Dipole polarizability of nanodiamonds and related structures

Diamond & Related Materials 55 (2015) 64–69

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Dipole polarizability of nanodiamonds and related structures Denis Sh. Sabirov a,⁎, Eiji Ōsawa b a b

Institute of Petrochemistry and Catalysis, Russian Academy of Sciences, Ufa 450075, Russia NanoCarbon Research Institute, AREC, Faculty of Textile Science and Technology, Shinshu University, Ueda 386-8567, Japan

a r t i c l e

i n f o

Article history: Received 5 December 2014 Received in revised form 5 March 2015 Accepted 6 March 2015 Available online 11 March 2015 Keywords: Nanodiamond Fluorinated nanodiamond Hydroxylated nanodiamond Nitrogen vacancy centers Dipole polarizability Density functional theory

a b s t r a c t In the present work, mean dipole polarizabilities of the nanodiamonds and the related nanostructures modified with nitrogen vacancy centers or surface functional groups F and OH have been studied by DFT methods. Linear correlation between the mean polarizability and the size of nanodiamonds has been justified. It opens opportunities for prediction of polarizabilities of larger nanodiamond particles without high-cost quantum-chemical calculations. Introducing nitrogen vacancy centers slightly increases the resulting mean polarizability due to the repulsive interactions of lone pairs of N atoms (in A, B, and N3 centers) or unpaired electron (C center). Mean polarizabilities of polyfluorinated and polyhydroxylated nanodiamonds have been calculated for model structures with isolated, focal, and compact arrangements of the F/OHs on the surface. The enhancement of polarizability over the expected additive value has been found for nanodiamonds with compact placement of F/OH and in the case of the polyhydroxylated nanodiamond with the isolated OH groups. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Recent years are marked with a growing interest in carbon materials. New allotropes such as fullerenes and carbon nanotubes are not the only source of novel insights into this field. Classic graphite and diamond acquire unusual physicochemical properties when converted into their nanoscale forms, graphene and nanodiamonds, respectively. Nanodiamonds (NDs) stay apart from the mentioned nanocarbon allotropes because these mainly contain sp3-hybridized carbon atoms. The shape, size, and structure of NDs depend on the way of their production (detonation or high-pressure and high-temperature synthesis [1–3]), so physical properties of NDs are consequently varied. This causes diversity of their applications: as additives in polymer composites [4], platforms for drug delivery [5], fluorescent markers for biomedical applications [6], materials for solar cells [7] and electronic devices (see also reviews, considered on the applications [2,3,8,9]). In addition to the production strategies, physicochemical properties of NDs may be tuned by surface modifications [10–17], so NDs are chemically functionalized in many ways by coating with hydrogen [11,14,16], fluorine atoms [11,12], or hydroxyl groups [14,17]. Another way of changing the ND properties comes from the inside, viz. the defects in their bulk structure. Here, nitrogen vacancy centers are the most widespread and experimentally well studied [18]. For better understanding of physicochemical properties, NDs have been theoretically treated in terms of diverse approaches (semiempirical [19], ab initio [19], tight-binding DFT [20–24], molecular dynamics ⁎ Corresponding author. E-mail address: [email protected] (D.S. Sabirov).

http://dx.doi.org/10.1016/j.diamond.2015.03.009 0925-9635/© 2015 Elsevier B.V. All rights reserved.

[25,26], and DFT [27–31]). These works scrutinized stability [27,28], electronic properties [28], vacancies [18,31], aggregation [20,23,24], and chemical transformations of NDs [21,22,25,26]. Only one previous work reported on the dipole polarizability of NDs [19]. However, this property has a wide-range influence upon the physicochemical properties in the other carbon nanostructures. As previously shown, polarizability is sensitive to chemical modifications and plays an important role in understanding fullerene chemistry and materials science [32–40]. It defines many physical properties (e.g., refraction and dispersion interaction) and explains diverse physical processes in fullerene-containing systems such as aggregation, quenching of electronically-excited states, wetting, and Rayleigh scattering. Moreover, it is recognized as a criterion of usability of fullerene derivatives as screening cages or electron acceptors for molecular devices and organic solar cells [34,36–38]. In a fundamental aspect, changes in chemical structure of fullerenes and their derivatives lead to positive and negative deviations from the appropriate additive schemes. This finding is considered as one of the important tools for the design of novel fullerene-based compounds [33]. On the other hand, measurements of polarizability require rather sophisticated experimental procedures and special preparation of the samples (sufficient amounts and purity). This is rarely possible in the case of novel carbon nanostructures such as nanodiamonds. At the same time, data on polarizability of nanodiamonds could be useful for understanding of their physical properties (optical and electro-optical effects, scattering, aggregation, etc. [1,2, 7–9]) and provide novel insights to their applications. To the best of our knowledge, polarizability of nanodiamonds and their modified forms has been poorly studied. In the only theoretical work cited above [19], mean polarizabilities of hydrocarbons with

