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

Self Cite

Thermodynamic potential of a neutral two-dimensional (2D) Coulomb fluid, confined to a large domain with a smooth boundary, exhibits at any (inverse) temperature β a logarithmic finite-size correction term whose universal prefactor depends only on the Euler number of the domain and the conformal anomaly number c = −1. A minimal free boson conformal field theory, which is equivalent to the 2D symmetric two-component plasma of elementary ±e charges at coupling constant Γ = βe 2 , was studied in the past. It was shown that creating a non-neutrality by spreading out a charge Qe at infinity modifies the anomaly number to c(Q, Γ ) = −1 + 3Γ Q 2 . Here, we study the effect of non-neutrality on the finite-size expansion of the free energy for another Coulomb fluid, namely the 2D one-component plasma (jellium) composed of identical pointlike e-charges in a homogeneous background surface charge density. For the disk geometry of the confining domain we find that the non-neutrality induces the same change of the anomaly number in the finite-size expansion. We derive this result first at the free-fermion coupling Γ ≡ βe 2 = 2 and then, by using a mapping of the 2D one-component plasma onto an anticommuting field theory formulated on a chain, for an arbitrary coupling constant.

“…[33] and developed further in Refs. [34,36,37,38,39,40]. The relation between the two methods was established in Ref.…”

Section: Mapping Onto 1d Fermionsmentioning

confidence: 99%

Finite-Size Effects in Non-neutral Two-Dimensional Coulomb Fluids

Šamaj¹

2017

Self Cite

“…[25] and developed further in Refs. [26,27,30,31,32]. For γ a positive integer, the partition function (2.1) can be expressed in terms of two sets of anticommuting variables {ξ…”

Section: Formalism Of Anticommuting Variablesmentioning

confidence: 99%

“…[5,18,19,23,29]. In spite of the fact that sparse Coulomb fluids have poor screening properties, the standard sum rules hold in semi-infinite and finite geometries [31,32,33].…”

Section: Introductionmentioning

confidence: 99%

Amplitude Function of Asymptotic Correlations Along Charged Wall in Coulomb Fluids

Šamaj¹

2016

Self Cite

In classical semi-infinite Coulomb fluids, two-point correlation functions exhibit a slow inverse-power law decay along a uniformly charged wall. In this work, we concentrate on the corresponding amplitude function which depends on the distances of the two points from the wall. Recently [L. Samaj, J. Stat. Phys. 161, 227 (2015)], applying a technique of anticommuting variables to a 2D system of charged rectilinear wall with "counter-ions only", we derived a relation between the amplitude function and the density profile which holds for any temperature. In this paper, using the Möbius conformal transformation of particle coordinates in a disc, a new relation between the amplitude function and the density profile is found for that model. This enables us to prove, at any temperature, the factorization property of the amplitude function in point distances from the wall and to express it in terms of the density profile. Presupposing the factorization property of the amplitude function and using specific sum rules for semi-infinite geometries, a relation between the amplitude function of the charge-charge structure function and the charge profile is derived for many-component Coulomb fluids in any dimension.

“…[43,44,47,48,49]. The main advantage is that the one-body and two-body densities are expressible explicitly in terms of averaging over the anticommuting variables, · · · ≡ DψDξ e S · · · /Z N (γ).…”

Section: Mapping the Model To 1d Fermions For Disc Geometrymentioning

confidence: 99%

“…The 2D system of counter-ions between symmetrically and asymmetrically charged lines for the cylinder geometry was solved in Refs. [47] and [49], respectively. The validity of certain sum rules for the case of single charged line was verified in Ref.…”

Section: Introductionmentioning

confidence: 99%

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

Šamaj

2015

Self Cite

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.