Charged rod-like polymers are not able to bind all their neutralizing counterions: a fraction of them evaporates, while the others are said to be condensed. We study here counterion condensation and its ramifications, both numerically by means of Monte Carlo simulations employing a previously introduced powerful logarithmic sampling of radial coordinates and analytically, with special emphasis on the strong-coupling regime. We focus on the thin rod or needle limit that is naturally reached under strong Coulombic couplings, where the typical intercounterion spacing a' along the rod is much larger than its radius R. This regime is complementary and opposite to the simpler thick rod case where a' ≪ R. We show that due account of counterion evaporation, a universal phenomenon in the sense that it occurs in the same clothing for both weakly and strongly coupled systems, allows one to obtain excellent agreement between the numerical simulations and the strong-coupling calculations.

When a Coulombic fluid is confined between two parallel charged plates, an exact relation links the difference of ionic densities at contact with the plates, to the surface charges of these boundaries. It no longer applies when the boundaries are curved, and we work out how it generalizes when the fluid is confined between two concentric spheres (or cylinders), in two and in three space dimensions. The analysis is thus performed within the cell model picture. The generalized contact relation opens the possibility to derive new exact expressions, of particular interest in the regime of strong coulombic couplings. Some emphasis is put on cylindrical geometry, for which we discuss in depth the phenomenon of counter-ion evaporation/condensation, and obtain novel results. Good agreement is found with Monte Carlo simulation data.

We present a comprehensive study of the two dimensional one component plasma in the cell model with charged boundaries. Starting from weak couplings through a convenient approximation of the interacting potential we were able to obtain an analytic formulation to the problem deriving the partition function, density profile, contact densities and integrated profiles that compared well with the numerical data from Monte-Carlo simulations. Additionally, we derived the exact solution for the special cases of Ξ = 1, 2, 3, . . . finding a correspondence between those from weak couplings and the latter. Furthermore, we investigated the strong coupling regime taking into consideration the Wigner formulation. Elaborating on this, we obtained the profile to leading order, computed the contact density values as compared to those derived in an earlier work on the contact theorem. We formulated adequately the strong coupling regime for this system that differed from previous formulations. Ultimately, we calculated the first order corrections and compared them against numerical results from our simulations with very good agreement; these results compared equally well in the planar limit, whose results are well known. I. INTRODUCTIONIn this work we present a thorough analysis of the condensation phenomenon of counter-ions around a charged disk. The model under consideration is a two-dimensional (2D) system formed by an impenetrable disk of charge Q 1 surrounded by ions dispersed freely in a larger disk with external charged boundary Q 2 and no dielectric discontinuities between the regions delimited by the geometry. The problem resembles an annulus with particles moving freely between the inner and outer radii as in fig. 1. The N ions have, respectively, a charge −q in such a way that neutrality yields,(1.1)We will assume a point-like geometry for the free charges, which is not a problem due to electrostatic repulsion alone. This model is seemingly the one component plasma (2D-OCP) with a small variation. First the neutralizing charge is not distributed homogeneously in the background and second the inner core is impenetrable. * jp.mallarino50@uniandes.edu.co † gtellez@uniandes.edu.co This is the two-dimensional analog of the Manning counter-ion condensation phenomenon around charged cylinders [22][23][24]. Unlike the three-dimensional (3D) situation where the Coulomb potential shapes as 1/ |r|, the partition function for two-dimensional Coulomb systems is written as a product of contributions which, in some cases, may be computed exactly. That and the logarithmic nature of the potential motivated the theoretical computation of the abundant static and dynamic properties of electrolytes for two-dimensional systems.The interaction between two unit charges separated by a distance r is given by the two-dimensional Coulomb potential − log(r/L), where L is an irrelevant arbitrary length scale. We are interested in the equilibrium thermal properties of the system at a temperature T . As usual, we define β = 1/(k B T ) where k B is the...

We present a comprehensive study for common second order PDE’s in two dimensional disc-like systems and show how their solution can be approximated by finding the Green function of an effective one dimensional system. After elaborating on the formalism, we propose to secure an exact solution via a Fourier expansion of the Green function, which entails solving an infinitely countable system of differential equations for the Green–Fourier modes that in the simplest case yields the source-free Green distribution. We present results on non separable systems—or such whose solution cannot be obtained by the usual variable separation technique—on both annulus and disc geometries, and show how the resulting one dimensional Fourier modes potentially generate a near-exact solution. Numerical solutions will be obtained via finite differentiation using Finite Difference Method (FDM) or Finite Element Method (FEM) with the three-point stencil approximation to derivatives. Comparing to known exact solutions, our results achieve an estimated numerical relative error below 10−6. Solutions show the well-known presence of peaks when r=r′ and a smooth behavior otherwise, for differential equations involving well-behaved functions. We also verified how the Green functions are symmetric under the presence of a “weight function”, which is guaranteed to exist in the presence of a curl-free vector field. Solutions of non-homogeneous differential equations are also shown using the Green formalism and showing consistent results.

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