Analysis of diamond nanocrystal formation from multiwalled carbon nanotubes

A systematic analysis is presented of the nanocrystalline structures generated due to the intershell C-C bonding between adjacent concentric graphene walls of multiwalled carbon nanotubes (MWCNTs). The analysis combines a comprehensive exploration of the entire parameter space determined by the geometrical characteristics of the individual graphene walls comprising the MWCNT with first-principles density-functional theory calculations of intershell C-C bonding and structural relaxation by molecular-dynamics simulation of the resulting nanocrystalline structures. We find that these structures can provide seeds for the nucleation of the cubic-diamond and hexagonal-diamond phase in the form of nanocrystals embedded in the MWCNTs. The resulting lattice structure is determined by the chirality and relative alignment of adjacent graphene walls in the MWCNT. These crystalline phases are formed over the broadest range of nanotube diameters and for any possible combination of zigzag, armchair, or chiral configurations of graphene walls. The key parameter that determines the size of the generated nanocrystals is the chiral-angle difference between adjacent graphene walls in the MWCNT.

Figure 1

(Color online) Three possible alignments, (a), (b), and (c), between two adjacent graphene planes and respective local structures generated by the formation of interplanar C–C bonds. In each case, different planar views are displayed in a Cartesian frame of reference. Dark gray (blue online) and black spheres are used to denote C atoms on the two adjacent graphene planes, respectively, that are bonded to each other.

Figure 2

(Color online) (a) Top view of relative alignment maps for two pairs of armchair nanotubes, (30,30)@(35,35) and (50,50)@(55,55), arranged concentrically. The dotted lines are used to mark the regions (a total of ten) where a certain alignment, type I or type II, is favored. [(b) and (c)] Front views of the (50,50)@(55,55) DWCNT in (b) a region where type-II alignment is favorable and (c) a region where type-I alignment is favorable. [(d) and (e)] Top views of the (50,50)@(55,55) DWCNT with close views of the local structures generated by intershell C-C bond formation in (d) one of the regions shaded dark gray (colored blue online) in (a) where type-II alignment is favorable and (e) one of the regions colored black in (a) where type-I alignment is favorable. (f) Angle $\alpha$ used for determining the possible alignment type according to the criteria described in the text. (g) Misalignment angle between nearest-neighbor atoms in different adjacent concentric nanotube graphene walls as a function of the polar angle $\theta$ in a cylindrical frame of reference for the sections marked in (a).

Figure 3

(Color online) Relative alignment maps for DWCNTs formed by two chiral nanotubes with a difference in the chiral angles, $\Delta \chi$, varying from $3.88°$ to $16.8°$ and an average diameter, $D$, of approximately 8.9 nm. The dark gray (blue online) and black colored regions correspond to atoms with intershell alignments as depicted in Figs. 1a, 1b, respectively, which favor the formation of hexagonal-diamond and cubic-diamond seeds, respectively. Only the outer nanotubes are shown for clarity. The DWCNTs have indices (a) (87,29)@(88,40) with $D=8.88\text{nm}$ and $\Delta \chi =3.89°$; (b) (80,40)@(98,28) with $D=8.97\text{nm}$ and $\Delta \chi =6.89°$; (c) (85,34)@(77,55) with $D=8.99\text{nm}$ and $\Delta \chi =8.40°$; (d) (99,9)@(90,36) with $D=8.80\text{nm}$ and $\Delta \chi =11.80°$; (e) (99,11)@(86,43) with $D=8.91\text{nm}$ and $\Delta \chi =13.90°$; and (f) (72,48)@(105,15) with $D=8.87\text{nm}$ and $\Delta \chi =16.83°$.

Figure 4

(Color online) Relative alignment maps for DWCNTs with indices (a) (58,58)@(70,56) with $D=8.56\text{nm}$ and $\Delta \chi =3.67°$; (b) (94,94)@(110,88) with $D=13.45\text{nm}$ and $\Delta \chi =3.67°$; (c) (112,0)@(112,16) with $D=9.46\text{nm}$ and $\Delta \chi =6.59°$; and (d) (165,0)@(161,23) with $D=13.59\text{nm}$ and $\Delta \chi =6.59°$. Only the outer nanotubes are shown for clarity and the gray shading (coloring online) scheme is consistent with that of Fig. 3. Comparing (a) with (b) and (c) with (d) shows that the sizes of the gray shaded (colored online) regions are independent of the DWCNT average diameter and depend only on the difference in the chiral angle, $\Delta \chi$, of the two nanotubes in each DWCNT.

