Langmuir 2005, 21, 9231-9237
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Simulation of Electric Double Layers Undergoing Charge Inversion: Mixtures of Mono- and Multivalent Ions M. Quesada-Pe´rez,† A. Martı´n-Molina,‡ and R. Hidalgo-A Ä lvarez*,§ Departamento de Fı´sica, Universidad de Jae´ n, Escuela Universitaria Polite´ cnica, 23700 Linares, Jae´ n, Spain, and Laboratoire de Physique Statistique de l’Ecole Normale Supe´ rieure Associe´ e au CNRS et aux Universite´ s Paris VI et Paris VII, 24 rue Lhomond, 75231 Paris Cedex 05, France, and Grupo de Fı´sica de Fluidos y Biocoloides, Departamento de Fı´sica Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain Received March 4, 2005. In Final Form: July 18, 2005 In this paper, the electric double layer (EDL) of a charged plane in the presence of mixtures of 1:1 and 3:1 electrolytes has been investigated through Monte Carlo (MC) simulations using a nonrestrictive primitive model of EDL. In particular, the charge inversion in colloids (attributable to an accumulation of counterions on the surface) can be better understood by means of the simulations performed here. Moreover, two mechanisms proposed for charge inversion are probed: The formation of a strongly correlated layer (SCL) of multivalent counterions and excluded volume effects (to which we will also refer as ion size correlations). Our results are in agreement with the behavior found experimentally for some model colloids with increasing the concentration of monovalent salt in the presence of trivalent ions, which clearly supports the relevance of ion size correlations. In contrast, certain disagreement with predictions of SCL theories is reported.
1. Introduction There is a resurgent interest in studying the electric double layer of colloidal particles with multivalent ions. Several fascinating (and somehow counterintuitive) phenomena occur in the presence of such ions. For instance, trivalent and tetravalent ions (spermidine and spermine) are believed to play a key role in maintaining DNA in a compact state. The electrostatic attraction between likecharged macromolecules and the reversal of the sign of the electrophoretic mobility of charged colloids (intimately related to charge inversion) are also attributed to the presence of high-valence counterions. Such phenomena are not only interesting processes in strongly correlated systems but also have potential applications (e.g., gene therapy in the case of DNA). The reader interested in these topics is referred to recent review articles.1-4 The traditional approach to describing the EDL has been the Gouy-Chapman model, whose keystone is the Poisson-Boltzmann (PB) equation. This classical model could be applied to the analysis of some of the abovementioned experimental findings, such as the mobility reversal, introducing a specific (but usually unspecified) interaction.5 However, it is widely known the PB theory fails to describe EDLs with multivalent ions (see ref 4 and studies of EDL performed by Boda et al.6,7 as well as references cited in all of them). Consequently, its ap* Corresponding author. Telephone: (+34) 958243213. Fax: (+34) 958243214. E-mail:
[email protected]. † Universidad de Jae ´ n. ‡ CNRS. § Universidad de Granada. (1) Gelbart, W. M.; Bruinsma, R. F.; Pincus, P. A.; Parsegian, V. A. Phys. Today 2000, 53, 38. (2) Levin, Y. Rep. Prog. Phys. 2002, 65, 1577. (3) Grosberg, A. Y.; Nguyen, T. T.; Shklovskii, B. I. Rev. Modern Phys. 2002, 74, 329. (4) Quesada-Pe´rez, M.; Gonza´lez-Tovar, E.; Martı´n-Molina, A.; Lozada-Cassou, M.; Hidalgo-A Ä lvarez, R. ChemPhysChem 2003, 4, 234. (5) Ottewill, R. H.; Shaw, J. N. J. Colloid Interface Sci. 1968, 26, 110. (6) Boda, D.; Henderson, D.; Plaschko P.; Ronald Fawcett, W. Mol. Simul. 2004, 30, 137. (7) Henderson, D.; Gillespie, D.; Nagy, T.; Boda, D. J. Chem. Phys. 2005, 122, 084504.
