 Nano Express
 Open Access
 Published:
Magical Mathematical Formulas for Nanoboxes
Nanoscale Research Letters volume 16, Article number: 39 (2021)
Abstract
Hollow nanostructures are at the forefront of many scientific endeavors. These consist of nanoboxes, nanocages, nanoframes, and nanotubes. We examine the mathematics of atomic coordination in nanoboxes. Such structures consist of a hollow box with n shells and t outer layers. The magical formulas we derive depend on both n and t. We find that nanoboxes with t = 2 or 3, or walls with only a few layers generally have bulk coordinated atoms. The benefits of lowcoordination in nanostructures is shown to only occur when the wall thickness is much thinner than normally synthesized. The case where t = 1 is unique, and has distinct magic formulas. Such lowcoordinated nanoboxes are of interest for a myriad variety of applications, including batteries, fuel cells, plasmonic, catalytic and biomedical uses. Given these formulas, it is possible to determine the surface dispersion of the nanoboxes. We expect these formulas to be useful in understanding how the atomic coordination varies with n and t within a nanobox.
Introduction
Nanoboxes were originally synthesized circa 2002 [1, 2]. A nanobox is distinct from a nanocage in that the latter has porous walls. Also, both are distinct from a nanoframe, in that the nanoframe is a structure (frame) consisting of the lowcoordinated outline of the cluster. Such anisotropic, polyhedral structures may be created from galvanic displacement reactions [3, 4]
where a nanocluster with metal A is sacrificially hollowed out by an aqueous solution of metal B, which has a higher reduction potential and creates the hollow solid of element B. Half reactions occur at the anode and cathode of an electrochemical cell, resulting in the full combined reaction as above [5]. In some instances, scientists have combined galvanic displacement with void formation via Kirkendall Fickian diffusion of metals and vacancies [6]. Models for this activity exist for specific cases and in situ electron microscopy experiments have been reported [7, 8]. Other synthetic methods include chemical etching [9], ion exchange [10], and metal–organic frameworks (MOFs) [11, 12]. A recent review of synthesis methods mentions that anisotropic clusters have yet to be made in the size region \(2<D<20\) nm, hindering the progress of nanocage fabrication in this important size domain [13].
Such hollow structures have low coordination, making them of interest for batteries [12], fuel cells [14], plasmonic [15], catalytic [16], and biomedical applications [17]. Previous analysis shows that for catalytic applications, a coordination approach applies [18], while for energy storage, there are only some hints with density functional theory (DFT) results indicating that select facets are important [19]. We use a previously derived method from adjacency matrix analysis [20, 21] to discover the atomic coordination of a box with n shells and a wall thickness of t layers. This analysis shows that a nanobox with t = 2 or 3 has bulk coordination and as such the benefits of low coordination are present only for nanoboxes with thinner walls than generally believed necessary. The methods we use quantifies the atomic coordination through magic numbers and formulas for thirteen types of nanoboxes.
Methods
Key to our analysis by coordination methods, is the creation of an adjacency matrix from the atomic coordinates of the nanobox. Such a matrix is created as follows. We define i and j as nearest neighbors, and separate them from the rest by requiring that the bond length \(r_{ij} < r_c\) where \(r_c\) is a threshold value, appropriate for the nanobox. The value for \(r_c\) must be less than the distance for second nearest neighbors and varies with the crystal structure [21]. For bcc crystals, \(r_c < 1.15\cdot r_{{\rm min}}\), where \(r_{{\rm min}}\) is the smallest bond length. Thus,
describes the adjacency matrix for the cluster, and
describes the Euclidean matrix for the box. We use the Euclidean matrix to determine the diameter, D, (nm) for the nanoboxes.
