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. However, our work Downloaded by UNIV OF GUELPH LIBRARY on August 13, 2009 also results in faster convergence to bulk values than those

Published on July 28, 2009 http://pubs.acs.org | doi: 10.1021/jp905015u reported most recently (see Table 1). employs neutral rather than charged cubes or other polyhedra It is worthwhile stressing at this stage (and we will return to (see Computational Methods section) resulting in substantially this later) that specific ion MCs change in magnitude when faster convergence, as shown in Table 1. The MC values we larger and larger structures of the same symmetry are examined. report in Table 1 are those for the central ions in each structure; For example, in a series of sodium chloride cubic structures these are numerically equal to the overall or weighted average the central ion MCs steadily increase, approaching a value of MC as detailed in the Computational Methods section. The 1.74756460 which matches the NaCl bulk (infinitely large) smallest or seed cube for rocksalt structures consists of eight structure MC. The fact that central ion MCs approach the bulk ions, one at each corner. Each ion has the same environment, value for larger and larger structures (vide supra) allows for and thus the same MC, which we calculated as 1.45602993. fast computation of MCs on a computer by using an algorithm For larger cubes, these eight ions remain at the central interior, and their MCs progressively and rapidly increase to the accepted that computes the lattice sums. 5 Another connection between bulk and specific ion MCs bulk MC of NaCl. Convergence to an accuracy of 1 part in 10 naturally arises from the fact that in any finite crystal, there is is achieved with a just 10 ions along each edge, and accuracy a range of specific ion MCs (see Introduction). We will refer to to 12 decimal places is achieved with larger cubes. The convergence is also equally fast for CsCl as shown in Table 1. the weighted average of these as MCwa for the structure under survey. For CsCl, we used expanding neutral rhombohedra rather than cubes in order meet the criteria for convergence; namely the ) + + + MCwa MC(i)p(i)/n MC( j)p( j)/n MC(k)p(k)/n ... absence of charge, dipole, or quadrupole moments. Also, (2) because the inner eight ions in CsCl rhombohedra are in two different environments (meaning two MCs), it was necessary where p(i), p(j), ... are the number of ions in a particular to take an average of these to match the literature values. environment, MC(i), MC(j), etc. are the MCs for each ion of a The method was also applied to zincblende and accurately particular type, and n is the total number of ions in the particle. reproduced the documented MC (1.6381). Madelung Constants of Nanoparticles and Nanosurfaces J. Phys. Chem. C, Vol. 113, No. 33, 2009 14795

Figure 1. Variation of specific ion MCs with size for various generations of neutral cubic NaCl clusters; (a) ions closest to body center, (b) ions at face-centers, (c) ions at edge-centers, and (d) ions at corners. In (a) the value for the smallest neutral cubic cluster (8 ions, MC ) 1.45603) has been omitted so that the variation among larger clusters is more apparent.

The same procedure was used to obtain all specific ion MCs the surface Madelung maps of smaller and larger “ideal” MgO in the structures under survey. We focus on MgO in the particles, and so the surface of bulk MgO will also have a similar remainder of this paper. In its bulk form this ionic solid has the appearance. rocksalt structure. Although it is well known that nanostructures The lower stability areas constitute a significant fraction of often have a relaxed structure compared to bulk materials (and the maps. Understanding and exploiting the pattern of surface this will be discussed below), we first focus on the some MCs has promise in the design of patterned nanosurfaces. For individual ion MCs for the {100} surface and the body center example, MgO {100} has been used as a template for the growth ion of “ideal” (rocksalt) MgO nanocrystals. The individual ion of nanostructered metal assemblies.16 Free-electron metals bind MCs for (a) body center, (b) face center, (c) edge center, and to the surface of oxides through the Coulomb attraction between (d) corner ions are plotted as a function of particle size in Figure the surface ions and their screening charge density in the metal.17 1. This shows that the specific ion MCs progress toward limiting Thus, the initial deposition of quantum dots on the mapped values. The limiting surface values (Figure 1b-d) indeed agree surface should occur at sites with the smallest local MCs and Downloaded by UNIV OF GUELPH LIBRARY on August 13, 2009

