J. Phys. Chem. C 2009, 113, 14793–14797
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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 Madelung constants (MCs) play an important role in understanding phenomena related to the electrostatic potentials of crystals. The main focus of this paper is to investigate the degree to which MCs for highly ionic materials can be used to understand and predict the properties of the ionic surface and the stability of magnesium oxide nanotubes. The Madelung constant is defined as a single ion value: n
MC(i) ) -
∑ c(i)c(j)/r(ij)
(i * j)
(1)
j)1
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 the surrounding ions, r(ij) is their interionic distance, and n is the total number of ions in the particle. For an infinite array where all ions are equivalent (n ) ∞ in eq 1), there is only one MC. It is the value reported in reference works and is used in lattice energy calculations. An infinite series based on eq 1 is only conditionally convergent, and evaluation has required specialized methods. This has fascinated many physicists and mathematicians over the past century.1 Although this problem has persisted for almost 100 years, there continues to the present day significant interest in developing faster and more efficient methods (vide infra). In any real material (n * ∞), rather than there being a single MC value, there will instead be a range of MCs corresponding to ions in different environments. It is these MCs which are the prime focus in this paper. The most common approach for the evaluation of MCs was developed by Ewald2 and achieves fast convergence to bulk values by setting Gaussian charge distributions around each ion and using compensating Gaussians to handle charge issues. Nevertheless, several reservations to the use of the Ewald method have surfaced. For example, Crandall and Buhler3 pointed out that the use of various error functions demands unwieldy computations, particularly when high precision is * To whom correspondence should be addressed. E-mail: mbaker@ uoguelph.ca. † City University of New York. ‡ University of Guelph.
desired. Similarly Harrison4 and Tyagi1 reported that the evaluation of error functions can be computationally problematic and requires considerable effort to implement. Others have reported that the method fails for slab nanosheet (2D + h) geometries and simple nanostructures.5 When there is an interest in exploring the rate of convergence to a bulk value, an alternative approach that is often used is called “the method of expanding cubes” (EC) which takes various forms.6-8 Most recently, papers by Gaio and Silvestrelli (GS)8 and by Harrison4 have focused on improving EC convergence rates obtained by adopting spherical or cubic 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 issue.6,7 In the present work we develop a modified and considerably simplified version of the EC approach (vide infra) which requires no charge corrections. We also present results from extremely rapid computations of all single-ion MCs in structures containing tens of thousands of ions, and also nanostructures. We then use the surface-ion MCs calculated in this way to construct complete Madelung maps of surfaces, as reported below. The stability of an ion in a finite structure can be gauged by examining the magnitude of its MC relative to those of other ions. Larger positive values correlate with greater stability. A corner ion will therefore have a smaller MC than a central ion in any given cubic structure because of its smaller coordination number. Surface ion MCs have in some cases been reported in the literature9-12 with a view to assessing variations in the relative activities of particular sites. In this context there has been a sustained interest in the MCs of ionic surfaces.13,14 For example, MCs have been used to rationalize the fact that step, edge and kink sites are generally more reactive than those situated on the terraces of both bulk and nanosurfaces. A complete evaluation of all surface-ion MCs in nanostructures of different sizes and shapes is thus of interest. In this paper, we will report for the first time (to our knowledge) a complete cataloging of surface-ion MCs and assess their significance. To date there has been little or no progress in determining the MCs 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
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TABLE 1: Madelung Constants as a Function of Particle Size ions per particlea
this work NaCl
this work CsCl
1000 9261 27 000 68 921 125 000 1 000 000 1 030 301 8 120 601 accepted bulk MC value
1.7475 1.7476 1.7476 1.7476 -
1.7611
1.7476
GS NaClb
GS CsClb
Harrison NaClc
Harrison CsClc
1.7505
1.7654
1.6650
1.7958
1.7483
1.7634
1.7826
1.7513
1.7477 1.7476 1.7476
1.7525 1.7627 1.7627
1.7628 1.7457 1.7476
1.7610 1.7613 1. 7627
1.7625 1.7626 1.7627 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 MC with those determined by density functional theory (DFT) calculations. Computational Methods Individual ion MCs were determined by computing a sum of all Coulombic interactions of the chosen ion with all other ions in the structure. The usual assumptions of point charges with a closest near-neighbor distance of unity were followed. Bulk MCs for cubic crystals were calculated by computing the individual MC of the specific ions in progressively larger and larger cubes. If central ions are chosen, one finds that as the particle grows in size the central ion MC rapidly approaches the average or “bulk” MC found in reference work for the crystal type being studied. The situation for surface ions will be discussed below (see Results and Discussion). Previous workers using this approach considered a succession of cubes containing an odd number of ions along each edge. However, such cubes are charged, which is contrary to one of the conditions required for convergence (see Results and Discussion). Thus to make the method work efficiently, a charge correction was employed.4-8 In our work, cubes with an eVen number of ions along each edge were used which obviates the need for a charge correction. It is important to stress that the calculations in this paper were therefore implemented using full charges for all ions unlike many other methods which employ partial charges. This method also results in faster convergence to bulk values than those reported most recently (see Table 1). It is worthwhile stressing at this stage (and we will return to this later) that specific ion MCs change in magnitude when larger and larger structures of the same symmetry are examined. For example, in a series of sodium chloride cubic structures the central ion MCs steadily increase, approaching a value of 1.74756460 which matches the NaCl bulk (infinitely large) structure MC. The fact that central ion MCs approach the bulk value for larger and larger structures (vide supra) allows for fast computation of MCs on a computer by using an algorithm that computes the lattice sums. Another connection between bulk and specific ion MCs naturally arises from the fact that in any finite crystal, there is a range of specific ion MCs (see Introduction). We will refer to the weighted average of these as MCwa for the structure under survey. 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 particular environment, MC(i), MC(j), etc. are the MCs for each ion of a particular type, and n is the total number of ions in the particle.
