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 usingCoulomb sums. The magnitude of these values was tracked through a succession of progressively largerstructures having the same symmetry. A significantly faster convergence to limiting bulk values than obtainedpreviously was achieved when the structures used in the convergence were constrained to be electricallyneutral. Evaluation of specific ion MCs for all surface ions allows the construction of surface Madelungmaps. Calculated MCs for MgO nanotubes correlate well with the minimum total energies from DFT methods.

Introduction

Madelung constants (MCs) play an important role in under-standing phenomena related to the electrostatic potentials ofcrystals. The main focus of this paper is to investigate the degreeto which MCs for highly ionic materials can be used tounderstand and predict the properties of the ionic surface andthe stability of magnesium oxide nanotubes. The Madelungconstant is defined as a single ion value:

where MC(i) is the Madelung constant for an individual ion(i), c(i) is the charge of this ion, the c(j)’s are charges on thesurrounding ions, r(ij) is their interionic distance, and n is thetotal number of ions in the particle.

For an infinite array where all ions are equivalent (n ) ∞ ineq 1), there is only one MC. It is the value reported in referenceworks and is used in lattice energy calculations. An infinite seriesbased on eq 1 is only conditionally convergent, and evaluationhas required specialized methods. This has fascinated manyphysicists and mathematicians over the past century.1 Althoughthis problem has persisted for almost 100 years, there continuesto the present day significant interest in developing faster andmore efficient methods (vide infra). In any real material (n *∞), rather than there being a single MC value, there will insteadbe a range of MCs corresponding to ions in different environ-ments. It is these MCs which are the prime focus in this paper.

The most common approach for the evaluation of MCs wasdeveloped by Ewald2 and achieves fast convergence to bulkvalues by setting Gaussian charge distributions around each ionand using compensating Gaussians to handle charge issues.Nevertheless, several reservations to the use of the Ewaldmethod have surfaced. For example, Crandall and Buhler3

pointed out that the use of various error functions demandsunwieldy computations, particularly when high precision is

desired. Similarly Harrison4 and Tyagi1 reported that theevaluation of error functions can be computationally problematicand requires considerable effort to implement. Others havereported that the method fails for slab nanosheet (2D + h)geometries and simple nanostructures.5

When there is an interest in exploring the rate of convergenceto a bulk value, an alternative approach that is often used iscalled “the method of expanding cubes” (EC) which takesvarious forms.6-8 Most recently, papers by Gaio and Silvestrelli(GS)8 and by Harrison4 have focused on improving ECconvergence rates obtained by adopting spherical or cubicexpansions around a central core, and applying a suitable Q/Rcharge correction. Earlier investigators using the EC approachemployed partial charges for surface ions to deal with the chargeissue.6,7 In the present work we develop a modified andconsiderably simplified version of the EC approach (vide infra)which requires no charge corrections. We also present resultsfrom extremely rapid computations of all single-ion MCs instructures containing tens of thousands of ions, and alsonanostructures. We then use the surface-ion MCs calculated inthis way to construct complete Madelung maps of surfaces, asreported below.

The stability of an ion in a finite structure can be gauged byexamining the magnitude of its MC relative to those of otherions. Larger positive values correlate with greater stability. Acorner ion will therefore have a smaller MC than a central ionin any given cubic structure because of its smaller coordinationnumber. Surface ion MCs have in some cases been reported inthe literature9-12 with a view to assessing variations in therelative activities of particular sites. In this context there hasbeen a sustained interest in the MCs of ionic surfaces.13,14 Forexample, MCs have been used to rationalize the fact that step,edge and kink sites are generally more reactive than thosesituated on the terraces of both bulk and nanosurfaces. Acomplete evaluation of all surface-ion MCs in nanostructuresof different sizes and shapes is thus of interest. In this paper,we will report for the first time (to our knowledge) a completecataloging of surface-ion MCs and assess their significance. Todate there has been little or no progress in determining the MCsof ionic nanotubes. In this paper we briefly consider the MCs

* To whom correspondence should be addressed. E-mail: mbaker@uoguelph.ca.

† City University of New York.‡ University of Guelph.

MC(i) ) -∑j)1

n

c(i)c(j)/r(ij) (i * j) (1)

J. Phys. Chem. C 2009, 113, 14793–14797 14793

10.1021/jp905015u CCC: $40.75 2009 American Chemical SocietyPublished on Web 07/28/2009

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for MgO nanotubes and compare the predicted stabilities usingMC with those determined by density functional theory (DFT)calculations.

