Theoretical Study of Size Effects on Surface Chemical Properties for

Oct 21, 2014 - Nanodiamond has displayed some unique physical and chemical properties compared to bulk diamond, which broadens its applications in ...
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Theoretical Study of Size Effects on Surface Chemical Properties for Nanoscale Diamond Particles Tian Yuan and Karin Larsson* Department of Chemistry − Ångström Laboratory, Uppsala University, Uppsala 75121, Sweden ABSTRACT: Nanodiamond has displayed some unique physical and chemical properties compared to bulk diamond, which broadens its applications in various areas. However, a more detailed picture of nanodiamond quantum confinements is still missing from a theoretical point of view. This investigation presents a study where the effects of onedimension (i.e., diamond thin films) and three-dimension (i.e., nanodiamond particles) confinement on surface reactivity, and properties, have been calculated using density functional theory (DFT) and tight binding density functional theory (DFTB) methods. Surface specific parameters like (i) surface C−H bond length, (ii) atomic charges, (iii) H adsorption energy, (iv) highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), (v) band gap, and (vi) Fukui functions were thereby carefully calculated and compared. For both the one-dimensional diamond thin films of different surface planes, quantum confinements have strong influences on these factors from thickness of 0.2 to ∼1 nm, while for thin films thicker than 1 nm, the values stabilize around a plateau value. For three-dimensional situations, these factors were found to change within a range of nanodiamond diameter of 0.4 to ∼2 nm, followed by oscillations around specific values as well. These results reveal that nanoscale diamond quantum confinements exist for a nanodiamond particle of a diameter smaller than 2 nm, but not for larger particle sizes. It must here be stressed that all surface specific parameters did independently show the existence of the here presented size ranges for quantum confinement.

1. INTRODUCTION Diamond has attracted much research interest since the first artificial synthesis of diamond using chemical vapor deposition or high-pressure high-temperature methods.1 Its unique chemical and mechanical properties, and tunable surface structures, have been observed to improve the possibility for applications within, e.g., renewable energy,2 biotechnology,3,4 and electronics5 As the nanotechnology synthesis methodology has developed quite fast during the past few years, nanodiamond particles have been successfully synthesized by using detonation methods.6 However, a better control of the synthesis of nanoscale diamond particles has been introduced and improved, and nanodiamond particles can today display an even bigger interest within material science and have therefore caused extensive research interests (e.g., in the field of fluorescent nanodiamond powders for biomedical imaging7,8). Nanodiamond particles are nowadays synthesized using various techniques: (i) by applying high temperature and high pressure, in transformation graphite to diamond,9 (ii) detonation of other carbon sources in an oxygen-deficient atmosphere,10 (iii) milling of diamond microcrystals,11 and (iv) by using plasma assisted chemical vapor deposition (CVD) techniques.12 The sizes of nanodiamond particles synthesized can reach single digit nanometer scale (less than 10 nm), down to 2 nm, still maintaining a quite narrow size distribution.13 When materials reaches nanoscale, which is only several atoms involved in formation of structures, the electronic, physical, and chemical properties can be very different from bulk materials. The well-known quantum mechanical term quantum confinement (i.e., quantum size effects) is commonly used to define © 2014 American Chemical Society

these differences, and it describes the changes of particles behaviors at single digit nanometer dimensions. In semiconductor industry, quantum size effects can modify the material electron transmission properties tremendously and thereby give rise to various de novo applications for otherwise traditional materials.14 However, in other applications like biomedicine or drug delivery, quantum size effects of diamond must be avoided since the good chemical stability of the bulk material should be maintained to avoid toxicity,15 but still keeping the sizes in a small scale for passing physiological membranes.16 In order to tailor quantum size effects for nanodiamond (ND) particles and for different application purposes, a deeper study of the quantum size effect on the chemical and electronic ND properties is of an utter importance to perform. In order to seek for a better understanding of the nanoscale diamond particle structures and chemical properties, numerous experimental and theoretical studies have been conducted. High resolution transition electron microscopy (HRTEM) was implemented to investigate surface chemistry of nanodiamond particles synthesized by detonation. They found that nanodiamond particles were mostly in the form of sp3 carbon, but the chemical properties were unique (high acidity, high surface charge, and colloidal stability).17 In addition, Drummond et al. presented some electron emission possibilities for H-terminated nanodiamond particles by performing Monte Carlo simulaReceived: July 24, 2014 Revised: October 20, 2014 Published: October 21, 2014 26061

