Spherical versus Faceted Anatase TiO2 Nanoparticles: A Model Study

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Spherical vs Faceted Anatase TiO Nanoparticles: A Model Study of Structural and Electronic Properties Gianluca Fazio, Lara Ferrighi, and Cristiana Di Valentin J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b06384 • Publication Date (Web): 12 Aug 2015 Downloaded from http://pubs.acs.org on August 18, 2015

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The Journal of Physical Chemistry

Spherical vs Faceted Anatase TiO2 Nanoparticles: A Model Study of Structural and Electronic Properties Gianluca Fazio, Lara Ferrighi, Cristiana Di Valentin*

Dipartimento di Scienza dei Materiali, Università di Milano Bicocca via R. Cozzi 55 20125 Milano Italy

Abstract TiO2 nanoparticles are fundamental building blocks of many TiO2-based technologies. However, most of the computational studies simulate either bulk or surface titania. Structural and electronic properties of nanoparticles are expected to differ much from extended systems. Moreover, nanoparticles of different size and shape may also present peculiar features. In this study we compare nanocrystals and nanospheres of various sizes (up to a diameter of 3 nm) in order to highlight analogies and differences. In particular, we focus the attention on the surface-to-bulk sites ratio, on the surface sites coordination distribution, on the atomic distortions or curvature and on the surface energies, from the structural point of view. Regarding the electronic properties, we investigate the difference between Kohn-Sham and fundamental gaps of these finite-sized systems, the frontiers orbitals space distribution, ionization potentials and electron affinities and finally the densities of states projected on the various coordination sites present in the nanoparticles. This detailed analysis proves that faceted and spherical nanoparticles present different structural and electronic properties which make each of them better suited for different uses and applications.

*

Corresponding author: [email protected]; +390264485235 ACS Paragon Plus Environment

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1. Introduction Titanium dioxide nanoparticles are fundamental building blocks in many technological applications,1 especially in those involving light irradiation and photochemical processes, such as photovoltaics, photoelectrochemistry and photocatalysis.2,3,4,5 However, more recently, titania nanoparticles have attracted also the interest of the biomedical community.6 Typically, TiO2 nanoparticles are grown via sol-gel synthesis.7 Several studies during the last few years have proven that the shape and size of titanium dioxide nanoparticles can be efficiently and successfully tailored by controlling the conditions of preparation and by using adhoc surface chemistry.8-12 Below a certain nanoparticle dimension, anatase phase is found to be more stable than rutile.13 The formation rate and the shape of anatase nanoparticles in aqueous solution are dependent on the pH value.14 Another growth determining factor is the particle density during synthesis: an excessive dilution may cause a partial dissolution of titania nanocrystals leading to the formation of spherical nanoparticles.6 The latter, analogously to nanotubes and nanorods, are characterized by a high curvature profile which is expected to trigger higher binding properties. In fact, high-curvature nanoparticles present many undercoordinated sites which are very reactive.6 Undercoordinated sites are particularly important for nanoparticles smaller than 20 nm since the fraction of surface atoms becomes comparable to that of bulk atoms. Surface-to-bulk aspect ratios intimately depend also on the shape of the nanoparticle. Additionally, the coordination and structure of Ti surface sites, which are the surface binding sites for molecular and adsorbate species, are expected to be highly affected by the size and the shape of the nanoparticles. Commonly, computational first-principles studies are devoted to either bulk of surface slabs of TiO2. The object of this work is to systematically investigate and compare structural and electronic properties of faceted and spherical anatase nanoparticles of different size in order to get insight into the intimate differences between these two commonly observed nanocrystals shapes. State-of-the-art density functional methodologies will be used in order to accurately describe the quantum size effects on the electronic structure of the semiconducting oxide system. In particular, the hybrid functional HSE06,15 which is considered to be the best suited to describe semiconducting oxides, will be compared to the more popular, at least in the quantum chemical community, B3LYP.16,17 Faceted nanoparticles have been the object of a number of previous first-principles studies,18-24 with full relaxation for a maximum diameter size considered of 2.7 nm. Spherical nanoparticles have only been studied by means of force field approaches,25 which, however, lack any information on the electronic properties. Here we will consider nanocrystals/nanospheres diameter sizes up to 3 nm.

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The paper is organized as follows: in Section 2 the Computational Details are presented; in Section 3.1 we will discuss the Structural Properties of the nanoparticles in terms of shape (3.1a), size, morphology (3.1.b), distortions with respect to bulk (3.2.c) and surface energies (3.2.d); in Section 3.2 we will analyze the Electronic Properties of the nanoparticles in terms of Kohn-Sham vs fundamental gaps (3.2.a), frontier orbitals (3.2.b) and total and projected densities of states (3.2.c); Section 4 is devoted to a summary and to the conclusions of the work.

