Counter-ions at charged walls: Two-dimensional systems

Šamaj, L.; Trizac, Emmanuel

doi:10.1140/epje/i2011-11020-1

Cited by 19 publications

(30 citation statements)

References 51 publications

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“…The large-distance asymptotic behaviour of the pressure for the unconstrained model (47) is universal, however, the prefactor to 1/d 2 is not consistent with the mean-field prediction (7) but has to be renormalized by a temperature-dependent function. This remark on the failure of mean-field at asymptotic distances, corroborates previous 2D results obtained for symmetric lines [25]. It goes against common expectation that mean-field should hold at large d [27,28], the underlying argument being that the small density of ions far from the plate may effectively drive the system into a weakly-coupled regime, in the corresponding distance range.…”

Section: Resultssupporting

confidence: 89%

“…-First, for the "unconstrained" version of the model with counter-ions moving freely in space between the lines, we generalize the exact result at Γ = 2 for symmetric lines [25] to asymmetrically charged lines, see Fig. 1.…”

Section: Introductionmentioning

confidence: 92%

“…For Γ = 2γ (γ a positive integer), the technique of anticommuting variables [25,22,23] allows us to express Q N as the integral over Grassman variables with the action that couples certain composite operators by interaction strengths…”

Section: General Formalism For Cylinder Geometrymentioning

confidence: 99%

“…In a recent work [25], we solved exactly the free-fermion point Γ = 2 by using a technique of anticommuting variables, with the result…”

Section: Introductionmentioning

confidence: 99%

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Counter-Ions Between or at Asymmetrically Charged Walls: 2D Free-Fermion Point

Šamaj

Trizac

2014

J Stat Phys

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This work contributes to the problem of determining effective interaction between asymmetrically (likely or oppositely) charged objects whose total charge is neutralized by mobile pointlike counter-ions of the same charge, the whole system being in thermal equilibrium. The problem is formulated in two spatial dimensions with logarithmic Coulomb interactions. The charged objects correspond to two parallel lines at distance d, with fixed line charge densities. Two versions of the model are considered: the standard "unconstrained" one with particles moving freely between the lines and the "constrained" one with particles confined to the lines. We solve exactly both systems at the free-fermion coupling and compare the results for the pressure (i.e. the force between the lines per unit length of one of the lines) with the mean-field Poisson-Boltzmann solution. For the unconstrained model, the large-d asymptotic behaviour of the free-fermion pressure differs from that predicted by the mean-field theory. For the constrained model, the asymptotic pressure coincides with the attractive van der Waals-Casimir fluctuational force. For both models, there are fundamental differences between the cases of likely-charged and oppositely-charged lines, the latter case corresponding at large distances d to a capacitor.

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Section: Resultssupporting

confidence: 89%

Section: Introductionmentioning

confidence: 92%

Section: General Formalism For Cylinder Geometrymentioning

confidence: 99%

“…In a recent work [25], we solved exactly the free-fermion point Γ = 2 by using a technique of anticommuting variables, with the result…”

Section: Introductionmentioning

confidence: 99%

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Counter-Ions Between or at Asymmetrically Charged Walls: 2D Free-Fermion Point

Šamaj

Trizac

2014

J Stat Phys

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“…In other words, from eq. (3.13), 14) thus recovering the celebrated Manning condensation fraction in the thermodynamic limit. As a result, j ⋆ will be intrinsically related to the number of condensed ions; this will be further clarified in the following sections.…”

Section: Figurementioning

confidence: 79%

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

Mallarino

Téllez

2015

Phys. Rev. E

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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...

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Counter-Ions Near a Charged Wall: Exact Results for Disc and Planar Geometries

Šamaj

2015

J Stat Phys

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Macromolecules, when immersed in a polar solvent like water, become charged by a fixed surface charge density which is compensated by "counter-ions" moving out of the surface. Such classical particle systems exhibit poor screening properties at any temperature and the trivial bulk regime (far away from the charged surface) with no particles, so the validity of standard Coulomb sum rules is questionable. In the present paper, we concentrate on the two-dimensional version of the model with the logarithmic interaction potential. We go from the finite disc to the semi-infinite planar geometry. The system is exactly solvable for two values of the coupling constant Γ : in the Poisson-Boltzmann mean-field limit Γ → 0 and at the free-fermion point Γ = 2. We show that the finite-size expansion of the free energy does not contain universal term as is usual for Coulomb fluids. For the coupling constant being an arbitrary positive even integer, using an anticommuting representation of the partition function and many-body densities we derive a sequence of sum rules. As a result, the contact density of counter-ions at the wall is available for the disc. The amplitude function, which characterizes the asymptotic inverse-power law behavior of the two-body density along the wall, is found to be related to the particle density profile. The dielectric susceptibility tensor, calculated exactly for an arbitrary coupling and the particle number, exhibits the anticipated disc value in the thermodynamic limit, in spite of zero contribution from the bulk region. Some of the results obtained in the Poisson-Boltzmann limit are generalized to an arbitrary Euclidean dimension.

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Counter-ions at charged walls: Two-dimensional systems

Cited by 19 publications

References 51 publications

Counter-Ions Between or at Asymmetrically Charged Walls: 2D Free-Fermion Point

Counter-Ions Between or at Asymmetrically Charged Walls: 2D Free-Fermion Point

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

Counter-Ions Near a Charged Wall: Exact Results for Disc and Planar Geometries

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