madelung constants of

280 Journal of Chemical Education

_Vol. 87 No. 3 March 2010

_pubs.acs.org/jchemeduc

_r2010 American Chemical Society and Division of Chemical Education, Inc.

10.1021/ed800080k Published on Web 02/09/2010

In the Classroom

Teaching Nanochemistry: Madelung Constants ofNanocrystalsMark D. Baker*Department of Chemistry and Biochemistry, University ofGuelph,Guelph,OntarioN1G2W1Canada*[email protected]

A. David BakerDepartment of Chemistry and Biochemistry, Queens College, City University of New York, New York,New York 11367

The burgeoning field of nanoscience is placing increasingdemands on teaching strategies in the undergraduate curriculum.It is no longer acceptable to teach that any form of a puresubstance has the same properties. The very essence of nano-science is that some of these properties change, often drama-tically, when the nanosized regime is entered.

At an introductory level, it is important to convey tostudents a rationale for changes in properties as size decreases.While it is useful to point out that the fraction of surface speciesincreases with decreasing size, additional analysis is needed toprovide students with a better perspective on the nanoworld.Students in first- and second-year courses invariably do notappreciate the electronic structures of molecules and solids at adeep enough level to understand the ramifications of quantum-size effects displayed, for example, by quantum dots and wires, soa different approach is appropriate.

A method for introducing size-dependent phenomenabased on ideas that students meet in their first semesters ofundergraduate chemistry is presented. Crystal structures areusually introduced via considerations of ionic materials, and itis therefore relatively straightforward to use ionic nanostructuresto introduce aspects of nanoscience. We build our discussion oncalculations of the lattice energies of binary ionic solids. Latticeenergies are useful quantities linked to melting points, which areknown to undergo profound changes as a function of particlesize (1, 2).

Lattice Energies of Nanoparticles

The lattice energy (LE) of bulk ionic solids can be deter-mined by the Born-Land�e equation

LE ¼ kNAzþz- e2

r1-

1n

� �(1Þ

where z denotes the charge on the ions, r is the equilibriumanion-cation distance, A is the Madelung constant, e is thecharge on an electron, n is given by the Pauling method (3),N isthe Avogadro constant, and k is 1/(4πε0), where ε0 is thepermittivity of free space.

Size-dependent changes in lattice energies constitute themain theme of this article. We focus on the term A, which is theMadelung constant (MC) for a bulkmaterial. In essence, theMCis the electrostatic energy per ion per unit charge for a bulk

material normalized to the closest anion-cation distance (videinfra). The term A for any given crystal type decreases fornanosized particles, so it is not a constant but is size dependent.We refer to these smaller A values as A* and refer to A* as the“Madelung factor” (MF) rather than the “Madelung constant”.A* should converge to A with increasing size. In introductorytexts, an understanding of theMC is often framed by consideringthe Coulombic interactions operating within the sodium chlor-ide structure. In the solid state, students appreciate that there areboth attractive and repulsive interactions between anions andcations ions in the crystal. These are summed using the structureof the unit cell to give A. As Grosso et al. (4) discussed in thisJournal, the summation is often presented using the “first sixterms” of an appropriate series to illustrate the procedure. Forexample, for sodium chloride type lattices, the expression is

A� ¼ þ 6=ffiffiffi1

p- 12=

ffiffiffi2

pþ 8=

ffiffiffi3

p- 6=

ffiffiffi4

pþ 24=

ffiffiffi5

p- 24

ffiffiffi6

p

(2ÞIt is sometimes implied or misunderstood that, if taken toinfinity, this series would converge to the MC. However, thisis not a convergent series because the sum oscillates wildly as theseries is extended (4, 5). Special methods are needed to make theseries converge to theMC.One of these discussed by Grosso et al.is to use partial charges for ions at the periphery of thesummation volume, based on the notion familiar to studentsthat any given surface ion in a unit cell “belongs” in part toneighboring cells.

Determinations of the MF for a nanocrystal using thisapproach is however problematic because in reality partial chargesare not appropriate for nanostructures. For example, in the casewhere a nanostructure is identical to a unit cell, all ions must beconsidered to have full charges. There are no neighboring cells!All of the ions would be part only of the nanostructurecorresponding to the unit cell and so would have full charges.We present a way to deal with this issue, thus, avoiding the use ofpartial charges for ionic nanostructures.

