Structural and Phonon Properties of Bundled Single-and Double-Wall

Article pubs.acs.org/JPCC

Structural and Phonon Properties of Bundled Single- and DoubleWall Carbon Nanotubes Under Pressure A. L. Aguiar,†,‡ Rodrigo B. Capaz,§ A. G. Souza Filho,*,† and A. San-Miguel*,‡ †

Departamento de Física, Universidade Federal do Ceará, 60455-900 Fortaleza, Ceará, Brazil Université de Lyon, F-69000, France, Université Lyon 1, Laboratoire PMCN, CNRS, UMR 5586, F-69622 Villeurbanne Cedex, France § Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ 21941-972, Brazil ‡

ABSTRACT: In this work, we report a theoretical study of the structural and phonon properties of bundled single- (SWNTs) and double-walled carbon nanotubes (DWNTs) under hydrostatic compression. Our results confirm drastic changes in volume of SWNTs in a high-pressure regime as assigned by a phase transition from circular to collapsed phase which are strictly dependent on the tube diameter. For the DWNTs, those results show first a transformation to a polygonized shape of the outer tube and subsequently the simultaneous collapse of the outer and inner tube, at the onset of the inner tube ovalization. Before the DWNT collapse, phonon calculations reproduce the experimentally observed screening effect on the inner tube pressure-induced blue shift for both radial and tangential modes. Furthermore, the collapse of nanotube bundles induces a sudden redshift of tangential components in agreement with experimental reports. The Gz band analysis of the SWNT collapsed tubes shows that the flattened regions of the tubes are at the origin of their G-band signal. This explains the observed graphite-type pressure evolution of the G band in the collapsed phase and provides in addition a means for the identification of collapsed tubes using Raman spectroscopy.

I. INTRODUCTION Carbon nanotubes are key materials for developing nanoelectromechanical systems because they present a unique combination of outstanding electronic and mechanical properties. Cross-sectional deformations of single-walled carbon nanotubes (SWNTs) under pressure have been extensively studied by means of experimental tools1−5 and theoretical models.6−12 On the other hand, the behavior of double-walled carbon nanotubes (DWNTs) under pressure has been considerably less studied.13−17 DWNTs may be better candidate materials than SWNTs for the engineering of nanotube-based composite materials because of their geometry, in which the outer tube ensures the chemical coupling with the matrix and the inner tube acts as mechanical support for the whole system.18 Some theoretical studies have been performed to study the pressure dependence of the cross-sectional shape of SWNTs, depending on the chirality and diameter. By starting from an almost perfect circular cross section, the cross-sectional shape of the nanotube becomes oval or polygonized as the pressure is increased, evolving later to a collapsed state with a peanut shape.11,19−22 On the basis of elasticity theory, Sluiter et al. have proposed a diameter-dependent phase diagram for the crosssectional shape of SWNTs under pressure.23 Some authors reported that the radial deformation of SWNTs for diameters smaller than 2.5 nm is reversible from the collapsed state; however, the deformation of larger diameter tubes could be © 2012 American Chemical Society

irreversible, and the collapsed state is metastable or even absolutely stable without pressure application.11,23 Many experimental studies have suggested that phase transition in nanotubes could be dependent on their metallic character or on the surrounding chemical environment used for transmitting the pressure.4,24−27 However, theoretical calculations suggest that phase transitions of SWNTs under pressure are mainly dependent on the cube diameter (pc ∼ d−3 t ) of the tubes and not on the chirality.11,28 Even if there is a huge dispersion of results concerning the pressure transition values from theoretical models, there is an overall agreement between different calculations on the existence of two phase transitions (circular−oval and oval−peanut) and that the critical pressures for those transitions depend on nanotube diameter. The pressure evolution of DWNTs has been much less studied, and in opposition to SWNTs, a detailed understanding of their cross-sectional evolution is still under discussion due to their complexity regarding the role of inner and outer tubes on the mechanical stability. Ye et al. have suggested that the critical collapse pressure of DWNTs is essentially determined by inner tube stability, and so the collapse pressure of DWNTs is close to what is expected for an inner tube.29 Other authors suggested that the collapse pressure of DWNTs is completely different from that expected for the corresponding SWNTs Received: September 19, 2012 Published: September 28, 2012 22637

dx.doi.org/10.1021/jp3093176 | J. Phys. Chem. C 2012, 116, 22637−22645

The Journal of Physical Chemistry C

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Figure 1. Top panels: Enthalpy vs pressure curves calculated for (a) (8,8), (b) (18,0), and (c) (24,0) bundled SWNTs and graphite (open black squares). The pressure evolution for (10,0) SWNT is similar to (8,8) SWNT showed in (a), and the circular−oval and oval−peanut transitions are found close to 1.5 and 9.6 GPa, respectively. Bottom panels: We show the enthalpy difference (ΔH) between SWNT phases and hexagonal graphite with the pressure. The transitions (horizontal dashed lines) between circular (black circles), oval (down red triangles), polygonal (up red triangles), and peanut (blue squares) are compared with hexagonal graphite.