Computational Methods

Individual ion MCs were determined by computing a sum ofall Coulombic interactions of the chosen ion with all other ionsin the structure. The usual assumptions of point charges with aclosest near-neighbor distance of unity were followed. Bulk MCsfor cubic crystals were calculated by computing the individualMC of the specific ions in progressively larger and larger cubes.If central ions are chosen, one finds that as the particle growsin size the central ion MC rapidly approaches the average or“bulk” MC found in reference work for the crystal type beingstudied. The situation for surface ions will be discussed below(see Results and Discussion). Previous workers using thisapproach considered a succession of cubes containing an oddnumber of ions along each edge. However, such cubes arecharged, which is contrary to one of the conditions requiredfor convergence (see Results and Discussion). Thus to makethe method work efficiently, a charge correction was employed.4-8

In our work, cubes with an eVen number of ions along eachedge were used which obviates the need for a charge correction.It is important to stress that the calculations in this paper weretherefore implemented using full charges for all ions unlikemany other methods which employ partial charges. This methodalso results in faster convergence to bulk values than thosereported most recently (see Table 1).

It is worthwhile stressing at this stage (and we will return tothis later) that specific ion MCs change in magnitude whenlarger and larger structures of the same symmetry are examined.For example, in a series of sodium chloride cubic structuresthe central ion MCs steadily increase, approaching a value of1.74756460 which matches the NaCl bulk (infinitely large)structure MC. The fact that central ion MCs approach the bulkvalue for larger and larger structures (vide supra) allows forfast computation of MCs on a computer by using an algorithmthat computes the lattice sums.

Another connection between bulk and specific ion MCsnaturally arises from the fact that in any finite crystal, there isa range of specific ion MCs (see Introduction). We will refer tothe weighted average of these as MCwa for the structure undersurvey.

MCwa ) MC(i)p(i)/n + MC( j)p( j)/n + MC(k)p(k)/n + ...

(2)

where p(i), p(j), ... are the number of ions in a particularenvironment, MC(i), MC(j), etc. are the MCs for each ion of aparticular type, and n is the total number of ions in the particle.

This weighted average steadily increases with size, reaching alimiting value that is identical to the bulk (infinitely large) value.

For each structure, an algorithm was developed to generatethe ion x, y, z coordinates (and thus rij, see eq 1) of a suitableseed structure and its higher homologues. For example, the seedstructure for NaCl is a cube having two ions along each edge.Higher homologues logically contain more ions along each edgeof progressively larger cubes with the constraint that they wereuncharged. These can then be used to evaluate individual ionMCs (eq 1) and track the rapidity of convergence to a limitingor bulk value. The computation of individual ion MCs wasdetermined by using nested loops to evaluate eq 1. Thedeterminations even for particles with 100 000 ions run in afew seconds on a PC. The Fortran 77 programs that generatedMCs used double precision arithmetic.

In the case of the MgO nanotubes, the geometries developedby Bilabegovic15 were used, and we adopt the same terminologyto describe the structures. For example, 4 × 4 and 6 × 8 referto nanotubes with 4 and 6 ions in the faces of the tubes withlengths of 4 and 8 ions, respectively.

Results and Discussion

Prior to a discussion of the MCs for nanotubes and surfaces,it is instructive to consider the “ideal” bulk rocksalt structures.As mentioned in the Introduction, there has been recent interestin this topic,8 showing that rapid convergence to bulk valuescould be effected using the EC method. However, our workemploys neutral rather than charged cubes or other polyhedra(see Computational Methods section) resulting in substantiallyfaster convergence, as shown in Table 1. The MC values wereport in Table 1 are those for the central ions in each structure;these are numerically equal to the overall or weighted averageMC as detailed in the Computational Methods section. Thesmallest or seed cube for rocksalt structures consists of eightions, one at each corner. Each ion has the same environment,and thus the same MC, which we calculated as 1.45602993.For larger cubes, these eight ions remain at the central interior,and their MCs progressively and rapidly increase to the acceptedbulk MC of NaCl. Convergence to an accuracy of 1 part in 105

is achieved with a just 10 ions along each edge, and accuracyto 12 decimal places is achieved with larger cubes. Theconvergence is also equally fast for CsCl as shown in Table 1.For CsCl, we used expanding neutral rhombohedra rather thancubes in order meet the criteria for convergence; namely theabsence of charge, dipole, or quadrupole moments. Also,because the inner eight ions in CsCl rhombohedra are in twodifferent environments (meaning two MCs), it was necessaryto take an average of these to match the literature values.

