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Madelung Constants of Nanoparticles and Nanosurfaces

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Madelung Constants of Nanoparticles and Nanosurfaces J. Phys. Chem. C 2009, 113, 14793–14797 14793 Madelung Constants of Nanoparticles and Nanosurfaces A. D. Baker† and M. D. Baker*,‡ Department of Chemistry and Biochemistry, Queens College, City UniVersity of New York, Flushing, New York 11367, and Department of Chemistry, UniVersity of Guelph, Guelph, Ontario N1G 2W1, Canada ReceiVed: May 28, 2009; ReVised Manuscript ReceiVed: June 25, 2009 Specific ion Madelung Constants (MCs) were calculated for ionic nanostructures and nanosurfaces using Coulomb sums. The magnitude of these values was tracked through a succession of progressively larger structures having the same symmetry. A significantly faster convergence to limiting bulk values than obtained previously was achieved when the structures used in the convergence were constrained to be electrically neutral. Evaluation of specific ion MCs for all surface ions allows the construction of surface Madelung maps. Calculated MCs for MgO nanotubes correlate well with the minimum total energies from DFT methods. Introduction desired. Similarly Harrison4 and Tyagi1 reported that the Madelung constants (MCs) play an important role in under- evaluation of error functions can be computationally problematic standing phenomena related to the electrostatic potentials of and requires considerable effort to implement. Others have + crystals. The main focus of this paper is to investigate the degree reported that the method fails for slab nanosheet (2D h) 5 to which MCs for highly ionic materials can be used to geometries and simple nanostructures. understand and predict the properties of the ionic surface and When there is an interest in exploring the rate of convergence the stability of magnesium oxide nanotubes. The Madelung to a bulk value, an alternative approach that is often used is constant is defined as a single ion value: called “the method of expanding cubes” (EC) which takes various forms.6-8 Most recently, papers by Gaio and Silvestrelli 8 4 n (GS) and by Harrison have focused on improving EC MC(i) )-∑ c(i)c(j)/r(ij)(i * j) (1) convergence rates obtained by adopting spherical or cubic j)1 expansions around a central core, and applying a suitable Q/R charge correction. Earlier investigators using the EC approach employed partial charges for surface ions to deal with the charge where MC(i) is the Madelung constant for an individual ion issue.6,7 In the present work we develop a modified and (i), c(i) is the charge of this ion, the c(j)’s are charges on the considerably simplified version of the EC approach (vide infra) surrounding ions, r(ij) is their interionic distance, and n is the total number of ions in the particle. which requires no charge corrections. We also present results For an infinite array where all ions are equivalent (n ) ∞ in from extremely rapid computations of all single-ion MCs in eq 1), there is only one MC. It is the value reported in reference structures containing tens of thousands of ions, and also works and is used in lattice energy calculations. An infinite series nanostructures. We then use the surface-ion MCs calculated in Downloaded by UNIV OF GUELPH LIBRARY on August 13, 2009 based on eq 1 is only conditionally convergent, and evaluation this way to construct complete Madelung maps of surfaces, as Published on July 28, 2009 http://pubs.acs.org | doi: 10.1021/jp905015u has required specialized methods. This has fascinated many reported below. 1 physicists and mathematicians over the past century. Although The stability of an ion in a finite structure can be gauged by this problem has persisted for almost 100 years, there continues examining the magnitude of its MC relative to those of other to the present day significant interest in developing faster and ions. Larger positive values correlate with greater stability. A * more efficient methods (vide infra). In any real material (n corner ion will therefore have a smaller MC than a central ion ∞), rather than there being a single MC value, there will instead in any given cubic structure because of its smaller coordination be a range of MCs corresponding to ions in different environ- number. Surface ion MCs have in some cases been reported in ments. It is these MCs which are the prime focus in this paper. the literature9-12 with a view to assessing variations in the The most common approach for the evaluation of MCs was developed by Ewald2 and achieves fast convergence to bulk relative activities of particular sites. In this context there has 13,14 values by setting Gaussian charge distributions around each ion been a sustained interest in the MCs of ionic surfaces. For and using compensating