Molecular Physics, Vol. 104, No. 9, 10 May 2006, 1475–1486
Monte Carlo simulations of primitive models for ionic systems using the Wolf method CARLOS AVENDAN˜Oy and ALEJANDRO GIL-VILLEGAS*z yFacultad de Quı´ mica, Universidad de Guanajuato, Noria Alta s/n, Guanajuato, Gto., 36050, Me´xico zInstituto de Fı´ sica, Universidad de Guanajuato, Lomas del Bosque 103, Colonia Lomas del Campestre, Leo´n 37150, Me´xico (Received 15 September 2005; in final form 16 December 2005) Thermodynamic and structural properties of primitive models for electrolyte solutions and molten salts were studied using NVT and NPT Monte Carlo simulations. The Coulombic interactions were simulated using the Wolf method [D. Wolf, Phys. Rev. Lett. 68, 3315 (1992); D. Wolf, P. Keblinnski, S. R. Phillpot, and J. Eggebrecht, J. Chem. Phys. 110, 8254 (1999)]. Results for 1 : 1 and 2 : 1 charge ratio electroneutral systems are presented, using the restricted and non-restricted primitive models, as well as a soft PM pair potential for a monovalent salt [J.-P. Hansen and I. R. McDonald, Phys. Rev. A 11, 2111 (1975)] that has also been used to model 2 : 12 and 1 : 20 asymmetric colloidal systems, with size ratios 1 : 10 and 2 : 15, respectively [B. Hribar, Y. V. Kalyuzhnyi, and V. Vlachy, Molec. Phys. 87, 1317 (1996)]. We present the predictions obtained for these systems using the Wolf method. Our results are in very good agreement with simulation data obtained with the Ewald sum method as well as with integral-equation theories results. We discuss the relevance of the Wolf method in the context of variable-ranged potentials in molecular thermodynamic theories for complex fluids.
1. Introduction Electrolyte solutions are present in industrial process and living organisms. Because of their important applications, the interest in molecular-based theories to understand and quantify electrolyte solutions properties, using statistical mechanics and molecular simulations, has been one of the main and growing areas of scientific development over the past decades. The restricted and non-restricted primitive models (RPM and PM, respectively) for electrolyte solutions, that is, ions regarded as rigid spheres in a structureless dielectric solvent, have been fundamental systems in order to understand a wide range of properties of ionic fluids. There exists a vast amount of analytical and numerical information for these systems [1–4], that has been the first stage in the study of phase properties of ionic systems [5–14] and in the development of more complex models for electrolyte solutions [15, 16]. Primitive models are fundamental inputs for theories and models where a more detailed description of the complex phenomena arising in electrolytes, colloids, molten salts, etc., are taken into account [17–27]. Specifically, our understanding of thermodynamic *Corresponding author. Email:
[email protected]
properties of asphaltenes based on methods derived for associating fluids [28–31] and primitive models has been of interest recently [32, 33]. A fundamental aspect in obtaining robust theories and equations of state for complex fluids based on primitive models is the generation of information through molecular simulations. One of the technical problems that arises in the computer simulation of electrolyte solutions is the correct modelling of the longrange Coulombic interactions. Standard methods such as the Ewald sum (ES) and Onsager reaction-field (RF) methods [34, 35] are the most common procedures to deal with electrostatic interactions, although over the years other alternative methods have been developed that improve in different ways the ES and RF methods [36–42]. Whereas the ES method is exact, a well-known shortcoming of this method is that it is expensive in computing time, and optimized algorithms must be used in order to apply this method to more complex systems [41, 42]. On the other hand, the RF method requires one to know the dielectric constant of the surrounding media, information that is not known a priori and that requires to be calculated in a self-consistent way [43, 44]. Wolf and co-workers [45, 46] have developed an alternative route to deal with the long-range nature of electrostatic interactions. This method was originally
Molecular Physics ISSN 0026–8976 print/ISSN 1362–3028 online # 2006 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/00268970600551155
