Counterions at Charged Cylinders: Criticality and Universality beyond Mean-Field Theory

Abstract

The counterion-condensation transition at charged cylinders is studied using Monte-Carlo simulation methods. Employing logarithmically rescaled radial coordinates, large system sizes are tractable and the critical behavior is determined by a combined finite-size and finite-ion-number analysis. Critical counterion localization exponents are introduced and found to be in accord with mean-field theory both in 2 and 3 dimensions. In 3D the heat capacity shows a universal jump at the transition, while in 2D, it consists of discrete peaks where single counterions successively condense.PACS numbers: 64.60. Fr, 82.35.Rs, 61.20.Ja Many biopolymers, such as DNA, actin, tubulin, fdviruses, are charged and stiff. On length scales smaller than the persistence length, they can be represented by straight, charged cylinders and oppositely charged ions (counterions) are attracted via an electrostatic potential that grows logarithmically with radial distance. As the ion confinement entropy also exhibits a logarithmic dependence, it was suggested early by Onsager that a counterion delocalization transition occurs at a critical cylinder charge (or equivalently at a critical temperature) [1]. This argument is strictly valid only for a single ion since it neglects cooperativity due to inter-ionic repulsions. Nevertheless, it was corroborated by mean-field (MF) studies [1,2,3], which demonstrate that below a critical temperature, a fraction of counterions stays bound or condensed in the vicinity of the central cylinder even in the limit of infinite system size; while above the critical temperature, all counterions de-condense to infinity. This counterion-condensation transition (CCT) dramatically affects a whole number of static and dynamic quantities for charged polymers [1]. It has been observed with different polymers by varying the medium dielectric constant [4,5] or the polymer charge density [5,6]; the counterion distribution around DNA strands has been directly measured recently using anomalous scattering techniques [7]. Since its discovery, the CCT has been at the focus of numerical [8,9] and analytical [10] studies. Under particular dispute has been the connection between CCT and the celebrated Kosterlitz-Thouless transition of logarithmically interacting particles in 2 dimensions [11]. Also, the precise location of the CCT critical point remains subject of ongoing experimental investigations [5,6].As is well known from bulk critical phenomena, fluctuations and correlations typically make non-universal and universal quantities deviate from mean-field theory (MFT) below the upper critical dimension [12]. Surprisingly, the MFT prediction for the CCT critical temperature has not been questioned in literature and apparently assumed to be exact. Likewise, the existence of scaling relations and critical exponents associated with the CCT has not been considered, neither on the MFT level and (consequently) also not in the presence of correlations.In this paper we pose the questions: i) what is the critical temperature of the CCT, ...

Electrostatic interactions in strongly coupled soft matter

“…This qualitative picture is actually supported by existing analytical results of mean-field [89,90] and strong-coupling [71] theory; both limiting theories give the same threshold of ξ * = 1. Recent numerical simulations [91] show that this threshold is in fact universal and holds in all ranges of the coupling parameter Ξ. Moreover, the threshold counterion-condensation process at charged cylinders exhibits a set of scaling relations, which are characterized by universal exponents [91].…”

“…Recent numerical simulations [91] show that this threshold is in fact universal and holds in all ranges of the coupling parameter Ξ. Moreover, the threshold counterion-condensation process at charged cylinders exhibits a set of scaling relations, which are characterized by universal exponents [91].…”

Counterion density profile around a charged disk: From the weak to the strong association regime

“…To our problem, we acknowledge the published works from Naji and Netz [26,27] and Burak and Orland [2], which have covered thoroughly the basics of the two dimensional construction of the cell model. They observed that condensation obeys the well-known threshold at ξ = 1 where below this value there is none.…”

“…Here the familiarized reader may have noticed that for the two dimensional systems it is accustomed to use Γ for the notation of the coupling constant [3,6,7,29,30]; the previous defined coupling equates the standardized one as Ξ = Γ/2, but we have kept Ξ motivated by a discussion with the three dimensional case. Finally the third parameter, defined as ∆ = log(D/R), which measures in log-scale the size of the system.To our problem, we acknowledge the published works from Naji and Netz [26,27] and Burak and Orland [2], which have covered thoroughly the basics of the two dimensional construction of the cell model. They observed that condensation obeys the well-known threshold at ξ = 1 where below this value there is none.…”

“…Additionally, if we were to look at a system with a fixed number of particles going from a temperature above the critical temperature for condensation, i.e. ξ < 1, and allow it undergo a cooling process, the energy, heat capacity and other quantities will present successive transitions due to iterative localization phenomena occurring when a counter-ion is condensed [2,26,27].1 The dimensionless parameters in 3D are as follows: ξ = l b λq, Ξ = ξ 2 λ q R , with l b the Bjerrum length, λ the cylinder's line charge, R its radii and q the counter-ion charge. …”