Abstract
Size, shape, overall composition, and surface functionality largely determine the properties and applications of metal nanoparticles. Aside from welldefined metal clusters, their composition is often estimated assuming a quasispherical shape of the nanoparticle core. With decreasing diameter of the assumed circumscribed sphere, particularly in the range of only a few nanometers, the estimated nanoparticle composition increasingly deviates from the real composition, leading to significant discrepancies between anticipated and experimentally observed composition, properties, and characteristics. We here assembled a compendium of tables, models, and equations for thiolprotected gold nanoparticles that will allow experimental scientists to more accurately estimate the composition of their gold nanoparticles using TEM image analysis data. The estimates obtained from following the routines described here will then serve as a guide for further analytical characterization of assynthesized gold nanoparticles by other bulk (thermal, structural, chemical, and compositional) and surface characterization techniques. While the tables, models, and equations are dedicated to gold nanoparticles, the composition of other metal nanoparticle cores with facecentered cubic lattices can easily be estimated simply by substituting the value for the radius of the metal atom of interest.
Graphical abstract
Introduction
Gold nanoparticles are everywhere! Aside from the curious and beautiful historic uses as colloidal additives to stain Roman glass in the fourth century and the discovery of the wondrous and different properties of colloidal gold by Michael Faraday in the mid nineteenth century (Tweney et al. 2004), gold nanoparticles and nanoclusters have penetrated almost every facet of science. Each year there are numerous educational and critical reviews on the use and study of gold nanoparticles for topics including in vitro diagnostics (Aillon et al. 2009; Almeida et al. 2011; Azzazy et al. 2006; Johnston et al. 2010; Khlebtsov and Dykman 2011; Mulder et al. 2009; Rosi and Mirkin 2005; Wolinsky and Grinstaff 2008), cancer diagnostics and therapy (Bhattacharyya et al. 2011; Chikkaveeraiah et al. 2012; Dreaden et al. 2011; Gindy and Prud’homme 2009; Jain et al. 2012; Kennedy et al. 2011; Lal et al. 2008; Perfezou et al. 2012; Wang and Thanou 2010; Yong et al. 2009; Zhang et al. 2013), biological and chemical sensors (Askim et al. 2013; Howes et al. 2014; Kim et al. 2012; Perfezou et al. 2012; Pingarron et al. 2008; Sepulveda et al. 2009; Stewart et al. 2008), catalysis (Crooks et al. 2001; Hou and Cronin 2013; Panigrahi et al. 2007; Sarina et al. 2013), gold metaatoms for metamaterials (Ross et al. 2016), selfassembly (Bishop et al. 2009; Boeker et al. 2007; Grzelczak et al. 2010; Lin et al. 2006; Ofir et al. 2008), intrinsic chirality (BenMoshe et al. 2013; Gautier and Bürgi 2009; GuerreroMartínez et al. 2011; Wang et al. 2013; Xia et al. 2011), and this list could go on. Mind you, 20 years after the beautiful and simple BrustSchiffrin synthesis methods (Brust et al. 1994, 1995) were published we are finding it increasingly difficult to summarize the contents of review articles, not even individual papers, on gold nanoparticles. Our group, working on understanding interactions between functionalized gold nanoparticles and soft condensed matter, even added a few to this list of reviews, summarizing studies on these magnificent nanomaterials as versatile additives in liquid crystal phases (Hegmann et al. 2007; Qi and Hegmann 2008; Shivakumar et al. 2011; Stamatoiu et al. 2012). The search for “gold nanoparticle” in Thomson Reuters Web of Science shows an everincreasing number of papers (several thousand), and just looking at the last couple of years, a Google search leads to a mindboggling number of over 1.8 million hits. Numerous established chemical suppliers as well as smaller startup companies now sell gold nanoparticles, but the majority of laboratories it seems still enjoy synthesizing their own, partially perhaps for educational reasons, mainly most likely for their need of specific surface functionalization toward specific application or researchdriven size or shape requirements. We are not going to summarize these synthetic efforts here, largely because, as you can easily imagine, multiple review articles heretofore did exactly that already (Alexandridis 2011; Crooks et al. 2001; Ganguli et al. 2010; Gopidas et al. 2003; Grzelczak et al. 2008; Lohse and Murphy 2013; Lu et al. 2009; Mourdikoudis and LizMarzan 2013; Shan and Tenhu 2007; Walther and Mueller 2013; Zhao et al. 2013; Zhou et al. 2009).
The goal of this compendium of gold nanoparticle tables, which list and compare models to more precisely calculate size and composition, is to be there for the experimentalist once the synthesis is done, and when the characterization of the just prepared precious gold nanoparticles begins. Several groups have recently shown that precise nanoclusters (magicnumbered or not) can be made exclusively, or isolated from batches with initially wider size and shape distribution, with great reproducibility (vide infra). These synthesis pathways become more and more refined, as indicated by the increasing number of articles describing new clusters. Aided by highresolution Xray diffraction, mass spectrometry, singleparticle (a combination of low dose and aberrationcorrected) transmission electron microscopy (TEM) (Azubel et al. 2014), electrophoretic mobility calculations and electromigration (Pyell 2010), thermogravimetric analysis (TGA), elemental analysis, NMR, smallangle Xray scattering, as well as an array of surface characterization techniques (Auger, AFM, XPS, etc.) (Baer et al. 2010), these gold nanoclusters can now be fully characterized and their composition unambiguously determined. However, most laboratories and research endeavors do not require the rigor and use of welldefined gold nanoclusters. In these cases, average size and welldefined surface chemistries are more critical as is the determination of the overall, yet average composition for a gold nanoparticle sample with a given size and likely shape distribution. The functions these nanoparticles need to perform, for example as plasmonic additives, in drug delivery, in biosensing, as surfaceenhanced Raman probes, among many others, do nevertheless require a precise knowledge of the nanoparticle composition. Reproducibility is a great concern for biological and medical applications as well as various other uses in device technologies, affecting performance, reliability, and last but not least intellectual property (IP). To assist in this process and create a practical goto guide to more precisely determine the core and, in part, ligand shell composition of synthesized nanoparticles, we collected and calculated compositions and best approximations and assembled these datasets based on the overall nanocluster shape. With more and more refined and higherresolution transmission electron microscopy (TEM) instrumentation available on the market, experimentalists should be in a position to more accurately determine their nanoparticle core composition using the nanoparticle shape revealed by TEM and using the datasets and calculations collected in the tables to come.
The ligand shell is slightly more complicated. Thiolateprotected gold nanoparticles and nanoclusters dominate the literature by a large margin, and the presence of (RS–Au(I)–SR)^{−} and [RS(Au(I)–SR)_{2}]^{−} “staple” and bridge motifs (Pensa et al. 2012) (better described as Au(0)thiyl surface bonding (Reimers et al. 2016)) largely governed by the synthesis method as well as the size and shape of the particle or cluster complicates a precise prediction or calculation of the full composition of a given thiolatecapped gold nanoparticle sample. With analytical methods such as NMR (before and after I_{2} decomposition; i.e., oxidation of thiolates to disulfides), TGA, Xray photoelectron spectroscopy (XPS) to the rescue, this hurdle can be overcome once the nanoparticle core composition is determined with some degree of precision.
