Pair potentials counterion-polyion

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. 

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. 

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). 

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... 

J. Ray and G. S. Manning, Macromolecules, 30, 5739 (1997). Effect of Counterion Valence and Polymer Charge Density on the Pair Potential of Two Polyions. 

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,... 

Figure 7 shows a 3D view of the reduced electrostatic potential eij/lknT around a 64 base-pair DNA at a polymer concentration of 1.2 mM nucleotide residues where if/ is the electrostatic potential. It is calculated by averaging over the configurations of counterions used in Figures 1, 2, and 3. Note that electric fields due to the polyion are completely shielded by counterions. Further, the reduced electrostatic potentials around a 64 base-pair DNA at various polymer concentrations are plotted in Figure 8 as functions of the radial coordinate r measured from the DNA cylinder at its center and in... 

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