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tions.18 They claimed that when sizes were below 1 nm, the band gap should change to 1−2 eV smaller than for bulk diamond. But for sizes above 1 nm, the band gap became larger than bulk diamond. This observations opens up the possibility to use ND particles as cold cathode material.19 Moreover, Barnard reported that the electronic structures of nonterminated ND particles are dependent on a combination of particles size and crystalline shape (i.e., morphology) with tightbinding density functional theory (DFTB).20 The main goal with the present work has been to theoretically outline the (i) geometrical structure, (ii) surface reactivity, and (iii) chemical property of various dimensionality of the nanoscale diamond. Quantum confinement in one dimension was studied for thin single crystalline diamond films, and quantum confinement in three dimensions was studied for nanodiamond particles. The surface sites of nanodiamond particles were 100% saturated with H. These calculations have been based on a multiscale modeling approach, where density functional theory (DFT), density functional tight binding (DFTB), and semiempirical methods were used with the purpose to cover a large range of sizes, while at the same time maintaining the accuracy in the study.

The tight-binding DFT method DFTB focuses more on the tightly binding parts of the models and can thereby be regarded as an approximate density functional theory method. The selfconsistent density functional defines a reference density, which is calculated by using weakly confined neutral atoms and further optimized by using already known charge densities and effective potentials (from both solids and molecules). By using these approximations, the computational costs can be reduced while maintaining reliable total energies and geometries results for nanodiamond particles.29 The Slater−Koster library is CH, which is designed for carbon and hydrogen containing structures, which works well with molecular geometry calculations. Corrections for van der Waals force interactions were added by using semiempirical solutions within the DFTB calculations.30 A combination of these three theoretical methods has in the present investigation been used with the purpose to study various chemical and electrical properties of nanodiamond particles of various sizes, from diameter of 0.4−3 nm. The DFT method was used for the 1D thin film cases and for diamond particle sizes up to 1.4 nm in diameter. Because of practical difficulties in the simulations of even larger particles, the DFTB was introduced for diameters spanning from 0.4 to 3 nm. The adsorption energy is a common and efficient tool in studying surface reactivity

2. COMPUTATIONAL METHODOLOGY The present study is based on different levels of theoretical methods: DFT (density functional theory), semiempirical theory, and tight-binding DFT. For these calculations, the following programs were used: DMol3,21 VAMP,22 and DFTB23,24 by Accelrys Inc. For the first-principles DFT calculations, a local numerical orbital basis set was generated as values on an atomic-centered spherical polar mesh, with a specific orbital cutoff of 0.5 Å (for a very accurate integration). A doubling of the number of functions, in addition to an introduction of polarization pfunctions (for hydrogen atoms) and d-functions (for nonhydrogen atoms), was used to get a more flexible basis set. The quality of the basis set is imperative for a good description of the system, especially when weaker bonds are present.23 The exchange and correlation parts of the Hamiltonian were approximated using the Perdew−Burke−Ernzerhof (PBE) generalized gradient approximation (GGA). This will, in contrast to the local density approximation (LDA), give a much better energy evaluation. The LDA method tends to overbind electrons in a molecule (or solid).25 An energy cutoff value of 380 eV was used and 3K points in total according to earlier test calculation for similar systems.26 The van der Waals forces interactions and hydrogen bonding were dealt with by using dispersion corrections within the DMol3 module.27,28 The semiempirical calculations were based on approximations made from library, which made it possible to use much larger models as compared with the first-principles DFT calculations. For these semiempirical calculations, the Hamiltonian was set to neglect differential diatomic orbitals (NDDO), and the spins were restricted Hartree−Fock, in order to pair all electrons and fill available orbitals. The multiplicity parameter was auto, for which VAMP would perform the calculation from a spin-unrestricted condition. The spin option was RHF, which is short for restricted Hartree− Fock, for better calculation accuracy with adequate computational time. In addition, the simplest basis set was used, and the nonvalence electrons were treated as a frozen core in order to save computational power.

