Size-Dependent Bond Dissociation Enthalpies in Single-Walled Carbon Nanotubes

a School of Science, University of Greenwich, Central Avenue, Chatham Maritime, Kent, 5 ME4 4TB, UK b Department of Chemistry, University College London, 20 Gordon Street, London, WC1H 0AJ, UK c Department of Natural Sciences, Middlesex University, Hendon Campus, The Burroughs, London, NW4 4BT, UK 10 d School of Chemistry, University of Lincoln, Brayford Pool, Lincoln, LN6 7TS, UK *Correspondence to: c.zeinalipouryazdi@greenwich.ac.uk


Introduction
Single-Walled Carbon Nanotubes (SWCNT) are used in many nanotechnology and commercial applications because of their unusual optical, electrical, mechanical and chemical properties. [1] Some of the wide range of applications include microelectronics, medicinal 25 therapy, biosensors, gas-sensors, computing and one of their broadest use is in composite materials, to enhance their mechanical strength. [2] SWCNTs can produce materials such as nanosheets [3] and nanofibers [4,5] with very strong eleastic properties. SWCNTs filled with transition metal chains have been suggested as one-dimensional nanocables. [6] These applications rely in part on the strong elastic properties, the thermodynamic stability and 30 inertness of SWCNTs and therefore, understanding these properties is important on a microscopic level. There are recent theoretical studies on structural and electronic effects of finite size zigzag or armchair carbon nanotubes of various diameters and lengths [7], their charge polarisation [8], the effect of doping [9] and the role of tube chirality on the diffusion of water. [10] There have also been studies of the properties of hybrid SWCNT with other materials, mostly carbon materials, transition metals and even confined water molecules. [11] However, 5 there are relatively few computational studies of the thermodynamic of CNTs as a function of their radius, diameter, chirality, functionalisation and doping. To the best of our knowledge there is currently no other study of the size-dependent BDE of CNTs.
Previously, computational studies have shown that the elastic constant of zig-zag and arm-chair CNTs is smaller than that of graphene.
[12] The elastic properties (Young's, shear and bulk modulus) of functionalised CNTs have been studied with periodic density functional theory (DFT) calculations for functionalised (-NH, -NH 2 , -CH 2 , -CH 3 , -OH) organic fragments. [13] The decrease of the elastic properties is roughly proportional to the strength of the binding of the functional groups and it can be 30% for high concentrations of 15 adsorbates. [13] Generalized gradient approximation (GGA) based DFT calculations show that the Young's modulus of SWCNT increase as a function of their radius from 0.946 TPa for the CNT(3,3) to 1.040 TPa of the CNT (9,9) [14], which indicates that the C-C bond becomes stronger as the diameter of the CNT increases from 0.2 nm to 0.6 nm. [14] Furthermore, hybrid exchange and correlation functional based calculations (i.e. B3LYP/6-31G) of Raman and IR 20 spectra show that in 14 zig-zag SWCNTs there is an increasing frequency in their A 2u (bending along radial axis) and E 1u (twisting along radial axis) bands, as a function of the tube diameter [15], which suggests that the elastic properties scale linearly with tube diameter. [15] The adsorption of H 2 has been studied on the CNT(10,0) using GGA DFT calculations, which found that the barrier for dissociative chemisorption is 79 kcal mol -1 and that the reaction does 25 not happen spontaneously at 0 K.[16] Therefore, the use of dopants such as Li + ion is necessary to use CNTs as hydrogen storage materials. [17] Energetic materials for propellants can be stored inside CNTs and are stabilised due to charge transfer from the CNT to the molecule. [18] Although these computational studies have addressed how the elastic and therefore the IR and Raman spectroscopic properties change as a function of the tube diameter and length, they have not given an explanation of the observed trends on the microscopic level and on the basis of the BDE of SWCNTs.
In this periodic DFT study we report the BDE and electronic properties of SWCNTs with range of lengths and diameters. In particular, we have studied the size-dependent bond dissociation enthalpy of the C-C bond (BDE CC ) and the partial-density-of-states (PDOS) of 5 various arm-chair and zig-zag SWCNTs in the length range of 3.6 nm and the radius range of 1.0 nm in order to explain the size-dependent effect of the BDE and consequently, the elastic properties of SWCNT. 10