Since we create nearest neighbor adjacency matrices, we know the coordination number \(\hbox {cn}_i\) of vertex i by summing the elements of \({{\mathbf{A}}}(i,:)\). Our structure consists of \(n+1\) shells numbered 0, 1, ..., n, with t outer layers. Let \(N_{{\rm cn}_i}(n,t)\) be the number of atoms with coordination \(\hbox {cn}_i\) where \(1\le \hbox{cn}_i\le \hbox{cn}_M\) with \(\hbox {cn}_M\) the maximal coordination in the nanobox. Then the total number of atoms in the nanobox is given by
The surface atoms in the outer shell (or interior) of the nanobox, n have a set of bondings less than the bulk coordination. Thus the maximal coordination for surface atoms is \(\hbox {cn}_s < \hbox{cn}_M\), and the number of surface atoms is
This holds if all the nonsurface vertices have coordination larger than \(\hbox {cn}_s\), which is true for all fcc, bcc, and hcp clusters. We determine the \(N_{{\rm cn}_i}(n,t)\) by counting the columns of the adjacency matrix whose sum is \(\hbox {cn}_i\). Note that our cluster coordinate algorithm is built by shells, so that each subsequent shell contains all the previous lower values of n. In addition, the number of bonds in the box is
where \(N_{{\rm B}}(n,t)\) is the number of bonds and \(\hbox {cn}_M\) is the maximum coordination. The factor of 1/2 comes about because of the pairwise nearest neighbor bonding.
Since we know that these equations depend on n, t as a polynomial of degree at most 3, we can compute \(N_{{\rm cn}_i}(n,t)\) for 4 consecutive values of n, say \(n=n_0+j\), j = 0, 1, 2, 3. A simple interpolating polynomial will then give the polynomial coefficients. It has to be verified that by increasing \(n_0\), which is usually equal to 1, the formulas do not change. If the formulas become stable from \(n_0\) on, then they hold for all \(n\ge n_0\). To get the exact rational coefficients, one needs to solve the Vandermonde system for the coefficients in exact arithmetic.
Note that in the magical formulas for nanoboxes we have that \(n > t\) so that therefore contrary to any expectation, filling up the box by an appropriate choice of t will not recreate the original magic formulas for the complete solid clusters. These magic formulas are useful for modeling the mesoscale properties of clusters and boxes, or cages. Complete sets of formulas were originally derived for nineteen cluster types. In this manuscript, we derive magical formulas for thirteen types of nanoboxes.
In the magical formulas below, we find that bulk coordination may appear for either t = 2 or t = 3 layers of shell thickness. Most are for layers where t = 2; the exceptions are the fcc cube, the cuboctahedron, the icosahedron, and the bcc cube and truncated cube. In the latter, bulk coordination only appears for t = 3 layers. For the data below, the tables of the magical formulas are accompanied by a figure of a ‘halfbox’ to show the interior of the nanoboxes. Alongside is a colorbar indicating the coordination and number of such in parentheses.
Results and Discussion
In order to delineate the applicability of magic formulas, we outline how catalytic behavior may depend on coordination and such formulas. We define G as the size dependent Gibbs energy of the cluster. Because of adatoms being bonded to the outer shell atoms there is an increase in G that is called the adsorption energy and is denoted as \(\Delta G\). This can be split up over different coordination types of the atoms on the outer shell bonding to adatoms. For example, a kink atom adds to the adsorption energy with an amount \(\Delta G_{k}\). Similarly an edge atom adds \(\Delta G_{e}\), while a facet atom contributes \(\Delta G_f\) then [18]:
where \(N_o\) is the number of atoms in the outer shell of the indicated type. The total number of atoms in the outer shell bonded to adatoms is defined as \(N_s=N_f+N_e+N_k\), resulting in:
with the Gibbs energy fraction expressed through the edge and kink sites which have explicit coordinations for specific structures. This demonstrates that magic formulas have a role in surface reactions, through edge and kink coordinations and their formulas. Note that Eq. (8) applies to adsorption to ontop sites, otherwise not all adatoms will be bonded to atoms in the outer shell. In such a model, the kink sites have magic formulas that are constant with the number of shells, n, edge sites have formulas that are linear with n, and facet sites have formulas that are quadratic with n. More specifically, the kink sites are the lowest coordinated formulas, the edge sites are the second lowest coordinated, and facet sites have cn = 8 for (100) facets and cn = 9 for (111) facets.
Two fundamental relationships on a perparticle basis can be applied. For the Gibbs energy and adsorption constant, \(K_{{\rm a}}\), it holds:
where R is the gas constant and T is the temperature in Kelvin. In addition, Brønsted–Evans–Polanyi relationships are widely used in homogeneous and heterogeneous catalysis [18, 22] using a relationship for reaction constants k and equilibrium constants K as follows:
where g and \(\alpha\) (Polanyi parameter) are constants. The Polanyi parameter is unitless and a proper fraction, as given originally by Brønsted [23]. We then have:
where
and
This analysis shows that determining a catalytic model necessitates a method of calculating the Gibbs energy. Known catalytic reactions such as the twostep and Langmuir–Hinshelwood mechanisms have been considered [24].