2. Computational Details All the calculations were performed with the CRYSTAL1426 package where the Kohn–Sham orbitals are expanded in Gaussian type orbitals (the all-electron basis sets are O 8-411(d1), Ti 86411 (d41) and H 511(p1)). Some test calculations have been performed with a diffuse basis set (Ti diffuse 86-411(d411), O diffuse 8-4111(d1), H diffuse 5111(p1)) on all the surface atoms and the hydrogen atoms, showing negligible effects which will be discussed in the following. The HSE0615 and B3LYP16,17 hybrid functionals have been used throughout this work. The values of the optimized lattice parameters are 3.766 Å and 3.789 Å for a and 9.663 Å and 9.777 Å for c, respectively for HSE06 and B3LYP (see Table 1 for experimental values).27 To describe the (101) surface, we used a slab of ten triatomic layers with 60-atoms and a unit cell periodicity along the [ 101 ] and [010] directions; no periodic boundary conditions were imposed in the direction perpendicular to the surface. The k-space sampling for the surface geometry optimization included 43 k-points. Nanoparticles have been treated as molecules in the vacuum without any periodic boundary conditions. Simulated total densities of states (DOS) of the nanoparticles have been obtained through the convolution of Gaussian peaks (σ = 0.01 eV) centered at the Kohn-Sham energy eigenvalue of each orbital. Projected densities of states (PDOS) have been obtained by using the coefficients in the linear combination of atomic orbitals (LCAO) of each molecular orbital: summing the squares of the coefficients of all the atomic orbitals centered on a certain atom type results, after normalization, in the relative contribution of each atom type to a specific eigenstate. Then, the various projections are obtained from the convolution of Gaussian peaks with heights that are proportional to the relative contribution. The zero energy for all the DOS is set to the vacuum level, corresponding to an electron at an infinite distance from the surface. Similarly, the extended X-ray adsorption fine structure (EXAFS) simulated spectra have been constructed with the Gaussian convolution of peaks (σ = 0.0005 Å) centred at the distance lengths between each Ti atom and other atoms (O or Ti) from its first, second and third coordination


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shells. Projections have been performed by taking into account groups of titanium atoms with the same coordination sphere. For the 3D plots of the molecular orbitals we have used an isovalue of 0.0005 a.u. for all the orbitals except for very localized ones for which we have used a 0.001 a.u. isovalue. Geometric surfaces of the nanocrystals have been designed as the smallest Wulff-shape decahedron, with a specific angle θ, uniquely determined by the cell parameters (θ = tan-1(c/a)), between (101) and (001) planes. We can then determine the values of the parameters A, B and H (see Figure 1) for that specific decahedron. The surface area, Sgeom, is then calculated as:

For nanospheres, the geometric surfaces are those of a sphere of radius RM which has been used to truncate the bulk. Connolly surfaces (SConn) for nanoparticles have been created using the algorithm developed by Connolly.28,29 Firstly, we build the surface resulting from the overlap of all the atomic Van der Waals spheres; secondly, a probe sphere of a certain chosen radius (i.e. 3.0 Å) is rolled on that surface and the contact points are used to form arcs which smooth the Van der Waals surface. The resulting surface is the Connolly surface.

3. Results and Discussion 3.1 Structural Properties 3.1.a Nanoparticles shapes Nanoparticles have been carved from the optimized anatase TiO2 bulk, as calculated with both the HSE06 and B3LYP functionals (see Table 1). A Ti atom was set at the origin of the coordinate axis and an overall D2d point group symmetry was kept when cutting the nanoparticle. Faceted nanoparticles or nanocrystals (NC) have been cut from the bulk anatase crystal according to the minimum energy shape predicted by Barnard et al.13 for dimensions below 10 nm, which is a decahedral shape where the two lowest energy anatase surfaces, (101) and (001), are exposed. The horizontal sides of the decahedron are defined A and B with A>B (see Figure 1). A/B ratio was determined to be typically around 0.32.30 When cutting the nanocrystal, excess atoms were removed and monocoordinated oxygen atoms were saturated with H atoms. This sequence of operations resulted in stoichiometric nanocrystals saturated with few dissociated water molecules, as described in Figure 1: (TiO2)159 • 4 H2O (NCS) and (TiO2)260 • 6 H2O (NCL). Spherical nanoparticles or nanospheres (NS) have been obtained by carving a sphere of radius RM from the bulk anatase crystal. Then, all the two-fold Ti atoms on the surface were removed, while three- and four-fold Ti atoms were coordinated to hydroxyl groups. Analogously, ACS Paragon Plus Environment

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monocoordinated O atoms were saturated with H atoms. These operations result in the creation of stoichiometric nanospheres saturated with a bunch of water molecules, as described in Figure 1: (TiO2)223 • 26 H2O (NSS) and (TiO2)399 • 32 H2O (NSL). 3.1.b Nanoparticles size and morphology A representation of the nanocrystals and nanospheres together with their stoichiometry and coordination data are provided in Figure 1, as well as in Table 2 and 3. In particular, in Figure 1 the surface-to-bulk % ratio for each nanobject is indicated and the sketch representations for the parameters (A, B, H and RM) defining the size of the nanostructures are shown. Further details on the spatial dimensions are provided in Table 2 for both the HSE06 and B3LYP functionals. We may notice that B3LYP calculations produce somewhat larger nanoparticles, in particular, regarding the NCs, the dimension along the z-axis is slightly elongated, in line with a longer lattice parameter c (see Table 1). In Table 2, the nanoparticles surface (SConn) and volume (VConn) as determined with the Connolly method28,29 (see Computational Details) are reported. This type of surface and its inner volume are conceived for three dimensional molecular objects which involve protruding atoms and take atomic Van der Waals radii into account. The equivalent diameter (DConn) is the diameter of a corresponding ideal sphere which has the same volume (VConn) of the nanoparticles. This parameter allows direct comparison between NCs and NSs. Noteworthy is that the size, in terms of number of atoms and equivalent diameter, of NCL is about that of NSS. The average size of the nanoparticles is between 2 and 3 nm, which is consistent with real small nanocrystallites.31,32,33 The coordination/undercoordination of all the atomic species in the designed nanoparticles is quantitatively detailed by the data in Table 3. The number and percentage of each coordinationtype atomic species is given, ranging from O3c to OH and from Ti6c to Ti4c(OH). The position of the atoms of different coordination in the nanoparticles is then visually shown by the colour coding in Figure 2. We can note that the percentage of Ti5c atoms is much larger in the nanocrystals than in the nanospheres, with an increasing trend for smaller particles. On the contrary, the balance between O3c and O2c is not that different when comparing NCs with NSs. Small nanocrystals (NCS) have about the same contribution of fully coordinated O atoms than larger nanospheres (NSL) (70.2% vs 69.2%). The number of OH groups is larger in the nanospheres since those are required to achieve a minimum four-fold coordination that we have set as necessary for chemical stability. It is noteworthy that nanocrystals of different size (NCS and NCL) present the same atomic species at the analogous positions: four-fold Ti atoms at corners between the upper and the lower part of the decahedron, five-fold Ti atoms at the edges and on the (101) lateral surface and finally