First, however, we will provide an overview of the various polyhedral shapes relevant for gold nanoparticles. Most gold nanoparticles assumed to be quasispherical are in fact Platonic, Archimedean, or Catalan solids. Polyhedral gold nanoclusters are classified as icosahedra and facecentered cubic (fcc) polyhedra. The stable Ino’s and Marks’ decahedra are nonspherical shapes and are best described as ellipsoids. The icosahedral or Ino’s as well as Marks’ decahedralbased gold nanoclusters (with icosahedral structure considering the triangular faces and fcc structure when considering the rectangular faces) represent more molecularlike structures. The fccbased gold nanoparticles have more bulk (plasmonic) structures.
There are five platonic solids constructed by regular polygonal faces (Fig. 1), with tetrahedron, cube, and octahedron combined known as fcc unit cell substructures. Magicnumbered gold nanoclusters have regular icosahedral shapes.
Two or three regular polygonal faces are needed to construct Archimedean solids, and truncating Platonic solids can compose them. As a result, Archimedean solids that are truncated from either tetrahedron, cube, or octahedron have fcc structures; other Archimedean solids have icosahedral structures as graphically shown in Fig. 2.
The process of obtaining these Archimedean solids by truncation is graphically shown in Fig. 3. The remaining Catalan solids are defined as dual solids of Archimedean solids. As their faces are not regular polygonal, it is expected that the cores of metal clusters could not be Catalan solid structures; however, ligand shells of Archimedean metal clusters could have Catalan solid structures (Fig. 4).
Tables, models, and calculations
Spherical versus icosahedral model
The number of gold atoms N _{Au} is commonly calculated assuming a quasispherical gold nanoparticle shape using Eq. 1 (Leff et al. 1995):
where V _{Au} is the volume of the Au atom (V _{Au} = 17 Å^{3}), r and D are the radius and the diameter of the gold nanoparticle, respectively, and r _{Au} is the radius of the gold atom (r _{ Au } = 1.44 Å) with r = (2n + 1)r _{Au}, where n is the number of full gold atoms along the radius of the nanoparticle as shown in Fig. 5.
As the size of the gold nanoparticles change (decrease or increase), and polyhedral shapes of specific clusters are now increasingly synthetically accessible, the use of this simple model becomes, as we will see, more and more problematic. Figure 6 shows that with the progression from a larger to a smaller nanoparticle (or cluster) the assumption of a quasispherical nanoparticle leads to a larger and larger discrepancy in composition.
For the calculation we first introduce the radius of the circumscribed sphere for an icosahedron R _{cs}. For a Au_{55} cluster, this leads to R _{cs} = 7.2 Å and a diameter of D = 1.44 nm as shown in Fig. 7. For magicnumbered gold clusters, whose overall shape is best described as regular icosahedral, the following equations give the number of gold atoms in a regular icosahedron N _{ico} (Eq. 2), the radius of the circumscribed sphere for an icosahedron R _{cs} (Eq. 3), and the magic number M _{ N } (Eq. 4):
where V _{ico} is the volume of the icosahedron, V _{Au} the volume of the gold atom, L _{ico} the edge length of the icosahedron, and where R _{cs} = (2n + 1)r _{Au} as mentioned earlier. For the magicsized clusters with n = 1–5, this results in a composition of these clusters as shown in Fig. 8.
Table 1 lists the values and Fig. 9 graphically shows the obvious discrepancies between a quasispherical model and the regular icosahedral shape and the comparison of the number of gold atoms obtained for both shapes, where the values of N _{cs} (the number of gold atoms obtained from the radius of the circumscribed sphere) and N _{ico} (the number of gold atoms in the regular icosahedron) divided by the magic number M _{ N } should be close to 1 for a match between experiment and calculation. As one can see, this number quickly and rather drastically deviates when a quasispherical model is used. For example, for nanoparticles in the size range of 2–3 nm, which are frequently described in the literature, the quasispherical model overestimates the number of gold atoms by a factor of over 1.7.
The magnitude of deviation between the number of gold atoms in a nanoparticle or nanocluster varying with the use of either the quasispherical or more accurate polyhedral model largely depends on the shape of individual particles. Highresolution transmission electron microscopy (HRTEM) is nowadays more than capable of revealing precise nanoparticle shapes and sizes, particularly when coupled with TEM tomography. Exact calculation of the nanoparticle composition should therefore be rather straightforward using the equations for the various polyhedral shapes provided in Section S1 of the Electronic Supplementary Material (ESM). Table S2A expands on the comparison between the quasispherical model and models of many other polyhedral shapes that are, or could be, formed by gold (or other coinage metal) nanoparticles or clusters for a specific radius of the circumscribed sphere of R _{c} = 10.08 Å (related to a Au_{147} cluster with regular icosahedral shape). We again provide a measure of the goodness of fit between the quasispherical and the given polyhedral model by the ratio between the number of gold atoms obtained from each model N _{c}/N _{ v }, where N _{c} is again the number of gold atoms contained within the circumscribed sphere and N _{ v } is the number of gold atoms calculated from the volume of the polyhedron. For Catalan solids N _{ve} is the number of gold atoms calculated from the volume of the sphere with vertex radius (N _{ve} = V _{ Rv }/V _{Au}) calculated from the volume of the sphere with vertex radius in Catalan solids (V _{ Rv }). A related table showing the discrepancies between the quasispherical model and specific polyhedral shapes assuming a radius of the circumscribed sphere of R _{c} = 7.2 Å (related to a Au_{55} cluster with regular icosahedral shape) is given in the ESM (Section S2, Table S2B).
Decahedral model
Jiang et al. (2003) Now that we have general sense of the influence of the nanoparticle or nanocluster shape, we will look at specific and commonly found polyhedral nanocluster shapes and calculate the composition of the clusters depending on the specific subtype and size. Specifically, we will look at decahedra, Archimedean cubes, and Archimedean icosahedra. Table 2 provides a complete list of pentagonal decahedra, Ino’s decahedra, and Marks’ decahedra by generation (layers of gold atoms around the center atoms) giving the number of gold atoms at the surface, the total number of gold atoms of the cluster, the parent cluster that is covered with another layer of gold atoms as well as their calculated heights and widths. Pentagonal decahedra are composed of ten faces of icosahedra. Ino’s and Marks’ decahedra are created by truncating pentagonal decahedra. Thus, these decahedra have icosahedral structures on triangular faces and fcc structures on rectangular faces.
A more condensed view of these values is given in Table 3, also providing additional generation 4 (G4) clusters. The number of gold atoms calculated assuming an ellipsoidal shape (N _{ e }) of the overall cluster is given in Eq. 5 (h, r _{Au}, and w are defined in Table 2).
Divided by the precise number of gold atoms in the cluster, the ratio of N _{ e }/N shows how close an elliptical particle shape assumption would be as the size of the cluster increases, especially in the absence of highresolution TEM images or Xray diffraction data that would allow the experimentalist to deduce the exact particle shape and composition.
Archimedean icosahedra model
Table 4 lists the same information for Archimedean icosahedra starting with the smallest, firstgeneration (G1) Au_{13} cluster. Among them, we also find several of the magicsized gold nanoclusters with icosahedral shape such as Au_{13}, Au_{55}, Au_{147}, among others, as shown in Fig. 9.