FCC Nanoboxes
Face centered cubic structures are the most common form for nanoclusters and nanoboxes. This is the structure of the metals with interesting properties, such as the noble metals with plasmonic properties, and the catalytic precious metals. Since gold has a high reduction potential of 1.50 V (see Eq. 1) versus the standard hydrogen electrode (SHE) [5], it is one of the easiest metals to synthesize as a nanobox or nanocage. Gold nanoboxes or nanocages have been formed in cubic [1], cuboctahedron [25], icosahedron and decahedron [26], octahedron [27] and tetrahedron [28] shapes.
We can determine the approximate size of these nanoboxes by using a coordination approach for the nearest neighbor bond length r(cn) [29],
Here \(r_{{\rm B}}\) is the bulk bond length for gold (0.2884 nm) and \(\langle cn \rangle _c\) is the average coordination of the cluster. We find a linear relationship between D and n, the number of cluster shells, as shown in Table 1:
We use nanoboxes with t = 3, as the formulas vary with t, and we wish to achieve some bulk coordination. For the calculation of D(n), we use the maximum distance between atoms in the cluster, derived from the Euclidean matrix. Note that D(n) is an empirical formula, derived from data (vary n and calculate D), and as such is not proven.
These relationships produce diameters in agreement with other data, from DFT. For the solid cuboctahedra with N equal to 55, 561, and 923 we get diameters of 1.12 nm, 2.85 nm, and 3.43 nm. This compares favorably with published DFT results for 55 atoms of 1.1 nm [30], for 561 atoms, 2.7 nm [31], and for 923 atoms, 3.5 nm [30]. The magical formulas for some fcc nanoboxes are tabulated below (Tables 2, 3, 4, 5, 6, 7, 8).
Diamond and Simple Cubic Nanoboxes
The diamond cubic lattice structure is formed by an allotrope of carbon as well as the elements silicon and germanium. Also, some cubic compounds form this structure, as cubic iron oxide, tetrahedral diamond maghemite \(\gamma\)Fe_{2}O_{3}. The bond length for Fe–O in tetrahedral diamond maghemite \(\gamma\)Fe_{2}O_{3} = 0.186 nm [32]. This leads to the diameter of diamond clusters D(n) as below:
According to reference [12], microboxes of cubic iron oxide formed and had interesting lithium storage capabilities. We are not aware of a complete coordination model for energy storage, but as mentioned above, DFT results indicate that activity may depend on facet orientation [19]. No such model of storage dependence on coordination exists presently as we have for catalysis. From equation (16) above (created using t = 4), a microbox requires approximately n = 1600 shells for diamond maghemite. Magical formulas for the diamond and simple cubic lattice structures are listed below (Tables 11, 12).
HCP Nanoboxes
See Table 16.
The Case t = 1
The special case t = 1 is unique and as such has distinct magical formulas. We examine this case for some of the above nanoboxes. Nanoboxes with ultrathin walls have been formed with cubic [33], octahedral [16], and icosahedral shapes [34]. According to the magical formulas below, the cubic nanobox with t = 1 has the lowest coordination. Platinum has a relatively high reduction potential of 1.18 V versus the SHE, so it can be formed by galvanic replacement, see Eq. (1) [5]. However, the oxidation reduction reaction (ORR) properties of some of these platinumbased nanocages indicate that structures with (111) facets as opposed to (100) facets have better ORR mass activities [35].
Thus the icosahedron with 20 (111) facets has the best ORR mass activity, followed by the octahedron, and lastly the truncated cube. This property of catalytic behavior from facet orientation taking precedence over coordination number is evidenced by the tabular data below. In other words, as mentioned in the following tables, the cube with (100) facets has the lowest magic coordination numbers with four and five, yet the octahedron and icosahedron with (111) facets and larger magic formulas have better ORR activity. This property is evidenced in nanoclusters as well, where DFT results confirm the dominance of the (111) facets [36], especially for PtNi alloys (Tables 17, 18, 19, 20, 21).