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five-fold Ti atoms involving an OH group on the top (001) surfaces. Nanospheres (NSS and NSL) are characterized by a larger percentage of undercoordinated Ti atoms (especially < 5c) which are more randomly distributed on the quasi-spherical surface, giving the Harlequin aspect of the representations in the bottom part of Figure 2.

3.1.c Structural distortions in nanoparticles vs bulk The structural distortion present in nanoparticles and in nanospheres are expected to play a fundamental role in determining the properties of these nanobjects. In particular, we have already mentioned that the high curvature characteristic of nanospheres (but also analogously present in nanorods or nanotubes) is a key factor for enhancing their reactivity and affinity towards chemical adsorbates or ligands. For this reason simulated EXAFS (X-ray absorption fine structure) spectra are precious tools to systematically investigate bond lengths shrinking or stretching with respect to bulk distances. To this aim, we have first determined the EXAFS spectrum for bulk anatase TiO2 as obtained by performing structural relaxation with the hybrid functional HSE06 (Figure 3a). We limit the range of investigated distances from 0 to 4 Å. Five lines are present which can be assigned to the first three coordination spheres of a selected Ti atom: the first two lines at the distance of about 2 Å (first coordination sphere) are the Ti-Oeq (eq = equatorial) and Ti-Oax (ax = axial) bonds which are well known to be slightly different in the D2D symmetry, with the former shorter than the latter.; then the third line (second coordination sphere) is the distance between the selected central Ti atom and the next-neighbouring Ti atoms (Ti---Ti) of about 3 Å; finally, the forth and the fifth lines (third coordination sphere) are the distances between the central Ti atom and the second shell of Ti and O atoms (Ti---Ti and Ti--O) of about 3.8-3.9 Å. The EXAFS spectra of the nanocrystals and of the nanospheres under investigation is more complex since, after relaxation, these systems become less ordered than bulk, with the various lines converted into peaks after convolution. We first analyzed the EXAFS spectra of the nanocrystals NCS and NCL, focusing on the first coordination sphere (Figure 3b and 3c). The Ti-O distance critically depends on the Ti coordination number. Thus, there are two main contributions (green and blue) which are associated to the Ti5c (shorter Ti-O bonds) and to Ti6c (almost bulk Ti-O distances). The ratio between axial and equatorial contribution is not the typical 4:2 as for Ti6c in the bulk because the number of Ti-Oax bonds is higher, as a consequence of their shape where H > A (elongated decahedron). The green curve (Ti5c) presents three features at low bond (about 1.8 Å), at bulk (about 2.0 Å) and at high bond (about 2.2 Å) distances. These features are due to the typical relaxation associated to two-fold


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coordinated oxygens and five-fold coordinated titanium atoms on the prevalent (101) surfaces, as previously reported.39 Finally, the red curve, to be associated to Ti4c species, presents a low peak at short distances (about 1.8 Å). As regards the EXAFS spectra of the nanospheres NSS and NSL, we observed a broader variety of features contributing to the peak of the first coordination sphere, in the bottom two panels of Figure 3 (Figure 3d and 3e). For both nanospheres there is a predominant contribution of Ti6c (blue line), which is more evident for the larger nanosphere. Then, there are various contributions at short Ti-O distances of about 1.8 Å due to all the undercoordinated species on the surface, such as four-, five-fold coordinated Ti atoms or Ti atoms bound to an OH group. For both types of nanoparticles, the Ti---Ti feature presents peaks centered at the bulk distance with some broadening due to surface, edge, corner species. The faceted nanoparticles are characterized by a sharper Ti---Ti peak because of the higher percentage of bulk-like atoms with respect to the nanospheres. The larger the size of the nanoparticles, the more the EXAFS peaks approach the bulk line. For the large nanosphere NSL , which has an average diameter (DConn) of about 3 nm, we notice a general compression of the Ti-O bonds accompanied by a slight average increase of the Ti---Ti distances, in line with the experimental EXAFS data observed for 3 nm size nanoparticles.34 3.1.d Surface energies In this Section we investigate the cost to form nanoparticles from bulk systems in terms of surface energies. The standard free energy of formation of nanocrystals is the sum of two terms:13 [1] As regards the surface formation term

, we assume that, in the case of large nanocrystals

for which the contribution of edges and corners is negligible, it can be obtained from the sum of the surface free energies of the exposed facets as follows:

[2] where M is the molar mass, ρ is the material density, e is the volume dilation induced by surface tension, q is the surface/volume ratio and fi are the weighting factors of each facet surface free energy γi, defined as:13 [3] where Si is the surface area of each i-th facet. For small nanocrystals, the surface free energy of edges and corners may become significant, thus an additional correction term is required as follows: ACS Paragon Plus Environment

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[4] In order to estimate

from DFT calculations, one must subtract the free energy of the bulk

material and of water molecules from the total free energy of the nanoparticle, as follows: [5] where x is the number of adsorbed water molecules and n is the number of TiO2 units in the nanoparticles. However, free energies computation from DFT calculations is very demanding since it requires to perform vibrational frequencies or phonons calculations. Therefore, in the present study, as it is generally done, we will approximated total free energies to total electronic energies: [6] On this basis, the nanoparticle surface energy γ corresponds to:

[7] The concepts just exposed above for faceted nanocrystals can be easily extended to spherical nanoparticles by using equations [5], [6] and [7] and considering that the surface is not made by facets, edges, and corner, but is approximately that of a sphere. The main critical point of this general approach is to have a proper definition of the surface area of the nanoparticle (SNP). We have used two conceptually different approaches to define this quantity, as detailed in the Section 2 (Computational Details). In both cases, protruding hydroxyls where not taken into account to define the surface because they create artificial juts. Briefly, the first approach is to determine the smallest geometrical solid which contains all the atoms of the nanoparticle under study, i.e. the smallest Wulff shape for nanocrystals and the sphere of radius RM for nanospheres. Then, we have calculated the geometrical surface area of the containing solid (Sgeom) and used it as the surface area of the nanoparticle (SNP = Sgeom). This first method will be used to compare surface energies of nanoparticles to the surface energy of regular anatase TiO2 surfaces, such as the lowest energy (101). The second approach is based on the Connolly method29 that takes into account also the roughness of the nanoparticles surface, by considering the Van der Waals volume of each atom. Although the resulting surface (SConn) is then smoothed by the method, it still accounts better for the geometrical irregularities of the nanoparticles surface, especially when they are spherical.35 This is evident comparing the two different surface types, obtained with the two approaches (the geometric in dark red and the Connolly in blue), in Figure 4. The Connolly surface is almost flat on the regular (101) and (001) surfaces of nanocrystals, while it is slightly wavy along their edges. In the case of


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nanospheres, it is wavy all around the nanoparticle, as expected, given the natural corrugation of this kind of nanoparticles. Once the nanoparticles surface area is defined, either as SNP = Sgeom or SNP = SConn, it is then possible to compute the surface energy γ form equation [7] and obtain γgeom or γConn. All the results for the geometrical and Connolly surface approaches are reported in Table 4 and in Table 5, respectively. In Table 4 we compare the surface energies (γgeom) of the various nanoparticles (both NCs and NSs) with that of the regular (101) anatase surface. As one would expect, they are all higher than for the flat surface, in line with what defined with equation [4]. For the larger nanocrystal, the increment of the surface energy γgeom with respect to the (101) surface is smaller, due to the smaller contribution of edges and corners. Note that for NCS and NCL the (101) facets represent the 9590% of the total geometric surface, respectively, and the presence of high energy edges and corners is mitigated by the energy release of dissociated water molecules on the (001) surface. Spherical nanoparticles present significantly higher surface energies than faceted nanocrystals. Analogous results are obtained with both the HSE06 and B3LYP functionals, although HSE06 surface energies are systematically larger (by 0.1 J/m2) than the B3LYP ones, in line with what observed for the regular surface. In Table 5 Connolly surface energies (γConn) are presented for all the nanoparticles. This method is better suited for a direct comparison of nanocrystals and nanospheres because exactly the same algorithm is used to obtain their surface area. The Connolly surface areas are somewhat larger than the geometrical ones, consequently all the corresponding surface energies result to be lower. The trends are the same as discussed above for Table 4. Finally, in Table 6 we report the number of undercoordinated atoms or OH groups per unit area (SConn), computed for the two nanocrystals and the two nanospheres. This type of information help in the rationalization of the trends observed for the surface energies. In fact nanospheres present a higher density of these type of sites than nanocrystals, that justifies their higher surface energies. As regards nanocrystals, it is interesting to observe that the density of defect species is about the same in total, but the density of more energetically costly Ti4c is larger in NCS than in NCL. This is one of the factors causing a higher γConn for NCS vs NCL.

3.2 Electronic Properties 3.2.a Kohn-Sham vs fundamental gaps of finite nanoparticles It is very important to compare the electronic properties of nanoparticles of different shape with those of bulk. First of all we discuss the energy gap between occupied and unoccupied states which ACS Paragon Plus Environment