plication would not contribute to a better insight into the origin of charge inversion and would lack significant predictive power. Charge inversion can be reasonably well understood in the strong coupling limit (multivalent counterions, highly charged macroions, and/or low temperatures),2,3,8 for which some authors have devised widely known models. According to these approaches, multivalent counterions form a strongly correlated layer on the macroion surface and the charge inversion process takes place because of a favorable gain in free energy. This is a very intuitive picture leading to analytical or semianalytical expressions. However, the situation is much more complicated for systems with weak electrostatic coupling and moderate or high concentrations of salt. Under such conditions, some theoretical studies claim that short-range correlations due to a finite ion size (not included in the PB theory) can be responsible for the failure of the classical approach as well as a driving force in phenomena like charge reversals.4,9,10 Recent atomic force microscope measurements support this hypothesis.11 Computer simulations are useful to throw light on these issues. Initially, simulations looked into suspensions with mono- and divalent ions (but only one electrolyte).8-10,12-17 More recently, many authors have revisited certain aspects of EDLs in the presence of these kinds of electrolytes. For instance, Boda et al. have analyzed the effect of the ion size,6,18 the effects of asymmetries ion diameters and (8) Messina, R.; Holm, C.; Kremer, K. Phys. Rev. E 2001, 64, 021405. (9) Attard, P. Adv. Chem. Phys. 1996, 92, 1. (10) Messina, R.; Gonza´lez-Tovar, E.; Lozada-Cassou, M.; Holm, C. Europhys. Lett. 2002, 60, 383. (11) Besteman, K.; Zevenbergen, M. A. G.; Heering, H. A.; Lemay, S. G. Phys. Rev. Lett. 2004, 93, 170802. (12) Torrie, G. M.; Valleau, J. P.; J. Chem. Phys. 1981, 73, 5807. (13) Torrie, G. M.; Valleau, J. P.; J. Phys. Chem. 1982, 86, 3251. (14) van Megen, W.; Snook, I. J. Chem. Phys. 1980, 73, 4656. (15) Degre`ve, L.; Lozada-Cassou, M.; Sa´nchez, E.; Gonza´lez-Tovar, E. J. Chem. Phys. 1993, 98, 8905. (16) Degre`ve, L.; Lozada-Cassou, M. Mol. Phys. 1995, 86, 759. (17) Terao, T.; Nakayama, T. Phys. Rev. E 2002, 63, 041401. (18) Boda, D.; Ronald Fawcett, W.; Henderson, D.; Sokolowski, S. J. Chem. Phys. 2002, 116, 7170.
10.1021/la0505925 CCC: $30.25 © 2005 American Chemical Society Published on Web 08/27/2005
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charges,19 and the consequences of an inhomogeneous dielectric coefficient discreteness of the solvent.7 Other authors have studied the discreteness of the surface charge,20-22 whereas the role of excluded volume correlations has also been elucidated with the help of simulations in an advisable manner by Messina and co-workers.10 These authors state that the entropy of the solution decreases by enlarging the ion size, which enhances the interparticle correlations. Suspensions with trivalent (and even more highly charged) counterions have also been studied.23-25 However, the case of mixtures of different electrolytes has been scarcely addressed. Although the phenomena mentioned above can take place with a background of monovalent electrolyte (e.g., DNA condensation in the presence of low concentrations of multivalent electrolyte and NaCl at physiological conditions). Related research works have been focused in the analysis of only one type of electrolyte. For instance, Deserno et al. have studied a model of a rodlike polyelectrolyte molecule immersed into monovalent or divalent electrolyte by comparing results obtained from integral equation theories with those from molecular dynamics (MD) simulations.26 Hence, in general, computer simulations have not paid so much attention to the EDL of suspensions with ions of different valence. Mukherjee and co-workers have investigated the effect of mixed valence (mono- and divalent) counterions on the overcharging of a DNA-like spherocylindrical macroion.25 Boda et al. have applied equilibrium MC simulations modeling a biological calcium channel in the presence of Na+ and Ca2+ ions.27 The MC simulations have also been employed by Delville and coworkers to study the competitive condensation of monovalent/divalent and monovalent/trivalent counterions confined between two charged