Counterion-mediated attraction

Ha., B.-Y. and Liu, A.J. (1997) Counterion mediated attraction between two like-charged rods. Phys. Rev. Lett., 79, 1289-1292. 

As is well known, the counterions distributed close to the plates work to shield their surface charges. Figure 6.3 shows, however, that there exist counterions at the middle of the inner region that do not belong to either of the plates. It is those counterions that result in an attraction, a counterion-mediated attraction, between... 

Ise N, Konishi T, Tata BVR. How homogeneous are homogeneous dispersions Counterion-mediated attraction between like-charged species. Langmuir 1999 15 4176-4184. 

Another striking experimental feature is that the attractions do not appear to lead to macroscopic phase separation. In this sense, the counterion-mediated attraction between the chains appears to have a different character from ordinary attractions that lead simply to phase separation at sufficiently high concentrations. Instead, the chains tend to form dense bundles of a fairly well-defined thickness [8,11]. The precise morphology of the bundles appears to depend sensitively on the persistence length of the polyelectrolyte, the chain length, and the concentration. In the case of dilute DNA, the bundles tend to be toroidal or rod-shaped. Other stiff polyelectrolytes tend to form rodlike bundles or networks of bundles. In each case, however, there is a well-defined cross-sectional thickness for the bundles. We will concentrate on the question of why there is a characteristic cross-sectional bundle diameter, rather than on the specific morphology of the bundles. 

The question of how counterions mediate attractions between like-charged chains has garnered much attention in the physics community because mean-field Poisson-Boltzmann theory predicts that like-charged rods should repel, regardless of the valency of the counterion [12], This implies that fluctuation and correlation effects that are neglected in Poisson-Boltzmann theory must be responsible for the attraction. 

A. Counterion-Mediated Attraction Between Like-Charged Plates... 

We have calculated the energy of an /V-rod bundle explicitly, using the one-loop approximation (the same approximation that we used for two rods) [30,46]. Because the repulsion weakens as the bundle size grows, we find that the repulsion does not limit the bundle size. Instead, the free energy decreases linearly with N, the number of rods in the bundle. Thus the counterion-mediated attraction is no different from more pedestrian attractions such as the dispersion attraction. This calculation does not include the translational... 

FIG. 1 Plot of free energy of polyvalent counterion-mediated interaction between two rods as a function of the angle between the rods for different rod separations, R (in units of the monomer size b). The two rods are parallel at 6 = 0. We then tilt one rod with respect to the other around the axis of separation between the two rods to a nonzero angle 0. The minimum at 6 = 0 that appears for short separations arises from the polyvalent counterion-mediated attraction. The minimum at 6 = 90° (which is stable at large separations and metastable at small separations) arises from the repulsion due to the net charge on the rods. For this calculation, T = 300 K, e = 80, and Z = 2. In addition, the length of the rods was taken to be 500 A (the persistence length of DNA). (From Ref. 47.)... 

Although most theorists have focused on the origin of the counterion-mediated attraction, the main focus of our work has been to explore the consequences of the interaction in many-chain systems. The charge fluctuation approach is particularly well suited to many-chain systems because it allows an analytical approach [30]. This is particularly important because the counterion-mediated interaction is not pairwise additive. We have found that, in equilibrium, the system should phase separate on a macroscopic scale into a dilute phase in coexistence with a concentrated phase of parallel chains. 

B.-Y. Ha. Modes of counterion density fluctuations and counterion-mediated attractions between like-charged fluid membranes. Preprint (2000). 

Ise, N. 2010. Like likes like Counterion-mediated attraction in macroionic and colloidal interaction. Physical Chemistry Chemical Physics 12, no. 35 10279. doi 10.1039/c000729c. 

The fact that the repulsive barrier does not evolve as we increase the volume fraction beyond a certain point seemingly indicates that at this point the addition of counterions does not lead to a decrease in the screening length. This evokes that the added counterions are located in space where there are no particles. This interpretation is somewhat reminiscent of the one implied by Sogami and Ise [41], who report the existence of an effective attraction between two macroions, because the counterions can be attracted simultaneously by neighboring particles. Such counterion-mediated attraction is only possible if the space between two particles is somewhat depleted of counterions. Our experimentally determined potentials are in fact very similar to the one reported by Sogami and Ise, such that we could conceive that both the effective attraction and the cessation of the evolution of the repulsive barrier are caused by the fact that counterions migrate to the space with a low particle concentration. This interpretation is obviously very speculative and our data do not exclude the possibility that the appearance of effective attractions is due to many body effects... 

B. Y. Ha and A. J. Liu. Counterion-mediated, non-pairwise-additive attractions in bundles of like-charged rods. Physical Review E 60 803—813 (1999). 

Like-charged polyelectrolytes can attract and condense into compact ordered states via counterion-mediated interactions. Recent examples include DNA toroids and F-actin bundles. Condensed divalent ions within F-actin... 

Such unusual experimental results indicate that the van der Waals forces, which do not change with the changing solvent quality, cannot be the major attractive forces for the blackberry formation. Instead, the presence of a moderate amount of charge on M0132 clusters is critical. Therefore, we should seriously consider the role of counterions and their mediated attractive forces. 

Ray, J. and Manning, G.S. (1994) An attractive force between two rodlike polyions mediated by the sharing of condensed counterions. Langmuir, 10,2450-2461. 

If the electrostatic interaction between two like-charged particles is attractive, we call the interaction inverted and if the interaction between oppositely charged particles is repulsive, we also call it inverted. Ionic force inversion is not a denial of Coulomb s law. If it exists, it must be a manifestation of many-body interactions. For example, if two like-charged polymers electrostatically attract each other, it can only be through the mediation of counterions and/or polarizable solvent molecules. Force inversion is a topic currently of widespread interest in polyelectrolyte physics (this volume, Chapter 5). We currently believe that inverted forces can arise through several distinct types of interactions, and it may not always be easy to pick... 

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