Figure 5

(Color online) Relative alignment maps for DWCNTs with indices (a) (56,0)@(54,18) with $D=5.08\text{nm}$ and $\Delta \chi =13.9°$; (b) (31,31)@(50,20) with $D=4.89\text{nm}$ and $\Delta \chi =13.9°$; and (c) (54,6)@(50,25) with $D=5.18\text{nm}$ and $\Delta \chi =13.9°$. Only the outer nanotubes are shown for clarity and the gray shading (coloring online) scheme is consistent with that of Fig. 3. Comparing (a), (b), and (c) with each other shows that the sizes of the gray shaded (colored online) regions are independent of the chirality of the individual nanotubes and depend only on the difference in the chiral angle, $\Delta \chi$, of the two nanotubes in each DWCNT.

Figure 6

Maximum number of hexagonal cells in an individual patch of a relative alignment map in DWCNTs as a function of the difference in the chiral angles, $\Delta \chi$, between the individual nanotubes of the DWCNTs and for various DWCNT average diameters. Black, gray, and open symbols are used to represent combinations of zigzag (Z) and chiral (C) walls (Z@C), armchair (A) and chiral walls (A@C), and two chiral walls (C@C), respectively. Diamonds, squares, and triangles correspond to DWCNTs with average external diameters $D_1=5.20\text{nm}$, $D_2=9.01\text{nm}$, and $D_3=13.64\text{nm}$, respectively.

Figure 7

(Color online) Structure resulting from intershell C-C bonding in a MWCNT according to first-principles DFT calculations. [(a)–(d)] Optimized atomic configuration of MWCNT with indices (6,0)@(15,0)@(24,0), i.e., with (6,0), (15,0), and (24,0) concentric inner [shaded very dark gray (colored in blue online)], middle [shaded dark gray (colored in green online)], and outer [shaded gray (colored in gray online)] nanotubes, respectively, depicting intershell C-C bonding. (a) Top view along the axial direction. (b) Cross-sectional view on the symmetry plane. Very dark gray (blue online), dark gray (green online), gray, and light gray (yellow online) spheres denote C atoms. The light-gray shaded (yellow-colored online) C atoms highlight the seed unit for the nucleation of a bulk phase of hexagonal diamond (lonsdaleite). (c) VED distribution on a plane perpendicular to the axial direction of the MWCNT. (d) VED distribution on a plane parallel to the axial direction of the MWCNT containing the atoms labeled C1, C2, and C3 in (a) and (b). The VED distributions in (c) and (d) highlight the intertube interactions and reveal the intershell C-C bonding and the accompanying shell distortions and structural relaxations [e.g., region enclosed by the triangle in (c)]. (e) Lonsdaleite lattice. (f) VED distribution for the lonsdaleite phase on the plane of C atoms labeled C4, C5, and C6 in (e). In (b) and (e), the numbers shown indicate interatomic distances or lattice parameters in $\text{Å}$. In (c), (d), and (f), the VED increases as the color changes from red (zero VED) to blue. For interpretation of the referneces to color in the VED distributions, the reader is referred to the online version of this article.