D.S. Sabirov, E. Ōsawa / Diamond & Related Materials 55 (2015) 64–69

tetrahedral symmetry from CH4 to ND-like C136H104 have been calculated with semiempirical and Hartree–Fock methods. Unfortunately, the mentioned work considered only highly symmetric species and did not include structures with low symmetry, vacancies, or surface modifications of NDs. To fill the lack of information about the ND polarizability, we have studied dipole polarizabilities of nanodiamonds and their modified forms in terms of two accurate and widely used density functional theory methods. 2. Computational details Construction of the nanodiamond molecules is based on common crystallographic notions. We have used well-known a, b, and c vectors for diamond crystals to specify translation vectors t as t ¼ x1 a þ x2 b þ x3 c;

ð1Þ

where integer coefficients xi were varied in the range between 2 and 5. We avoided xi = 1 because the “core” of the nanodiamond (C atoms lying in the depth of the structure) is not formed in this case. To generate the ND structure, each of the vectors t with Σ xi b 6 was multiplied on the matrix with Cartesian coordinates of the diamond elementary cell (8 atoms). All the generated carbon atoms with less than three neighbors were then deleted (after the addition of hydrogen, such C atoms would be transformed into the CH3 or CH2CH3 groups). Free valences were saturated with hydrogen atoms. The described algorithm allows obtaining initial ND geometries with \CH and N CH2 moieties on the surfaces. All the initial structures were optimized by the density functional theory methods PBE/3ζ [41] and B3LYP/Λ1 [42,43], implemented in the Priroda program [44]. The chosen methods PBE/3ζ and B3LYP/Λ1 reproduce structures and physicochemical characteristics of diverse hydrocarbons, fullerenes, and their OH/F-containing derivatives with high accuracy [33,35,36,38,40,45–50]. Additionally, these methods correctly reproduce the experimental mean polarizability of adamantane (15.9 Å3), obtained by the depolarized collision-induced light scattering technique [51]: the experimental and calculated values 16.3 (PBE/3ζ) and 16.6 Å3 (B3LYP/Λ1) well agree. After the DFT-optimizations and vibrational analysis using standard techniques, the components of polarizability tensors α were calculated in terms of the finite field approach as the second order derivatives of the total energy E with respect to the homogenous external electric field F (the field gradient and higher derivatives are zero): αi j ¼ −

∂2 E : ∂F i ∂F j

65

Table 1 Mean polarizabilities of NDs, calculated by the PBE/3ζ and B3LYP/Λ1 methods. Formula

C10H16 C14H20 C18H24 C26H32 C46H48 C81H84 C98H100 C111H110 C127H118 C141H136 C197H160 C225H182 a

Symmetry

Td D3d C2v T CS CS CS CS CS CS C3 CS

Number of C atoms CH

CH2

C

4 8 10 8 18 12 10 14 14 18 16 22

6 6 7 12 15 36 45 48 52 59 72 80

0 0 1 6 13 33 43 49 61 64 109 123

Percentage of core atomsa

αPBE, Å3

αB3LYP, Å3

0.0 0.0 5.6 23.1 28.3 40.7 43.9 44.1 48.0 45.4 55.3 54.7

16.3 22.0 27.8 39.2 66.1 113.0 136.6 155.0 171.3 198.5 253.4 296.0

16.6 22.6 28.7 40.7 69.6 122.3 148.2 169.4 187.7 218.5 280.5 328.1

φ = N(non-hydrogenated C) / N(hydrogenated C) × 100%.