The ratio of the number of gold atoms between calculated and ideal cluster in these Archimedean icosahedra (Table S3, Section S3) shows how the quasispherical model (using the diameter of the circumscribed (N _{c}), midscribed (N _{m}) or inscribed diameter (N _{i}) sphere), or using the number of gold atoms deduced from a polyhedral model (N _{ v }) deviates from the correct number of gold atoms for these clusters.
Archimedean cube model
Table 5 finally shows a list of Archimedean cubes from generation 1 to 6 (G1–G6). Again, the ratio of the number of gold atoms between calculated and ideal cluster in these Archimedean cubes (Table S4, Section S4) shows how quasispherical models (using the diameter of the circumscribed (N _{c}), midscribed (N _{m}) or inscribed diameter (N _{i}) sphere), or using the number of gold atoms deduced from a polyhedral model (N _{ v }) deviate from the correct number of atoms.
Other models: defects, shell structures, and “staples”
Thus far, we have provided calculations and models for regular polyhedral shapes of gold nanoparticles. However, these models do not include specifics of shell structures (the outer layer of gold atoms involved in thiolate ligand binding), but we can estimate the number of goldthiolate “staple” and bridge motifs by calculating the ligand density at the gold nanoparticle surface.
Particularly, the cores of chiral gold nanoparticles consist of nonregular polyhedral structures and usually exhibit defects in their crystal structures (Chen et al. 2015; Dolamic et al. 2012; Kimura et al. 2009; LeviKalisman et al. 2011; LopezAcevedo et al. 2010; Pei et al. 2009; Pelayo et al. 2015; Qian and Jin 2009; Takagi et al. 2015; TlahuiceFlores et al. 2013a, b; Weissker et al. 2014; Zeng et al. 2015). Such nonregular polyhedral as well as defect structures of nanoparticle cores are generally the origin for the observed nanoparticle chirality. The core structures of several prominent chiral gold nanoparticles are summarized in Table 6.
Particularly, the cores of the smaller chiral gold nanoparticles are composed of connected regular polyhedra such as continuous tetrahedra and/or icosahedra. The core of Au_{68}(3MBA)_{50} features 50 gold atoms formed from an Archimedean icosahedral structure (icosidodecahedron) with defects (Pelayo et al. 2015). Similarly, the cores of both Au_{102}(pMBA)_{32} and Au_{133}(SPhp–tBu)_{52} were formed from rhombicosidodecahedron also with defects (Chen et al. 2015; LeviKalisman et al. 2011; Pelayo et al. 2015; Zeng et al. 2015). The core structure of Au_{68}(3MBA)_{50} appears to be most closely related either to an Ino’s decahedron with defects or to a nonregular polyhedral Au_{53} as shown in Fig. 10 (Pelayo et al. 2015). Several other chiral gold nanoparticles have regular polyhedra cores (entries highlighted by # in Table 6). For example, the core of Au_{144}(SR_{3})_{60} formed from Au_{114} (Qian and Jin 2009; TlahuiceFlores et al. 2013a) finds its best match in the Archimedean icosahedra model as rhombicosidodecahedron Au_{115} in Table 4.
To estimate the number of gold atoms on the surface we have to calculate the ligand density (ρ _{L}). Table 7 shows the surface area of the nanoparticle core (S _{c}) as well as the ligand density for gold nanoparticles. The ligand density of [Au_{25}(S–CH_{2}CH_{2}Ph)_{18}]^{−} and Au_{144}(S–CH_{3})_{60} with Archimedean icosahedra cores, calculated using simple Eq. 6 (N _{L} is the number of thiolate ligands), is close to 15 Å^{2}, which is equal to the maximum packing density of thiolates on gold nanoparticle (Kimura et al. 2009).
The Au_{144}(SCH_{3})_{60} cluster reported by Jin et al. (Qian and Jin 2009) has 30–S–Au–S–“staple” motifs with 30 gold and 60 sulfur atoms within the shell structure (Weissker et al. 2014). Using the same approach, we calculated that the ligand density of the Au_{24}(Sadamantane)_{16} cluster(Pelayo et al. 2015) with an Archimedean cube core was 19 Å^{2}, similar to the surface area of thiols calculated for planar gold surfaces (i.e., selfassembled monolayers on gold, SAMs (Love et al. 2005)) at 21 Å^{2}. This suggests that the surface of cores with Archimedean cube structure would act more like a bulk gold surface than cores with Archimedean icosahedra shape that are more faceted, which makes sense.
The cores of Au_{54}(S–C_{18}H_{37})_{30} and Au_{55}(S–C_{18}H_{37})_{31} were formed from Ino’s decahedron 39 (Negishi et al. 2012; Tsunoyama et al. 2010). The ligand density of Ino’s decahedron 39 (ρ _{L} = 18.2 Å^{2}) is situated between the thiol ligand density of Archimedean icosahedra gold nanoparticles (ρ _{L} = 15 Å^{2}) and flat gold surfaces (ρ _{L} = 21 Å^{2}). The reason for this is that the decahedra surfaces are formed from a combination of icosahedral core (particle like) on triangular faces and cubic core (bulk like) on rectangular faces.
The core structure of the Au_{187}(S–C_{12}H_{25}) ^{15}_{68} has been elucidated by density functional theory (DFT) calculations as a Marks’ decahedron Au_{153} (TlahuiceFlores 2015). It is conceivable that Au_{187}(SC_{12}H_{25})_{68} clusters have either a truncated cuboctahedron 135 or a cuboctahedron 147 core structure considering the models listed in Table 5. The ligand densities of the Au_{187}(S–C_{12}H_{25})_{68} cluster were calculated for each possible core (Table 7).
The ligand densities of a Marks’ decahedral Au_{153}, a truncated cuboctahedron 135, and a cuboctahedron 147 amount to 16.6, 14.1, and 15.9 Å^{2}, respectively. The values of the ligand densities obtained for the truncated cuboctahedron 135 and the cuboctahedron 147 suggest that thiolate ligands are more tightly packed on these clusters than thiolates on gold SAMs (21 Å^{2}). Thus, the core of the Au_{187}(S–C_{12}H_{25})_{68} should be neither a truncated cuboctahedron 135 nor a cuboctahedron 147, because the ligand density of a particle core with Archimedean cube structure should be closer to the surface area of thiolates on a flat gold SAM. Thus, the core structure of the Au_{187}(S–C_{12}H_{25})_{68} should be based on a Marks’ decahedral Au_{153} as determined by the authors experimentally.
Qian et al. (2012) reported on the core of a Au_{333}(SCH_{2}CH_{2}Ph)_{79} cluster formed from fcc Au_{293}. We can suggest other possible core structures from the models summarized in the tables. The ligand densities of cores with Archimedean cube structure such as subtruncated cuboctahedron 297, cuboctahedron 309, subtruncated octahedron 314 were calculated to be 22 Å^{2}, closely matching with the surface area of thiolates on gold SAMs. The ligand densities of cores with decahedral structure such as Ino’s decahedron 309 and Marks’ decahedron 318 are 15 Å^{2}, which is close to that of the Au_{187}(S–C_{12}H_{25})_{68} cluster formed from a Marks’ decahedral Au_{153}. The authors considered subtruncated cuboctahedron 297 or cuboctahedron 309 as core structure of the Au_{333}(S–CH_{2}CH_{2}Ph)_{79} cluster, but a subtruncated octahedron 314, an Ino’s decahedron 309, and a Marks’ decahedron 318 should be reconsidered as the most likely core structures based on ligand density values. These examples show how the tables, calculations, and consideration of ligand densities can be used to determine the core structure of gold nanoparticles. The following part will provide a quick howto guide.
Using the tables
Now that we have the tabulated data for the various models, it is time to put them to the test. First, we provide a pointbypoint procedure how to use these tables for a given nanoparticle sample. Several examples can be found in the ESM (Section S5). Equipped with TEM images (even better highresolution TEM images or TEM tomography data) that should allow the experimentalist to determine the shape(s) or closest match to one of the regular polyhedra, Archimedean icosahedra, Archimedean cubes, Ino’s or Marks’ decahedra, the following steps should lead to a close match between real and calculated core composition (a simplified flowchart of this procedure is shown in Fig. 11):