Figure 8

(Color online) Structure resulting from intershell C-C bonding in a MWCNT according to first-principles DFT calculations. (a)–(e) Optimized atomic configuration of a (6,0)@(15,0)@(24,0) MWCNT with (6,0), (15,0), and (24,0) concentric inner [shaded very dark gray (colored in blue online)], middle [shaded dark gray (colored in green online)], and outer [shaded gray (colored in gray online)] nanotubes, respectively, depicting intershell C-C bonding. (a) Top view along the axial direction. (b) Only a small region from the original supercell in (a) is shown highlighting the seed unit for the nucleation of a cubic-diamond crystal. Very dark gray (blue online), dark gray (green online), and gray spheres denote C atoms. (c) VED distribution on a plane perpendicular to the axial direction of the MWCNT. (d) VED distribution on another plane perpendicular to the axial direction of the MWCNT. (e) VED distribution on a plane containing the atoms labeled C7, C8, and C9 in (a) and (b). The VED distributions in (c), (d), and (e) highlight the intertube interactions and reveal the intershell C-C bonding and the accompanying shell distortions and structural relaxations. (e) Cubic-diamond lattice. (f) VED distribution for the cubic-diamond phase on the plane of C atoms labeled C12, C13, and C14 in (f). In (b) and (f), the numbers shown indicate interatomic distances or lattice parameters in $\text{Å}$. In (c), (d), (e), and (g), the VED increases as the color changes from red (zero VED) to blue. For interpretation of the references to color in the VED distributions, the reader is referred to the online versiion of this article.

Figure 9

(Color online) Representative MD-relaxed local structures formed by the creation of intershell C-C bonds in a DWCNT with indices (63,63)@(81,54) and $\Delta \chi =6.6°$. (a) Regions of the DWCNT where these relaxed structures are created, marked with black and blue circles, and local coordinate vectors used for structural analysis and visualization. Only the outer nanotube of the DWCNT is shown for clarity and the gray shading (coloring online) scheme is consistent with that of Fig. 3. [(b) and (c)] Local structures that can act as seeds for the nucleation of (b) the cubic-diamond phase and (c) the hexagonal-diamond phase, extracted from the DWCNT. [(d) and (e)] Different planar views, (i), (ii), (iii), and (iv), of the structures given in (b) and (c), respectively, using the local coordinates defined in (a). In (b)–(e), light and dark gray (orange and green online) spheres are used to represent carbon atoms belonging to the outer and inner nanotube of the DWCNT, respectively.

Figure 10

(Color online) Representative MD-relaxed local structures formed by the creation of intershell C-C bonds in a DWCNT with indices (99,9)@(90,36) and $\Delta \chi =11.8°$. (a) Regions of the DWCNT where these relaxed structures are created, marked with black and blue circles, and local coordinate vectors used for structural analysis and visualization. Only the outer nanotube of the DWCNT is shown for clarity and the gray shading (coloring online) scheme is consistent with that of Fig. 3. [(b) and (c)] Local structures that can act as seeds for the nucleation of (b) the cubic-diamond phase and (c) the hexagonal-diamond phase, extracted from the DWCNT. [(d) and (e)] Different planar views, (i), (ii), (iii), and (iv), of the structures given in (b) and (c), respectively, using the local coordinates defined in (a). In (b)–(e), light and dark gray (orange and green online) spheres are used to represent carbon atoms belonging to the outer and inner nanotube of the DWCNT, respectively.

Figure 11

(Color online) Representative MD-relaxed local structures formed by the creation of intershell C-C bonds in a DWCNT with indices (110,0)@(95,38) and $\Delta \chi =16.1°$. (a) Regions of the DWCNT where these relaxed structures are created, marked with black and blue circles, and local coordinate vectors used for structural analysis and visualization. Only the outer nanotube of the DWCNT is shown for clarity and the gray shading (coloring online) scheme is consistent with that of Fig. 3. [(b) and (c)] Local structures that can act as seeds for the nucleation of (b) the cubic-diamond phase and (c) the hexagonal-diamond phase, extracted from the DWCNT. [(d) and (e)] Different planar views, (i), (ii), (iii), and (iv), of the structures given in (b) and (c), respectively, using the local coordinates defined in (a). In (b)–(e), light and dark gray (orange and green online) spheres are used to represent carbon atoms belonging to the outer and inner nanotube of the DWCNT, respectively.

Figure 12

Crystalline structures embedded locally in a (50,50)@(55,55)@(60,60) MWCNT due to intershell C-C bonding that provide seeds for the nucleation of (a) cubic diamond and (b) hexagonal diamond. In both (a) and (b), a top view of the MWCNT with the embedded nanocrystal is shown on the left and three different close planar views of the nanocrystalline regions are shown on the right, corresponding from top to bottom to the planes $\left(\theta ,z\right)$, $\left(r,\theta \right)$, and $\left(r,z\right)$ in a cylindrical-coordinate system.