ΔEadsorption = Etotal − Eradical − E H

(1)

where Etotal and Eradical are the total energies for a 100% Hterminated diamond nanoparticle (DNP) and one DNP where one of the H adsorbates have been removed (leaving a radical surface site). In addition, EH describes the energy for a single H radical. Atomic charges were calculated by Mulliken charges, which performed a projection of plane wave states onto localized basis.31,32 Bond lengths were also compared in both one- and three-dimensional models. pDOS, short for partial density of states, was utilized to describe the number of states and their relevant energy level with proportion of individual atoms with their according orbitals. Highest unoccupied molecular orbital (HUMO) energy level and lowest occupied molecular orbital (LOMO) calculated by DMol3 and DFTB level were compared as well as a function of nanodiamond sizes in three-dimensional models. Band gap values were calculated by DMol3 and DFTB in order to evaluate electronic properties. Other parameters that illuminate the surface chemical reactivity are the Fukui functions. There are three types of Fukui functions; f+ reflects the surface site’s susceptibility for a nucleophilic attack, and f+ reflects the surface site’s susceptibility for an electrophilic attack. The average value of f(+) and f(−) gives information about the susceptibility for a radical attack ⎛ ∂ρ(r ) ⎞ ⎟ f (r ) = ⎜ ⎝ ∂N ⎠v(r)

(2)

where ρ(r) is the electron density and N is the number of electrons. These Fukui functions have been calculated using the DFT method, and various graphs were in the present study generated in order to visualize the distribution of reactive surface sites.33,34 The Fukui functions were calculated using the DFT-based program DMol3. 26062

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3. MODELS In studying the effect of one-dimensional quantum confinement on surface properties, infinite diamond (111) and (100) surface planes were generated from bulk diamond (Figure 1). The

Figure 2. A three-dimensional model of a nanodiamond particle, terminated with H and with a diameter of 1.6 nm. The most abundant surface planes, (100) and (111), are marked with red and yellow circles, respectively.

Figure 1. Supercell models of H-terminated (a) diamond (100) and (b) diamond (111) thin films of thicknesses 7 Å.

4. RESULTS AND DISCUSSION 4.1. General. Among the three different theoretical methods used in the present study, the DFT one is by far the most accurate one when it comes to processes involving bond breakage and formation (e.g., when studying chemical processes on surfaces). On the contrary, this high level method makes it practically impossible to study larger systems. In the present study, the DFT method has been chosen for the simulations of smaller diamond particles, while semiempirical and tight-binding DFT have been used for the larger systems. It was thereby very important to ensure that this combination of methods gave reliable results, so the present investigation has been initiated by evaluating the methods and methodology used. The size effect (1D for diamond (111) and (100) slabs and 3D for diamond particles) on various parameters will be presented in section 4.2. They include structural indicators (C− H bond length), reaction energetic indicators (H adsorption energy), and electronic indicators (atomic charges, highest orbital molecular orbital (HOMO) levels, lowest unoccupied molecular orbital (LUMO) levels, band gaps, and Fukui functions). 4.2. Evaluation of Various Theoretical Methods. Before the calculations of the size dependency of various diamond clusters, an evaluation of different theoretical methods had to take place. This was performed in two steps, where the first step included smaller cluster sizes (diameter from 0.4 to 1.4 nm) and the second step larger ones (1.6 to 3 nm). For the smaller cluster sizes, the DFT, semiempirical, and tight-binding DFT methods were used and compared. As an indicator for surface reactivity, H adsorption energies, chemisorbed to otherwise completely H-terminated nanodiamond particles, were thereby used. As can be seen in Figure 3, the trends in H adsorption onto an (111) facet, as a function of particle size, are identical for the different methods used: DFT, semiempirical, and tight-binding DFT methods. For all of these methods, the absolute values of H adsorption energies were observed to decrease with an increasing particle size (up to 1.4 nm). Hence, the particle reactivity will decrease with an increase in size at this nanoscale level. Experimentally, the binding energy of H to C is 413 kJ/