DFT calculations
We performed periodic DFT calculations using the VASP 5.4.1 code. [19,20] The projector augmented-wave method has been used to represent the core states. [21,22] Exchange and correlation (XC) effects were considered within the generalized gradient approximation (GGA) using the revised Perdew-Burke-Ernzerhof (revPBE) XC functional. [23] The SWCNT, 15 were optimised inside an orthorhombic cell with a 20 Å vacuum gap between the CNTs and an axial vacuum gap of 15Å as shown in Fig. 1. For the 30 Å and 35 Å in length SWCNT, the unit cell width was 45 Å and 50 Å, respectively.
We have used a Γ-point[24] centered 2x2x2 Monkhorst-Pack grid for all calculations but the convergence of the energy was tested also with a 3x3x3 MP grid which gave energies to 30 within 0.0001 eV. The coordinates of the SWCNT were generated in Nanotube Modeler [25] and  Table 1. The dangling bonds in all SWCNT were saturated with one hydrogen atom per carbon atom. The cut-off energy for the energy of the planewaves was 600 eV. Geometry optimizations were performed with a residual force threshold on each atom of 0.01 eV Å -1 using the conjugate-gradient 5 algorithm. The convergence criterion for electronic relaxation was 10 -4 eV. The initial charge density was obtained by superposition of atomic charges. Dispersion corrections were included via the zero-damping DFT-D3 correction method of Grimme as implemented in VASP. [27] 2.2 Calculation of bond dissociation enthalpies 10 The change in enthalpy for the bond formation of an SWCNT from its atoms was calculated based on the following equation, , where and is the number of C and H atoms, respectively. , and are the total energies of the SWCNT, of an isolated carbon atom in its triplet state (-1.626 eV) and a 15 hydrogen atom in its doublet state (-1.178 eV) and the zero-point vibrational energy per carbon atom ( ) derived from a full electron B3LYP/STO-3G(d,f) calculation of the ZPV C of circum-circum-coronene (C 96 H 12 ) given by, The enthalpy of atomization of the SWCNT was calculated based on the following 20 relationship, , where and are the average bond-dissociation energies of the C-H bond and C-C bond, respectively. The change in enthalpy for the bond formation of the SWCNT is equal to the negative atomization enthalpy, 25 (4).
Therefore, after the combination of equations 1, 3 and 4, the average BDE per C-C bond in an arbitrary SWCNT becomes, The was estimated based on the energy required to dissociate a hydrogen atom from coronene ( = ) forming the corresponding radical ( = ) given by, For the average of coronene we have used the value 454 kJ mol -1 , which was calculated at B3YP/cc-pVDZ(5d,7f) level of theory. 5 The ZPV of C-H was calculated with the following equation, ,where and are the zero-point vibrational energies of coronene and coronene radical in its triplet state calculated at B3LYP/cc-pVDZ(5d,7f), respectively.

