on these profiles. 

FIG. 6 Distance of counterions from 64 base-pair DNA polyion or sum of their distances from both ends of the polyion sorted in increasing order and averaged over a number of uncorrelated counterion configurations collected during the simulation. The abscissa is also taken as the ordinal number n of counterions sorted in increasing order of their distance from the polyion. 

---