Dispersion
Given the importance of edge and kink sites relative to facet ones with regard to catalytic activity, we have determined the surface dispersion for some of the nanoboxes we study. The (100) facets have cn = 8 while the (111) facets have cn = 9. This may provide insight into the reasons for the individual polyhedral activity when compared among the nanoboxes. In Fig. 1 below, we plot the surface dispersion \(D_{{\rm s}} = (N_{{\rm e}} + N_{{\rm k}}) / N_{{\rm S}}\cdot 100\%\). In this relationship \(N_{{\rm k}}\) is the number of kink or corner sites and \(N_{{\rm e}}\) the number of edge sites. As can be seen in Figure 1, nanoboxes with (111) surfaces as opposed to (100) surfaces have higher dispersion, giving credence to the preference of catalytic activity of the (111) facet.
Conclusion
In summary, we have presented the first detailed mathematical description of magical formulas for nanoboxes. The case of the shell thickness, t = 1 is distinct from \(t>1\) and we tabulate the data for some of these cases. The formulas for the coordination, number of atoms, and number of bonds are all enumerated. We find that bulk coordination appears for layers where t = 2 or 3, and as such is much thinner than normally synthesized. The benefits of low coordination are only achieved for very thin walls. We expect these results to be useful for modeling and experimental work.
Availability of data and materials
The dataset(s) supporting the conclusions of this article may be obtained from the corresponding author.
Abbreviations
 bcc:

Body centered cubic
 fcc:

Face centered cubic
 hcp:

Hexagonal close packed
 DFT:

Density functional theory
 SHE:

Secondary hydrogen electrode
References
Sun Y, Xia Y (2002) Shapecontrolled synthesis of gold and silver nanoparticles. Science 298:2176–2179
Sun Y, Mayers BT, Xia Y (2002) Templateengaged replacement reaction: a onestep approach to the largescale synthesis of metal nanostructures with hollow interiors. Nano Lett 2:481–485
Oh MH, Yu T, Yu SH et al (2013) Galvanic replacement reactions in metal oxide nanocrystals. Science 340:964–968
Gao Z, Ye H, Wang Q et al (2020) Template regeneration in galvanic replacement: a route to highly diverse hollow nanostructures. ACS Nano 14:791–801
Xia X, Wang Y, Ruditskiy A, Xia Y (2013) 25th Anniversary article: galvanic replacement: a simple and versatile route to hollow nanostructures with tunable and wellcontrolled properties. Adv Mater 25:6313–6333
Yin Y, Rioux RM, Erdonmez CK et al (2004) Formation of hollow nanocrystals through the nanoscale Kirkendall effect. Science 304:711–714
Jana S, Chang JW, Rioux RM (2013) Synthesis and modeling of hollow intermetallic Ni–Zn nanoparticles formed by the Kirkendall effect. Nano Lett 13:3618–3625
Chee SW, Tan SF, Baraissov Z, Bosman M, Mirsaidov U (2017) Direct observation of the nanoscale Kirkendall effect during galvanic replacement reactions. Nat Commun 8(1224):1–8
Sui Y, Fu W, Zheng Y et al (2010) Synthesis of Cu_{2}O nanoframes and nanocages by selective oxidative etching at room temperature. Angew Chem Int Ed 49:4282–4285
Wu HL, Sato R, Yamaguchi A et al (2016) Formation of pseudomorphic nanocages from Cu_{2}O nanocrystals through anion exchange reactions. Science 351:1306–1310
Dang S, Zhu QL, Zu Q (2018) Nanomaterials derived from metal–organic frameworks. Nat Rev Mater 3(17075):1–14
Zhang L, Wu HB, Madhavi S, Hng HH, Lou XW (2012) Formation of Fe_{2}O_{3} microboxes with hierarchical shell structures from metal–organic frameworks and their lithium storage properties. J Am Chem Soc 134:17388–17391
Shi Y, Lyu Z, Zhao M, Chen R, Nguyen QN, Xia Y (2020) Noblemetal nanocrystals with controlled shapes for catalytic and electrocatalytic applications. Chem Rev. https://doi.org/10.1021/acs.chemrev.0c00454
Tian X, Zhao X, Su YQ et al (2019) Engineering bunched Pt–Ni alloy nanocages for efficient oxygen reduction in practical fuel cells. Science 366:850–856
Lee KS, ElSayed MA (2006) Gold and silver nanoparticles in sensing and imaging: sensitivity of plasmon response to size, shape, and metal composition. J Phys Chem B 110(39):19220–19225