Published on July 28, 2009 http://pubs.acs.org | doi: 10.1021/jp905015u with previously calculated values for surface ions in bulk then at the next lowest MC sites and so on. The predictions of materials.9-14 Note that the body center ion (a) is far less this work should be testable in principle by STM and AFM sensitive to particle size than those on the surface. The former imaging,16 although atomic-resolution mapping of quantum dot closely approaches the limiting value for a particle with 1000 arrays on ionic (template) nanosurfaces has so far not been ions whereas the surface ion values approach to the limit much reported. Nevertheless, anistotropy of this type has been more slowly. observed for the epitaxial deposition of Pt on a MgO {100} Each type of ion approaches a limiting (bulk) MC in a unique surface.18 In this case, maps of pressure in the interfacial layer fashion. For edge-center ions (Figure 1c), the MCs oscillate with were calculated in order to interpret the experimental data. The increasing particle size, while for corner- and face-center ions maps given (see particularly Figure 14, ref 18) bear a remarkable there are no oscillations. Furthermore, the MCs for corner ions resemblance to the MC map shown in Figure 3. (Figure 1d) gradually decrease (become less stable) with As mentioned above, variations in structure are expected for increasing particle size, while the values for face center ions nanostructures of different sizes. Of course, in the simplest case (Figure 1b) gradually increase. It is also noteworthy that the a cubic structure is possible only if the total number of ions MCs oscillate along an edge for any structure whereas no such present is the cube of an integer. However, even nanostructures oscillations are observed along a face diagonal, as shown in which do meet this criterion are not necessarily cubic. Deter- Figure 2. mining the most stable geometry of a particular nanostructure In Figure 3 we show a complete MC map of the 400 ions on is therefore important.15 the {100} surface of an 8000-ion particle containing 20 ions We conclude by focusing attention on MgO nanotubes. These along each edge. The MCs for each ion on the surface are color- have been recently synthesized19,20 and have been the subject coded. The values range from 1.3534 (corner) to 1.7161. The of intense scrutiny. The stability of small MgO nanotubes has details of this surface map are interesting. The surface exhibits recently been studied by Bilabegovic´15 using DFT methods. Our a central cross of more stable ions surrounded by areas of lower calculated values of the MCwa for 10 × N nanotubes are shown stability near the corners. This cross motif is also apparent on in Figure 4. It is noteworthy that there is a smooth progression 14796 J. Phys. Chem. C, Vol. 113, No. 33, 2009 Baker and Baker

Figure 4. Progression of MCwa for 10 × N nanotubes (see text).

TABLE 2: Total Energies (eV) and MCwa for N × 4 Nanotubes a b nanotube total energy (eV) MCwa 4 × 4 -464.6981 1.524971175 6 × 4 -465.1345 1.545662590 8 × 4 -465.2519 1.552265740 10 × 4 -465.2947 1.555036750 a Reference 15. b This work.

rocksalt structure. Similar behavior was observed for all the Figure 2. Variation of Surface MC along (a) an edge and (b) a face nanotubes studied here. We now consider the N × 4 nanotubes diagonal of a cubic neutral NaCl cluster having 18 ions on edge. The 15 × × × coordinate (0,0,0) is a corner ion. studied by Bilabegovic´ namely 4 4, 6 4, 8 4, and 10 × 4. The total energy and the MCwa (this work) for these tubes are collated in Table 2. There is a very good correlation between these values. Indeed, a plot of MC versus total energy is linear with a correlation coefficient R of 0.999 and a p value of 0.00095 where the p value in the analysis is the probability that the linear relationship can be rejected. In this case it is less than 1 in 10000.