Gaussians to handle charge issues. example, MCs have been used to rationalize the fact that step, Nevertheless, several reservations to the use of the Ewald edge and kink sites are generally more reactive than those method have surfaced. For example, Crandall and Buhler3 situated on the terraces of both bulk and nanosurfaces. A pointed out that the use of various error functions demands complete evaluation of all surface-ion MCs in nanostructures unwieldy computations, particularly when high precision is of different sizes and shapes is thus of interest. In this paper, we will report for the first time (to our knowledge) a complete * To whom correspondence should be addressed. E-mail: mbaker@ cataloging of surface-ion MCs and assess their significance. To uoguelph.ca. † City University of New York. date there has been little or no progress in determining the MCs ‡ University of Guelph. of ionic nanotubes. In this paper we briefly consider the MCs 10.1021/jp905015u CCC: $40.75 2009 American Chemical Society Published on Web 07/28/2009 14794 J. Phys. Chem. C, Vol. 113, No. 33, 2009 Baker and Baker TABLE 1: Madelung Constants as a Function of Particle Size ions per particlea this work NaCl this work CsCl GS NaClb GS CsClb Harrison NaClc Harrison CsClc 1000 1.7475 1.7611 9261 - 1.7505 1.7654 1.6650 1.7958 27 000 1.7476 1.7625 68 921 - 1.7483 1.7634 1.7826 1.7513 125 000 1.7476 1.7626 1 000 000 1.7476 1.7627 1 030 301 - 1.7477 1.7525 1.7628 1.7610 8 120 601 1.7476 1.7627 1.7457 1.7613 accepted bulk MC value 1.7476 1.7627 1.7476 1.7627 1.7476 1. 7627 a The number of ions for the odd cubes studied by GS and by Harrison were evaluated form the cubic side lengths, L ) second, where d is the cation-anion closest distance, given in Table 1 of ref 8. b Reference 8. c Calculated by GS on the basis of the Harrison method.4 for MgO nanotubes and compare the predicted stabilities using This weighted average steadily increases with size, reaching a MC with those determined by density functional theory (DFT) limiting value that is identical to the bulk (infinitely large) value. calculations. For each structure, an algorithm was developed to generate the ion x, y, z coordinates (and thus rij, see eq 1) of a suitable Computational Methods seed structure and its higher homologues. For example, the seed structure for NaCl is a cube having two ions along each edge. Individual ion MCs were determined by computing a sum of Higher homologues logically contain more ions along each edge all Coulombic interactions of the chosen ion with all other ions of progressively larger cubes with the constraint that they were in the structure. The usual assumptions of point charges with a uncharged. These can then be used to evaluate individual ion closest near-neighbor distance of unity were followed. Bulk MCs MCs (eq 1) and track the rapidity of convergence to a limiting for cubic crystals were calculated by computing the individual or bulk value. The computation of individual ion MCs was MC of the specific ions in progressively larger and larger cubes. determined by using nested loops to evaluate eq 1. The If central ions are chosen, one finds that as the particle grows determinations even for particles with 100 000 ions run in a in size the central ion MC rapidly approaches the average or few seconds on a PC. The Fortran 77 programs that generated “bulk” MC found in reference work for the crystal type being MCs used double precision arithmetic. studied. The situation for surface ions will be discussed below In the case of the MgO nanotubes, the geometries developed (see Results and Discussion). Previous workers using this by Bilabegovic´15 were used, and we adopt the same terminology odd approach considered a succession of cubes containing an to describe the structures. For example, 4 × 4 and 6 × 8 refer number of ions along each edge. However, such cubes are to nanotubes with 4 and 6 ions in the faces of the tubes with charged, which is contrary to one of the conditions required lengths of 4 and 8 ions, respectively. for convergence (see Results and Discussion). Thus to make - the method work efficiently, a charge correction was employed.4 8 Results and Discussion In our work, cubes with an eVen number of ions along each edge were used which obviates the need for a charge correction. Prior to a discussion of the MCs for nanotubes and surfaces, It is important to stress that the calculations in this paper were it is instructive to consider the “ideal” bulk rocksalt structures. As mentioned in the Introduction, there has been recent interest therefore implemented using full charges for all ions unlike 8 many other methods which employ partial charges. This method in this topic, showing that rapid convergence to bulk values could be effected using the EC method.
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