model thicknesses were within the range 0.2−4 nm, with the corresponding number of carbon layers from 3 to 45. These surface planes were chosen since they are the most abundant ones for synthesized nanodiamond particles.17 In addition, the (100) surfaces where not considered 2 × 1-reconstructed since the corresponding (100) facets on the nanodiamond particles were not observed to be 2 × 1-reconstruced (see below). Moreover, the carbon atoms in the lowest atomic layer were saturated with hydrogen atoms in order to simulate a continuation into the bulk. For the same reason, these lower layer atoms were constrained in the geometry optimization procedure. All other atoms were allowed to move freely. A delocalized Cartesian optimizer algorithm was used for these geometry optimizations.21 In addition, a vacuum distance of 10 Å was used in the calculations with the purpose to minimize the interactions between slabs in the z-direction. For the three-dimensional models, nanodiamond particles were built with the diameter ranging from 0.4 to 3 nm. In constructing these particles, the thermodynamically most stable cluster was formed (i.e., with the smallest surface-to-bulk ratio, and with low-energy surface planes). It has earlier been shown experimentally that the variations in band gap, as a function of NDP size, are very similar for the different crystalline shapes (octahedral, truncated octahedral, cuboctahedral, and cuboid).20 Based on these earlier results, it was decided that the specific forms of the diamond particles were not of any major importance for the present study. In fact, the spherical were used (see Figure 2), which are furthermore observed to be abundant in NDP synthesis with chemical vapor deposition (CVD).35 The smallest particle contained only 25 atoms, while the largest one contains 23 033 atoms. The dangling bonds on the particle surfaces were saturated to 100%. All cluster atoms were allowed to move freely in the calculations, resulting in more realistic geometrical structures (i.e., including both relaxation and reconstruction). An H-terminated diamond particle with a diameter of 1.6 nm is shown in Figure 2. The most abundant surface planes, (100) and (111), are marked with red and yellow circles, respectively. A smaller percentage of the diamond particle surface consists of diamond (113) and diamond (110) facets. 26063

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Figure 3. H adsorption energies (Ead in kJ/mol) for H chemisorbed onto a diamond (111) facet on diamond particles of different sizes. A comparison has been made for DFT, semiempirical, and tight-binding DFT methods.

Figure 4. C−H bond lengths (Å) for the terminating H species onto diamond (111) (black) and (100) (red) surfaces. The diamond slab thicknesses are in the range 0.2−4 nm.

mol.36 As can be seen in Figure 3, the semiempirical and DFT methods gave the numerical results that were very close to the tabulated ones. In the work by Qu et al.,37 bond dissociation energies (BDEs) were calculated for a selected number of small, amino-ring-containing hydrocarbon molecules. Results obtained using either a DFT, or a DFTB, method was especially compared. It was shown that the scatterings in calculated dissociation energy, obtained for each of the two methods used, were both smaller than 12.5 kJ/mol. However, the differences obtained between the DFT and DFTB methods were found to be much larger and could be up to 100 kJ/mol. That observation is similar to the results in the present study, where it is shown that the adsorption energy for chemisorbed H will differ by approximately 150 kJ/mol when using DFTB instead of DFT. In summary, all three methods are useful in studying trends when calculating H adsorption energies as a function of diamond particle size. However, it is only DFT and the semiempirical VAMP calculations that render numerical values of high quality. 4.3. One-Dimensional Quantum Effects: Diamond (111) and (100) Surfaces. 4.3.1. Introduction. As a result of the H adsorption energy calculations for smaller diamond particle sizes, the more accurate quantum mechanical method DFT and the semiempirical calculations were found to give the most accurate numerical result. And as stated above, also the tight-binding DFT (DFTB+) method resulted in a very good trend when studying size dependencies. It has here been decided to at first study the one-dimensional quantum effect on various film thicknesses of diamond (111) and (100) thin films. For these situations, the diamond slab thickness was increased from 0.2 to 4 nm, and it was possible to use the most accurate DFT under periodic boundary conditions. 4.3.2. Bond Lengths for Surface Terminating H Species. In order to evaluate the influence of film thickness on the surface geometrical structure, surface C−H bond lengths have been calculated for both diamond (111) and diamond (100). As can be seen in Figure 4, the bond lengths for the H-terminating species to the diamond (111) and (100) surfaces (i.e., C−H) were decreasing with film thicknesses. This is an evidence for a