Size-dependent properties of SWCNT as a function of their length
To the best of our knowledge, the trends of the bond dissociation enthapy per C-C bond (BDE CC ) as a function of SWCNT length and radius, have not been previously reported. We have used equations 1-7 to calculate this thermodynamic property for various arm-chair and zig- 15 zag SWCNT in which the dangling bonds were saturated with H-atoms. These results are tabulated in table 1 along with the chirality, point group (P.G.), symmetry group (S.G.), length (l), radius (r), number of C-C bonds (n CC ) and the number of C and H atoms (n C , n H ). 20 Table 1. Chirality, point group (P.G.), symmetry group (S.G.), length (l), radius (r), number of C-C bonds (n CC ), the number of C (n C ) and H (n H ) atoms, length (l), radius (r) and the bond dissociation enthalpy per C-C bond is (BDE CC ) of the various armchair and zig-zag SWCNTs. Also shown the average BDE of C-C and C=C. [29] BDE CH C-C n/a n/a n/a n/a n/a n/a n/a 377 C=C n/a n/a n/a n/a n/a n/a n/a 728 In order to study the BDE CC as a function of SWCNT length we have choosen the zig-zag SWCNT(6,0) and arm-chair SWCNT (3,3), which have about the same radius, 4.861 Å and 4.009 Å, respectively, and had a relatively small number of carbon atoms so that longer tube lengths could be simulated. These SWCNT have relatively small diameter and therefore systems in 5 which these have lengths as 4 nm could be studied. The bond dissociation enthalpy per C-C bond is proportional to 1/length of the CNT as this is shown in Fig. 2

