Attraction of Like-Charged Walls with Counterions Only: Exact Results for the 2D Cylinder Geometry
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Attraction of Like-Charged Walls with Counterions Only: Exact Results for the 2D Cylinder Geometry Ladislav Šamaj1 Received: 26 June 2020 / Accepted: 18 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study a 2D system of identical mobile particles on the surface of a cylinder of finite length d and circumference W , immersed in a medium of dielectric constant ε. The two end-circles of the cylinder are like-charged with the fixed uniform charge densities, the particles of opposite charge −e (e being the elementary charge) are coined as “counterions”; the system as a whole is electroneutral. Such a geometry is well defined also for finite numbers of counterions N . Our task is to derive an effective interaction between the end-circles mediated by the counterions in thermal equilibrium at the inverse temperature β. The exact solution of the system at the free-fermion coupling Γ ≡ βe2 /ε = 2 is used to test the convergence of the pressure as the (even) number of particles increases from N = 2 to ∞. The pressure as a function of distance d is always positive (effective repulsion between the like-charged circles), decaying monotonously; the numerical results for N = 8 counterions are very close to those in the thermodynamic limit N → ∞. For the couplings Γ = 2γ with γ = 1, 2, . . ., there exists a mapping of the continuous two-dimensional (2D) Coulomb system with N particles onto the one-dimensional (1D) lattice model of N sites with interacting sets of anticommuting variables. This allows one to treat exactly the density profile, two-body density and the pressure for the couplings Γ = 4 and 6, up to N = 8 particles. Our main finding is that the pressure becomes negative at large enough distances d if and only if both like-charged walls carry a nonzero charge density. This indicates a like-attraction in the thermodynamic limit N → ∞ as well, starting from a relatively weak coupling constant Γ in between 2 and 4. As a by-product of the formalism, we derive specific sum rules which have direct impact on characteristics of the long-range decay of 2D two-body densities along the two walls. Keywords Coulomb fluid · Electric double layer · Counterions only · Free-fermion point · Sum rules · Like-charge attraction
Communicated by Yariv Kafri.
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Ladislav Šamaj [email protected] Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, 84511 Bratislava, Slovakia
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L. Šamaj
1 Introduction The study of equilibrium statistical mechanics of classical (i.e. nonquantum) systems of particles interacting pairwisely by Coulomb potential is of particular importance in condensed matter and soft matter physics. In the real 3-dimensional (3D) space of practical interest, the Coulomb potential in vacuum of dielectric constant ε = 1 has in Gauss units the standard form φ(r) = 1/r with r being the modulus of r. The definition of the Coulomb potential can be extended to any dimension ν = 1, 2, . . . as the solution of the Poisson equation Δφ(r) = −sν δ(r),
(1.1)
where
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