Downloaded by UNIV OF GUELPH LIBRARY on August 13, 2009 Summary and Conclusions Published on July 28, 2009 http://pubs.acs.org | doi: 10.1021/jp905015u In this paper we have shown that bulk (infinite size) MCs are calculated with high accuracy by finding the limiting value of the central ion MCs of any structure as the size is increased, especially if only uncharged structures are considered. This method is substantially faster than other procedures recently reported. Using a similar approach, surface ion MCs were also computed for nano and bulk particles. For the first time we present a complete MC maps of MgO {100} surfaces which show an interesting anisotropy in common with maps of interfacial pressure for epitaxial growth on this surface. We further show that the weighted average MCs for MgO nanotubes correlate with the known minimum total energies calculated using DFT methods. Figure 3. Madelung map of {100} surface for an 8000-ion particle. MCs are color coded as follows. Purple, 1.3534; red, 1.5525-1.5880; Acknowledgment. - - - Both authors thank their parent institutes green, 1.5923 1.6036; yellow, 1.6165 1.6604; orange, 1.6722 1.6800; for the provision of sabbatical leaves during the 2008-2009 gray, above 1.6800. academic year. A.D.B. acknowledges the receipt of PSC-CUNY Grant No. 69597-00 38 which supported his research efforts. from the 10 × 2 tube (MCwa ) 1.497208) to the limiting value M.D.B. gratefully acknowledges funding from The Natural (MCwa ) 1.601524). The latter value never reaches the literature Sciences and Engineering Research Council of Canada. This value for bulk NaCl no matter how long the tube is. The bulk paper is dedicated to the memory of our late parents, Arthur value (1.74756460) is observed only in the case of the cubic and Catherine Baker. Madelung Constants of Nanoparticles and Nanosurfaces J. Phys. Chem. C, Vol. 113, No. 33, 2009 14797

References and Notes (13) Barbier, A.; Stierle, A.; Finocchi, F.; Jupille, J. J. Phys. Condens. Matter 2008, 20, 184014. (1) Tyagi, S. Prog. Theor. Phys. 2005, 114, 517. (14) Sauer, J.; Ugliengo, P.; Garrone, E.; Saunders, V. R. Chem. ReV. (2) Ewald, P. P. Ann. Phys. 1921, 64, 253. 1994, 94, 2095. (3) Crandall, R. E.; Buhler, J P. J. Phys. A: Math. Gen. 1987, 20, 5497. (15) Bilabegovic´, G. Phys. Rev B. 2004, 70, 045407. (4) Harrison, W. A. Phys. ReV.B2006, 73, 212103. (16) Walter, M.; Frondelius, P.; Honkala, H.; Ha¨kkinen, H. Phys. ReV. (5) Lamba, S. Phys. Stat. Sol. B 2004, 241, 3022. Lett. 2007, 99, 96102. (6) Borwein, D.; Borwein, J.; Taylor, K. J. Math. Phys. 1985, 16, 1457. (17) Scho¨nberger, U.; Andersen, O. K.; Methfessel, M. Acta Metall. (7) Grosso, R. P., Jr.; Fermann, J. T.; Vining, W. J. J. Chem. Educ. Mater. 1992, 40, S1. 2001, 78, 1198. (18) Olander, J.; Lazzari, R.; Jupille, J.; Mandili, B.; Goniakowski, J. (8) Gaio, M.; Silvestrelli, L. Phys. ReV.B2009, 79, 012102. Phys. ReV.B2007, 76, 075409. (9) Andre´s, J.; Beltra´n, A.; Moliner, V.; Longo, E. J. Mater. Sci. 1995, (19) Yang, Q.; Sha, J.; Wang, L.; Wang, Y.; Ma, X.; Wang, J.; Yang, 30, 4852. D. Nanotechnology 2004, 1004, 15. (10) Stefanovich, E. V.; Truong, T. N. J. Chem. Phys. 1995, 102, 5071. (20) Zhan, J.; Bando, Y.; Hu, J.; Goldberg, D. Inorg. Chem. 2004, 2462, (11) Pacchioni, G.; Clotet, A.; Ricart, J. M. Surf. Sci. 1994, 315, 337. 43. (12) Bocquet, F.; Nony, L.; Loppacher, C.; Glatzel, T. Phys. ReV.B 2008, 78, 35410. JP905015U Downloaded by UNIV OF GUELPH LIBRARY on August 13, 2009 Published on July 28, 2009 http://pubs.acs.org | doi: 10.1021/jp905015u