Abstract
We calculated the nonresonant Raman spectra of peapods to determine the concentration of fullerenes inside single-walled carbon nanotubes. We focus on peapods with large diameters for which molecules can adopt a double helix configuration or a two-molecule layer configuration. Our calculations are performed within the framework of the bond-polarizability model combined with the spectral moment’s method. The changes in the Raman spectra as a function of filling rate and the configuration of molecules inside the nanotubes are identified and discussed. Our calculations support the experimental method proposed by Kuzmany to evaluate the concentration of molecules inside nanotubes.
1. Introduction
Single walled carbon nanotubes (SWCNT) can encapsulate small and large molecules such as fullerenes [1–5]. A peapod consists of a SWCNT in which fullerene molecules are inserted. This hybrid system between fullerene and SWCNT has generated a lot of interest for future electronic applications. Carbon peapods have been proposed as possible candidates for novel nanometer scale devices and many efforts are devoted to the synthesis of high-quality 1D fullerene crystals inside SWCNTs [1, 4]. Due to their original one-dimensional nanosized structure and their tunable electronic properties, peapods have several potential applications as high temperature superconductors [6], memory elements [7], and nanometer-sized containers for chemical reactions [3].
The structure of fullerenes inside SWCNTs is strongly dependent on the nanotube diameter, so that even small changes in the SWCNT diameter can alter the geometry of fullerene arrays. From an experimental point of view, structural analysis of peapods can be performed using transmission electron microscopy [1, 4], electron diffraction [5, 8], X-ray scattering [9], and/or Raman spectroscopy [4, 9, 10]. From a theoretical point of view, the organization of fullerenes inside nanotubes has been studied using several models: energy calculations with a simple van der Waals potential [11, 12], Monte Carlo, and molecular dynamic based methods [13–15]. It was shown that molecules inside SWCNTs with a wide range of diameters (1.25–2.71 nm) can form ten different packing arrangements [13, 14]. According to the theoretical calculations, molecules can arrange in a chiral array inside the SWCNTs although such a double helical packing arrangement appears to be less prominent than the zigzag one. Nevertheless, Khlobystov et al. [16] observed the double helical array of molecules inside a SWCNT and a two-molecule layer inside a double-walled carbon nanotube (DWCNT) by high-resolution TEM. The helix formation of encapsulated ’s corresponds to a weaker binding, which was shown to lead to a reversible filling-removal process of the fullerenes [17]. encapsulation was successfully used as a starting point for in-the-tube chemistry [18] and because of the experimental stimuli, the enclosed peas transform into SWCNTs within the outer tubes, thus producing double-wall carbon nanotubes [19].
Raman spectroscopy is one of the promising tools to characterize carbon nanotubes and related nanomaterials. Raman experiments of peapods were performed by several groups [10, 20–24]. The data are mostly analyzed on the basis of the theoretical predictions stated for individual SWCNTs. The analysis of these experimental results leads to the well-established conclusion: an upshift (downshift) of the peapod radial breathing-like mode (PRBLM) with respect to the RBM of empty SWCNTs occurs for host tube diameters close or smaller (larger) than 1.37 nm [i.e., the diameter of the () SWCNT].
Kuzmany et al. [10] have reported a detailed study of the Raman spectra in peapods as a function of the temperature and the line excitation. In particular, they estimated the concentration of molecules inside the tubes using the measured scattering intensity from the fullerenes relative to that from the tubes. In addition, the absolute concentration of each sample can be evaluated from the Raman measurements (see Table of [10]) because the filling rate of the reference sample was determined from electron energy loss spectroscopy (EELS) to be 60%.
In previous works [11, 25], we studied the different possible configurations of molecules inside SWCNTs with diameter lower than 2.15 nm. We predicted that molecules inserted inside a SWCNT adopt a linear arrangement for a tube diameter lower than 1.45 nm (called linear peapods) whereas a zigzag configuration is preferred for larger diameters up to 2.15 nm (called zigzag peapods). Raman spectra of free peapods and packed in bundles have been calculated for linear and zigzag configurations within the bond-polarizability model [11, 25–27]. Our results were in qualitative agreement with the experiments. In particular, we suggested that (i) the presence of two PRBLMs [22, 23] could be related to a low filling rate and (ii) the downshift of the PRBLM in peapods with large diameters could be the fingerprint of the formation of zigzag chains [11].
Here, the nonresonant Raman spectra of peapods with large diameters (2.15–2.28 nm) are calculated within the framework of the bond-polarizability model combined with the spectral moment’s method [28]. This work therefore extends our previous calculations to a larger range of diameters in which molecules can adopt a double helix (Figure 1(a)) or a two-molecule layer (Figure 1(b)) configuration [29]. Although the nonresonant approach cannot predict the variation of the line intensities with the excitation energies, our predictions are useful to follow their evolution as a function of different parameters such as the filling rate of molecules inside SWCNT. The objectives of our work are threefold. First, we list the ground state configuration of molecules inside a nanotube as a function of its diameter. Then, we report for each configuration their associated Raman responses. Finally, we evaluate, for a large range of diameters, the reliability and the transferability of the experimental method proposed by Kuzmany to estimate the relative concentration in peapods.
