Electrostatic Attraction between Two Charged Surfaces: A (N,V,T) Monte Carlo Simulation R. J.-M. Pellenq, † J. M. Caillol, ‡ and A. Delville* ,† CRMD, CNRS (UMR6619), 1b rue de la Fe ´ rollerie, 45071 Orle ´ ans Cedex 02, France, and MAPMO, CNRS (UMR6628), De ´ partement de Mathe ´ matiques, UniVersite ´ d’Orle ´ ans, BP6759, 45067 Orle ´ ans Cedex 02, France ReceiVed: April 14, 1997; In Final Form: August 1, 1997 X We have performed (N,V,T) Monte Carlo simulations in order to study the stability of two parallel charged surfaces (lamellae) neutralized by exchangeable counterions. By varying each parameter characterizing the interface, i.e., the interlamellar separation, the surface charge density of the lamellae, the dielectric constant of the solvent, and the radius and charge of the counterions, we have determined the stability of a wide class of physical situations. The ion-ion, lamella-ion, and lamella-lamella interactions are described within the context of the “primitive model”. We give evidence that, despite the intrinsic simplicity of the primitive model, the phase diagram of such a system exhibits complex patterns. We have determined 2D contour maps of the equation of state in order to localize attracto/repulsive domains and optimize adhesive properties of the interface. This study concerns a large variety of lamellar materials, including hydrated cement and clays, and pillared and organo clays. At low dielectric constant, we have also found evidence of an attractive regime for lamellae neutralized by monovalent counterions. I. Introduction The stability of charged colloids 1,2 [latex, mineral oxides (clays, silica, ...), synthetic (polyacrylate, latex, ...) or natural (DNA, polysaccharides, ...) polyelectrolytes 3 ] is generally discussed on the basis of the Deryaguin-Landau-Verwey- Overbeek (DLVO) 4-5 theory, which includes short-ranged van der Waals attraction and long-ranged electrostatic repulsion between the colloidal particles. This effective repulsion results from the entropy of colloidal suspensions which is estimated on the basis of the overlap between the ionic diffuse layers surrounding each polyion. 1,6-7 In the context of the primitive model, 8 calculations 9-10 or simulations 11-12 have already shown a net attraction between moderately coupled interfaces (i.e., at weak interlamellar separation) neutralized by divalent counte- rions. However, these so-called attractive correlation forces 9-12 are ignored in the Poisson-Boltzmann treatment of charged interfaces, although this approximation underlies the DLVO theory. In this article, we show the existence of complex phase diagrams from which the stability of charged lamellar materials [clays, organo, and pillared clays, 13,14 cement, lamellar oxides (aluminates, V 2 O 5 , silicates, ...)] can be predicted. We also report evidence of a net attraction between lamellae neutralized by monoValent counterions. In this case, the condition of strong coupling, necessary to observe such a behavior, is obtained by reducing the dielectric constant of the solvent. This result should provide new guidelines for the understanding of the stability of mixed systems, 13 i.e., mixtures such as charged minerals (clays) neutralized by amphiphillic counterions (quaternary ammonium) in the presence of intercalated organic matter (organic solvent molecules, adsorbed reactant, ...). For such systems, the dielectric constants of both the lamellae and the intercalates are similar. Therefore, van der Waals attraction is strongly reduced and cannot contribute significantly to the pressure. As shown by our Monte Carlo simulations, the net interlamellar attraction results then mainly from electrostatic interactions, due to ionic correlations. We performed (N,V,T) Monte Carlo simulations 15 of the distribution of counterions neutralizing two infinite lamellae. We assume equality between the solvent and lamellar dielectric constant (no image charge). 16 The lamellae are structureless with uniform surface charge density, and no salt is added to the interface. In this case, the electrostatic interactions between parallel charged lamellae are optimal since the Debye screening length cannot be defined. The interactions between charged entities (counterions and lamellae) are described on the basis of the primitive model, 8 which includes contact hard core repulsion and long-ranged electrostatic interactions. Monte Carlo simulations are performed either within classical 3D Euclidian space 6,7,12,16-18 or on a hypersphere 19 (see Figure 1). The main advantage of this closed geometry 19-20 is to provide an exact analytical expression of Coulomb potential, without using long-range corrections or Ewald summations. 15 We have recently used Monte Carlo simulations on the hypersphere as a test of the approximate Euclidian approach, showing the equivalence of both procedures, 19 even for strongly coupled interfaces. Readers are referred to this article 19 which gives more details on the numerical procedures used for simulations within the hyperspherical geometry. * Corresponding author. † CRMD. E-mail: pellenq@cnrs-orleans.fr (for R. J.-M. Pellenq) and delville@cnrs-orleans.fr (for A. Delville). ‡ MAPMO. E-mail: caillol@labomath.univ-orleans.fr. X Abstract published in AdVance ACS Abstracts, September 15, 1997. Figure 1. Scheme illustrating the calculation procedures in Euclidian (a, left) and hyperspherical (b, right) geometries. 8584 J. Phys. Chem. B 1997, 101, 8584-8594 S1089-5647(97)01273-X CCC: $14.00 © 1997 American Chemical Society