Determination of the molecular structure and (m,n) index assignment to the constituent walls of MWCNTs with undetermined diameters and chirality

Each carbon nanotube (CNT) has its own mathematical representation due to its hexagonal lattice structure. The subjects of research are multi-wall carbon nanotubes (MWCNTs) and determining their structural parameters: innermost and outermost diameters, chiral indices m and n, number of walls and their unit cell parameters. Within this paper low frequency region and corresponding high frequency parts of Raman spectra of three experimentally produced CNTs are considered, as well as use of Python programming for the most accurate (m,n) assignment. Determining the chirality of these samples enables calculation of other structural properties which are performed hereby. Furthermore, this author’s work enables future studies on the samples, as are calculation of different topological indices using the graph representation and the chirality of the studied CNT samples.


INTRODUCTION
Carbon nanotubes (CNTs) are one of the several allotropes of carbon in nanodimensions (1 nm = 10 -9 m) with highly outstanding properties. Since graphene is a 2D building unit of all carbon allotropes such as fullerenes, CNTs, nanoribbons, and so on, CNTs may be geometrically observed as wrapped 2 up graphene structure having seamless cylindrical shape. Each nanotube has its own mathematical representation due to its hexagonal lattice structure [1], [2]. The geometric structure analysis of carbon nanotubes has been quite a challenging task, particularly if the subject of research is multi-wall carbon nanotubes (MWCNTs) [1]- [3], [5]- [8]. Knowing CNTs structural parameters (diameter, chiral angle, chiral indices m and n) is basically knowing their properties, which is essential for any research in the field of CNTs and their application. There are several excellent tools as are HRTEM, ED, RRS and others that suggest some models of (m,n) assignment for single-wall carbon nanotubes (SWCNTs), as well as for MWCNTs [5]. However, precise determination of the CNTs molecular structure features becomes extremely complicated for multi-walled tubes, even when there are just few walls (layers) [6], [7]. Thorough and overall analysis and use of experimental results combined with recent theoretical background may lead to successful estimation of its structural elements.
There are different experimental methods of producing CNTs, and based on several factors, the obtained tubes may be either SWCNTs or MWCNTs. The CNT with the lowest reported diameter value experimentally produced is known to have the diameter Å 4 = k d [10]. It is the narrowest attainable nanotube that can still remain energetically stable. Such nanotubes may be the innermost constituent layer of MWCNTs. In contrast to CNTs with larger diameters, whose conductivity nature depends on their diameter and chirality, these smallest nanotubes are always metallic, regardless of the chirality [10]. Within this paper Raman spectra of three experimentally produced commercial CNTs of undetermined diameter, chirality, and number of walls, are considered. They are assigned the following nomenclatures: CNT1, CNT2, and CNT3. Due to the fact that their properties are tightly connected and dependent on their atomic structure, detailed analyses with regard to determining their diameters is performed, as well as calculating their chiral indices m and n, and furthermore other parameters, hence estimating the number of walls of each nanotube. This research was strictly focused to determination of outermost and innermost diameters, as much as to assigning the corresponding chiral indices to the tubes' walls, estimation of the number of other inner diameters, which would further implicate the number and the conductive nature of the walls. Authors strongly suggest future studies on the samples, as are performing additional EDP analysis to enhance and confirm the accuracy of applied methods, as 3 well as estimation of distance based topological indices (Wiener, Harary, Balaban, Sum-Balaban, Gutman, etc.), since it is known that they are related to some properties of the corresponding molecules.
The latter is possible by using the graph representation and the chirality of the studied samples [8].