The method was also applied to zincblende and accuratelyreproduced the documented MC (1.6381).

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 1.76119261 - 1.7505 1.7654 1.6650 1.795827 000 1.7476 1.762568 921 - 1.7483 1.7634 1.7826 1.7513125 000 1.7476 1.76261 000 000 1.7476 1.76271 030 301 - 1.7477 1.7525 1.7628 1.76108 120 601 1.7476 1.7627 1.7457 1.7613accepted 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 isthe 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

14794 J. Phys. Chem. C, Vol. 113, No. 33, 2009 Baker and Baker

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The same procedure was used to obtain all specific ion MCsin the structures under survey. We focus on MgO in theremainder of this paper. In its bulk form this ionic solid has therocksalt structure. Although it is well known that nanostructuresoften have a relaxed structure compared to bulk materials (andthis will be discussed below), we first focus on the someindividual ion MCs for the {100} surface and the body centerion of “ideal” (rocksalt) MgO nanocrystals. The individual ionMCs for (a) body center, (b) face center, (c) edge center, and(d) corner ions are plotted as a function of particle size in Figure1. This shows that the specific ion MCs progress toward limitingvalues. The limiting surface values (Figure 1b-d) indeed agreewith previously calculated values for surface ions in bulkmaterials.9-14 Note that the body center ion (a) is far lesssensitive to particle size than those on the surface. The formerclosely approaches the limiting value for a particle with 1000ions whereas the surface ion values approach to the limit muchmore slowly.

Each type of ion approaches a limiting (bulk) MC in a uniquefashion. For edge-center ions (Figure 1c), the MCs oscillate withincreasing particle size, while for corner- and face-center ionsthere are no oscillations. Furthermore, the MCs for corner ions(Figure 1d) gradually decrease (become less stable) withincreasing particle size, while the values for face center ions(Figure 1b) gradually increase. It is also noteworthy that theMCs oscillate along an edge for any structure whereas no suchoscillations are observed along a face diagonal, as shown inFigure 2.

In Figure 3 we show a complete MC map of the 400 ions onthe {100} surface of an 8000-ion particle containing 20 ionsalong each edge. The MCs for each ion on the surface are color-coded. The values range from 1.3534 (corner) to 1.7161. Thedetails of this surface map are interesting. The surface exhibitsa central cross of more stable ions surrounded by areas of lowerstability near the corners. This cross motif is also apparent on

the surface Madelung maps of smaller and larger “ideal” MgOparticles, and so the surface of bulk MgO will also have a similarappearance.

The lower stability areas constitute a significant fraction ofthe maps. Understanding and exploiting the pattern of surfaceMCs has promise in the design of patterned nanosurfaces. Forexample, MgO {100} has been used as a template for the growthof nanostructered metal assemblies.16 Free-electron metals bindto the surface of oxides through the Coulomb attraction betweenthe surface ions and their screening charge density in the metal.17

Thus, the initial deposition of quantum dots on the mappedsurface should occur at sites with the smallest local MCs andthen at the next lowest MC sites and so on. The predictions ofthis work should be testable in principle by STM and AFMimaging,16 although atomic-resolution mapping of quantum dotarrays on ionic (template) nanosurfaces has so far not beenreported. Nevertheless, anistotropy of this type has beenobserved for the epitaxial deposition of Pt on a MgO {100}surface.18 In this case, maps of pressure in the interfacial layerwere calculated in order to interpret the experimental data. Themaps given (see particularly Figure 14, ref 18) bear a remarkableresemblance to the MC map shown in Figure 3.

As mentioned above, variations in structure are expected fornanostructures of different sizes. Of course, in the simplest casea cubic structure is possible only if the total number of ionspresent is the cube of an integer. However, even nanostructureswhich do meet this criterion are not necessarily cubic. Deter-mining the most stable geometry of a particular nanostructureis therefore important.15

We conclude by focusing attention on MgO nanotubes. Thesehave been recently synthesized19,20 and have been the subjectof intense scrutiny. The stability of small MgO nanotubes hasrecently been studied by Bilabegovic15 using DFT methods. Ourcalculated values of the MCwa for 10 × N nanotubes are shownin Figure 4. It is noteworthy that there is a smooth progression

Figure 1. Variation of specific ion MCs with size for various generations of neutral cubic NaCl clusters; (a) ions closest to body center, (b) ionsat 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) hasbeen omitted so that the variation among larger clusters is more apparent.