1.
Zoom into the TEM image showing isolated, nonaggregated gold nanoparticles as much as possible without sacrificing (shape) resolution,

2.
Using your imaging software (with analytical capabilities) measure the nanoparticle shape’s features such as the various circumscribed, midscribed, or inscribed diameters (or radii),

3.
Find the closest match in all provided tables (if the particular size is not listed, use the equations provided in Tables 2, 4, and 5), focusing particularly on those matching most closely the specific shape(s) visible in your TEM images (if you see a specific polygonal faces, consider the values in the square brackets for the specific polygon),

4.
Once this search is narrowed to the closest match, compare the N _{ x }/N values, with the lowest number giving the best match (N _{ x } stands for: N _{ve}, N _{c}, N _{ m }, N _{i}, or N _{ e } or N _{ v } which is the number of gold atoms calculated from the volume of a polyhedron with the radius of an edgescribed sphere, from a volume of a sphere with circumscribed diameter, from a volume of a sphere with midscribed diameter, from a volume of a sphere with inscribed diameter, from the volume of an elliptically shaped particle, or the volume of a given regular polyhedron, respectively),

5.
The closest match between shape and the lowest number of N _{ x }/N should give the nearest composition for the gold nanoparticle (nanocluster) core composition, and finally,

6.
Use this number (n) for the Au_{n} particle (or numbers if multimodal size distribution is observed), compare to the quasispherical model (Eq. 1) to see discrepancy, and elucidate full composition including ligand shell (number of thiolates) using methods including, but not limited to NMR, elemental analysis and TGA. Consider arguments of ligand density as described in the previous section.
In Figs. 12, 13, and 14, we also provide nanocluster generation trees for decahedra, Archimedean icosahedra, and Archimedean cubes, which should help understand connections between clusters and core structures as well as facilitate making the most suitable choices when analyzing TEM images.
Discussion
As one can appreciate, the approach pursued here, culminating in the presented nanoparticle tables, equations, and models, solely relies on geometrical considerations not precise electronic and atomic structural information, which are only accessible for welldefined clusters whose structure was solved by Xray diffraction (Chen et al. 2015; Jadzinsky et al. 2007; Zeng et al. 2014, 2015) or highresolution singleparticle TEM (Azubel et al. 2014) (aberrationcorrected TEM, vide supra). Numerous groups have made significant and important progress in applying density functional theory and other numerical as well as computational approaches to determine gold nanoparticle sizes, structures, and energetics (Barnard 2010; Barnard and Chen 2011; Barnard and Curtiss 2006; Barnard et al. 2009; McKenna 2009; Negishi et al. 2015) and focus increasingly on the very challenging task of elucidating the structure of the thiolate ligand shell (Barnard 2013; Xu et al. 2015). A closer look at these modeling and simulation data on various gold nanoparticle sizes and shapes, however, reveals that the current geometrically derived data tables and models capture these and that the implementation of both approaches endows experimental scientists with a powerful tool for the elucidation of nanoparticle composition. In addition, the geometrical models can much faster survey a greater number of nanoparticles and nanoclusters (including nanoparticles with larger diameters and many more gold atoms in the core) much faster. Practically speaking, the presented equations and tables are easily adjustable (via the radius of the metal atom) for the determination of the composition of other metal nanoparticles with fcc lattices (Pd, Pt, Ni (Lin et al. 2011) as well as coinage metals Ag, Cu); perhaps even alloytype metal nanoparticles if elemental composition is determined upfront. The predicted sequences of preferred shapes by Guisbiers et al. should here be tremendously helpful (Guisbiers et al. 2014). Of the 130 + clusters included in the current tables and models, several have not been experimentally observed for gold nanoparticles as of yet, and specific shapes observed for other transition metals are not included (e.g., tetrahedral for Pd nanoparticles (Barnard 2012)). Expansion to other shapes is a focus of future work, and numerical and theoretical methods recently presented by Barnard et al. will be used as guide for metal nanorods (Gonzalez et al. 2013). Finally, the section on shell structures is currently limited to the most frequently used thiolates and not considering other ligand motifs such as amines or phosphines among others. Thinking about the vast number of thiolate ligands reported in the literature, steric considerations are extremely difficult to include in any model system (Burgi 2015; Hakkinen 2012), especially since more and more sophisticated functions expected from gold nanoparticles require functional ligands with, for example, luminescent properties, binding capabilities to proteins, chirality, drug delivery, and many more.
Conclusions
Centered specifically around geometrical considerations, this compendium of tables, models, and equations serves as an easytouse, straightforward guide for experimental scientists synthesizing thiolprotected gold nanoparticles in the laboratory to assist them in calculating the nanoparticle composition based on geometric information deduced from TEM imaging and image analysis. The majority of research thrusts and applications focusing on thiolprotected metal nanoparticles do not require the precision of welldefined metal clusters, although synthetic approaches to obtain such clusters are tirelessly pursued and refined. Nevertheless, predicting and analytically confirming the composition of all other metal nanoparticles as accurately as possible is critical for fundamental and applied research alike. A nanoparticle’s composition significantly affects its properties and defines its function, irrespective of its use in applications ranging from drug delivery and cancer diagnostics to metamaterials and chiral discriminators. With the anticipated transformation of these tables, models, and equations to a webbased tool (that would also permit viewing of model clusters from various perspectives), we trust that experimental scientists will be provided with an invaluable, helpful, and expandable tool for the elucidation of metal nanoparticle compositions.
Abbreviations
 L :