with the CS symmetry point group, except C197H182, which has a C3 symmetry (Fig. 1). Additionally, we have studied more symmetrical hydrocarbons C10–C46, starting from adamantane C10H16 (Td), the first member of the diamondoid family. The content of the core carbon atoms is lower in this series (up to 28.3%). However, as we found, the joint consideration of large and small diamond-like species is justified. Their calculated mean polarizabilities fit to the general trend of increasing polarizability while going from smaller to larger ones. Since calculating polarizability of large nanostructures by pure-DFT techniques requires high computational load, a correlation between the structure and mean polarizability was sought. We found that the mean polarizabilities, calculated by both DFT methods, are linearly correlated with the number of carbon atoms in ND (NC) (Fig. 2): PBE=3ζ

α Cx Hy

  2 ¼ 6:340 þ 1:297NC R ¼ 0:999

B3LYP=Λ1

α Cx Hy

  2 ¼ 4:141 þ 1:447N C R ¼ 0:999 :

ð4Þ ð5Þ

ð2Þ

Tensors α were calculated in the arbitrary coordinate system. Their independent values were used for the calculation of the mean polarizabilities: α¼

 1 α þ α yy þ α zz : 3 xx

ð3Þ

Cartesian coordinates of all the ND structures, optimized by the PBE/ 3ζ method, are available as Supplementary material. 3. Results and discussion 3.1. ND polarizability Using the described algorithm, we obtained several optimized ND structures that contain from 81 to 225 carbon atoms per molecule (Table 1). In this range, the percentage of the non-hydrogenated (core) carbon atoms is within 40–55%. The structures are characterized

Fig. 1. Typical structures of the ND structures under study. Surfaces are colored for clarity. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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studied the effect of different NVCs on the resulting mean polarizability of NDs by the PBE/3ζ method (Fig. 3). For this purpose, we have used the C196H160 structure (it is a ~10 × 10 × 10 Å cube-like ND), in which NVC can be placed in the center of the structure and sufficiently remote from the ND surface. The types and structures of NVC-NDs C197 − xH160Ny are described in Table 2 and Fig. 3. In terms of the relevant additive schemes [33,58-60], changes in the mean polarizabilities of C197 − xH160Ny can be estimated according to the following equation:

expected

Δα ðx;yÞ

Fig. 2. Correlation between the mean polarizability of ND and the number of carbon atoms in the ND structure.

It is noteworthy that correlations (4) and (5) cover diamond-like structures with different symmetries. The linear correlation between α and number of carbon atoms in NDs was first discovered by Bishop and Gu [19] with semiempirical and Hartree–Fock quantum-chemical methods and postulated for Td-symmetrical hydrocarbons. In the present study, the linearity has been justified by DFT methods and extended to the general case (the structures with different symmetries and different numbers of hydrogen atoms). Note that the B3LYP/Λ1 method gave slightly higher ND polarizabilities in comparison with the PBE/3ζ values. This situation is inherited from the case of adamantane, the simplest compound of the series (αB3LYP/Λ1 N αPBE/3ζ for C10H16; Table 1). The difference in the computed mean polarizabilities between two methods is small for most of the compounds considered but recorded ~30 Å3 for the largest ND structure studied (C225H182). However, both methods reveal a common tendency of linear increase in the mean polarizability with the increase of the ND's size. Hence, we can recommend correlations (4) and (5) for lower and upper estimates of mean polarizability of ND structures larger than C225H182. We should mention that the behavior of polarizability in nanodiamonds with increasing size essentially differs from the fullerenes and related nanostructures. As shown previously, the mean polarizability of fullerenes increases non-linearly with the size and shows a positive deviation from additivity [33]. The same trend is seen in aromatic hydrocarbons [49], polyene [52], and polyyne chains [53]. In the present study, we have considered NDs, constructed with only sp3-hybridized carbon atoms. Thus, the sp 3 -carbon nanostructures demonstrate additivity of polarizability in a wide size range in contrast to the spn-carbon nanostructures with n b 3. Other models of ND particles consider that the structures have an sp3-carbon core, covered with sp2 + x - and sp2-hybridized carbon atoms (x b 1) [54]. We propose that their polarizability can be considered as a sum of polarizabilities of sp3-C, sp2 + x-C, and sp2-C parts. The first term linearly increases with the particle size whereas the other terms give a positive deviation from linearity.