larger surface reactivity for thinner diamond layers. More precisely, the diamond (111) and (100) surface reactivities were found to leveling out at a diamond film thickness of about 1 nm. Hence, for one-dimensional quantum confinement, there was only a size dependency observed for diamond (111) and (100) film thicknesses of less than 1 nm. The more pronounced scattering of C−H bond lengths for diamond (100) is most probably due to surface reconstruction, which has earlier been found to be more severe for the (100) surface, as compared with diamond (111).26,38 In addition, it was obvious from Figure 4 that the C−H bond is stronger for the diamond (100) surface compared to the diamond (111) (i.e., the C−H bond lengths are shorter for the (100) surface). When increasing the film thickness from 0.2 to 1 nm, the C−H bond length for the (100) surface increased by 0.020 Å (from 1.083 to 1.103 Å). The corresponding bond length variation for the diamond (111) surface was 0.012 Å (from 1.103 to 1.115 Å). These observations are confirming the observations and conclusions presented in an earlier paper by Petrini et al.38 4.3.3. Adsorption Energies for Surface Terminating H Species. In studying the effect of 1D quantum confinement for diamond thin film slabs, H adsorption energies were in the present study calculated for both diamond (111) and diamond (100). It has here been assumed that the adsorption energy of a small species like H is a clear indication of surface reactivity. The H adsorption energy for chemisorption of H onto diamond (111) and (100) has in Figure 5 been show for various thicknesses of the diamond surface planes. All calculations have been done by using the most accurate DFT method. As it is displayed in Figure 5, there will be a large difference when increasing the diamond film thickness from 0.2 to 1 nm, and onward, for diamond (111) and (100) thin films. After 1 nm the reactivity will level out, which means that surface reactivities are not size dependent for film thicknesses larger than 1 nm. However, between 0.2 and 1 nm in film thickness, there is a large size dependency for both of these surface planes. For the (111) surface it will decrease from −497 to −469 kJ/mol when going from a film thickness of 0.2 to 0.6 nm. For the (100) surface it will decrease from −609 to −595 kJ/mol when going from a film thickness of 0.2 to 1 nm. Hence, 26064

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to use the DFTB methods even for particle sizes up to 3 nm. As was the situation for the one-dimensional confinement, the bond lengths for the H-terminating species to the diamond surfaces (i.e., C−H) where decreasing with smaller diamond particle sizes. This is again an evidence of larger surface reactivities for smaller diamond dimensions. More precisely, the diamond (111) and (100) surface C atoms, on the nanodiamond particles, showed reactivities that decreased with particle sizes up to approximately 1.4 nm, after which it was found to level out. Concerning different surface facets on NDP, the diamond (100) surface facet did also show a much more pronounced reactivity and quantum size confinement as compared with the (111) surface facet. When using the DFTB method, an increase in particle diameter from 0.4 to 1.4 nm resulted in an enlargement of the (100) C−H bond length by 0.026 Å (from 1.083 to 1.109 Å). The corresponding bond length variation for the diamond (111) surface was 0.010 Å (from 1.106 to 1.116 Å). By using the DFT method, the corresponding increase in particle diameter resulted in an enlargement of 0.022 Å for the (100) surface C−H bond length (from 1.101 to 1.1023 Å). The corresponding bond length variation for the diamond (111) surface was 0.016 Å (from 1.110 to 1.126 Å).

Figure 5. H adsorption energies (Ead in kJ/mol) for H chemisorbed onto the diamond (111) and (100) surfaces and for different thicknesses in the range 0.2−4 nm.

