Influence of the Relative Orientation of Two Charged Anisotropic Colloidal Particles on Their Electrostatic Coupling: A (N,V,T) Monte Carlo Study S. Meyer, P. Levitz, and A. Delville* CRMD, CNRS, 1b rue de la Fe ´ rollerie, 45071 Orle ´ ans Cedex 02, France ReceiVed: March 29, 2001; In Final Form: July 5, 2001 Monte Carlo simulations are performed in the Canonical ensemble to determine the equilibrium configurations of the counterions and co-ions in the vicinity of two charged hard disks at fixed separations. Derivation of the electrostatic energy and the ionic configurational entropy is used to determine the relative stability of two charged platelets as a function of their separation and relative orientation. An effective pair potential is derived from the free energy variation and may be used in the framework of the one component plasma to model suspensions of charged platelets. Finally, the variation of the free energy of two charged colloids in the presence of salt may be reproduced by Yukawa potential with a residual charge equal to 7% of the nominal electric charge of the colloids. I. Introduction The long-ranged repulsion between charged colloidal particles neutralized by monovalent counterions is generally well under- stood on the basis of the overlap of their diffuse layers. 1-4 However suspensions of charged anisotropic colloids were recently the subject of numerous theoretical 5-10 and experi- mental 11-24 studies since their mechanical and dynamical behavior is far from being fully understood. As an example, dilute suspensions of clay platelets were shown to display large heterogeneities of density at low ionic strength 24 and a plateau, as occurring for a gas/liquidlike first-order transition, at high ionic strength. 13-18 Both observations require some effective attraction between the clay particles although their swelling pressure is characteristic of a material composed of strongly repulsive particles. In the framework of the primitive model, 25 the net force acting on charged colloids results from a balance between the attractive contributions (long-range van der Waals and electrostatic forces) and the repulsive contributions (contact forces). For colloids neutralized by monovalent counterions, correlation forces remain negligible 4,26,27 and the balance between these attractive/ repulsive contributions is always a repulsion resulting from the overlap of the diffuse layers of condensed counterions sur- rounding each particle. For anisotropic colloids, one may expect to find relative orientations of the colloids minimizing this overlap of the diffuse layers, leading to a net stabilization of the colloidal suspensions. This mechanism is expected to occur for suspensions of Laponite clays in the presence of salt, since the thickness of their diffuse layer then reaches the same order of magnitude as their spatial extent. We already suggested that the pseudoplateau displayed by their equation of state results from the reorganization 13 of the repulsive clay particles. A first- order transition was predicted by Onsager, 28 but it requires a parallel alignment of the neighboring clay platelets 29,30 and thus increases the overlap of their diffuse layer. By contrast, previous Monte Carlo simulations have shown a net reduction of the electrostatic energy of two charged clay platelets oriented perpendicularly 6,13 to each other, but the resulting overlap of their diffuse layers was not yet quantified. The purpose of this study is thus to quantify both the energetic and the entropic contributions to the free energy of two lamellar charged colloids neutralized by monovalent counterions in order to determine their most stable configuration as a function of their separation. This analysis will help to determine the path, allowing the approach of two clay particles by minimizing their mutual repulsion. 43 The resulting free energy will also be useful as an effective pair potential for modeling such suspensions of anisotropic charged colloids in the framework of the one component plasma, 31 without requiring explicitly the knowledge of the distribution of their counterions whose contribution is already included in the effective potential. Finally, the same pair potential may be useful to analyze the energy barrier implied in the rotation of a single clay particle within the Wigner cell determined by its immobile first neighboring clay particles, explaining the origin of the long time scale implied in this local motion 18 and the aging 19 behavior reported recently by dynamic light scattering experiments on Laponite aqueous suspensions. II. Material and Methods (A) Modeling Laponite Clay Suspensions. Laponite is a synthetic clay resulting from the sandwiching of one octahedral layer of magnesium oxide between two layers of silicon oxides. Because of the substitution of some magnesium cations of the octahedral network by lithium cations, negative charges are localized within the equatorial plane of the clay network and are neutralized by exchangeable sodium counterions. In the aqueous dilute regime and at pH 10, Laponite clays behave as isolated disks 13 (diameter 300 Å, thickness 10 Å) each bearing 1000 elementary charges. Thus, as long as electrostatic forces are considered, Laponite clay in alkaline pH 11 is modeled as hard disks bearing 1000 electric charges uniformly distributed within each equatorial plane. No lateral charges are used, since the sites localized at the periphery of the Laponite particle are not ionized under alkaline pH. 11,44 All ions are further described in the framework of the primitive model, 25 i.e. by charged hard spheres with an ionic diameter of 4.5 Å to mimic solvated sodium counterions. 32 The ionic diameter of anions is set equal * Corresponding author. E-mail: delville@cnrs-orleans.fr. 10684 J. Phys. Chem. B 2001, 105, 10684-10690 10.1021/jp011193v CCC: $20.00 © 2001 American Chemical Society Published on Web 09/29/2001