Edge length of polyhedron
 V :

Volume of polyhedron
 S :

Surface area of polyhedron
 R _{c} :

Circumscribed radius
 R _{m} :

Midscribed radius
 R _{i} :

Inscribed radius
 V _{ Rc} :

Volume of sphere with circumscribed radius (R _{c})
 V _{ Rm } :

Volume of sphere with midscribed radius (R _{m})
 V _{ Ri } :

Volume of sphere with inscribed radius (R _{ r })
 S _{ Rc } :

Surface area of sphere with circumscribed radius (R _{c})
 S _{ Rm } :

Surface area of sphere with midscribed radius (R _{m})
 S _{ Ri } :

Surface area of sphere with inscribed radius (R _{ r })
 a :

Edge length of dual solid in Catalan solids
 R _{v} :

Vertex radius
 R _{e} :

Edgescribed radius
 R _{i} :

Inscribed radius
 V _{ Rv } :

Volume of sphere with vertex radius (R _{v})
 V _{ Re } :

Volume of sphere with edgescribed radius (R _{e})
 V _{ Ri } :

Volume of sphere with inscribed radius (R _{ r })
 S _{ Rv } :

Surface area of sphere with vertex radius (R _{v})
 S _{ Re } :

Surface area of sphere with edgescribed radius (R _{e})
 S _{ Ri } :

Surface area of sphere with inscribed radius (R _{ r })
 n :

Generation of gold nanoparticle
 D :

Diameter of gold nanoparticle
 L :

Edge length of polyhedron (cluster)
 L _{ ico } :

Edge length of icosahedron
 r _{Au} :

Radius of gold atom = 1.44 Å
 R _{ cs } :

Circumscribed radius for an icosahedron
 R _{c} :

Circumscribed radius
 R _{m} :

Midscribed radius
 R _{i} :

Inscribed radius
 D _{c} :

Circumscribed diameter = 2R _{c}
 D _{m} :

Midscribed diameter = 2R _{m}
 D _{i} :

Inscribed diameter = 2R _{i}
 V _{NP} :

Volume of gold nanoparticle
 V _{Au} :

Volume of gold atom = 17 Å^{3}
 V _{ico} :

Volume of icosahedron
 N _{cs} :

Number of gold atoms in a circumscribed sphere for an icosahedron
 N _{ico} :

Number of gold atoms in a regular icosahedron
 M _{ N } :

Magic number for gold cluster
 V :

Volume of polyhedron (cluster)
 a :

Edge length of dual solid in Catalan solids
 R _{v} :

Vertex radius
 R _{e} :

Edgescribed radius
 N _{ Au } :

Number of gold atoms calculated assuming a quasispherical gold nanoparticle shape
 N _{ve} :

Number of gold atoms calculated from volume of sphere with vertex radius (N _{ve}) = V _{ Rv }/V _{Au}
 N _{es} :

Number of gold atoms calculated from volume of sphere with edgescribed radius (N _{es}) = V _{ Re }/V _{Au}
 N :

Number of gold atoms in cluster
 N _{ v } :

Number of gold atoms calculated from volume of polyhedron = V/V _{Au} (V _{Au}: volume of gold atom = 17 Å^{3})
 N _{c} :

Number of gold atoms calculated from volume of sphere with circumscribed radius (R _{c}) = V _{ Rc }/V _{Au}
 N _{ m } :

Number of gold atoms calculated from volume of sphere with midscribed radius (R _{m}) = V _{ Rm }/V _{Au}
 N _{ i } :

Number of gold atoms calculated from volume of sphere with inscribed radius (R _{i}) = V _{ Ri }/V _{Au}
 S _{c} :

Surface area of core
 N _{L} :

Number of ligands on the core surface
 ρ _{ L } :

Ligand density
 h :

Height of decahedron
 w :

Width of decahedron
 N _{ e } :

Number of gold atom calculated from volume of ellipsoid
 S _{Pe} :

Surface area of pentagonal decahedron
 S _{Ino} :

Surface area of Ino’s decahedron
 S _{Ma} :