these results give a multitude of information. At first, it shows that the surface is much more reactive than the diamond (111) surface. This has earlier also been shown both theoretically and experimentally.38,39 Second, the adsorption energy results show that the quantum size effects are more pronounced for the (111) surface, with an increase of 28 kJ/mol for a thickness difference of 0.4 nm. The diamond (100) surface only showed an energy difference of 14 kJ/mol for a thickness difference of 0.8 nm. Thus, the diamond (111) surface slab shows a quantum confinement effect for a size up to 1 nm, as compared with 0.6 nm for diamond (100). This observation correlates well with the more pronounced fluctuations of surface bond lengths for diamond (100) (as shown in section 4.3.2). When compared to the C−H bond length description in section 4.3.2, the H adsorption energy seems to indicate a threshold value at a lower particle size (0.5 nm instead of 1 nm). But in similar to the C−H bond lengths, both diamond surfaces showed the same behavior and the diamond (100) surface was also here found to be the most reactive one. 4.4. 3D Nanodiamond Particles with Diameters from 0.4 to 3 nm. 4.4.1. Introduction. Also, a three-dimensional nanodiamond particle has been studied in the present work, with a focus on the particle size effect on various surface-related properties. For practical reasons, two levels of theory have been used for the calculations: the more accurate DFT method and the somewhat less accurate DFTB method. The DFT method was used for particle diameters between 0.4 and 1.4 nm, and the DFTB method was used for particle size between 0.4 and 3 nm. Hence, the DFTB method served both as a method for larger sizes and to make a comparison with the more accurate DFT method. This comparison was made with the purpose to evaluate the usefulness of the DFTB method for this type of study. Moreover, since the surface (111) and (100) planes are abundant on the nanodiamond particles, the reactivity of these particle surfaces has been in focus in the present study. 4.4.2. Bond Lengths for NDP Surface Terminating H Species. The surface C−H bond lengths on the nanodiamond particles of a diameter between 0.4 and 1.4 nm were calculated by using both the DFT and DFTB methods for structural analysis. (Figure 5) As was the situation in section 4.2, calculations based on both methods resulted in identical trends for NDP sizes up to 1.4 nm. Hence, it was here found reliable

Figure 6. Surface C−H bond lengths for nanodiamond particles with a diameter ranging from 0.4 to 3 nm.

4.4.3. Surface H Adsorption Energies. Based on same arguments in section 4.2, different calculation methods had been compared and discussed for different NDP sizes with same trends. The adsorption energy for H chemisorbed to various surface planes on a nanodiamond particle has therefore been calculated using DFT and DFTB methods for energetic evaluations. As can be seen in Figure 7, there was a large difference when increasing the nanodiamond particle diameter from 0.4 to approximately 2 nm but not onward. When compared to the one-dimensional thin films, the adsorption energies will level out at approximately the same quantum size (approximately 0.7 nm in diameter) but with much more severe oscillations. The reasons for these oscillations are the more pronounced variations in structural geometries when changing the particle sizes. However, the oscillations for both diamond (111) and (100) surface-like sites were centered on values of approximately −425 kJ/mol (DFT) and −590 kJ/mol (DFTB), respectively. Hence, there is no indication that any of these surfaces ((111) or (100)) should be more reactive than the 26065

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Compared to the indicators of surface reactivity as presented above (e.g., C−H bond lengths and H adsorption energies), it is here much more difficult to identify the quantum confinement effect. A tendency for quantum size effect can be identified up to 1.2 nm in diamond particle diameter, after which the C charges and H charges on the diamond (111) and diamond (100) surface-like sites oscillates markedly. The C charges decreased to even more negative values, while the H charges increased to even more positive values. The charges of H was found to be very similar for diamond (111) and (100) facets, while if differed somewhat more for the C charges. Moreover, the surface C charges were observed to cover a larger region of charges than the H charges did: −0.118 to −0.198 e for diamond (111) surface-like sites and −0.114 to −0.459 e for diamond (100) surface-like sites. However, despite these minor differences in atomic charges, the trends in H and C charge variation with particle size were identical for both types of facets, with an equilibrium value that was reached around a diameter of 2 nm. These atomic charges can be closely correlated to the surface C−H bond lengths, which were increased from a diameter of 0.4 to 1.4 nm and then stabilized at approximately 1.113 Å for both diamond (111) and (100) (see Figure 6). Thus, the surface atomic C and H charges for diamond particle diameters above 1.4 nm matched the trend in C−H bond lengths for this specific region of particle sizes. 4.4.5. LUMO and HOMO Levels. For investigating electronic properties changes in detail, frontier energetic orbital energy levels are calculated. With an increase in nanodiamond particle sizes, clear changes in calculated lowest unoccupied molecular orbital (LUMO) level (see Figure 9) and highest occupied molecular orbital (HOMO) (see Figure 10) were observed. For comparison, the HOMO and LUMO levels for bulk diamond are inserted in Figure 9 as linear dashed lines. The experimental values for bulk diamond conduction band minimum and valence band maximum were 1.3 and −4.2 eV, respectively.40 As can be seen in Figure 9, the same trend was observed when using the DFT and DFTB methods for the smaller particles. The LUMO levels were found to decrease from 2.90 to 0.47 eV (DFT) and 6.49 to 1.69 eV (DFTB) when increasing the diameter from 0.4 to 1.4 nm, respectively. After 1.4 nm the LUMO levels obtained by DFTB decreased to 1.2076 eV. For bulk diamond, LUMO levels calculated by DFT and DFTB were 1.28 and 1.21 eV, which are very close to the experimentally obtained values. The LUMO level of nanodiamond started to converge to the bulk-like value at a diameter of 1 nm (using DFT) vs 1.8 nm (using DFTB). When calculating the HOMO levels for the nanodiamond particles, the DFT-based results shifted from −6.50 eV (diameter of 0.4 nm) to −3.96 eV (diameter of 1.4 nm). These values are to be compared with the bulk diamond HOMO level of −4.02 eV. Moreover, the HOMO levels of the NDP started to converge toward the bulk-like value at diameter of 1.4 nm. While for DFTB-based results, the HOMO values ranged from −6.79 eV (diameter of 0.4 nm) to 3.97 eV (diameter of 3 nm) and started to become bulk-like (−4.42 eV) at a diameter of 1.8 nm. Similar to all earlier observations in the present study, an identical trend in results was observed for the HOMO levels when comparing the DFT values with the DFTB ones. In summary, both the LUMO and HOMO levels for the nanodiamond particles were found to decrease/increase to approach bulk values. A clear quantum confinement effect was observed for both the HOMO and LUMO levels of the diamond particles, up to a diameter size of approximately 2 nm.