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for the bulk system, is the "band gap". Bulk anatase TiO2 is characterized by an indirect band gap, which is estimated, by the present setup and by the hybrid functional HSE06, to be 3.65 eV, in agreement with previously reported values.36 In the case of the nanoparticles, which can be considered as very large molecular (finite) systems, it is not fully correct to refer to band states and, thus, to band gaps. We, therefore, need to distinguish between very localized states (and refer to molecular orbitals, MOs) and states which are delocalized on several atoms of the nanoparticle (and refer to bands). In the case of the nanocrystals (NC), the highest occupied state and the lowest unoccupied state can be considered band states, if we follow the definitions above, therefore the Kohn-Sham band gap of the nanocrystals (EgKS) corresponds to the HOMO-LUMO gap (∆EHOMO-LUMO) and is about 0.3 eV larger than for the bulk system (3.94 eV for NCS and 3.90 eV for NCL), as reported in Table 7. For the frontier orbitals description see next Section 3.2.b. In the case of the nanospheres (NS), the highest occupied state is a molecular orbital fully localized on protruding OH groups (see next Section 3.2.b for further details). Therefore, the ∆EHOMO-LUMO cannot be considered to be the band gap of the nanospheres. To determine the EgKS for the spherical nanoparticles, we have selected a lower energy occupied states which is delocalized on more O atoms of the nanoparticle and can be considered a "band" state. Such state is the HOMO-4 (see Supporting Information for the 3D plots) that is used to compute the EgKS for nanospheres. Therefore, EgKS values are larger than ∆EHOMO-LUMO values, but very close to the bulk band gap (3.69 eV for NSS and 3.67 eV for NSL, see Table 7). In order to check whether these data are at convergence with respect to the basis set, we have added a diffuse function on each atom of the surface layer of the nanoparticles. We observe only a tiny difference between the results obtained with the two types of basis sets which proves that the convergence is reached. Finally, all the considerations above on the HSE06 results are valid also for the B3LYP ones, although all the energy gaps are systematically larger with this functional, as expected. Kohn-Sham gaps suffer of severe limitations being determined from one-electron KS eigenvalues. Thus, they generally differ from fundamental gaps (Eg), defined as the difference between the ionization potential, IP, and the electron affinity, EA, in solid state physics. This is a consequence of the derivative discontinuity, i.e. the finite "jump" the exchange correlation potential exhibits as the electron number crosses the integer number of electrons of the system (N).37,38 For solids, the use of hybrid functionals

leads to meaningful improvements in the prediction of

fundamental band gaps from KS gap values, because these functionals are based on a mixture of local exchange and nonlocal Fock exchange. For finite-sized systems (such as molecules and ACS Paragon Plus Environment

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nanoparticles) the use of hybrid schemes is not sufficient because the asymptotic potential, not present in the solids, crucially affects the energy of electron addition or removal. KS gaps from hybrid functional calculations for molecules or nanoparticles are most of the times considerably smaller than fundamental gaps and in better agreement with optical gaps. The difference between fundamental gaps and KS gaps is commonly denoted as the derivative discontinuity (DD). All these considerations are fully valid for the nanoparticles under investigation in the present study. If we analyze Table 8, we can compare HOMO-LUMO KS gaps with the fundamental gaps obtained from vertical ionization potentials (IPv) and vertical electron affinities (EAv). Fundamental gaps are always higher than KS gaps by about 1.5 eV, both with HSE06 and B3LYP functionals. IPv and EAv were obtained by removing from and adding to each nanoparticle one electron and freezing the geometry at the ground state minimum. The excess electron or the resulting hole occupy the LUMO and HOMO of the nanoparticle, respectively. Nanocrystals and nanospheres present similar electron affinities since the LUMO is in all cases a delocalized state involving the Ti atoms in the central portion of the nanoparticle (see next Section 3.2.b). On the contrary, ionization potentials considerably differ, with those of the nanospheres being lower than those of the nanocrystals. This is a consequence of the fact that the electron is removed from the HOMO state, which, in the case of the nanocrystals, is a delocalized state involving the oxygen atoms in the central portion of the nanoparticle, while, in the case of the nanospheres, is a highly localized state on the most external OH groups. This will be further discussed in the next Section 3.2.b. An increase in size causes an increase of EAv and IPv of approximately 0.1 eV for both NCs and NSs. We have checked that fundamental gap values are at convergence with respect to the basis set by performing the same calculations with one additional diffuse function on each atom of the surface layer of the nanoparticles.

3.2.b Frontier orbitals In this section we present the frontier orbitals as computed with the hybrid functional HSE06 for all the nanoparticles under investigation. Fully analogous results have been obtained for the hybrid functional B3LYP and they are reported in the Supporting Information. We first discuss the frontier orbitals of the nanocrystals (see Figure 5). HOMOs and LUMOs of NCS and NCL are totally analogous and in agreement with previous studies39 for nanocrystals of the same size of NCS. The HOMOs of NCS and of NCL resemble a delocalized band states, mostly involving the O 2p states (a mixture of 2px and 2py according to the row considered) of two central atomic layers of the nanocrystals, with a higher density on the atoms in the core. These layers are