Surface area of Marks’ decahedron
References
Aillon KL, Xie Y, ElGendy N, Berkland CJ, Forrest ML (2009) Effects of nanomaterial physicochemical properties on in vivo toxicity. Adv Drug Deliv Rev 61:457–466. doi:10.1016/j.addr.2009.03.010
Alexandridis P (2011) Gold nanoparticle synthesis, morphology control, and stabilization facilitated by functional polymers. Chem Eng Technol 34:15–28. doi:10.1002/ceat.201000335
Almeida JPM, Chen AL, Foster A, Drezek R (2011) In vivo biodistribution of nanoparticles. Nanomedicine 6:815–835. doi:10.2217/nnm.11.79
Askim JR, Mahmoudi M, Suslick KS (2013) Optical sensor arrays for chemical sensing: the optoelectronic nose. Chem Soc Rev 42:8649–8682. doi:10.1039/c3cs60179j
Azubel M et al (2014) Electron microscopy of gold nanoparticles at atomic resolution. Science 345:909–912. doi:10.1126/science.1251959
Azzazy HME, Mansour MMH, Kazmierczak SC (2006) Nanodiagnostics: a new frontier for clinical laboratory medicine. Clin Chem 52:1238–1246. doi:10.1373/clinchem.2006.066654
Baer DR, Gaspar DJ, Nachimuthu P, Techane SD, Castner DG (2010) Application of surface chemical analysis tools for characterization of nanoparticles. Anal Bioanal Chem 396:983–1002. doi:10.1007/s0021600933601
Barnard AS (2010) Modelling of nanoparticles: approaches to morphology and evolution. Rep Prog Phys 73:086502. doi:10.1088/00344885/73/8/086502
Barnard AS (2012) Mapping the shape and phase of palladium nanocatalysts. Catal Sci Technol 2:1485–1492. doi:10.1039/c2cy20017a
Barnard AS (2013) Modeling the impact of alkanethiol SAMs on the morphology of gold nanocrystals. Cryst Growth Des 13:5433–5441. doi:10.1021/cg401397y
Barnard AS, Chen Y (2011) Kinetic modelling of the shapedependent evolution of faceted gold nanoparticles. J Mater Chem 21:12239–12245. doi:10.1039/c1jm11677k
Barnard AS, Curtiss LA (2006) Predicting the shape and structure of facecentered cubic gold nanocrystals smaller than 3 nm. ChemPhysChem 7:1544–1553. doi:10.1002/cphc.200600107
Barnard AS, Young NP, Kirkland AI, van Huis MA, Xu HF (2009) Nanogold: a quantitative phase map. ACS Nano 3:1431–1436. doi:10.1021/nn900220k
BenMoshe A, Maoz BM, Govorov AO, Markovich G (2013) Chirality and chiroptical effects in inorganic nanocrystal systems with plasmon and exciton resonances. Chem Soc Rev 42:7028–7041
Bhattacharyya S, Kudgus RA, Bhattacharya R, Mukherjee P (2011) Inorganic nanoparticles in cancer therapy. Pharm Res 28:237–259. doi:10.1007/s1109501003180
Bishop KJM, Wilmer CE, Soh S, Grzybowski BA (2009) Nanoscale forces and their uses in selfassembly. Small 5:1600–1630. doi:10.1002/smll.200900358
Boeker A, He J, Emrick T, Russell TP (2007) Selfassembly of nanoparticles at interfaces. Soft Matter 3:1231–1248. doi:10.1039/b706609k
Brust M, Walker M, Bethell D, Schiffrin DJ, Whyman R (1994) Synthesis of thiolderivatised gold nanoparticles in a twophase liquid–liquid system. J Chem Soc Chem Commun. doi:10.1039/C39940000801
Brust M, Fink J, Bethell D, Schiffrin DJ, Kiely C (1995) Synthesis and reactions of functionalized gold nanoparticles. J Chem Soc Chem Commun. doi:10.1039/C39950001655
Burgi T (2015) Properties of the goldsulphur interface: from selfassembled monolayers to clusters. Nanoscale 7:15553–15567. doi:10.1039/c5nr03497c
Chen Y et al (2015) Crystal structure of barrelshaped chiral Au130(pMBT)50 nanocluster. J Am Chem Soc 137:10076–10079
Chikkaveeraiah BV, Bhirde AA, Morgan NY, Eden HS, Chen X (2012) Electrochemical immunosensors for detection of cancer protein biomarkers. ACS Nano 6:6546–6561. doi:10.1021/nn3023969
Crooks RM, Zhao MQ, Sun L, Chechik V, Yeung LK (2001) Dendrimerencapsulated metal nanoparticles: synthesis, characterization, and applications to catalysis. Acc Chem Res 34:181–190. doi:10.1021/ar000110a
Dolamic I, Knoppe S, Dass A, Burgi T (2012) First enantioseparation and circular dichroism spectra of Au38 clusters protected by achiral ligands. Nat Commun 3:798. doi:10.1038/ncomms1802
Dreaden EC, Mackey MA, Huang X, Kang B, ElSayed MA (2011) Beating cancer in multiple ways using nanogold. Chem Soc Rev 40:3391–3404. doi:10.1039/c0cs00180e
Ganguli AK, Ganguly A, Vaidya S (2010) Microemulsionbased synthesis of nanocrystalline materials. Chem Soc Rev 39:474–485. doi:10.1039/b814613f
Gautier C, Bürgi T (2009) Chiral gold nanoparticles. ChemPhysChem 10:483–492
Gindy ME, Prud’homme RK (2009) Multifunctional nanoparticles for imaging, delivery and targeting in cancer therapy. Expert Opin Drug Deliv 6:865–878. doi:10.1517/17425240902932908
Gonzalez AL, Noguez C, Barnard AS (2013) Mapping the structural and optical properties of anisotropic gold nanoparticles. J Mater Chem C 1:3150–3157. doi:10.1039/c3tc30313f
Gopidas KR, Whitesell JK, Fox MA (2003) Nanoparticlecored dendrimers: synthesis and characterization. J Am Chem Soc 125:6491–6502. doi:10.1021/ja029544m
Grzelczak M, PerezJuste J, Mulvaney P, LizMarzán LM (2008) Shape control in gold nanoparticle synthesis. Chem Soc Rev 37:1783–1791. doi:10.1039/b711490g
Grzelczak M, Vermant J, Furst EM, LizMarzán LM (2010) Directed selfassembly of nanoparticles. ACS Nano 4:3591–3605. doi:10.1021/nn100869j
GuerreroMartínez A, AlonsoGómez JL, Auguié B, Cid MM, LizMarzán LM (2011) From individual to collective chirality in metal nanoparticles. Nano Today 6:381–400
Guisbiers G, MeijaRosales S, Khanal S, RuizZepeda F, Whetten RL, JoseYacaman M (2014) Gold copper nanoalloy, “Tumbaga”, in the era of nano: phase diagram and segregation. Nano Lett 14:6718–6726. doi:10.1021/nl503584q
Hakkinen H (2012) The goldsulfur interface at the nanoscale. Nat Chem 4:443–455. doi:10.1038/nchem.1352
Hegmann T, Qi H, Marx VM (2007) Nanoparticles in liquid crystals: synthesis, selfassembly, defect formation and potential applications. J Inorg Organomet Polym Mater 17:483–508. doi:10.1007/S1090400791405