Madelung Constants of Nanoparticles and Nanosurfaces J. Phys. Chem. C, Vol. 113, No. 33, 2009 14795

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from the 10 × 2 tube (MCwa ) 1.497208) to the limiting value(MCwa ) 1.601524). The latter value never reaches the literaturevalue for bulk NaCl no matter how long the tube is. The bulkvalue (1.74756460) is observed only in the case of the cubic

rocksalt structure. Similar behavior was observed for all thenanotubes studied here. We now consider the N × 4 nanotubesstudied by Bilabegovic15 namely 4 × 4, 6 × 4, 8 × 4, and 10× 4. The total energy and the MCwa (this work) for these tubesare 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 correlationcoefficient R of 0.999 and a p value of 0.00095 where the pvalue in the analysis is the probability that the linear relationshipcan be rejected. In this case it is less than 1 in 10000.

Summary and Conclusions

In this paper we have shown that bulk (infinite size) MCsare calculated with high accuracy by finding the limiting valueof the central ion MCs of any structure as the size is increased,especially if only uncharged structures are considered. Thismethod is substantially faster than other procedures recentlyreported. Using a similar approach, surface ion MCs were alsocomputed for nano and bulk particles. For the first time wepresent a complete MC maps of MgO {100} surfaces whichshow an interesting anisotropy in common with maps ofinterfacial pressure for epitaxial growth on this surface. Wefurther show that the weighted average MCs for MgO nanotubescorrelate with the known minimum total energies calculatedusing DFT methods.

Acknowledgment. Both authors thank their parent institutesfor the provision of sabbatical leaves during the 2008-2009academic year. A.D.B. acknowledges the receipt of PSC-CUNYGrant No. 69597-00 38 which supported his research efforts.M.D.B. gratefully acknowledges funding from The NaturalSciences and Engineering Research Council of Canada. Thispaper is dedicated to the memory of our late parents, Arthurand Catherine Baker.

Figure 2. Variation of Surface MC along (a) an edge and (b) a facediagonal of a cubic neutral NaCl cluster having 18 ions on edge. Thecoordinate (0,0,0) is a corner ion.

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;green, 1.5923-1.6036; yellow, 1.6165-1.6604; orange, 1.6722-1.6800;gray, above 1.6800.

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

TABLE 2: Total Energies (eV) and MCwa for N × 4Nanotubes

nanotube total energy (eV)a MCwab

4 × 4 -464.6981 1.5249711756 × 4 -465.1345 1.5456625908 × 4 -465.2519 1.55226574010 × 4 -465.2947 1.555036750

a Reference 15. b This work.

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References and Notes

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2001, 78, 1198.(8) Gaio, M.; Silvestrelli, L. Phys. ReV. B 2009, 79, 012102.(9) Andres, J.; Beltran, A.; Moliner, V.; Longo, E. J. Mater. Sci. 1995,

30, 4852.(10) Stefanovich, E. V.; Truong, T. N. J. Chem. Phys. 1995, 102, 5071.(11) Pacchioni, G.; Clotet, A.; Ricart, J. M. Surf. Sci. 1994, 315, 337.(12) Bocquet, F.; Nony, L.; Loppacher, C.; Glatzel, T. Phys. ReV. B

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(13) Barbier, A.; Stierle, A.; Finocchi, F.; Jupille, J. J. Phys. Condens.Matter 2008, 20, 184014.

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(15) Bilabegovic, G. Phys. Rev B. 2004, 70, 045407.(16) Walter, M.; Frondelius, P.; Honkala, H.; Hakkinen, H. Phys. ReV.

Lett. 2007, 99, 96102.(17) Schonberger, U.; Andersen, O. K.; Methfessel, M. Acta Metall.

Mater. 1992, 40, S1.(18) Olander, J.; Lazzari, R.; Jupille, J.; Mandili, B.; Goniakowski, J.

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Madelung Constants of Nanoparticles and Nanosurfaces J. Phys. Chem. C, Vol. 113, No. 33, 2009 14797

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