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those with the largest xy surface area. Going from NCS to NCL we can notice a lower contribution of the undercoordinated external O2c 2p states. The LUMOs of NCS and of NCL are again similar to a delocalized band states, mostly involving the Ti 3dxy states of three central atomic layers of the nanocrystals, with the largest xy surface area. Again, going from NCS to NCL we can notice a lower contribution from the surface Ti atoms. For further discussion on this we refer to the next Section 3.2.c where total and projected densities of states will be presented and discussed. The nanospheres frontier orbitals are different in character (see Figure 6). Starting from the LUMOs of NSS and NSL, we observe that these are certainly delocalized states but, especially for the larger nanosphere, they involve only bulk Ti 3dxy states. In other words, for NSL, surface Ti atoms are not involved in the LUMO, which is different from what observed for nanocrystals. Even more different is the situation for the HOMOs. The highest occupied states are fully localized on surface hydroxyls, both in the case of NSS and NSL. We have already discussed in the previous section that this peculiar feature has a relevant effect on the KS gaps. For this reason we have also investigated deeper states and determined that the first delocalized state on several O atoms is the HOMO-4 (see Supporting Information) and used it to computed EgKS. 3.2.c Total and projected densities of states In this last section total and projected densities of states (DOS and PDOS) are presented and reported in Figure 7 (for NCs) and Figure 8 (for NSs) for the HSE06 calculations (B3LYP DOS and PDOS are reported in the Supporting Information). The projections on all the different types of species present in the nanoparticles are investigated. We first discuss the smaller nanocrystal NCS and observe that: the top of the valence band is essentially made by O3c and O2c on the (101) facets (inset in the upper left panel of Figure 7a); the bottom of the conduction band is mostly made up by Ti6c and Ti5c (inset in the bottom left panel of Figure 7b); below the bottom of the valence band a peak of OH groups clearly detaches. Comparing DOS and PDOS for NCS (left side of Figure 7a,b) with those for NCL (right side of Figure 7c,d) we can observe many similarities and only few differences. The O2c/O3c and the Ti6c/Ti5c ratios decrease, respectively, as one would expect. The OH peak below the valence band splits in two peaks because, in the larger nanocrystal, two inequivalent OH groups exist. For further details on the projections on OH groups we refer to the Supporting Information. For the smaller spherical nanoparticle NSS (left side of Figure 8a,b) we can clearly observe a predominant contribution OH groups to the top of the valence band (inset in the upper left panel of Figure 8a). At lower energies O2c 2p states appear and only at even lower energies the contribution of the O3c becomes predominant. The bottom of the conduction band (inset in the bottom left panel


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of Figure 8b) is mostly made by Ti6c with a consistent contribution also from Ti4c and Ti5c. Comparing DOS and PDOS for NSS with those for NSL (Figure 8a,b vs Figure 8c,d) we can observe some clear trends. The O2c/O3c and OH/O3c ratios, as well as the Ti5c/Ti6c ratio, diminish. This has a consequence also on the relative contributions to the top of the valence band (inset in the upper panel on the right of Figure 8c) in terms of a lower contribution of the O2c and OH species. For the bottom of the conduction band we observe an increased contribution of Ti4c (inset in the bottom right panel of Figure 8d). The corresponding DOS and PDOS with the B3LYP functional are reported in the Supporting Information. Besides the already discussed differences in the energy gaps, analogous observation than for HSE06 counterparts can be made. The use of an extended basis set on the atoms in the external layer of the nanoparticles does not cause major variations in the DOS and PDOS, exept that there is a higher degree of covalency with a larger contribution of Ti atoms to the valence band and a larger contribution of the O atoms to the conduction band (see Supporting Information).

4. Conclusions To summarize, in this work we have presented a comparative density functional (HSE06 and B3LYP) study of spherical vs faceted anatase TiO2 nanoparticles. Specific quantities have been also compared either to the anatase (101) surface or to the bulk system when appropriate. Not only different shape but also different sizes have been considered. The small (NCS) and large (NCL) nanocrystals are (TiO2)159 • 4 H2O and (TiO2)260 • 6 H2O, respectively; while the small (NSS) and large (NSL) nanospheres are: (TiO2)223 • 26 H2O and (TiO2)399 • 32 H2O, respectively. Nanoparticles sizes have been compared in terms of equivalent diameter derived from the Connolly volume while nanoparticles surfaces have been compared in terms of Connolly surfaces because these quantities allow the direct comparison of differently shaped nanoparticles. We note that the surface area of a nanocrystal of analogous size of a nanosphere (similar DConn) is comparatively larger. Surface-to-bulk sites ratios are larger for nanospheres than for nanocrystals, and of course, they decrease with increasing size. Nanocrystals are particularly rich of Ti5c species, nanospheres present a broader variety of undercoordinated sites. Structural distortions of the Ti-O bonds in the nanoparticles have been investigated by simulated EXAFS spectra. The peak corresponding to the first coordination sphere is broader for nanospheres than for nanocrystals, confirming a wider variety of undercoordinated species and a


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more disordered surface for the former. The larger the nanoparticles size, the more the EXAFS peaks approach the bulk system reference lines, as one would expect. The calculated surface energies for nanoparticles are always higher than for the regular (101) surface, but they are particularly higher for nanospheres. As it is true for all finite-size systems, also in the case of anatase titania nanoparticles, Kohn-Sham gaps consistently differ from fundamental gaps obtained from ionization potentials and electron affinities (by about 1.5 eV). This is a consequence of the derivative discontinuity, totally analogous to the case of molecules. To conclude, the detailed analysis presented in this work proves that faceted and spherical nanoparticles are characterized by different structural and electronic properties. Such differences may turn out to be crucial for different uses and applications of TiO2 nanoparticles. In particular, for applications which require high crystallinity and low surface reactivity, nanocrystals are expected to be better suited, while for applications which require high adsorbate binding energy for an efficient nanoparticles functionalization, nanospheres are expected to be a more useful choice.

ASSOCIATED CONTENT Supporting Information Figures showing frontiers orbitals from B3LYP calculations, HOMO-4 orbitals from HSE06 calculations, B3LYP DOS and PDOS, HSE06 and B3LYP DOS and PDOS with the extended basis sets. This material is available free of charge via the Internet at http://pubs.acs.org.