Determining the Madelung Factor for a Nanoparticle

In practice, our approach is to determine the Cartesiancoordinates for suitable building blocks, then develop algorithms(simple computer programs) to generate the Cartesian coordinates

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_Journal of Chemical Education 281

In the Classroom

for higher generations of these building blocks, thus, producingsuccessively larger nanoparticles. The final step is to calculateeach ion's Coulombic interactions with all surrounding ions andto sum the results. For example, in the case of a cation-anioninteraction where the coordinates are (x1, y1, z1) and (x2, y2, z2),respectively, the Coulombic energy, E, is given by

E ¼ c1c2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx2 - x1Þ2 þðy2 - y1Þ2 þðz2 - z1Þ2

q (3Þ

where c1 and c2 are charges on the ions. Another ion in thenanostructure is then chosen, for example, (x3, y3, z3), and all itsinteractions with other surrounding ions in the nanoparticle aredetermined and added to the first and so on until the interactionsof every ion in the structure with every other ion have beencalculated. In each case, for MC determinations, the closestanion-cation distance is taken as unity. This procedure countsevery interaction twice, and so the final computation requires adivision by a factor of 2. Computations were performed on apersonal computer using Fortran 77 with double precisionarithmetic. Our approach thus determines the MF for a nano-particle by summing all Coulombic interactions without re-course to partial charges. It is crucially important to note that toconverge to theMCwith increasing size, an appropriate buildingblock must be chosen. Expansion from the nanoparticle to thebulk is then done symmetrically about this unit. The buildingblock must be electrically neutral or close to neutral, and highergenerations must also include clusters that are neutral or close toneutral.

Madelung Factors for NaCl and CsCl Nanocrystals

The bulk Madelung constant (A) for NaCl is 1.74756459(to nine significant figures), but for nanostructures, the value ofA* is smaller. The smallest unit we considered was the cubicprimitive unit cell shown in Figure 1A. This contains 8 ions andis electrically neutral. If we take the Na-Cl distance as unity, thevalue of A* is analytically given by

A� ¼ -- 12=ð1Þþ 12=

ffiffiffi2

p þ - 4=ffiffiffi3

p

4¼ 1:45602993

(4ÞIn this expression, the sum is over the Na cations only giving theMadelung constant per ion. Thus, for example, the first term inthe summation is multiplied by 12 because each of the 4 sodiumions in the primitive cell has 3 nearest-neighbor chloride ions.The next generation of the primitive cell nanoparticle is theconventional NaCl unit cell, which is composed of 8 primitive

cells. In introductory texts, partial occupancies (corner ions =1/8, face center = 1/2, and edge = 1/4) are applied to this unitcell structure. This leads to the notion that the unit cell iselectrically neutral and is composed of 4 sodium ions and 4chloride ions. However, a model nanoparticle having the samestructure as the unit cell would in fact contain 13 Naþ ions and14 Cl- ions (or 13 Cl- and 14 Naþ ions, depending on thechoice of central ion) and thus would have a charge of either 1þor 1-. The A* value for this particle, calculated using ourprocedure of summing all Coulombic interactions among the27 ions, is 1.5665422. We performed calculations on progres-sively larger clusters based on expansions of the primitive cell,obtaining a value of A* for each cluster examined. As shown inTable 1, the value ofA* approaches theMC for higher and highergenerations and exceeds 99% of the magnitude of the MC forcubic clusters containing more than 27,000 ions. A* values forcubic NaCl nanostructures as a function of size (i.e., number ofedge ions) are also plotted in Figure 2.

Figure 1. Nanostructure units used to compute Madelung constants.Units were chosen so that for large arrays the Madelung factorsconverged to the bulk value. (A) The primitive cell for the NaCl structuretype. (B) Rhombohedral primitive cell (outlined in red) for the CsClstructure type. Note the relationship to the conventional unit cell.

Table 1. Madelung Factors for NaCl and CsCl Nanostructures

Number of Ions NaCl CsCl

8 1.45602990 1.42407794

27 1.56654220 1.53635802

64 1.62872320 1.60371963

125 1.65493080 1.63469997

216 1.67360860 1.65739380

343 1.68500570 1.67207384

512 1.69392760 1.68369691

729 1.70025850 1.69228817

1000 1.70547580 1.69948599

3375 1.72018721 1.72033870

8000 1.72729460 1.73086376

27,000 1.73421195 1.74140986

64,000 1.73760870 1.74670185

125,000 1.73962758 1.74988411

216,000 1.74096564 1.75200878

343,000 1.74191761 1.75352801

512,000 1.74262955 1.75466834

729,000 1.74318197 1.75555594

1,000,000 1.74362828 1.75626660

Figure 2. Madelung Factor (A*) versus number of edge ions for NaCland CsCl structure types.





