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

Abstract

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.

Amplitude Function of Asymptotic Correlations Along Charged Wall in Coulomb Fluids

“…Since this tensor contains also long-ranged correlations along the wall, it is likely that a more detailed exploration of the Möbius conformal transformation might reveal another relation between the amplitude function and the density profile which is complementary to the one derived in Ref. [33].…”

“…This equation can be checked on two exactly solvable 2D cases of the present counter-ion system [33]. The PB Γ → 0 limit yields f (x, x ′ ) of the long-range form [31] f (x,…”

“…[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].…”

“…In a recent paper about 2D charged rectilinear wall with counter-ions only [33], we used a technique of anticommuting variables [25] to derive a relation between the amplitude function and the density profile which holds for any coupling (temperature) of the fluid regime. Moreover, using the Möbius conformal transformation of particle coordinates in the partition function for a disc geometry, an exact formula for the dielectric susceptibility tensor was derived.…”

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

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