Figure 7. Surface hydrogen adsorption energies for nanodiamond particles of diameters from 0.2 to 3.0 nm.

other. Between 0.4 and approximately 1.4 nm in particle diameter, there is a large size dependency for both of these surface facets. When using the DFT method, the H adsorption energy increased from −451 to −421 kJ/mol for the diamond (111) facet and from −465 to −424 kJ/mol for the diamond (100) facet, when going from a particle diameter of 0.4 to 1.4 nm. For the same range of particle sizes, the DFTB method resulted in an H adsorption energy increase from −608 to −581 kJ/mol for diamond (111) and from −628 to −575 kJ/ mol for the diamond (100) facet. Hence, the H adsorption processes became less exothermic when increasing the particle diameter at these very narrow nanosize regions. There is a major difference between the adsorption energy results obtained for the diamond particles and films. The adsorption energies for the (111) and (100) surface-like particle sites were found to be very similar in numerical value, while they were very different for the one-dimensional diamond films (see section 4.3.2). 4.4.4. Surface Carbon and Hydrogen Charges. Another parameter that indicate the quantum size effect on surface reactivity and electronic properties is the atomic charge for the respective surface C and H atoms. As displayed in Figure 8, the trends in surface carbons and hydrogen charges have been calculated for diamond particle sizes from 0.4 to 3 nm using the DFTB method.

Figure 8. Surface carbon and hydrogen charges for (111) and (100) surface sites on nanodiamond particles with a diameter within the range 0.4−3 nm. 26066

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Figure 9. Demonstration of the lowest unoccupied molecular orbital (LUMO) energies (a), and highest occupied molecular orbital (HOMO) energies (b), as a function of diamond particle sizes. The band gaps (c) are shown for diameters from 0.4 to 3 nm. The DFT results are demonstrated by a solid black line and the DFTB results by a solid red line. The LUMO level for bulk diamond is demonstrated by a dashed blue line (DFT) and a dashed pink line (DFTB).

It shall be stressed that this threshold of diameter was also observed for the other surface reactivity indicators (i.e., C−H bond length, H adsorption energy, and surface atomic charges). The band gap results, as calculated by the difference between LUMO and HOMO energy values, are presented in Figure 9. It is also here obvious that the trend in band gaps, when varying the particle diameter, is identical when comparing the results obtained by the DFT method and the tight-binding DFT method. It can also be shown that the band gap values for the diamond particles are converging to the bulk values of 3.80 eV (DFT) vs 5.78 eV (DFTB). The experimental bulk band gap is 5.5 eV, and it is thus clear that the DFTB method will performed better in estimating band gaps. In general, it is known that periodic plane-wave density functional theory will underestimate the band gaps. However, it must here be emphasized that the present study is only focusing on the trend in band gap as a function of particle diameters. The results in Figure 9 show that the band gap decreases from 5.3 to 2.8 eV

