Testing a modified model of the Poisson–Boltzmann theory that includes ion size effects through Monte Carlo simulations Jose´ Guadalupe Ibarra-Armenta, a Alberto Martı´n-Molina b and Manuel Quesada-Pe´rez* a Received 11th July 2008, Accepted 15th October 2008 First published as an Advance Article on the web 20th November 2008 DOI: 10.1039/b811928g In this work we test the validity of a recent modified Poisson–Boltzmann (MPB) theory that includes ion size effects through a Langmuir-type correction. In particular, we will focus on an analytic charge–potential relationship accounting for such effects. Previous electric double layer (EDL) surveys have demonstrated that the inclusion of ions size in classical EDL theories, based on the Poisson–Boltzmann (PB) equation, can yield considerable improvements. In this sense, the theory we analyze assumes that, as a result of the excluded volume, the ion concentration close to the charged surface cannot exceed a fixed value determined by the close packing fraction. This leads to predictions of counterion concentrations (in this region) smaller than the corresponding PB values. In our opinion, it is worthwhile to test the validity of this novel theory. To this end, computer simulations appear as a useful tool for this kind of task. Our results prove that the above-mentioned analytical expression works fairly well for 1 : 1 electrolytes and large ions, and its predictions can be considerably improved with certain corrections in the estimation of some key parameters. However, it fails for multivalent electrolytes. 1. Introduction For many decades, the PB equation has been the cornerstone of the classical EDL theory. It has been extensively applied in colloid science, chemical physics as well as other research fields (e.g., semiconductor physics). Its conceptual and mathematical simpli- city is responsible for the success of this classical approach. However, certain limitations of this mean field theory were obvious from the beginning. For instance, Stern was the first who noticed that the PB equation predicts unrealistic concentrations of counterions in the vicinity of the charged interface as a result of neglecting the ion dimensions. He tried to overcome this deficiency introducing the notion of the so-called Stern layer. 1 However, the direct inclusion of ion size effects in the PB equation dates back to Bikerman, who added an excess term to the ideal chemical potential with this purpose. 2 Since then, different corrections to the chemical potential have been proposed. 3–9 Other authors have developed alternative approaches, such as lattice-based models, 10–13 different modified PB equations 14–17 or integral equation theories. 18–20 Nowadays ion size effects are still a burning issue in colloid science. For instance, some researchers have recently studied their role in the ion layering at high salt concentrations, 17 counterintuitive electrostatic phenomena (e.g. charge inver- sion 20 ) and the competition of different ions in solutions with electrolyte mixtures. 9 The deviations from the PB predictions of the charge–potential relationship have also been looked into in recent times. 21–23 In particular, Lo´pez-Garcı´a et al. have analyzed a MPB equation that takes the finite ion size into account through a Langmuir-type correction. 24 What is more, they have put forward an approximate analytical expression (including volume effects) for the charge–potential relationship. This could indeed be a valuable result since the surface charge density and the electric potential at the outer Helmholtz plane (OHP), usually known as diffuse potential, are key parameters in the description of the EDL. In any case, these authors claim that significant discrepancies with the classical predictions can appear even for typical sizes of hydrated ions (ranging mostly from 0.75 to 1 nm). 25 In addition, Lo´pez-Garcı´a et al. have also extended their results to a nonequilibrium problem, calculating the effect of ion size on the electrophoretic mobility of a rigid spherical particle. 26 In any case, one should keep in mind that the approach and the analytical expression proposed by Lo´ pez-Garcı´a et al. were developed under certain assumptions. For instance, as a consequence of the excluded volume, the ion concentration in the vicinity of the charged surface cannot exceed a given value (determined by close packing). For that reason the Maxwell–Boltzmann distribution is modified introducing a Langmuir-type correction (as mentioned above). This is certainly an approximation whose validity could deserve further studies. In fact, it leads to predictions in disagreement with other approaches and simulations results. More specifi- cally, monotonic ionic profiles with a saturation value next to the charged surface are reported from such approximation whereas other recent approaches predict non-monotonic profiles at high salt concentrations. 17 This feature has also been corroborated by computer simulations. 17,27 In addition, other theoretical and simulation results suggest that ion size a Departamento de Fı´sica, Escuela Polite ´cnica Superior de Linares, Universidad de Jae ´n, 23700, Linares Jae ´n, Spain b Grupo de Fı´sica de Fluidos y Biocoloides, Departamento de Fı´sica Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain This journal is  c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 309–316 | 309 PAPER www.rsc.org/pccp | Physical Chemistry Chemical Physics