In the Classroom

We next consider cesium chloride. Students should befamiliar with the unit cell having a pseudo-body-centered cubicstructure. If we consider a nanoparticle having the CsCl unit cellstructure, it would contain 9 ions: 1 ion at the center, and 8 ionsof opposite charge at the corners. Thus, the particle would have acharge ofþ7 or-7 depending on whether a cation or anion wasat the center. If we calculate theMF for such a nanoparticle, againsetting the anion-cation closest distance to unity, we arrive at

A� ¼ - 2

- 8=1þ 12=2ffiffiffi3

p� �

þ 12=

ffiffiffi8

pffiffiffi3

p !

þ 4=2

9

¼ - 2:60906091 (5ÞThis result shows that this nanoparticle is, not surprisingly inlight of the large charge imbalance, unstable. TheMF is negative!Expansion of this unit to produce larger stable nanostructures isclearly untenable and the MF does not converge to the bulkvalue. This is a crucial point for two reasons

• If one wishes to produce a large enough particle to determine thebulk MC, then this particular structure will not work

• Potential building blocks cannot be based on the unit cell

A highly charged building block will result in an inordinatenumber of repulsive potentials. Neutral building blocks aredesirable, and for CsCl, a convenient starting point is therhombohedral primitive cell shown in Figure 1B containing 4Csþ and 4 Cl- ions. It is noteworthy that the 8 ions are not inequivalent environments as they are in the NaCl primitive cell,and this will naturally persist in all higher generations. Aninstructive exercise for computer literate students is to devisean algorithm that will generate all the coordinates for highergenerations of this basic building block, to compute the MF as afunction of size, and show that the MF converges to the MC forcesium chloride. Alternatively, students can be provided with theinformation (algorithms and coordinate files) needed for thecomputations (see the supporting information). Once again,there is a large change in the MF in the nanosized regime. Thevariation of MF with number of ions for the CsCl structure isshown in Table 1 and Figure 2. The latter shows that theMFs forCsCl and NaCl actually cross on their way to convergence to theacceptedMCs for both substances. In practice,A* exceeds 99% ofthe bulk value at 27,000 ions.

The crossover in theNaCl/CsCl plots merits comment. Forlarger clusters, the dominant stabilizing factor is the coordinationnumber of the ions: in CsCl, it is 8 and inNaCl it is 6. Therefore,in a sufficiently large cluster, CsCl has more nearest-neighborattractions than does NaCl, resulting in a greater MF. However,in small clusters, other factors come into play. For example, in thetwo primitive cells (Figure 1), all the ions are three-coordinate, sorelative coordination numbers play no role. The normalizeddistances between ions are now the determining factors. Inparticular, in the cubic NaCl primitive cell, all the face diagonalshave the same length, which is 1.414a, where a is the shortestanion-cation distance. However, in the rhombohedral CsClprimitive cell, the face diagonals do not all have the same length.Some have the value 1.633a while others are 1.155a. This lattershort normalized distance significantly increases next nearest-neighbor repulsions in CsCl relative to NaCl, which results in asmaller MF for the CsCl primitive cell.

Madelung Factors for ZnS Nanocrystals

The semiconductor zinc sulfide has been widely studied interms of nanostructure applications (6, 7). The MFs for variousshapes and sizes of ZnS nanoparticles can be determined usingthe approach described above. A nanostructure corresponding tothe conventional unit cell would be unstable because, as withCsCl, there would be a large charge imbalance in the unit cell,which contains 14 ions of one type and only 4 of the opposite

Figure 3. (A) Nanostructure building block used to calculate Madelungfactors for spherical zinc blende structure types. This unit (10 ions)resembles the adamantine structure, which is symmetrically disposedaround the vacant octahedral hole in the close-packed sulfide lattice. (B)The next generation of this structure (30 ions), which corresponds to themodel typically used to depict the zinc blende structure in lecture classes.

Figure 4. Nanoclusters having the zinc blende symmetry can be neutralor charged. Shown here are the two smallest clusters that are electricallyneutral. They contain (A) 136 and (B) 1472 ions, respectively.

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_Journal of Chemical Education 283

In the Classroom

charge. As with CsCl, larger generations based on the conven-tional cubic unit cell do not yield MFs that converge to the MCfor ZnS (zinc blende).