RESULTS AND DISCUSSIONS
The analysis and the approaches to the considered nanotubes, are thoroughly described and discussed with regard to CNT1, CNT2, and CNT3.  is undetermined with regard to its diameter, chirality and number of walls. In Fig. 2 there are shown TEM and SEM images of the tubes, and it is evident that these nanotubes are of highly different diameters, which are undetermined, as is also their chirality.  The relation among the diameters and the interlayer distances is given by describes the relation among the number of layers N, the innermost diameter i d , the outermost diameter o d , and the average interlayer distance δ r .
The key role in the analysis was assigned to the Raman spectra within two frequency regions of each between the diameter of the tube and the RBM frequency ω RBM [5]. This work uses two equations (6) and (7), depending on the studied samples Raman data, since those relations are equivalent only within a limited range of diameters. While (6) shows the dependence of the innermost diameter i d and the outermost diameter o d on the corresponding frequencies, and is more accurate when small diameters are considered, (7) can be used for much larger range of diameters and is more useful when it comes to very large diameters (or extremely low frequencies). Hence it will be used whenever relation (6) is unusable or unreliable due to the size of the outermost diameter. One may notice that i d is obtained by the same equation as o d , with e C being 0. The latter is due to the fact that the parameter e C is used whenever there are environmental conditions around the nanotube [1], [5]. The innermost concentric nanotube within the MWCNT is not affected by such conditions, which is not the case with the outermost concentric tube. Obtaining possible (m,n) candidates to satisfy equation (2) and the number of these pairs was performed by using Python programming. The Python code is given in Table 1. The final choice of (m,n) assignment is made by the analysis of the G-peak. if((0.079*math.sqrt(n*n+m*m+n*m))>=((diameter-interval)) and (0.079*math.sqrt(n*n+m*m+n*m))<=((diameter+interval))): is not relevant for this research, one may notice that the ratio of their intensities ID/IG for all of the three samples is relatively high, which points to CNTs' defective structure. Regarding the G-mode only, a useful diameter dependence for the chiral CNTs (semiconducting or metallic), which was here used as a control method, is given by the equations (8) and (9) [5], [11], [12]. The number and shape of components in the G-peak are an excellent indicator of the conductive nature of the studied CNT sample. Table 2 will help the analysis when the chiral indices assignment to the samples is done [5]. The Raman spectra in both frequency regions for CNT1 are shown in Fig. 5 A, B. There is only one peak in the low frequency region, which indicates that the nanotube is a single-wall, i.e. (1) (1) Using (6), it is obtained (1) (1) 0.665 nm = = o d d . The determined diameter is lower than 1 nm and the most accurate results are expected for diameters 1 nm -2.5 nm, hence the determining possible chiral indices candidates was performed within somewhat broader diameter interval (1) (  Due to the broadness of the G-peak (see Fig. 5 B), the chiral indices candidate pairs need to satisfy the metallic condition MOD(2 ,3) 0 + = m n . Hence, only the five chiral indices pairs in red (see Table 3) are considered. However, the armchair type (5,5) would show one narrow and symmetric G-peak, and the chiral metallic (6,3), (7,1), and (8,2) would show two components of the G-peak, one being narrow (see Table 2), which here is not the case.
This leaves only one possibility, the zig-zag metallic tube (6,0), which is in high accordance with the broad and asymmetric G-peak of the nanotube CNT1.
In Figure 6 there is an illustration of CNT1 with determined diameter and chiral indices assignment.  , which accurately fits the equilibrium distance for tubes that are DWCNTs [5].
With regard to the G-peak and the chirality of each constituent SWNT, one expects to observe 4 (Ch@Ch), 3 (Ch@ACh or ACh@Ch) or 2 (ACh@ACh) components in the Raman spectrum data of any DWNT [5]. However, some components can appear at very close frequencies, and therefore cannot be easily differentiated. Hence, the number of observed components can be less than the number expected for certain sample structure (see Table 2). Considering CNT2, two identified components of the G-peak are located at the frequencies 1570.98 cm -1 and 1574 cm -1 . These frequencies enable the estimating of the diameters, and consequently the following was obtained: (2) (2), and the results of 16 combinations are derived from the pairs indicated in Table 4. Qualitative analysis of the G-peak indicated a broad component, hence a metallic chiral character of one layer and semiconducting chiral character of the other layer. Hence, the possible cases are either MC@SC or SC@MC, which corresponds to the implications from equations (8) and (9) [5], which greatly differs at the last two possibilities. It is highly expected that it would leave only one candidate combination.
In Fig. 8 there is an illustration of the two candidates with determined diameters and chiral indices assignment to the tube CNT2.

