Abstract
We complement a recent exact study by L. Šamaj on the properties of a guest charge Q immersed in a two-dimensional electrolyte with charges +1/−1. In particular, we are interested in the behavior of the density profiles and electric potential created by the charge and the electrolyte, and in the determination of the renormalized charge which is obtained from the long-distance asymptotics of the electric potential. In Šamaj’s previous work, exact results for arbitrary coulombic coupling β were obtained for a system where all the charges are points, provided β Q < 2 and β < 2. Here, we first focus on the mean field situation which we believe describes correctly the limit β→ 0 but β Q large. In this limit we can study the case when the guest charge is a hard disk and its charge is above the collapse value β Q > 2. We compare our results for the renormalized charge with the exact predictions and we test on a solid ground some conjectures of the previous study. Our study shows that the exact formulas obtained by Šamaj for the renormalized charge are not valid for β Q > 2, contrary to a hypothesis put forward by Šamaj. We also determine the short-distance asymptotics of the density profiles of the coions and counterions near the guest charge, for arbitrary coulombic coupling. We show that the coion density profile exhibit a change of behavior if the guest charge becomes large enough (β Q≥ 2−β). This is interpreted as a first step of the counterion condensation (for large coulombic coupling), the second step taking place at the usual Manning–Oosawa threshold β Q = 2.
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References
L. Šamaj, Anomalous effects of “guest” charges immersed in electrolyte: Exact 2D results, e-print cond-mat/0503201, to be published in J. Stat. Phys. 120:125 (2005).
A. Zamolodchikov and Al. Zamolodchikov, Ann. Phys. (N.Y.) 120:253 (1979).
C. Destri and H. de Vega, Nucl. Phys. B 358:251 (1991).
Al. Zamolodchikov, Int. J. Mod. Phys. A 10:1125 (1995).
S. Lukyanov and A. Zamolodchikov, Nucl. Phys. B 493:571 (1997).
V. Fateev, S. Lukyanov, A. Zamolodchikov and Al. Zamolodchikov, Nucl. Phys. B 516:652 (1998).
S. Lukyanov, Mod. Phys. Lett. 12:2543 (1997).
S. Lukyanov, Phys. Lett. B 408:192 (1997).
L. Šamaj and I. Travěnec, J. Stat. Phys. 101:713 (2000).
S. Alexander, P. M. Chaikin, P. Grant, G. J. Morales and P. Pincus, J. Chem. Phys. 80:5776 (1984).
E. Trizac, L. Bocquet and M. Aubouy, Phys. Rev. Lett. 89:248301 (2002).
L. Belloni, Colloids Surfaces A 140:227 (1998).
J.-P. Hansen and H. Löwen, Annu. Rev. Phys. Chem. 51:209 (2000).
Y. Levin, Rep. Prog. Phys. 65:1577 (2002).
G. Téllez and E. Trizac, Phys. Rev. E 68:061401 (2003).
G. Téllez and E. Trizac, Phys. Rev. E 70:011404 (2004).
T. Kennedy, J. Stat. Phys. 37:529 (1984).
T. Kennedy, Comm. Math. Phys. 92:269 (1983).
B. M. McCoy, C. A. Tracy and T. T. Wu, J. Math. Phys. 18:1058 (1977).
J. S. McCaskill and E. D. Fackerell, J. Chem. Soc. Faraday Trans. 2 84(2):161 (1988).
C. A. Tracy and H. Widom, Physica A 244:402 (1997).
C. A. Tracy and H. Widom, Commun. Math. Phys. 190:697 (1998).
G. Téllez and E. Trizac, Exact results for the cylindrical Poisson-Boltzmann.
E. Trizac and G. Téllez, Onsager-Manning-Oosawa condensation phenomenon, preprint (2005).
G. S. Manning, J. Chem. Phys. 51: 924 (1969).
F. Oosawa, Polyelectrolytes, Dekker, New York (1971).
A. L. Kholodenko and A. L. Beyerlein, it Phys. Rev. Lett. 74:4679 (1995).
E. Trizac, L. Bocquet, M. Aubouy and H. H. von Grunberg, Langmuir 19:4027 (2003).
J.-P. Hansen and P. Viot, J. Stat. Phys. 38:823) (1985).
L. Šamaj, J. Stat. Phys. 111:261 (2003).
Vl. S. Dotsenko and V. A. Fateev, Nucl. Phys. B 240:312 (1984).
Vl. S. Dotsenko and V. A. Fateev, Nucl. Phys. B 251:691 (1985).
L. Šamaj, private communication.
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Téllez, G. Guest Charges in an Electrolyte: Renormalized Charge, Long- and Short-Distance Behavior of the Electric Potential and Density Profiles. J Stat Phys 122, 787–798 (2006). https://doi.org/10.1007/s10955-005-8069-8
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DOI: https://doi.org/10.1007/s10955-005-8069-8