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Attraction of Like-Charged Walls with Counterions Only: Exact Results for the 2D Cylinder Geometry

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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 \(\varepsilon \). 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 \(\beta \). The exact solution of the system at the free-fermion coupling \(\varGamma \equiv \beta e^2/\varepsilon =2\) is used to test the convergence of the pressure as the (even) number of particles increases from \(N=2\) to \(\infty \). 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 \(N\rightarrow \infty \). For the couplings \(\varGamma =2\gamma \) with \(\gamma =1,2,\ldots \), 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 \(\varGamma =4\) 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 \(N\rightarrow \infty \) as well, starting from a relatively weak coupling constant \(\varGamma \) 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.

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Notes

  1. The explicit formulas for \(Q_N(2)\) and \(Q_N(3)\) up to \(N=10\) will be sent upon request by the author.

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Acknowledgements

The support received from the project EXSES APVV-16-0186 and VEGA Grant No. 2/0003/18 is acknowledged.

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  1. Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, 84511, Bratislava, Slovakia

    Ladislav Šamaj

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Šamaj, L. Attraction of Like-Charged Walls with Counterions Only: Exact Results for the 2D Cylinder Geometry. J Stat Phys 181, 1699–1729 (2020). https://doi.org/10.1007/s10955-020-02642-9

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