(a)
(b)
2. Models and Computational Method
The relaxation of the different structures is obtained from an energy minimization calculation as described in previous works [11].
The dynamical matrix of SWCNT and free molecule is calculated using the force constant models parametrized by Saito et al. [30] and Jishi et al. [31], respectively. The dynamical matrices associated with the coupling between the different subsystems (tube-tube, , or tube-) are described by the usual (12-6) Lennard-Jones potential, , where is the carbon atom-atom distance. We fixed and according to [12]. This model was successfully used to calculate the generalized phonon density-of-states of peapods [32]. We note that similar models were used successfully to model interactions in double-wall carbon nanotubes [24, 33, 34].
The Raman intensities of peapods are calculated within the framework of the nonresonant bond-polarizability model [35]. In this model, each bond is characterized by a longitudinal polarizability, , and a polarizability perpendicular to the bond, . Thus, the polarizability contribution of a particular bond can be written as follows:where and are the Cartesian directions (, , ) and is the unit vector along the bond which connects the atom and the atom covalently bonded. The derivatives of (1) with respect to the atomic displacement of the atom in the direction , , are linked to the Raman susceptibility of modes (see [36] for the detailed formalism) and are given bywhere and is the equilibrium bond distance. The values of these parameters () are usually fitted with respect to the experiments. Here, we use the same set of parameters as the one we used in the Raman calculation of the linear and zigzag peapods [11, 26].
3. Results and Discussion
In the framework of the spectral moment’s method, Raman frequencies are directly obtained from the position of the Raman lines in the calculated spectra. In all our calculations, the nanotube axis is along the -axis and a carbon atom is along the -axis of the nanotube reference frame. The laser beam is kept along the -axis of the reference frame. We consider that both incident and scattered polarization are along the -axis to calculate the -polarized spectra.
3.1. C60 Configurations inside Nanotubes
We have previously studied the different possible configurations inside SWCNT for diameters () below 2.15 nm. We found that molecules adopt two configurations: linear for nm and zigzag for nm. When the tube diameter increases up to 2.28 nm, the energy minimizations show that two other optimal configurations of molecules are possible: a double helix structure ( nm) and a two-molecule layer ( nm). Optimized structural parameters issued from the energy minimizations are listed in Table 1.
The optimum diameter of the helix can be obtained as , where is diameter and is the interlayer -SWCNT distance which can vary from 0.30 to 0.33 nm. These values are consistent with a general interaction for graphitic systems for which the gap distance is 0.33 nm. The double helix can be algebraically described as a single helix [13], whereas for larger tube diameters the structure consists of two-molecule layers with nm.
For both configurations (double helix and two-molecule layers), the interfullerene distance varies from 0.998 to 1.01 nm. This result is in good agreement with the previously reported peapod interball separation of 0.97 nm from electron-diffraction profiles [8] and 0.95 nm from high-resolution transmission electron microscopy (TEM) data [9].
3.2. Raman Spectra of C60 Completely Filled Nanotubes
The configuration of the guest molecules inside the nanotube and the Raman spectra of peapods are diameter dependent and do not significantly depend on the tube chirality [11, 27]. Thus, we can restrict our discussion without a loss of generality to zigzag tubes.
Raman spectra of peapods have been calculated for a double helix [] and a two-molecule layer [] configurations of molecules. To reach a 100% filling rate, 20 molecules have been encapsulated into the SWCNT against 32 molecules in the SWCNT. In both configurations, periodic conditions have been used along the tube axis and we considered a tube length of 12.71 nm, leading to a number of carbon atoms close to 3360 in the SWCNT and 3480 in the SWCNT. Thus, the ratio between the number of carbons in and the one in the tube, which corresponds to concentration in the tubes, is about 36% and 55% for and peapods, respectively. Results are displayed in Figure 2 for the -polarization and within the PRBLM range (Figure 2(a)), the region of the radial modes (Figure 2(b)), the intermediate range (Figure 2(c)), and the range of tangential modes [ mode for and G-modes for the tubes] (Figure 2(d)). For comparison purposes, we also included in Figure 2 the Raman responses of peapods where molecules adopt a linear chain [] and a zigzag chain [], together with a Raman spectrum of a free molecule.
(a)
(b)
(c)
(d)
For the high-frequency range above 1200 cm−1, the main lines of free molecules are located at 1219 (), 1403 (), 1468 (), and 1582 cm−1 (). These lines are slightly upshifted in peapods by 1-2 cm−1 as already reported in linear and zigzag peapods (see Figure in [11]). The tangential-like modes (TLM) in SWCNTs are almost not affected by the insertion of molecules inside the nanotube [28].