Figure 10. Map of Fukui functions for nanodiamond particles of diameters from 0.4 to 1.4 nm (as calculated by DFT). Left column: f(−), electrophilic susceptibility. Middle column: f(+), nucleophilic susceptibility. Right column: f(0), radical susceptibility. Diameters from up to down: 0.4, 0.6, 0.8, 1.0, 1.2, and 1.4 nm.

(DFT) and from 11 to 5 eV (DFTB), when increasing the diamond particle size from 0.4 to 3 nm. Furthermore, the DFT band gaps of the diamond particles were observed to converge to the bulk diamond value at a particle diameter of 0.8 nm, while the corresponding convergence took place at 1.8 nm when using the DFTB method. When compared to the threshold values for the other surface reactive indicators (1.4 nm for C−H bond lengths and adsorption energies, 2 nm for surface C and H charges), it could be concluded that 2 nm is an 26067

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diamond). Moreover, the Fukui function maps visualized the surface reactive sites of NDP and provided a profound supplement in explaining the nanodiamond particles surface reactivities toward electrophilic, nucleophilic, and radical approaches to the surface. In summary, all surface specific parameters that have been studied in the present work have independently been found to display pronounced quantum confinements for 1D nanoscalediamond (thin films) below 1 nm in thicknesses and for 3D nanodiamond particles diameters below ∼2 nm. Since the sizes of the NCD particles that are today being used for various applications are larger than 2 nm in diameter, it has in the present study been proved that the experimental nanocrystalline diamond particles can be modeled using periodic supercells including diamond low index planes.

upper limit of size for the quantum confinements for diamond particles. Similar trends have been observed in another theoretical approach as well, which were based on DFTB calculations.20 In that study, the HOMO and LUMO levels for various shapes of a nanodiamond particles were investigated and for various particle sizes. Even though the numerical results differ as compared to the here presented results, it was shown in ref 19 that there is an existing quantum confinement up to a diamond particle size of about 2 nm. These earlier obtained results are in perfect agreement with the results in the present study. It must here be stressed that the main purpose with the present study was partially to locate the upper diamond particle diameter size for quantum confinement. In addition to this goal, further effects by particle size on various particle properties have been elucidated in the present study. 4.4.6. Surface Reactivity Site Distribution. As presented in section 2, the Fukui functions (FFs) will be able to analyze the surface chemical reactivity by studying the frontier orbitals. Three types of FFs [f(+), f(−), f(0)] were utilized to visualize the susceptibility for electrophilic, nucleophilic, and radical attacks, respectively. The f(+) and f(−) Fukui functions will thereby provide a measure of degree of electronegativity and degree of electropositivity, respectively, at the various surface spots. The results for nanodiamond particles of diameters from 0.4 to 1.4 nm are shown in Figure 10. For each of the Fukui functions represented in Figure 10 [f(−) in column 1; f(+) in column 2: f(0) in column 3], identical isovalues was used for all particle sizes (i.e., as identified with the color scale). As can further be seen in Figure 10, surface sites with more or less identical surface reactivity do also show a large similarity in the three-dimensional protrusion of the Fukui function electron density. Hence, less reactive surface sites leads to less protrude electron densities, and vice versa. As can be seen in Figure 10, there is a clear trend when going from smaller to larger nanodiamond particles. All three types of surface reactivity parameters [f(+), f(−), f(0)] show that a smaller NDP is more reactive than a larger one. It is here worth mentioning that NDPs larger than 1 nm in diameter are not susceptible for a nucleophilic attack. The Fukui function maps are, hence, strongly correlating with the other surface reactivity indicators (i.e., H adsorption energy, C−H bond lengths, atomic charges, LUMO/HOMO levels and band gap). All of these indicators have in the present study been individually pointing at a major quantum confinement up to approximately a diamond particle diameter of 2 nm.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (K.L.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The project was funded by the EU FP7 network Vascubone with the grant agreement 242175. The calculation programs were provided by Accelrys. Inc.

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REFERENCES

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