Manning-Oosawa Counterion Condensation

Abstract

Counterion condensation is a basic feature of 2D electrostatics exhibited by highly charged rodlike polymers such as DNA. In the framework of the Poisson Boltzmann equation with salt, we show that such a polymer of radius a attracts a condensate of thickness RM=A(axi)1/2 where xi is the Debye length and A depends weakly on the polymer charge density q0. To leading order in 1/ln(xi/a), we derive the condensate structure and show that free ions follow universal density profiles independent of a and q0. Generalizing this approach we calculate ion profiles for finite concentration solutions.

Scaling and universality in the counterion-condensation transition at charged cylinders

“…Previous studies [9,10,11,13,24,43] show that the Manning condensed fraction, α M , may also be identified systematically within the Poisson-Boltzmann theory by employing an inflection-point criterion [24,43]. This can be demonstrated using the PB cumulative density (the number of counterions inside a cylindrical region of ra- dius r), n PB (r), obtained as…”

“…This enables us to investigate the critical limit of infinite system size (that is when the outer boundaries confining counterions tend to infinity) within tractable equilibration times in the simulations. The importance of taking very large system sizes becomes evident by noting that lateral finite-size effects, which mask the critical unbinding behavior of counterions, depend on the logarithm of system size in the cylindrical geometry [1,4,9,10,11,12,13,24,31,32,36,38,43,50,51], causing a quite weak convergence to the critical infinite-size limit.…”

“…Since its discovery, the CCT has been at the focus of numerical [23,24,25,26] and analytical [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44] studies. Under particular dispute has been the connection between CCT and the celebrated Kosterlitz-Thouless transition of logarithmically interacting particles in two dimensions [31,45,46,47].…”

The renormalized jellium model for spherical and cylindrical colloids

“…A useful measure of the condensate thickness is provided by the socalled Manning radius R M [27] that has been recently worked out in the infinite dilution limit and for low salt content [16,28]: in practice, the integrated charge per unit length q(r) around a rod has an inflection point at r = R M , when plotted as a function of log r. This is exactly the point where q(R M )ℓ B /e = 1. We expect a similar behavior for the renormalized jellium, given that in the vicinity of highly charged rods, the (largely dominant) counterion distribution should not be sensitive to the difference between a uniform background as in the renormalized jellium model, and coions as in the situation worked out in [16].…”

Electric-field-induced polarization of the layer of condensed ions on cylindrical colloids

“…These observations seem to validate the assumption in our model that the condensed ions are located right at the macromolecular surface. To within the realm of Poisson-Boltzmann theory, the spatial extent of the layer of condensed ions, including the core, is approximately equal to a/κ, where a is the core radius of the rod, and κ −1 is the Debye length which characterizes the extent of the diffuse double layer [25]- [28]. In the experiments in Refs.…”