NANOTUBE CNT3. ( , ) n m ASSIGNMENT RESULTS
Several (six) peaks may be noticed in the Raman spectrum in RBLM frequency region of CNT3 ( Fig. 9 A), which identifies this nanotube as MWCNT. Presence of both narrow and broad components in the G-mode range (Fig. 9 B) implicates both semiconducting and metallic layer in the structure of CNT3.
According to the corresponding frequency values in Fig. 9 A and equation (6) is extremely low and near the limit of possible calculation, therefore its calculation may have a high error bar or even be highly inaccurate. Hence, the outermost diameter is recalculated using equation (7) it is obtained that ( Raman intensities (a.u.)

A B
A Figure 9: A) Raman spectrum RBLM for CNT3; B) Raman spectrum G-mode for CNT3 To be able to decide which outermost diameter value is good, equation (8)  13 distance of 0.340 nm is closer to the determined 0.339 nm δ = r , than the distance of 0.341 nm that holds for the option (8,0)@@(32,27). However, these findings are not enough of a discrepancy at the latter option in order for it to be excluded.
In Fig. 10 there is an illustration of the two candidates with determined diameters and chiral indices assignment to the tube CNT3. Considering this nanotube CNT3 it is again strongly suggested an additional EDP analysis to be performed. This would differentiate the two candidates, since they have different m/n ratios of the innermost constituent tubes, and hence it is highly expected that the EDP would leave only one possible candidate.

EXPERIMENTAL
The general equations used for determining the CNTs structures are given as follows.
The morphological structures of the nanotubes were investigated by TEM analysis using a FEI Tecnai G2 Spirit TWIN with a LaB6 cathode. The nanotubes were observed by scanning electron microscopy, using JEOL 6340F (SEM, 10 kV). Structural characteristics of the carbon nanostructures were studied by means of Raman spectroscopy. Non-polarized Raman spectra were recorded by a confocal Raman spectrometer (Lab Ram ARAMIS, Horiba Jobin Yvon) operating with a laser excitation source emitting at 532 nm. The low frequency regions 50-350 cm -1 were taken into consideration, as well as the frequency regions 1200-1800 cm -1 from the Raman spectra of the CNTs. Python programming was applied to determine possible chiral indices assignment to the studied samples.

CONCLUSIONS
Based on the analysis and discussions in previous sections, it is achieved a determining of MWCNT samples by means of Raman spectroscopy data and Python programming. Although in some cases limitations occure, and particularly if the subject of research is nanotubes with more than two layers, the achievement of solely using one main tool (Raman spectroscopy) is indeed outstanding. Several important findings may be concluded: • Determination of three CNTs' molecular structure has been performed, as well as determination of the conductive nature of the CNTs' walls; fully for CNT 1 and CNT 2 , and partially for CNT 3 (as presented in Table 5); • The thorough analyses were made using a unique approach combining the CNTs' Raman spectra in RBLM and G-mode frequency regions as a tool with employment of Python programming. The results, although with some limitations, enable working on determining ctructural characteristics of carbon nanotubes when there is just Raman spectroscopy data available; • The performed calculations were in excellent agreement with the theoretical background and with the control methods; • The calculations can be further improved in terms of higher certainty and results exactness, therefore corresponding methods, as is EDP, are strongly suggested; • The results enable many applications, as well as providing full necessary information for graph theorists who work on distance-based topological indices, such as Wiener, Harary, Balaban, Sum-Balaban, Gutman, etc.

ACKNOWLEDGMENTS
This study was done within the following EU projects: COST Action CA17139 "European Topology Interdisciplinary Action", and COST Action CA17140 "Cancer nanomedicine -from the bench to the bedside".