ACKNOWLEDGMENTS We are grateful to Fulvio Paleari and Francesco Ferrari for fruitful discussion about programming issues and to Elisa Albanese and Lorenzo Ferraro for their constant technical help. The research leading to these results has received funding from the European Research Council under the European Union's HORIZON2020 Programme / ERC Grant Agreement n. [647020], from the Cariplo Foundation / Grant n. [2013-0615] and from the CINECA supercomputing center through computing LI03p_CBC4FC and IscrB_CMGEC4FC grants.


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Tables

Table 1 – Optimized cell parameters a and c (in Å), and a/c ratio of bulk anatase as obtained with the HSE06 and B3LYP functionals, together with the corresponding experimental values.

HSE06 B3LYP Exp.

a 3.766 3.789 3.782

c 9.663 9.777 9.502

a/c 2.569 2.580 2.512

Table 2 – Optimized geometrical parameters, at the HSE06 and B3LYP level of theory, for the nanoparticles as shown in Figure 1. The decahedron parameters A, B and H are reported as well as the B/A ratio for the faceted nanoparticles, whereas the cutting radius RM is reported for the nanospheres. The diameter DM = 2 × RM for nanospheres and the equivalent diameter (where V= ⅓ H (A2 + B2 + AB) is the geometrical volume of the nanocrystal) for nanocrystals are also given. SConn and VConn are the surface area and the volume estimated with the Connolly method (see Computational Details) while DConn is the equivalent diameter of a sphere with the same Connolly volume. Parameters are reported in nm, surfaces in nm2 and volumes in nm3.

Nanoparticle

A

B

H

B/A

RM

Deq or DM

SConn

VConn

DConn

HSE06 NCS (TiO2)159 • 4 H2O

1.56

0.39

2.83

0.25

1.79

18.4

5.7

2.22

NCL (TiO2)260 • 6 H2O

1.93

0.77

2.85

0.40

2.19

25.0

9.3

2.61

NSS (TiO2)223 • 26 H2O

1.22

2.44

21.4

8.1

2.49

NSL (TiO2)399 • 32 H2O

1.50

3.00

32.0

14.6

3.03

B3LYP NCS (TiO2)159 • 4 H2O

1.57

0.39

2.87

0.25

1.81

18.7

5.8

2.23

NCL (TiO2)260 • 6 H2O

1.94

0.78

2.89

0.40

2.21

25.4

9.4

2.62

NSS (TiO2)223 • 26 H2O

1.23

2.56

21.8

8.3

2.52

NSL (TiO2)399 • 32 H2O

1.51

3.02

32.5

15.0

3.06


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Table 3 – Number of O and Ti atoms with a specific coordination sphere in the HSE06 and B3LYP optimized structures (when different, B3LYP value is in bracket) and their percentage with respect to the total number of atoms. The cutoff radius for a Ti-O bond is 2.5 Å. NCS Number

NCL %

Number

NSS %

NSL

Number

%

Number

%

O atoms OH

8

2.5

12

2.3

52

11.0

64

7.7

O2c

88

27.3

120

22.8

124(128)

26.3

192(200)

23.1

O3c

226

70.2

394

74.9

296(292)

62.7

574(566)

69.2

Ti atoms Ti4c

4

2.5

4

1.5

24

10.7

36

9.0

Ti5c

76

47.8

106

40.8

46

20.6

64(72)

16.1

Ti6c

71

44.7

138

53.1

105

47.1

243(235)

60.9

24

10.8

32

8.0

8(12)

3.6

16

4.0

12(8)

5.4 8

2.0

Ti4c(OH) Ti5c(OH)

8

5.0

12

4.6

Ti6c(OH) Ti5c(OH)2 Ti6c(OH)2

4

1.8

Table 4 – Surface formation energy terms (in eV), geometrical surface areas (in nm2) and surface energies (in J/m2) for the various nanoparticles and for the (101) anatase surface (computed on 10 triatomic layers), as obtained with the HSE06 and B3LYP functionals.

TiO2 units

NCS

NCL

NSS

NSL

159

260

223

399

(101) Surface

HSE06 74.96

97.67

109.62

155.54

17.2

23.5

18.7

28.3

0.70

0.67

0.94

0.88

0.634

B3LYP 67.61

87.77

95.68

137.04

17.5

23.8

19.0

28.7

0.199

0.62

0.59

0.81

0.77

0.544


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Table 5 –Connolly surface areas (in nm2) and surface energies (in J/m2) for the various nanoparticles and for the (101) anatase surface (calculate for 10 triatomic layers), as obtained with the HSE06 and B3LYP functionals The Connolly radius is set to 3.0 Å.