Zhang L, Roling LT, Wang X et al (2015) Platinumbased nanocages with subnanometerthick walls and welldefined, controllable facets. Science 349(6246):412–416
Chen J, Wiley B, Li ZY et al (2005) Gold nanocages: engineering their structure for biomedical applications. Adv Mater 17:2255–2261
Murzin DY (2010) Kinetic analysis of cluster size dependent activity and selectivity. J Catal 276:85–91
Ganapathy S, Wagemaker R (2012) Nanosize storage properties in spinel Li_{4}Ti_{5}O_{12} explained by anisotropic surface lithium insertion. ACS Nano 6(10):4702–4712
Kaatz FH, Bultheel A (2019) Magic mathematical relationships for nanoclusters. Nanoscale Res Lett 14(1):150
Kaatz FH, Bultheel A (2019) Magic mathematical relationships for nanoclusters—errata and addendum. Nanoscale Res Lett 14(1):295
Bligaard T, Nørskov JK, Dahl S, Matthiesen J, Christensen CH, Sehested J (2004) The Brønsted–Evans–Polanyi relation and the volcano curve in heterogeneous catalysis. J Catal 224:206–217
Brønsted JN (1928) Acid and basic catalysis. Chem Rev 5(3):231–338
Murzin DY (2015) Cluster size dependent kinetics: analysis of different reaction mechanisms. Catal Lett 145:1948–1954
Lin ZW, Tsao YC, Yang MY, Huang MH (2016) Seedmediated growth of silver nanocubes in aqueous solution with tunable size and their conversion to Au nanocages with efficient photothermal property. Chem Eur J 22:2326–2332
Lu X, Tuan HY, Chen J, Li ZY, Korgel BA, Xia Y (2007) Mechanistic studies on the galvanic replacement reaction between multiply twinned particles of Ag and HAuCl_{4} in an organic medium. J Am Chem Soc 129:1733–1742
Liu X (2011) Cu_{2}O microcrystals: a versatile class of selftemplates for the synthesis of porous Au nanocages with various morphologies. RSC Adv 1:1119–1125
NaiQiang Y et al (2013) Preparation of gold tetrananocages and their photothermal effect. Chin Phys B 22:097502
Sun CQ (2007) Size dependence of nanostructures: impact of bond order deficiency. Prog Solid State Chem 35(1):1–159
Li H, Li L, Pedersen A, Gao Y, Khetrapal N, Jonsson H, Zeng XC (2015) Magicnumber gold nanoclusters with diameters from 1 to 3.5 nm: relative stability and catalytic activity for CO oxidation. Nano Lett 15(1):682–688. https://doi.org/10.1021/nl504192u
Kleis J, Greeley JP, Romero NA, Morozov VA, Falsig H, Larsen AH, Lu J, Mortensen JJ, Dułak M, Thygesen KS, Nørskov JK, Jacobsen KW (2011) Finite size effects in chemical bonding: from small clusters to solids. Catal Lett 141:1067–1071. https://doi.org/10.1007/s1056201106320
Fasiska EJ (1967) Structural aspects of the oxides and oxyhydrates of iron. Corros Sci 7:833–839
Xie S, Choi SI, Lu N et al (2014) Atomic layerbylayer deposition of Pt on Pd nanocubes for catalysts with enhanced activity and durability toward oxygen reduction. Nano Lett 14:3570–3576
Wang X, Choi SI, Roling LT et al (2015) Palladiumplatinum core–shell icosahedra with substantially enhanced activity and durability towards oxygen reduction. Nat Commun 6:7594
Zhao M, Wang X, Yang X, Gilroy KD, Qin D, Xia Y (2018) Hollow metal nanocrystals with ultrathin, porous walls and wellcontrolled surface structures. Adv Mater 30:1801956
Stamenkovic VR, Fowler B, Mun BS, Wang G, Ross PN, Lucas CA, Markovic NM (2007) Improved oxygen reduction activity on Pt3Ni(111) via increased surface site availability. Science 315:493–497
Acknowledgements
All computational modeling uses original code developed by the authors compatible with MATLAB 2020b.
Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or notforprofit sectors.
Author information
Authors and Affiliations
Contributions
All authors contributed to the final version of the manuscript and approved it for publication.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Kaatz, F.H., Bultheel, A. Magical Mathematical Formulas for Nanoboxes. Nanoscale Res Lett 16, 39 (2021). https://doi.org/10.1186/s11671021034728
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s11671021034728
Keywords
 Nanobox
 Nanocage
 Nanoframe
 Coordination
 Magic numbers
 Dispersion