Therefore, we constructed the various generations ofnanostructures by expanding spherically from the octahedralhole at the center of the zinc blende structure. In effect, thebasic building block was the adamantane-like structure shownin Figure 3A. The next generation of this is the structuretypically used as a lecture demonstration model for the zincblende structure (Figure 3B). Analysis of the results was morecomplex than with NaCl or CsCl because the building block ischarged, as are many of its higher generations, and the magni-tude of the charge varies significantly from cluster to cluster. Incontrast, for NaCl and CsCl, the primitive cell building blocksare neutral, and all expansions are neutral or have a singlecharge. To obtain meaningful comparisons of ZnS clusters, wecomputedMFs for those having minimal (or zero) charges. Thetwo smallest neutral nanostructures with the zinc blendestructure are illustrated in Figure 4. Students can generatefigures such as these by importing the x, y, and z coordinatesinto a molecular modeling program.

For the smaller clusters, we selected those having four orfewer charges; for larger clusters, we selected only those for whichthe number of charges was less than 0.5% of the total number ofions making up the cluster.When these constraints were applied,our data for spherically expanded zinc blende nanoparticles

produced a plot (Figure 5A) similar to those for NaCl and CsCland once again the bulkMCof zinc blende (1.638) is approachedfor large particles. In Figure 5B, the melting points of CdSnanoparticles of these sizes (2) are plotted (comparable data forNaCl andCsCl nanocrystals are as yet not available). CdS has thezinc blende structure in the bulk form and so students can seethat the trend in behavior is mirrored by the MFs. Notunexpectedly, the plots have different shapes in part becausethere is not a direct straightforward relationship betweenmeltingpoint and lattice energy. It also should be stressed that thesecomputations are based on model systems and that other factorsalso merit consideration. For example, a given nanoparticle doesnot necessarily have the bulk crystal structure, and interionicdistances are not necessarily the same. In addition, the degree ofcovalent bonding should be considered. In the case of ZnS, thiswill be important. These and other subtleties are the subjects ofintensive research work in this area (6, 7).

Conclusions

The major theme of this article is to use simple electrostaticideas to introduce lower-level undergraduates to size-dependentphenomena. The major points are

• The total Coulombic (potential) energy of a nanostructurecannot be determined using partial occupancy or fractionalcharge protocols.

• A complete summation of all attractions and repulsions, usingfull charges, gives the MF for a nanoparticle.

• The calculatedMFs converge smoothly to the acceptedMCs forthe bulk materials.

• This convergence allows students to compute MCs withoutusing partial charges, which are physically unrealistic for nano-particles.

• This method requires neutral or nearly neutral building units,meaning that standard unit cells are unsuitable if they incorpo-rate a large charge imbalance (as for CsCl and ZnS unit cells).

• The variation of MF with size provides a useful insight intothe large depression of melting point exhibited by nanostruc-tures.

This method enables undergraduate students to computetheMF for various ionic nanoparticles. In this article, we focusedon cubic, spherical, and rhombohedral clusters. However, stu-dents can use the methods described herein to address anynanoparticle shape that they can imagine, such as tubes, stars,crosses, and so forth. By using the coordinate files they generate,students can visualize any structure they create using a standardmolecular modeling program.

Acknowledgment

The authors gratefully acknowledge their home institutionsfor the privilege of sabbatical leaves, during which this work wasconceived and written. The article is dedicated to the memory ofour late parents, Arthur and Catherine Baker.

Literature Cited

1. Alivisatos, A. P. J. Phys. Chem. 1996, 100, 13226–13239.2. Breaux, G. A.; Benirschke, R . C.; Jarrold, M. F. J. Chem. Phys. 2004,

121, 6502–6507.3. Pauling, L. J. Am. Chem. Soc. 1927, 49, 765–790.

Figure 5. (A) The variation of Madelung factor with size for neutral ornearly neutral zinc blende structures. The exact diameter of the particle isgiven by the product of the size of cluster for a cubic crystal and the Zn-Sbond length. (B) Melting point versus size for CdS nanoparticles with thezinc blende structure (see also ref 1). Using the Cd-S bond length of 2.9Å, the scales in A and B can be interconverted.





In the Classroom

4. Grosso, R., Jr.; Fermann, J. T.; Vining,W. J. J. Chem. Educ. 2001, 78,1198–1202.

5. Bridgeman, W. B. J. Chem. Educ. 1969, 46, 592–593.6. Burnin, A.; Sanville, E.; BelBruno, J. J. J. Phys. Chem. A 2005, 109,

5026–5034.7. Pal, S.; Goswami, B.; Sarkar, P. J. Phys. Chem. C 2008, 112, 6307–

6312.

Supporting Information Available

Computer programs that enable students to determine the MF forvarious sizes and shapes of nanostructures will be provided upon request.These are available in compiled Basic and Fortran 77. Files containingCartesian coordinates of ions in various sized structures of NaCl, CsCl,and ZnS are also available. This material is available via the Internet athttp://pubs.acs.org.

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