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

Abstract

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.

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

“…II), followed in Sec. III by an analysis of the regime when Ξ < 1 (or Γ < 2), where we extend previous works by Burak and Orland [2], Naji and Netz [26,27], Varghese et al [32]. Then follows, in Secs.…”

“…For that matter, this system's behavior depends on Q 1 − N q, which can take arbitrary values. We may introduce the standardized dimensionless notation suggested by Naji and Netz [26,27] by rescaling all distances with the Gouy-Chapmann length (µ = R/ξ), i.e. r = r/µ, but this will only be important extending the analysis to the planar limit at which the Gouy length is significant; to this problem, all radial distances will be divided by either R or D. Substituting the Manning parameter as ξ = (β Q 1 q/2) and the coupling Ξ = β q 2 /2 (or Γ = βq 2 ) we obtain, using eq.…”

“…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. …”