Abstract
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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Notes
Note that whenever the pressure is of the form \(\beta P \propto 1/d^2\), it cannot depend on the absolute values of \(\sigma \) and \(\sigma '\), but only on the ratio \(\eta =\sigma /\sigma '\), for dimensional reasons. In the symmetric case, it is thus \(\sigma \) independent.
References
Andelman, D.: Introduction to electrostatics in soft and biological matter. In: Poon, W.C.K., Andelman, D. (eds.) Soft Condensed Matter Physics in Molecular and Cell Biology, vol. 6. Taylor & Francis, New York (2006)
Bordag, M., Mohideen, U., Mostepanenko, V.M.: New developments in the Casimir effect. Phys. Rep. 353, 1–205 (2001)
Buenzli, P.R., Martin, PhA: Microscopic origin of universality in Casimir forces. J. Stat. Phys. 119, 273–307 (2005)
Burak, Y., Andelman, D.: Test-charge theory for the electric double layer. Phys. Rev. E 70, 016102 (2004)
Carnie, S.L., Chan, D.Y.C.: The statistical mechanics of the electrical double layer: stress tensor and contact conditions. J. Chem. Phys. 74, 1293–1297 (1981)
Choquard, Ph: The two-dimensional one component plasma on a periodic strip. Helv. Phys. Acta 54, 332–332 (1981)
Cornu, F., Jancovici, B.: On the two-dimensional Coulomb gas. J. Stat. Phys. 49, 33–56 (1987)
Cornu, F., Jancovici, B.: The electrical double layer: a solvable model. J. Chem. Phys. 90, 2444–2452 (1989)
Forrester, P.J.: Exact results for two-dimensional Coulomb systems. Phys. Rep. 301, 235–270 (1998)
Henderson, D., Blum, L.: Some exact results and the application of the mean spherical approximation to charged hard spheres near a charged hard wall. J. Chem. Phys. 69, 5441–5449 (1978)
Jancovici, B.: Exact results for the two-dimensional one-component plasma. Phys. Rev. Lett. 46, 386–388 (1981)
Jancovici, B.: Inhomogeneous two-dimensional plasmas. In: Henderson, D. (ed.) Inhomogeneous Fluids, pp. 201–237. Dekker, New York (1992)
Jancovici, B., Šamaj, L.: Screening of classical Casimir forces by electrolytes in semi-infinite geometries. J. Stat. Mech., P08006 (2004)
Kanduč, M., Trulsson, M., Naji, A., Burak, Y., Forsman, J., Podgornik, R.: Weak- and strong-coupling electrostatic interactions between asymmetrically charged planar surfaces. Phys. Rev. E 78, 061105 (2008)
Lau, A.W.C., Levine, D., Pincus, P.: Novel electrostatic attraction from plasmon fluctuations. Phys. Rev. Lett. 84, 4116–4119 (2000)
Lau, A.W.C., Pincus, P., Levine, D., Fertig, H.A.: Electrostatic attraction of coupled Wigner crystals: finite temperature effects. Phys. Rev. E 63, 051604 (2001)
Martin, PhA: Sum rules in charged fluids. Rev. Mod. Phys. 60, 1075–1127 (1988)
Milton, K.A.: The Casimir Effect. World Scientific, London (2001)
Netz, R.R., Orland, H.: Beyond Poisson-Boltzmann: Fluctuation effects and correlation functions. Eur. Phys. J. E 1, 203–214 (2000)
Netz, R.R.: Electrostatics of counter-ions at and between planar charged walls:from Poisson-Boltzmann to the strong-coupling theory. Eur. Phys. J. E 5, 557–574 (2001)
Paillusson, F., Trizac, E.: Interaction regimes for oppositely charged plates with multivalent counterions. Phys. Rev. E 84, 011407 (2011)
Šamaj, L., Percus, J.K.: A functional relation among the pair correlations of the two-dimensional one-component plasma. J. Stat. Phys. 80, 811–824 (1995)
Šamaj, L., Wagner, J., Kalinay, P.: Translation symmetry breaking in the one-component plasma on the cylinder. J. Stat. Phys. 117, 159–178 (2004)
Šamaj, L., Trizac, E.: Counterions at highly charged interfaces: from one plate to like-charge attraction. Phys. Rev. Lett. 106, 078301 (2011)
Šamaj, L., Trizac, E.: Counter-ions at charged walls: two-dimensional systems. Eur. Phys. J. E 34, 20 (2011)
Šamaj, L., Trizac, E.: Critical phenomena and phase sequence in a classical bilayer Wigner crystal at zero temperature. Phys. Rev. B 85, 205131 (2012)
Shklovskii, B.I.: Screening of a macroion by multivalent ions: correlation-induced inversion of charge. Phys. Rev. E 60, 5802–5811 (1999)
dos Santos, A.P., Diehl, A., Levin, Y.: Electrostatic correlations in colloidal suspensions: density profiles and effective charges beyond the Poisson-Boltzmann theory. J. Chem. Phys. 130, 124110 (2009)
Mallarino, J.P., Téllez, G., Trizac, E.: Counter-ion density profile around charged cylinders: the strong-coupling needle limit. J. Phys. Chem. B 117, 12702 (2013)
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The support received from Grant VEGA No. 2/0049/12 is acknowledged.
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Šamaj, L., Trizac, E. Counter-Ions Between or at Asymmetrically Charged Walls: 2D Free-Fermion Point. J Stat Phys 156, 932–947 (2014). https://doi.org/10.1007/s10955-014-1053-4
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DOI: https://doi.org/10.1007/s10955-014-1053-4