This weighted average steadily increases with size, reaching a limiting value that is identical to the bulk (infinitely large) value. For each structure, an algorithm was developed to generate the ion x, y, z coordinates (and thus rij, see eq 1) of a suitable seed structure and its higher homologues. For example, the seed structure for NaCl is a cube having two ions along each edge. Higher homologues logically contain more ions along each edge of progressively larger cubes with the constraint that they were uncharged. These can then be used to evaluate individual ion MCs (eq 1) and track the rapidity of convergence to a limiting or bulk value. The computation of individual ion MCs was determined by using nested loops to evaluate eq 1. The determinations even for particles with 100 000 ions run in a few seconds on a PC. The Fortran 77 programs that generated MCs used double precision arithmetic. In the case of the MgO nanotubes, the geometries developed by Bilabegovic´15 were used, and we adopt the same terminology to describe the structures. For example, 4 × 4 and 6 × 8 refer to nanotubes with 4 and 6 ions in the faces of the tubes with lengths 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 interest in this topic,8 showing that rapid convergence to bulk values could be effected using the EC method. However, our work employs neutral rather than charged cubes or other polyhedra (see Computational Methods section) resulting in substantially faster convergence, as shown in Table 1. The MC values we report in Table 1 are those for the central ions in each structure; these are numerically equal to the overall or weighted average MC as detailed in the Computational Methods section. The smallest or seed cube for rocksalt structures consists of eight ions, 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 accepted bulk MC of NaCl. Convergence to an accuracy of 1 part in 105 is achieved with a just 10 ions along each edge, and accuracy to 12 decimal places is achieved with larger cubes. The convergence is also equally fast for CsCl as shown in Table 1. For CsCl, we used expanding neutral rhombohedra rather than cubes in order meet the criteria for convergence; namely the absence of charge, dipole, or quadrupole moments. Also, because the inner eight ions in CsCl rhombohedra are in two different environments (meaning two MCs), it was necessary to take an average of these to match the literature values. The method was also applied to zincblende and accurately reproduced the documented MC (1.6381).
Madelung Constants of Nanoparticles and Nanosurfaces
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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 in the structures under survey. We focus on MgO in the remainder of this paper. In its bulk form this ionic solid has the rocksalt structure. Although it is well known that nanostructures often have a relaxed structure compared to bulk materials (and this will be discussed below), we first focus on the some individual ion MCs for the {100} surface and the body center ion of “ideal” (rocksalt) MgO nanocrystals. The individual ion MCs for (a) body center, (b) face center, (c) edge center, and (d) corner ions are plotted as a function of particle size in Figure 1. This shows that the specific ion MCs progress toward limiting values. The limiting surface values (Figure 1b-d) indeed agree with previously calculated values for surface ions in bulk materials.9-14 Note that the body center ion (a) is far less sensitive to particle size than those on the surface. The former closely approaches the limiting value for a particle with 1000 ions whereas the surface ion values approach to the limit much more slowly. Each type of ion approaches a limiting (bulk) MC in a unique fashion. For edge-center ions (Figure 1c), the MCs oscillate with increasing particle size, while for corner- and face-center ions there are no oscillations. Furthermore, the MCs for corner ions (Figure 1d) gradually decrease (become less stable) with increasing particle size, while the values for face center ions (Figure 1b) gradually increase. It is also noteworthy that the MCs oscillate along an edge for any structure whereas no such oscillations are observed along a face diagonal, as shown in Figure 2. In Figure 3 we show a complete MC map of the 400 ions on the {100} surface of an 8000-ion particle containing 20 ions along each edge. The MCs for each ion on the surface are colorcoded. The values range from 1.3534 (corner) to 1.7161. The details of this surface map are interesting. The surface exhibits a central cross of more stable ions surrounded by areas of lower stability near the corners. This cross motif is also apparent on
the surface Madelung maps of smaller and larger “ideal” MgO particles, and so the surface of bulk MgO will also have a similar appearance. The lower stability areas constitute a significant fraction of the maps. Understanding and exploiting the pattern of surface MCs has promise in the design of patterned nanosurfaces. For example, MgO {100} has been used as a template for the growth of nanostructered metal assemblies.16 Free-electron metals bind to the surface of oxides through the Coulomb attraction between the surface ions and their screening charge density in the metal.17 Thus, the initial deposition of quantum dots on the mapped surface should occur at sites with the smallest local MCs and then at the next lowest MC sites and so on. The predictions of this work should be testable in principle by STM and AFM imaging,16 although atomic-resolution mapping of quantum dot arrays on ionic (template) nanosurfaces has so far not been reported. Nevertheless, anistotropy of this type has been observed for the epitaxial deposition of Pt on a MgO {100} surface.18 In this case, maps of pressure in the interfacial layer were calculated in order to interpret the experimental data. The maps given (see particularly Figure 14, ref 18) bear a remarkable resemblance to the MC map shown in Figure 3. As mentioned above, variations in structure are expected for nanostructures of different sizes. Of course, in the simplest case a cubic structure is possible only if the total number of ions present is the cube of an integer. However, even nanostructures which do meet this criterion are not necessarily cubic. Determining the most stable geometry of a particular nanostructure is therefore important.15 We conclude by focusing attention on MgO nanotubes. These have been recently synthesized19,20 and have been the subject of intense scrutiny. The stability of small MgO nanotubes has recently been studied by Bilabegovic´15 using DFT methods. Our calculated values of the MCwa for 10 × N nanotubes are shown in Figure 4. It is noteworthy that there is a smooth progression
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Figure 4. Progression of MCwa for 10 × N nanotubes (see text).