Hou W, Cronin SB (2013) A review of surface plasmon resonanceenhanced photocatalysis. Adv Funct Mater 23:1612–1619. doi:10.1002/adfm.201202148
Howes PD, Chandrawati R, Stevens MM (2014) Colloidal nanoparticles as advanced biological sensors. Science 346:53. doi:10.1126/science.1247390
Jadzinsky PD, Calero G, Ackerson CJ, Bushnell DA, Kornberg RD (2007) Structure of a thiol monolayerprotected gold nanoparticle at 1.1 Å resolution. Science 318:430–433
Jain S, Hirst DG, O’Sullivan JM (2012) Gold nanoparticles as novel agents for cancer therapy British. J Radiol 85:101–113. doi:10.1259/bjr/59448833
Jiang HY, Cai WS, Shao XG (2003) New lowest energy sequence of Marks’ decahedral LennardJones clusters containing up to 10,000 atoms. J Phys Chem A 107:4238–4243. doi:10.1021/jp0342327
Johnston HJ, Hutchison G, Christensen FM, Peters S, Hankin S, Stone V (2010) A review of the in vivo and in vitro toxicity of silver and gold particulates: particle attributes and biological mechanisms responsible for the observed toxicity. Crit Rev Toxicol 40:328–346. doi:10.3109/10408440903453074
Kennedy LC et al (2011) A new era for cancer treatment: goldnanoparticlemediated thermal therapies. Small 7:169–183. doi:10.1002/smll.201000134
Khlebtsov N, Dykman L (2011) Biodistribution and toxicity of engineered gold nanoparticles: a review of in vitro and in vivo studies. Chem Soc Rev 40:1647–1671. doi:10.1039/c0cs00018c
Kim HN, Ren WX, Kim JS, Yoon J (2012) Fluorescent and colorimetric sensors for detection of lead, cadmium, and mercury ions. Chem Soc Rev 41:3210–3244. doi:10.1039/c1cs15245a
Kimura K, Sugimoto N, Sato S, Yao H, Negishi Y, Tsukuda T (2009) Size determination of gold clusters by polyacrylamide gel electrophoresis in a large cluster region. J Phys Chem C 113:14076–14082
Lal S, Clare SE, Halas NJ (2008) Nanoshellenabled photothermal cancer therapy: impending clinical impact. Acc Chem Res 41:1842–1851. doi:10.1021/ar800150g
Leff DV, Ohara PC, Heath JR, Gelbart WM (1995) Thermodynamic control of gold nanocrystal size—experiment and theory. J Phys Chem 99:7036–7041. doi:10.1021/J100018a041
LeviKalisman Y, Jadzinsky PD, Kalisman N, Tsunoyama H, Tsukuda T, Bushnell DA, Kornberg RD (2011) Synthesis and characterization of Au102(pMBA)44 nanoparticles. J Am Chem Soc 133:2976–2982
Lin C, Liu Y, Rinker S, Yan H (2006) DNA tile based selfassembly: building complex nanoarchitectures. ChemPhysChem 7:1641–1647. doi:10.1002/cphc.200600260
Lin ZZ, Chen X, Yin C, Tang H, Hu YC, Ning XJ (2011) Theoretical prediction of the growth and surface structure of Pt and Ni nanoparticles. Europhys Lett 96:66005. doi:10.1209/02955075/96/66005
Lohse SE, Murphy CJ (2013) The quest for shape control: a history of gold nanorod synthesis. Chem Mater 25:1250–1261. doi:10.1021/cm303708p
LopezAcevedo O, Tsunoyama H, Tsukuda T, Aikens CM (2010) Chirality and electronic structure of the thiolateprotected Au38 nanocluster. J Am Chem Soc 132:8210–8218
Love JC, Estroff LA, Kriebel JK, Nuzzo RG, Whitesides GM (2005) Selfassembled monolayers of thiolates on metals as a form of nanotechnology. Chem Rev 105:1103–1169. doi:10.1021/cr0300789
Lu X, Rycenga M, Skrabalak SE, Wiley B, Xia Y (2009) Chemical synthesis of novel plasmonic nanoparticles. Ann Rev Phys Chem 60:167–192. doi:10.1146/annurev.physchem.040808.090434
McKenna KP (2009) Gold nanoparticles under gas pressure. Phys Chem Chem Phys 11:4145–4151. doi:10.1039/b821408p
Mourdikoudis S, LizMarzán LM (2013) Oleylamine in nanoparticle synthesis. Chem Mater 25:1465–1476. doi:10.1021/cm4000476
Mulder WJM, Strijkers GJ, Van Tilborg GAF, Cormode DP, Fayad ZA, Nicolay K (2009) Nanoparticulate assemblies of amphiphiles and diagnostically active materials for multimodality imaging. Acc Chem Res 42:904–914. doi:10.1021/ar800223c
Negishi Y, Sakamoto C, Ohyama T, Tsukuda T (2012) Synthesis and the origin of the stability of thiolateprotected Au130 and Au187 clusters. J Phys Chem Lett 3:1624–1628
Negishi Y et al (2015) A critical size for emergence of nonbulk electronic and geometric structures in dodecanethiolateprotected Au clusters. J Am Chem Soc 137:1206–1212
Ofir Y, Samanta B, Rotello VM (2008) Polymer and biopolymer mediated selfassembly of gold nanoparticles. Chem Soc Rev 37:1814–1823. doi:10.1039/b712689c
Panigrahi S et al (2007) Synthesis and sizeselective catalysis by supported gold nanoparticles: study on heterogeneous and homogeneous catalytic process. J Phys Chem C 111:4596–4605. doi:10.1021/jp067554u
Pei Y, Gao Y, Shao N, Zeng XC (2009) Thiolateprotected Au20(SR)(16) cluster: prolate Au8 core with new [Au3(SR)(4)] staple motif. J Am Chem Soc 131:13619–13621. doi:10.1021/ja905359b
Pelayo JJ, Whetten RL, Garzón IL (2015) Geometric quantification of chirality in ligandprotected metal clusters. J Phys Chem C 119:28666–28678
Pensa E et al (2012) The chemistry of the sulfurgold interface: in search of a unified model. Acc Chem Res 45:1183–1192. doi:10.1021/ar200260p
Perfezou M, Turner A, Merkoci A (2012) Cancer detection using nanoparticlebased sensors. Chem Soc Rev 41:2606–2622. doi:10.1039/c1cs15134g
Pingarron JM, YanezSedeno P, GonzalezCortes A (2008) Gold nanoparticlebased electrochemical biosensors. Electrochim Acta 53:5848–5866. doi:10.1016/j.electacta.2008.03.005
Pyell U (2010) Characterization of nanoparticles by capillary electromigration separation techniques. Electrophoresis 31:814–831. doi:10.1002/elps.200900555
Qi H, Hegmann T (2008) Impact of nanoscale particles and carbon nanotubes on current and future generations of liquid crystal displays. J Mater Chem 18:3288–3294. doi:10.1039/B718920f
Qian HF, Jin RC (2009) Controlling nanoparticles with atomic precision: the case of Au144(SCH2CH2Ph)(60). Nano Lett 9:4083–4087. doi:10.1021/nl902300y