Size-dependent properties of SWCNT as a function of radius
The radius of a CNT is a significant geometric parameter for CNTs as it determines the curvature of the hexagonal carbon sheet that is rolled into the tubular structure. The larger the radius or diameters of a CNT the flatter its surface. For CNTs with radius greater than 10 nm electronic properties similar to graphene would be expected. In Fig. 4 the BDE CC is plotted as a This suggests that arm-chair CNT have a larger Young's modulus, tensile strength, and bending modulus than zig-zag SWCNTs when their length is below 4 nm and their radius less than 1 nm.
The lines in Fig. 4 are perfectly linear based on the R 2 = 0.999 and R 2 = 0.994 for arm- 20 chair and zig-zag, respectively. For an infinitely long SWCNT the arm-chair CNT(n,n) and zigzag CNT(m,0) we find that the BDE CC becomes 480 kJ mol -1 and 479 kJ mol -1 , which means that on the basis of radius both arm-chair and zig-zag have the same BDE CC (taking into consideration the numerical limitations of the calculations) when their radius is large.
Furthermore, at small radii (radius = 1 nm) the BDE CC of arm-chair CNTs is similar to the BDE CC of zig-zag CNTs to within 2 kJ mol -1 . Therefore, the BDE CC of both arm-chair and zigzag CNTs is the same as a function of their radius or curvature.. 5 Also in Fig. 4 we observe that the BDE CC becomes greater as the radius of the CNTs increases, which means that the carbon nanotube becomes thermodynamically more stable. We explain this increase of the thermodynamic stability via PDOS plots in section 2 which show that the overlap of the p orbitals of the π-system is greater when the curvature is smaller (i.e larger radii). and CNT(m,0) ranged between 4.191 Å -9.605 Å and 4.861 Å -7.956 Å, respectively. 15 In Fig. 5 we have plotted the BDE CC as a function of the SWCNT radius in order to show the converging trend of this thermodynamic property. Similar to the conclusions reached from Fig. 4, we observe that a radius of 10 Å is not sufficient to observe convergence of the BDE CC .
By fitting these curves to a function of the form BDE CC = A -B/r 2 we have seen that at infinite 20 radius, the BDE CC of the arm-chair and zig-zag SWCNT become 480.2 kJ mol -1 and 479.2 kJ mol -1 , respectively, which are identical within the accuraccy of DFT calculations (~ 1-2 kJ mol -1 ) but we also find that the BDE CC becomes constant, to within 0.1 kJ mol -1 , for SWCNT, with radii greater than 7.3 nm. Therefore, bundles of SWCNT that have radii greater than 7 nm will have isotropic elastic properties which is an important design feature in the use of CNTs in various applications. The trends in the BDE CC of CNTs as a function of their diameter can be explained 10 considering that there are certain energy requirements to bend a planar graphene sheet and roll it into a nanotube. We have previously studied what the bending modulus is in graphene nanoribbons and found that energy is required to bend a graphene nanoribbon (GNR), which suggests that carbon materials with curved surfaces have higher energies than flat GNR. [30] The bending energy is considerably smaller than the shearing and compression energy as we have 15 previously shown for GNR of varying sizes. [30] When the GNR is bent there is less overlap between the C p z orbitals primarily, which causes the BDE CC to decrease as the curvature of the CNT increases. Furthermore, the largest BDE CC should be observed in flat graphene sheets. [28,34] This means that in MWCNT the outermost SWCNT will always have the largest elastic properties, such as Young's modulus, tensile strength, and bending modulus. (a-c)). For these plots, we choose two carbon atoms from the center of the SWCNTs. The 10 summation of p x , p y, and p z contributions around the Fermi energy (E F ) show that with increasing length, the gap between the highest occupied and the lowest unoccupied states near the E F decrease ( Fig. 6 (a-c)). We also observe that for the SWCNT(3,3) with 30Å length there are C p states at the E F , meaning that this SWCNT is conducting while SWCNT(3,3) with a length of 10Å, and 20Å are semiconducting. To clarify the chemical bonding in these two carbon atoms 15 we analyse the overlapping between their C p z orbitals. As shown in Fig. 6 (d-f), the C p z orbitals, as expected, are completely overlapped with each other but with the increasing length of the SWCNT(3,3) these overlapping C p z orbital signatures become broader. In particular, the area under the PDOS plots from -5.5 eV to -2.7 eV was found to be 0.55 eV, 0.57 eV and 0.63 eV for the 10Å, 20Å and 30Å SWCNT(3,3), respectively. An increased overlap between p z 20 orbitals means that the C-C bonds become stronger as the length of the CNT increases. This also explains the increase in BDE CC with the increase in the SWCNT lengths. This observation has significant applications to it as it suggests that CNT-composite materials that have the same mass(CNT)/mass(composite) the one with the lengthier CNTs will have in principle larger Young's modulus, tensile strength and bending modulus. The vertical dashed line at zero represents the Fermi energy (E F ). 5 In Fig. 7 we explore the effect of the radius of a SWCNT on the BDE CC by considering the PDOS of the SWCNT(6,0) and SWCNT(9,0) with a similar length of 25Å. The analysis on p z orbital in these systems show that as the radius increases the overlap between the p z -orbitals increases significantly, which is evident from the broad overlapping signatures of SWCNT(9,0) 10 as compared to SWCNT(6,0). The reason for the increased overlap between the p z -orbitals in SWCNT(9,0) is related to the decrease in curvature, which is comparatively more pronounced in SWCNT(6,0). We conclude that the increase in BDE cc with increasing radius of SWCNT is related to the greater p z -p z overlap and hence stronger chemical bonding between the carbon atoms. This is in agreement with a previous study where we have evaluated the BDE for various 15 carbon materials (e.g. fullerene, carbon nanocones, CNTs, graphene nanoribbon) and found the BDE increases as the carbon material becomes flatter. [28,34]

Conclusions
We present a detailed DFT study of the thermodynamic properties of Single-Walled 20 Carbon Nanotubes of various lengths, diameters and chirality. Our study shows that the BDE CC is inversely proportional to the curvature-squared (1/r 2 ) and inversely proportional of the SWCNT length (1/l). We derive quantitative relationships from which the BDE can be calculated as a function of size and radius of the SWCNT. This suggests that SWCNT have stronger elastic properties as their length increases due to the emergence of metallic properties and as their radius increases due to larger overlap of the p orbitals. This clearly shows that there is a size-effect of the thermodynamic properties of SWCNTs when their length and diameter is less than 4 and 1 nm, respectively, which should be considered in the design of materials that use SWCNT of such dimensions. Lastly, we calculate that the BDE in SWCNTs is intermediate between the BDE of a C-C and C=C bond, confirming partial double bond of the carbon framework. 5

Conflicts of interest
There are no conflicts to declare.