Painleve III and Manning's Counterion Condensation

Kholodenko, Arkady L.; Beyerlein, Adolph L.

doi:10.1103/physrevlett.74.4679

Cited by 29 publications

(31 citation statements)

References 12 publications

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“…At this point the total charge Q guest+condens of the guest particle plus the condensed counterion layer is such that βQ guest+condens = 2, as it can be seen from the first term of equation (2.12). We recover here another characteristic of the Manning-Oosawa counterion condensation [25,26,27]: the layer of condensed counterions reduces the bare charge of the guest charge to βQ guest+condens = 2.…”

Section: The Mean Field Limit: Poisson-boltzmann Equationmentioning

confidence: 99%

Guest Charges in an Electrolyte: Renormalized Charge, Long- and Short-Distance Behavior of the Electric Potential and Density Profiles

Téllez

2005

J Stat Phys

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We complement a recent exact study by L.Šamaj on the properties of a guest charge Q immersed in a two-dimensional electrolyte with charges +1/ − 1. In particular, we are interested in the behavior of the density profiles and electric potential created by the charge and the electrolyte, and in the determination of the renormalized charge which is obtained from the long-distance asymptotics of the electric potential. InŠamaj's previous work, exact results for arbitrary coulombic coupling β were obtained for a system where all the charges are points, provided βQ < 2 and β < 2. Here, we first focus on the mean field situation which we believe describes correctly the limit β → 0 but βQ large. In this limit we can study the case when the guest charge is a hard disk and its charge is above the collapse value βQ > 2. We compare our results for the renormalized charge with the exact predictions and we test on a solid ground some conjectures of the previous study. Our study shows that the exact formulas obtained byŠamaj for the renormalized charge are not valid for βQ > 2, contrary to a hypothesis put forward byŠamaj. We also determine the short-distance asymptotics of the density profiles of the coions and counterions near the guest charge, for arbitrary coulombic coupling. We show that the coion density profile exhibit a change of behavior if the guest charge becomes large enough (βQ ≥ 2 − β). This is interpreted as a first step of the counterion condensation (for large coulombic coupling), the second step taking place at the usual Manning-Oosawa threshold βQ = 2.

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Section: The Mean Field Limit: Poisson-boltzmann Equationmentioning

confidence: 99%

Guest Charges in an Electrolyte: Renormalized Charge, Long- and Short-Distance Behavior of the Electric Potential and Density Profiles

Téllez

2005

J Stat Phys

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“…The scaling relations (42) and (45) indicate that S PB n takes a general scale-invariant form with respect to ζ and ∆ as…”

Section: The Cct Order Parametersmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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

2006

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Counterions at charged rod-like polymers exhibit a condensation transition at a critical temperature (or equivalently, at a critical linear charge density for polymers), which dramatically influences various static and dynamic properties of charged polymers. We address the critical and universal aspects of this transition for counterions at a single charged cylinder in both two and three spatial dimensions using numerical and analytical methods. By introducing a novel Monte-Carlo sampling method in logarithmic radial scale, we are able to numerically simulate the critical limit of infinite system size (corresponding to infinite-dilution limit) within tractable equilibration times. The critical exponents are determined for the inverse moments of the counterionic density profile (which play the role of the order parameters and represent the inverse localization length of counterions) both within mean-field theory and within Monte-Carlo simulations. In three dimensions, we demonstrate that correlation effects (neglected within mean-field theory) lead to an excessive accumulation of counterions near the charged cylinder below the critical temperature (condensation phase), while surprisingly, the critical region exhibits universal critical exponents in accord with the mean-field theory. Also in contrast with the typical trend in bulk critical phenomena, where fluctuations are strongly enhanced in lower dimensions, we demonstrate, using both numerical and analytical approaches, that the mean-field theory becomes exact for the 2D counterion-cylinder system at all temperatures (Manning parameters), when number of counterions tends to infinity. For finite particle number, however, the 2D problem displays a series of peculiar singular points (with diverging heat capacity), which reflect successive de-localization events of individual counterions from the central cylinder. In both 2D and 3D, the heat capacity shows a universal jump at the critical point, and the energy develops a pronounced peak. The asymptotic behavior of the energy peak location is used to locate the critical point, which is also found to be universal and in accordance with the mean-field prediction.

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“…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].…”

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confidence: 99%

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

2005

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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, ...

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Painleve III and Manning's Counterion Condensation

Cited by 29 publications

References 12 publications

Guest Charges in an Electrolyte: Renormalized Charge, Long- and Short-Distance Behavior of the Electric Potential and Density Profiles

Guest Charges in an Electrolyte: Renormalized Charge, Long- and Short-Distance Behavior of the Electric Potential and Density Profiles

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

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

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