10630
Langmuir 2006, 22, 10630-10634
Planar Electric Double Layer for a Restricted Primitive Model Electrolyte at Low Temperatures† L. B. Bhuiyan* Laboratory of Theoretical Physics, Department of Physics, UniVersity of Puerto Rico, San Juan 00931-3343, Puerto Rico
C. W. Outhwaite Department of Applied Mathematics, UniVersity of Sheffield, Sheffield S3 7RH, U.K.
D. Henderson Department of Chemistry and Biochemistry, Brigham Young UniVersity, ProVo, Utah 84602-5700 ReceiVed February 14, 2006. In Final Form: April 10, 2006 Monte Carlo simulation and the modified Poisson-Boltzmann theory are used to investigate the planar electric double layer for a restricted primitive model electrolyte at low temperatures. Capacitance as a function of temperature at low surface charge is determined for 1:1, 2:2, 2:1, and 3:1 electrolytes. Negative adsorption can occur for 1:1 electrolytes at low surface charge with low electrolyte concentration. The 1:1 electrolyte diffuse layer potential as a function of surface charge displays a maximum at low densities. At high densities, the diffuse layer potential is negative with a negative slope. The Gouy-Chapman-Stern theory fails in this low-temperature regime, whereas the modified Poisson-Boltzmann theory is fairly successful in this regard.
1. Introduction The molten salt/electrode double layer capacitance of molten salts near the position of zero charge increases with temperature.1 In contrast, the electrolyte/electrode capacitance always apparently decreases at standard temperatures. This seemingly anomalous capacitance behavior has been resolved by recent Monte Carlo (MC)2-4 and theoretical work5-9 on the restricted primitive model (RPM) electrolyte next to a planar electrode at small surface charge. At very low temperatures, the electrolyte capacitance increases in an analogous fashion to that of molten salts while decreasing at higher temperatures in accordance with the Gouy-Chapman (GC) theory. The anomalous capacitance behavior has been one of the more interesting results to come out of research into the low-temperature regime in recent years and justifies a look at other related equilibrium properties of the double layer. This regime has hitherto not received adequate attention in double layer literature because traditional studies have been carried out at the habitual T ≈ 298 K (room temperature). In this article, we continue to investigate the predictions of the modified Poisson-Boltzmann (MPB) theory10-12 at low tem†
Part of the Electrochemistry special issue. * Corresponding author. E-mail:
[email protected].
(1) March, N. H.; Tosi, M. P. Coulomb Liquids; Academic: London, 1984. (2) Boda, D.; Henderson, D.; Chan, K. Y. J. Chem. Phys. 1999, 110, 5346. (3) Boda, D.; Henderson, D.; Chan, K. Y.; Wasan, D. T. Chem. Phys. Lett. 1999, 308, 473. (4) Henderson, D. J. Mol. Liq. 2001, 92, 29. (5) Holovko, M.; Kapko, V.; Henderson, D.; Boda, D. Chem. Phys. Lett. 2001, 341, 363. (6) Resko-Zygmunt, J.; Sokołowski, S.; Henderson, D.; Boda, D. J. Chem. Phys. 2005, 122, 084504. (7) Bhuiyan, L. B.; Outhwaite, C. W.; Henderson, D. J. Chem. Phys. 2005, 123, 034704. (8) Reszko-Zygmunt, J.; Sokołowski, S.; Pizio, O. J. Chem. Phys. 2005, 123, 016101. (9) Di Caprio, D.; Stafiej, J.; Borkowska, Z. J. Electroanal. Chem. 2005, 582, 41.
peratures, in particular, the capacitance C of 2:1, 3:1, and 2:2 salts and the diffuse layer potential and adsorption for 1:1 salts. Monte Carlo data for the capacitance has recently been obtained by Valisko et al.13 We have also done MC simulations to obtain some “exact” results for the diffuse layer potential drop and adsorption. Comparisons of the MPB results are made with all of these MC simulations.
2. Theory 2.1. Modified Poisson-Boltzmann Theory. The MPB theory for the planar electric double layer has been presented elsewhere.10-12 Here we give the relevant equations for a RPM electrolyte next to a planar, uniformly charged, hard wall, with the electrolyte being characterized by ions i of charge ei and diameter a moving in a dielectric medium of permittivity . The electrode has a uniform surface charge density σ and is taken to have the same permittivity of the electrolyte so that there is no imaging. At a perpendicular distance x into the solution from the wall, the mean electrostatic potential ψ(x) satisfies
d2ψ dx2
)-
4π
∑zi|e|Figi(x) i
ψ(x) ) ψ(0) -
4πσx
x> 0