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based on the behaviour of neighbour ions surrounding another central ion, for an ionic crystal. When the energy of the ionic crystal is obtained by adding the pair Coulombic interactions, it gives the result that the sum is conditionally convergent. Wolf and co-workers [45, 46] have shown that by performing this sum in a spherical way, if the potential energy due to the charge inside the sphere is neutralized by a charge of opposite sign located at the surface of the sphere, then the convergence of the sum can be accelerated. The physical basis of this approach is the screening of the ion–ion pair interaction due to the presence of the other ions. This method has the main advantage of requiring a simulation time of the same order of a standard periodic boundary condition simulation for a short-ranged potential. In this article, we present the application of the Wolf method to model primitive models of electrolyte solutions with different charge and size ratios. These results reproduce very accurately simulated states using the Ewald sum method. For several of the systems studied here we present comparison with integralequation theory results, like the Hypernetted Chain equation theory and the Mean Spherical Approximation. Although there are very efficient methods to speed up Ewald sums [42, 42], it is useful to show the performance of the Wolf method studying basic systems as the primitive models of electrolyte solutions, as an alternative route to model more complex systems where computational simulation methods are fundamental tools, such as charged systems in molecular biology [47, 48]. At the same time, we want to point out the relevance of this method in theoretical approaches based on perturbation theories to describe phase diagrams of complex fluids. 2. Wolf method The total energy Etot for a system of M ions is given by Etot ¼
M X M 1X qi qj ; 2 i¼1 j6¼i¼1 rij
ð1Þ
where qi is the charge of ion i and rij is the distance between ions i and j. Equation (1) is a conditionally convergent sum when M ! 1, i.e. the result of the sum depends on the order followed to add pair interactions. Ewald [34, 35, 49] solved this problem considering a system of N ions located in a cell, which is replicated periodically in all directions. Then, in order to screen the Coulombic long-range interactions, a continuous charge density distribution is added to the original system in such a way that the sign of the charge of this distribution is opposite to the sign of each one of the original charges. The corresponding energy of this new composite
system is now a convergent sum of pair interactions that can be evaluated directly in real space. To recover the original system, the effect of the continuous charge distribution is cancelled in the reciprocal space of the periodic arrangement of cells. The energy associated with this charge density in k-space is expressed as a Fourier sum, which is also absolutely convergent. By a proper selection of a screening parameter related to the density charge distribution and the number of k vectors required for the evaluation of the Fourier sum, a computational scheme can be implemented to simulate the properties of charged systems in an optimal time that, however, is considerably longer than the time required to perform a simulation for a system interacting with a short-ranged pair potential [35, 49]. According to the Wolf method [45, 46, 50], the conditional convergence of the pairwise r1 summation can be understood in the following way. Considering the radial symmetry of the Coulombic interaction, the sum of charge pair interactions can be done using spherical shells of radius Rc . The absence of electroneutrality in these shells contributes to the non-convergence of the sum in equation (1). However, the evidence that the effective pair potential between ions is short ranged when we consider an electrolyte composed of many ions, suggests performing the calculation of the energy of the system using an effective pair potential under the restriction of the electroneutrality of the spherical shells. Based on these facts, the energy of the system is obtained using a spherically truncated sum, corrected by a charge-neutralizing potential, which is assumed to be localized exactly at the sphere surface. In this way, the net charge of the complete system (charges within Rc plus charges over the spherical surface of radius Rc ) is null. In other words, this method reduces the problem to the calculation of the energy of an electroneutral finite system. Following this approach, the actual value of the energy of the system, EðRc Þ, is related to the spherically truncated sum, Etot ðRc Þ, and the energy of the neutralizing charge located over the sphere, Eneutr ðRc Þ, by the following expression EðRc Þ Etot ðRc Þ Eneutr ðRc Þ;
ð2Þ
where Etot ¼
N X 1X qi qj 2 i¼1 j6¼i¼1 rij
ð3Þ
ðrij