¼ yα N −xα C ;

ð6Þ

where αN (0.82 Å3) and αC (0.98 Å3) are the atomic polarizabilities of nitrogen and carbon, respectively (calculated with the same method). As αN b αC and y b x, all the NVCs should decrease the resulting mean must be negpolarizability. It means that the expected values Δα expected (x,y) takes into account only changes in ative (Table 2). Note that Δα expected (x,y) the polarizability due to the replacement of carbon atoms with nitrogen. In contrast to the additive scheme, pure DFT calculations indicate an increase in the mean polarizability for all types of NVCs. The respective changes in polarizability ΔαDFT (x,y), calculated by D FT

D FT

D FT

Δα ðx;yÞ ¼ α C197−x H160 Ny −α C197 H160 ;

ð7Þ

DFT are positive values (Table 2). Positive signs of Δα(x,y) are caused by the repulsive interactions of lone electron pairs of neighboring nitrogen atoms in A, B, and N3 centers. In the case of C-center, an increase in polarizability is due to the readily polarizable unpaired electron, located on the carbon atom of the NVC. On the other hand, all the studied NVCs weakly enhance the mean polarizability of nanodiamonds. Indeed, the DFT are equal to 0.11–4.66 Å3 or 0.04– calculated enhancements Δα(x,y) 1.83% of mean polarizability in the intact ND (C197H160).

3.2. Polarizability of ND with nitrogen vacancy centers Typical defects of bulk diamond may arise in NDs as well [55]. Nitrogen vacancy centers (NVCs) of three types (usually designated as A, B, C, and N3 [56,57]) are the most widespread among them. Their electronic structures and stabilities were computationally studied in the previous works [18,31,57] but not their polarizabilities. Therefore, we have

Fig. 3. Nanodiamonds with nitrogen vacancy centers. Nitrogen atoms are highlighted in blue. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

D.S. Sabirov, E. Ōsawa / Diamond & Related Materials 55 (2015) 64–69

67

Table 2 Mean polarizabilities of C196H160 with nitrogen vacancy centers, calculated by the PBE/3ζ method. Type of NVC

Formula

x

y

, Å3 Δαexpected (x,y)

3 ΔαDFT (x,y), Å

3 FT α DC197−x H160 Ny , Å

A B Ca N3a Initial ND

C195H160N2 C192H160N4 • C195H160N • C193H160N3 C197H160

2 5 2 4 0

2 4 1 3 0

−0.31 −1.60 −1.14 −1.45 0

+1.33 +0.11 +4.66 +0.83 0

254.70 253.48 258.03 254.20 253.37

a

Paramagnetic species.

3.3. Polarizability of fluorinated and hydroxylated NDs As fluorinated and hydroxylated NDs are experimentally studied [11,12,14], we have proposed several types of NDs, derived from the aforementioned C197H160 nanostructure and coated with F atoms or OH groups. The modified NDs with only one carbon atom functionalized by two groups were used for auxiliary calculations. Such functionalization leads to NDs with NCX2 moieties in the structure C197H158X2 (X = F or OH). The mean polarizabilities of C197H158X2, calculated by the PBE/3ζ method, are slightly higher than the respective values of the parent C197H160. This is demonstrated by α2X values that indicate changes in the mean polarizability resulting from the substitution of two hydrogen atoms by X: D FT

D FT

α 2X ¼ α C197 H158 X2 −α C197 H160 :

ð8Þ

Calculations according to Eq. (8), lead to α2F = 0.54 and α2OH = 1.83 Å3. These two values can be used as the increments for the calculation of additive mean polarizabilities as follows: add

D FT

α C197 H160−2n X2n ¼ α C197 H160 þ nα 2X :