In contrast, significant changes are observed below 1200 cm−1. In the region of the radial modes, line located at 492 cm−1 in the free spectrum shows a splitting into two components at (500, 513), (499, 505), (495, 497), and (495, 498) cm−1 in , , , and , respectively. This splitting was also obtained for a model consisting of a single molecule capped inside a nanotube, suggesting that it is related to the interactions between molecules and the nanotube host, and not to the inter- interactions. Below 200 cm−1, the radial breathing mode in , , and SWCNTs, respectively, located at 168, 102, and 98 cm−1, is upshifted in the corresponding peapods to 174, 103, and 100 cm−1. In , the PRBLM displays a double-structure. This feature is made of a low-frequency (high-frequency) component located at 126 cm−1 (135 cm−1), downshifted (upshifted) with respect to the position of the RBM located at 129 cm−1 in the SWCNT. This shift and splitting depend on the configuration of molecules inside SWCNT on one side and on the inter- interactions on the other side.
3.3. C60 Filling Rate in Nanotubes
In the case of a peapod sample, it seems reasonable to consider that all the nanotubes may not be fully filled with molecules. Here, we consider the hypothesis of a partial filling of the tubes with long (quasi-infinite) chains. We also assume that the molecules tend to cluster inside nanotubes. Indeed, this should correspond to a low energy configuration of the system as the energy is lowered by the attractive interactions. This hypothesis is supported by the observations reported in [37].
Calculated -polarized Raman spectra of (double helix chain of ) and (two-molecule layer) peapods are reported in Figure 3 as a function of five filling rates (20, 40, 60, 80, and 100%). The TLM range is not reported in this figure because we observed that this range slightly depends on the degree of filling of the SWCNT.
(a)
(b)
(c)
For both configurations, a single PRBLM is predicted whatever the tube filling. For the empty tubes, the RBM is located at 102 and 98 cm−1 for and SWCNTs, respectively. For a high level of filling, it is upshifted at 103 and 100 cm−1 in and peapods, respectively. As expected, the intensity of line located at 270 cm−1 in increases when the filling factor increases.
At this stage, we investigate the evolution on the ratio between the Raman scattering intensity from molecules and that from the tubes as a function of concentration inside the tubes. To make the comparison with the experimental results easier, we proceeded into three steps described as follows. (i) We performed an average of the Raman spectra over the peapod orientations with regard to the laboratory frame. Raman spectra are calculated in the VV-polarization for unoriented peapod samples. (ii) We calculated the intensity ratio between the Raman lines of molecule [, , , , , and ] and the PRBLM or G-mode. (iii) We normalized these calculated intensity ratio with respect to the same intensity ratio calculated for the 60% filling factor sample. This normalization was chosen to comply with Kuzmany’s choice.
The relative concentrations displayed in Figure 4 have been derived according to this procedure for each Raman mode in , , , and . These peapods correspond to a configuration where molecules adopt a linear, zigzag, double helix, and two-molecule layer orientations, respectively. As expected, for all the investigated peapod diameters, the relative concentrations calculated for each mode are close. The relative concentrations calculated for infinite peapods increase when its diameter increases. Indeed, for a filling factor ~20%, the relative concentration is close to 0.21, 0.26, 0.3, and 0.4 for a diameter of 1.35 [], 1.76 [], 2.19 [], and 2.27 nm [], respectively.
Kuzmany et al. [10] performed a detailed Raman analysis of concentration in peapod bundles and a host SWCNT of 1.36 nm diameter. In order to compare our results with theirs, we calculated the average intensity ratio, also normalized on the 60% filling rate intensities, between modes of in the infinite bundle of and the PRBLM [, , , , and mode] or the G-mode of the () nanotube []. For a ~70% concentration, the average relative concentrations are found between 0.97 and 1.04 following the different phonons modes, in good agreement with the experimental value evaluated [10] 1.13. For a 20% concentration (corresponding to the L43 sample (EELS concentration ) in [10]), the average relative concentration is calculated around , in good agreement with the experimental relative concentration evaluated around 0.19.
4. Conclusions
In this paper, we investigated on peapods for a large range of diameters (2.15–2.28 nm). The optimal configurations of molecules are derived using a convenient Lennard-Jones potential. We found that molecules adopt a double helix arrangement in SWCNTs with diameters between 2.16 and 2.23 nm, whereas a layer of two molecules is preferred for larger diameters up to 2.28 nm. For the obtained configurations, the nonresonant Raman spectra have been calculated as a function of the tube diameter and filling rate using the bond-polarizability model combined with the spectral moment’s method. The variation of the average intensity ratio between Raman-active modes and the nanotube ones, as a function of the concentration molecules, has been analyzed and a general good agreement is found between calculations and measurements. This good agreement supports the experimental method proposed by Kuzmany et al. to evaluate concentration inside SWCNTs.
Finally, to improve the comparison between our models and the experimental data, calculations of the nonresonant Raman spectra of peapods with some structural defects on its wall, as experimentally observed [19, 38], are currently in progress.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Discussions with J.-L. Sauvajol and J.-L. Bantignies from Charles Coulomb Laboratory (University of Montpellier, France) were much appreciated. The work was supported by the University of Moulay Ismail Research Grant (13-2016).