Qian H, Zhu Y, Jin R (2012) Atomically precise gold nanocrystal molecules with surface plasmon resonance. Proc Natl Acad Sci USA 109:696–700
Reimers JR, Ford MJ, Halder A, Ulstrup J, Hush NS (2016) Gold surfaces and nanoparticles are protected by Au(0)–thiyl species and are destroyed when Au(I)–thiolates form. Proc Natl Acad Sci USA 113:E1424–E1433. doi:10.1073/pnas.1600472113
Rosi NL, Mirkin CA (2005) Nanostructures in biodiagnostics. Chem Rev 105:1547–1562. doi:10.1021/cr030067f
Ross MB, Mirkin CA, Schatz GC (2016) Optical properties of one, two, and threedimensional arrays of plasmonic nanostructures. J Phys Chem C 120:816–830. doi:10.1021/acs.jpcc.5b10800
Sarina S, Waclawik ER, Zhu H (2013) Photocatalysis on supported gold and silver nanoparticles under ultraviolet and visible light irradiation. Green Chem 15:1814–1833. doi:10.1039/c3gc40450a
Sepulveda B, Angelome PC, Lechuga LM, LizMarzan LM (2009) LSPRbased nanobiosensors. Nano Today 4:244–251. doi:10.1016/j.nantod.2009.04.001
Shan J, Tenhu H (2007) Recent advances in polymer protected gold nanoparticles: synthesis, properties and applications. Chem Commun. doi:10.1039/b707740h
Shivakumar U, Mirzaei J, Feng X, Sharma A, Moreira P, Hegmann T (2011) Nanoparticles: complex and multifaceted additives for liquid crystals. Liq Cryst 38:1495–1514. doi:10.1080/02678292.2011.605477
Stamatoiu O, Mirzaei J, Feng X, Hegmann T (2012) Nanoparticles in liquid crystals and liquid crystalline nanoparticles. Top Curr Chem 318:331–393. doi:10.1007/128_2011_233
Stewart ME, Anderton CR, Thompson LB, Maria J, Gray SK, Rogers JA, Nuzzo RG (2008) Nanostructured plasmonic sensors. Chem Rev 108:494–521. doi:10.1021/cr068126n
Takagi N et al (2015) How can we understand Au8 cores and entangled ligands of selenolate and thiolateprotected gold nanoclusters Au24(ER)20 and Au20(ER)16 (E = Se, S; R = Ph, Me)? A theoretical study. J Am Chem Soc 137:8593–8602
TlahuiceFlores A (2015) New insight into the structure of thiolated gold clusters: a structural prediction of the Au187(SR)(68) cluster. Phys Chem Chem Phys 17:5551–5555. doi:10.1039/c4cp05695g
TlahuiceFlores A, Black DM, Bach SBH, JoseYacaman M, Whetten RL (2013a) Structure & bonding of the goldsubhalide cluster IAu144Cl60[z]. Phys Chem Chem Phys 15:19191–19195. doi:10.1039/c3cp53902d
TlahuiceFlores A, JoseYacaman M, Whetten RL (2013b) On the structure of the thiolated Au15 cluster. Phys Chem Chem Phys 15:19557–19560. doi:10.1039/c3cp53837k
Tsunoyama R, Tsunoyama H, Pannopard P, Limtrakul J, Tsukuda T (2010) MALDI mass analysis of 11 kDa gold clusters protected by octadecanethiolate ligands. J Phys Chem C 114:16004–16009
Tweney RD, Mears RP, Spitzmüller C (2004) Replicating the practices of discovery: Michael Faraday and the interaction of gold and light. In: Gorman M, Tweney RD, Gooding D, Kincannon A (eds) Scientific and technological thinking. Lawrence Erlbaum Associates, Mahwah, NJ, pp 137–158
Walther A, Mueller AHE (2013) Janus particles: synthesis, selfassembly, physical properties, and applications. Chem Rev 113:5194–5261. doi:10.1021/cr300089t
Wang M, Thanou M (2010) Targeting nanoparticles to cancer. Pharmacol Res 62:90–99. doi:10.1016/j.phrs.2010.03.005
Wang Y, Xu J, Wang Y, Chen H (2013) Emerging chirality in nanoscience. Chem Soc Rev 42:2930–2962. doi:10.1039/c2cs35332f
Weissker HC et al (2014) Information on quantum states pervades the visible spectrum of the ubiquitous Au144(SR)(60) gold nanocluster. Nat Commun 5:3785. doi:10.1038/ncomms4785
Wolinsky JB, Grinstaff MW (2008) Therapeutic and diagnostic applications of dendrimers for cancer treatment. Adv Drug Deliv Rev 60:1037–1055. doi:10.1016/j.addr.2008.02.012
Xia Y, Zhou Y, Tang Z (2011) Chiral inorganic nanoparticles: origin, optical properties and bioapplications. Nanoscale 3:1374–1382
Xu WW, Gao Y, Zeng XC (2015) Unraveling structures of protection ligands on gold nanoparticle Au68(SH)32. Sci Adv 1:e1400211. doi:10.1126/sciadv.1400211
Yong KT, Roy I, Swihart MT, Prasad PN (2009) Multifunctional nanoparticles as biocompatible targeted probes for human cancer diagnosis and therapy. J Mater Chem 19:4655–4672. doi:10.1039/b817667c
Zeng C, Chen Y, Li G, Jin R (2014) Magic Size Au64(ScC6H11)32 nanocluster protected by cyclohexanethiolate. Chem Mater 26:2635–2641
Zeng C, Chen Y, Kirschbaum K, Appavoo K, Sfeir MY, Jin R (2015) Structural patterns at all scales in a nonmetallic chiral Au133(SR)52 nanoparticle. Sci Adv 1:e1500045
Zhang Z, Wang J, Chen C (2013) Nearinfrared lightmediated nanoplatforms for cancer thermochemotherapy and optical imaging. Adv Mater 25:3869–3880. doi:10.1002/adma.201301890
Zhao P, Li N, Astruc D (2013) State of the art in gold nanoparticle synthesis. Coordin Chem Rev 257:638–665. doi:10.1016/j.ccr.2012.09.002
Zhou J, Ralston J, Sedev R, Beattie DA (2009) Functionalized gold nanoparticles: synthesis, structure and colloid stability. J Colloids Interface Sci 331:251–262. doi:10.1016/j.jcis.2008.12.002
Acknowledgments
This work was financially supported by the National Science Foundation (NSF, DMR1506018), the Ohio Third Frontier (OTF) program for Ohio Research Scholars “Research Cluster on Surfaces in Advanced Materials” (T.H.), and the Japan Society for the Promotion of Science (JSPS, postdoctoral scholarship for T.M.).
Author information
Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Mori, T., Hegmann, T. Determining the composition of gold nanoparticles: a compilation of shapes, sizes, and calculations using geometric considerations. J Nanopart Res 18, 295 (2016). https://doi.org/10.1007/s1105101635877
Received:
Accepted:
Published:
Keywords
 Gold nanoparticle
 Nanocluster
 Nanoparticle size
 Nanoparticle shape
 Nanoparticle composition
 Modeling and simulation