TABLE 2: Total Energies (eV) and MCwa for N × 4 Nanotubes nanotube 4×4 6×4 8×4 10 × 4 a
Figure 2. Variation of Surface MC along (a) an edge and (b) a face diagonal of a cubic neutral NaCl cluster having 18 ions on edge. The coordinate (0,0,0) is a corner ion.
total energy (eV)a -464.6981 -465.1345 -465.2519 -465.2947
MCwab 1.524971175 1.545662590 1.552265740 1.555036750
Reference 15. b This work.
rocksalt structure. Similar behavior was observed for all the nanotubes studied here. We now consider the N × 4 nanotubes studied by Bilabegovic´15 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. Summary and Conclusions 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; green, 1.5923-1.6036; yellow, 1.6165-1.6604; orange, 1.6722-1.6800; gray, above 1.6800.
from the 10 × 2 tube (MCwa ) 1.497208) to the limiting value (MCwa ) 1.601524). The latter value never reaches the literature value for bulk NaCl no matter how long the tube is. The bulk value (1.74756460) is observed only in the case of the cubic
Acknowledgment. Both authors thank their parent institutes for the provision of sabbatical leaves during the 2008-2009 academic year. A.D.B. acknowledges the receipt of PSC-CUNY Grant No. 69597-00 38 which supported his research efforts. M.D.B. gratefully acknowledges funding from The Natural Sciences and Engineering Research Council of Canada. This paper is dedicated to the memory of our late parents, Arthur and Catherine Baker.
Madelung Constants of Nanoparticles and Nanosurfaces References and Notes (1) Tyagi, S. Prog. Theor. Phys. 2005, 114, 517. (2) Ewald, P. P. Ann. Phys. 1921, 64, 253. (3) Crandall, R. E.; Buhler, J P. J. Phys. A: Math. Gen. 1987, 20, 5497. (4) Harrison, W. A. Phys. ReV. B 2006, 73, 212103. (5) Lamba, S. Phys. Stat. Sol. B 2004, 241, 3022. (6) Borwein, D.; Borwein, J.; Taylor, K. J. Math. Phys. 1985, 16, 1457. (7) Grosso, R. P., Jr.; Fermann, J. T.; Vining, W. J. J. Chem. Educ. 2001, 78, 1198. (8) Gaio, M.; Silvestrelli, L. Phys. ReV. B 2009, 79, 012102. (9) Andre´s, J.; Beltra´n, 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 2008, 78, 35410.
J. Phys. Chem. C, Vol. 113, No. 33, 2009 14797 (13) Barbier, A.; Stierle, A.; Finocchi, F.; Jupille, J. J. Phys. Condens. Matter 2008, 20, 184014. (14) Sauer, J.; Ugliengo, P.; Garrone, E.; Saunders, V. R. Chem. ReV. 1994, 94, 2095. (15) Bilabegovic´, G. Phys. Rev B. 2004, 70, 045407. (16) Walter, M.; Frondelius, P.; Honkala, H.; Ha¨kkinen, H. Phys. ReV. Lett. 2007, 99, 96102. (17) Scho¨nberger, U.; Andersen, O. K.; Methfessel, M. Acta Metall. Mater. 1992, 40, S1. (18) Olander, J.; Lazzari, R.; Jupille, J.; Mandili, B.; Goniakowski, J. Phys. ReV. B 2007, 76, 075409. (19) Yang, Q.; Sha, J.; Wang, L.; Wang, Y.; Ma, X.; Wang, J.; Yang, D. Nanotechnology 2004, 1004, 15. (20) Zhan, J.; Bando, Y.; Hu, J.; Goldberg, D. Inorg. Chem. 2004, 2462, 43.
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