lamellae.28 Diehl and Levin have recently proposed a new dynamical definition of the effective colloidal charge for aqueous colloidal suspensions containing monovalent and multivalent counterions, which is particularly applicable to MC and MD simulations,29 whereas Lobaskin et al. have studied the mechanism of colloidal destabilization in the presence of highly asymmetric electrolytes.30 Concerning charge inversion in model colloids, Martı´n-Molina et al. examined experimentally the effect of the monovalent salts (in electrolyte mixtures) on the mobility reversal observed in suspensions of model polystyrene particles.31 Some of their results were justified by an integral equation theory including ion size correlations. Nevertheless, some matters, such as the failure of integral equation theories, remain unresolved. This work is focused on the EDL structure in the presence of mono- and multivalent (particularly trivalent) ions in aqueous medium. More specifically, the charge inversion found under such conditions is investigated by (19) Valisko´, M.; Henderson, D.; Boda, D. J. Phys. Chem. B 2004, 108, 16548. (20) Ravindran, S.; Wu, J. Langmuir 2004, 20, 7333. (21) Messina, R.; Holm, C.; Kremer, K. Eur. Phys. J. E 2001, 4, 363. (22) Messina, R. Physica A 2002, 308, 59. (23) Quesada-Pe´rez, M.; Martı´n-Molina, A.; Hidalgo-A Ä lvarez, R. J. Chem. Phys. 2004, 121, 8618. (24) Tanaka M.; Grosberg, A. Y. Eur. Phys. J. E 2002, 7, 371. (25) Mukherjee, A. K.; Schmitz K. S.; Bhuiyan L. B. Langmuir 2004, 20, 11802. (26) Deserno, M.; Jime´nez-A Ä ngeles, F.; Holm, C.; Lozada-Cassou, M. J. Phys. Chem. B 2001, 44, 10983. (27) Boda, D.; Varga, T.; Henderson, D.; Busath, D. D.; Nonner, W.; Gillespie, D.; Eisenberg, B. Mol. Simul. 2004, 30, 89. (28) Delville, A.; Gasmi, N.; Pellenq, R. J. M.; Caillol, J. M.; Van Damme, H. Langmuir 1998, 14, 5077. (29) Diehl, A.; Levin, Y. J. Chem. Phys. 2004, 121, 12100. (30) Lobaskin, V.; Qamhieh, K. J. Phys. Chem. B 2003, 107, 8022. (31) Martı´n-Molina, A.; Quesada-Pe´rez, M.; Galisteo-Gonza´lez, F.; Hidalgo-A Ä lvarez, R. J. Phys. Condens. Matter 2003, 15, S3475.
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means of MC simulations. Our main goal is to find out the mechanisms behind the mentioned phenomenon. In this task, the comparison with previous experimental results will be important. In addition, our simulation data can also provide valuable information to test SCL models. Certain relevant predictions of such theories will be probed as well. 2. Model and Monte Carlo Simulations The system under consideration consists of a mixture of 1:1 and 3:1 electrolytes in the presence of a planar negatively charged wall. Obviously, charge inversion can also occur for colloids with curved surfaces (illustrative examples are those mentioned in the Introduction). However, the surface curvature is not expected to modify the qualitative aspects regarding the mechanism of this phenomenon. The case of trivalent counterions has been chosen as a representative situation in which mobility reversals and charge inversions are easily observable (in experiments and simulations). The primitive model, in which small ions are treated as charged hard spheres immersed in a dielectric continuum (included through its relative permittivity only), constitutes a basis for the simulation of this solution. All monovalent species are assumed to have the same hydrated ion diameter (0.65 nm). However, the diameter of trivalent counterions (d1) is supposed to be 0.90 nm. Unlike previous theoretical and simulation works, the values chosen here are typical of realistic hydrated monovalent cations and anions, and trivalent cations, respectively.32,33 The interaction energy between mobile ions is given by
u(r bij) )
ZiZje2 r > (ai + aj) 4π0r rij ij
u(r bij) ) ∞ rij < (ai + aj)
(1)
where Zi and ai are the valence and the radius of ion i respectively, e is the elementary charge, 0 is the permittivity of free space, r ()78.5) the relative permittivity of the solvent (corresponding to water at a temperature of bij| is 298 K), b rij is the center-to-center vector, and rij ) |r the distance between ions i and j, whereas the interaction energy of ion i with the charged wall is
{
|σ0|Ziezi zi > ai 20r u(r b i) ) zi < ai ∞
(2)
ri is its position where zi is the z-coordinate of particle i, b vector and σ0 (