ð9Þ

Eq. (9) means that the coating pattern does not influence the resulting mean polarizability. Indeed, in our previous works on fullerene polarizability, we used the analogous approach and demonstrated that mean polarizabilities of positional isomers of halo-, cyclopropa-, and epoxyfullerenes are almost equal [33]. In the present work, we calculated three types of coating patterns of NDs: with isolated, compact, or focal arrangement of 18 X groups on the ND surface (Fig. 4). Additionally, their mean polarizabilities have been estimated by Eq. (9), and the D FT respective differences Δα between α add C197 H160−2n X2n and α C197 H160−2n X2n values have been calculated (Table 3). DFT-calculations show that the mean polarizabilities of hydroxylated and fluorinated species C197H142X18 violate additivity (Δα ≠ 0). Herewith, depending on the nature of X and type of coating pattern, Δα may be either positive or negative. The largest (and the most explainable) deviation is observed in the case of C197H142F18 and C197H142(OH)18 with compact arrangement of the decorating atoms/ groups (9.49 and 13.33 Å3, respectively). This enhancement originates from the repulsion of electronic clouds of the neighboring F or OH groups as well as considerable distortions of the subsurface ND structure. Positive deviation of 10.12 Å3, comparable with the above, is typical for C197H142(OH)18 with isolated hydroxyls. In the other cases, the pure-DFT mean polarizabilities are lower than the respective additive values. For better understanding of the behavior of polarizability of the functionalized NDs, further studies are needed. 4. Conclusions In the present work, mean polarizabilities of the nanodiamonds and the related structures (NDs with nitrogen vacancy centers and functionalized NDs) have been studied by DFT methods. General linear correlation between the mean polarizability and the size of ND has been obtained for ND structures with different symmetries and sizes. It can

Fig. 4. Hydroxylated and fluorinated NDs C197H142X18 with isolated (a, b), focal (c, d), and compact (e, f) arrangement of X groups.

be used for the prediction of polarizabilities of larger ND species without demanding quantum-chemical calculations. Mean polarizabilities of NDs with nitrogen vacancy centers A, B, C, and N3 are slightly higher than the values expected from the replacement of carbon atoms by nitrogen. This enhancement is due to the repulsive interactions between lone pairs of N atoms of NVCs (A, B, and N3 centers) or the unpaired electron (C-center). Though the enhancement found here is small, it may be greater if multiple NVCs emerge in the ND structure. Mean polarizabilities of fluorinated and hydroxylated NDs have been calculated for model structures with isolated, focal, and compact arrangements of the F/OHs on the ND's surface. The excess of mean polarizability over the expected additive values has been found for NDs with compact placement of F/OHs and in the case of hydroxylated ND with isolated OH groups. This behavior is distinctly different from the polarizabilities of fullerene derivatives and depends on the positional relationship of the functional groups on the surface. However, we cannot explain this behavior at the moment.

Table 3 Mean polarizabilities of the functionalized nanodiamonds, calculated by the PBE/3ζ method and estimated according to the additive scheme (Å3). NDa

α add C197 H160−2n X2n

FT α DC197 H160−2n X2n

Δαb

C197H158F2 isol-C197H142F18 comp-C197H142F18 foc-C197H142F18 C197H158(OH)2 isol-C197H142(OH)18 comp-C197H142(OH)18 foc-C197H142(OH)18

– 258.77 258.77 258.77 – 269.84 269.84 269.84

253.91 253.58 268.26 256.46 255.20 279.96 313.17 267.92

– −5.19 +9.49 −2.31 – +10.12 +13.33 −1.92

a Isolated, compact, and focal arrangements of the functional groups are designated as isol, comp, and foc, respectively. b FT add Δα ¼ α DC197 H160−2n X2n −α C197 H160−2n X2n .

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Prime novelty statement Using density functional theory, the additivity of dipole polarizability has been justified for the diverse ND structures. As accurately calculated, the surface and core modifications of NDs (OH/F covering or introducing nitrogen vacancy centers) lead to the changes in polarizability with additivity violation. The obtained correlations may be used for the prediction of polarizability of larger ND structures as well as for calculating polarizability-dependent properties (Rayleigh scattering constants, molecular refraction, etc.).

Acknowledgments Denis Sabirov is grateful to the Presidium of the Russian Academy of Sciences for the financial support (Program No. 24 ‘Foundations of basic research of nanotechnologies and nanomaterials’). Eiji Ōsawa acknowledges Interdisciplinary Program on Material Efficiency of the G8 Research Councils Initiative on Multilateral Research Funding for USJapan-Russia Joint Proposal, ‘Nanodiamond-based Nanospacer Lubricants’ 2011-2015.

Appendix A. Supplementary data Supplementary data to this article can be found online at http://dx. doi.org/10.1016/j.diamond.2015.03.009.

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