NCS

NCL HSE06

NSS

NSL

18.5

25.0

21.4

32.0

0.65

0.625

0.82

0.78

B3LYP 18.7

25.4

21.8

32.5

0.58

0.55

0.70

0.67

Table 6 – Number of undercoordinated sites per unit area (sites/nm2) for HSE06 optimized nanostructures and (101) surface. The surface area is calculated with the Connolly method with radius = 3.0 Å. Undercoordinated sites per unit area

(in sites/nm2)

Surface site

NCS

NCL

NSS

NSL

OH

0.43

0.48

2.43

2.00

O2c

4.77

4.79

5.79

6.00

Ti4c

0.22

0.16

1.12

1.13

Ti5c

4.12

4.23

2.15

2.00

1.12

1.00

0.37

0.50

Ti4c(OH) Ti5c(OH)

0.43

0.48

Ti6c(OH)

5.14 5.14

0.56

Ti5c(OH)2

0.25

Ti6c(OH)2 Total

(101) Surface

0.19 9.97

10.14

13.73


12.88

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Table 7 – Relative energy values, with respect to the vacuum level, of the top of the valence band (Top VB), of the HOMO orbital and of the bottom of the conduction band (Bottom CB); energy differences between HOMO and LUMO levels (∆EHOMO-LUMO ); Kohn-Sham energy gap (EgKS). All HSE06 and B3LYP values are in eV. (+) refers to values obtained with the additional diffuse functions on all the surface atoms (see Computational Details). . HSE06 NCS

NCS (+)

NCL

NSS

NSS (+)

NSL

Bulk

Top VB

-8.06

-8.23

-8.08

-7.70

-7.88

-7.65

HOMO

-8.06

-8.23

-8.08

-7.48

-7.65

-7.43

Bottom CB

-4.12

-4.35

-4.18

-4.01

-4.20

-3.97

∆EHOMO-LUMO

3.94

3.88

3.90

3.47

3.45

3.45

EgKS

3.94

3.88

3.90

3.69

3.68

3.67

3.65 Bulk

B3LYP NCS

NCS (+)

NCL

NSS

NSS (+)

NSL

Top VB

-8.01

-8.19

-8.03

-7.88

-8.06

-7.68

HOMO

-8.01

-8.19

-8.03

-7.63

-7.83

-7.49

Bottom CB

-3.91

-4.15

-3.98

-3.92

-4.12

-3.81

∆EHOMO-LUMO

4.10

4.04

4.05

3.71

3.70

3.68

EgKS

4.10

4.04

4.05

3.96

3.94

3.86

3.81

Table 8 – Vertical electronic affinities (EAv) and ionization potentials (IPv); fundamental gaps (Eg) and Kohn-Sham (∆EHOMO-LUMO) gaps, and their difference (derivative discontinuity, DD). All values are in eV and reported for both the HSE06 and B3LYP functionals. (+) refers to values obtained with additional diffuse functions on all surface (see Section 3.2.a and the Computational Details for further information).

HSE06 EAv

IPv

Eg

B3LYP ∆EHOMO-

DD

EAv

IPv

Eg

LUMO

∆EHOMO-

DD

LUMO

NCS

3.41

8.93

5.52

3.94

1.58

3.23

8.80

5.57

4.10

1.47

NCS (+)

3.64

9.09

5.45

3.88

1.57

3.48

8.98

5.50

4.04

1.46

NCL

3.52

9.18

5.66

3.90

1.76

3.41

8.78

5.37

4.05

1.32

NSS

3.39

8.38

4.99

3.47

1.52

3.32

8.41

5.09

3.71

1.38

NSS (+)

3.61

8.55

4.94

3.45

1.49

3.54

8.61

5.07

3.70

1.37

NSL

3.45

8.47

5.03

3.45

1.58

3.30

8.36

5.06

3.68

1.38


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Figures

Figure 1 – Top: Space-filling representation of the faceted anatase nanoparticles and the corresponding schematic representation of the Wulff-shape decahedron with its parameters A, B and H. Bottom: Space-filling representation of the spherical anatase nanoparticles and the corresponding schematic representation of the sphere with radius RM. The HSE06 optimized nanostructures are shown. The surface-to-bulk % ratio and the stoichiometry of each nanoparticle are indicated on the side of the model.


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Figure 2 – The position of the Ti atoms with different coordination sphere within the various nanoparticles is visually shown by the colour coding indicated on the right side.


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Figure 3 – EXAFS spectra computed with the HSE06 functional for the anatase bulk (top panel, a), the nanocrystals NCS and NCL (central panels, b and c) and the nanospheres NSS and NSL (bottom panels, d and e). The total distribution of Ti-O, Ti---Ti and Ti---O distances is reported in black, while the distribution of the same distances for each type of Ti is color coded according to the legend on the right. See Section 3.1.c and the Computational Details for further information.


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Figure 4 – 3D plots of the geometric surfaces (in dark red) and Connolly surfaces (in blue) for all the anatase nanoparticles under investigation. See Section 3.1.d and the Computational Details for further information).


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Figure 5 – Side (upper) and top (lower) views of the electronic density plot for the frontier orbitals of the two nanocrystals, as obtained with the HSE06 functional and an isosurface value of 0.0005 au.


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Figure 6 – Side (upper) and top (lower) views of the electronic density plot for the frontier orbitals of the two nanospheres, as obtained with the HSE06 functional and an isosurface value of 0.0005 au (0.001 au for the highly HOMO localized states, see Computational Details).


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Figure 7 – Total (DOS) and projected (PDOS) density of states on different O and Ti atoms of anatase nanocrystals, as calculated with the HSE06 functional. Magnified portions of the diagram are shown in the insets. A 0.01 eV Gaussian broadening was used. The zero energy is set to the vacuum level.


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Figure 8 – Total (DOS) and projected (PDOS) density of states on different O and Ti atoms of anatase nanospheres, as calculated with the HSE06 functional. Magnified portions of the diagram are shown in the insets. A 0.01 eV Gaussian broadening was used. The zero energy is set to the vacuum level.


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Spherical versus Faceted Anatase TiO2 Nanoparticles: A Model Study

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