constants ax and ay = √3ax (the ay/ax ratio is kept fixed). The lattice constant along the z axis (az) is chosen to contain 5 (8) unit cells of the zigzag (armchair) tubes. We perform a sequence of small and controllable steps of unit cell volume reduction for each fixed nanotube phase studied (circular, polygonized, oval, peanut, etc.). Each step is of the order of ΔV/V0 = −0.01%, where V0 is the ambient pressure volume for each nanotube system in the circular phase. For each fixed volume V, we search for the atomic positions and lattice constants ax and az that minimize the internal energy U(V). The pressure is obtained by p = −ΔU/ΔV, and the enthalpy is H = U + pV. To remove large enthalpy slopes of SWNT phases, we have calculated the enthalpy curve of hexagonal graphite as the enthalpy differences were rescaled, so then the SWNT transitions can be well differentiated. Phonon frequencies and eigenvectors are directly obtained by the diagonalization of the force constant matrix. The matrix elements are calculated by using finite difference techniques. We focus our vibrational analysis on the radial breathing mode (RBM), the longitudinal G-band (denoted Gz) and the transversal G-band (denoted Gt), whose displacements are along and perpendicular to the tube axis, respectively. In chiral SWNTs, these two components are not strictly along the two high symmetry directions, and the G+ and G− peaks exhibit a mixed longitudinal optical (LO) and transversal optical (TO) nature with the degree of mixing depending on chiral angle.41 Therefore, RBM, Gz, and Gt projected density of states (PDOS) are constructed by projecting the phonon eigenvectors onto the corresponding axial and tangential (with opposite phases for atoms in different sublattices) displacement fields, respectively. Those probe eigenvectors are normalized, and they are

when they are considered separately, and the pressure value still depends on the 1/d*t 3 scale law but with a suitable choice of an average diameter dt*.30,31 In this paper we report a study of the vibrational properties of carbon nanotube bundles under pressure, a subject that has not been theoretically well investigated in the literature up to now. This paper is organized as follows. First, we will describe in Section II the methodology used to model a nanotube bundled structure. In Section III.A, we calculated the structural evolution of SWNT and DWNT bundles under pressure. We focused on the structural stability of the bundled structure by calculating the critical pressure for collapsing, and we compare the behavior of the DWNT bundle with its corresponding SWNT. In Section III.B, we explored the vibrational properties of SWNT and DWNT bundles under pressure by calculating RBM and tangential phonon modes before and after the nanotube collapse. We close our paper with conclusions in Section IV.

II. METHODOLOGY To access mechanical and vibrational properties of carbon nanotubes under pressure, we initially perform zero-temperature structural minimizations of SWNT and DWNT bundles. The carbon−carbon bonding within each CNT is modeled by a reactive empirical bond order (REBO) potential proposed by Brenner.32,33 Pairwise Lennard-Jones potentials are added to model the nonbonding van der Waals terms (ε/kb = 44 K, σ = 3.39 Å), which are essential to describe intertube interactions in bundles.11 We studied zigzag and armchair C bundles of CNTs in triangular arrangements within orthogonal unit cells containing two SWNTs or DWNTs each, with lateral lattice 22638



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Figure 2. (a) p−V curves and (b) enthalpy functions for the pressure-induced sequence of phase transitions in the (10,0)@(18,0) DWNT. Different cross-sectional shapes correspond to different colors in the plots and are identified in the insets. (c) We show the enthalpy difference (ΔH) between DWNT phases and hexagonal graphite with the pressure. The transitions (horizontal dashed lines) between A−B and B−D phases are compared using enthalpy rescaled by a hexagonal graphite curve.

phase transition. This can be seen more clearly when p−V plots are constructed (not shown here) where each transition is marked by a discontinuous change in volume. For the (8,8) SWNT bundle, the transition pressure from circular to oval (socalled p1) occurs close to 1.3 GPa and from oval to peanut (socalled p2) around 2.6 GPa (see Figure 1a). For the (18,0) SWNT bundle (see Figure 1b), we found phase transitions at 1.2 GPa (circular to polygonal, p′1) and 1.5 GPa (polygonal to peanut, p′2). Finally, for the (24,0) SWNT bundle the polygonal to peanut transition (p′2) is found at 0.6 GPa (see Figure 1c). We see that the small diameter (8,8) SWNT bundle undergoes a circular → oval → peanut phase transition sequence as the pressure increases, whereas for the intermediate diameter (18,0) SWNT the sequence is circular → polygonal → peanut. For the large diameter (24,0), we obtain simply a polygonal → peanut transition, since the tubes are already polygonized even at zero pressure. As expected, the critical pressures are strongly diameterdependent.11,19,21,28,36,37 Actually, many authors have pointed out that the critical pressure (p1) for the circular−oval 11 Some transition scales with d−3 t , where dt is tube diameter. authors also found that the critical pressure for the oval−peanut 28,38 transition (p2) also scales with d−3 Our results for bundled t . SWNTs shown in Figure 1 point out that the transition to the peanut geometry (p2 and p2′ ) follows approximately the same law p2 = C/d3t with C = 4.4 nm3 GPa. This scaling law was observed for all studied zigzag and armchair bundled tubes. However, the pressure transition values p1 and p′1 for circular to intermediate phase (oval and polygonal, respectively) show a slight dependence on the inverted tube diameter, but they do not follow the d−3 t scaling law. We also have modeled several DWNT bundles under pressure and choose the (10,0)@(18,0) DWNT (this notation means that the (10,0) tube is inside the (18,0) tube) to compare its structural stability with those of their constituent

orthogonal to each other. For zigzag (armchair) SWNT bundles, the Gz probe vectors are parallel (perpendicular) to C−C bonds, while Gt are perpendicular (parallel) to C−C bonds. We should note here that we are not looking at the Raman active zone center phonons, but the phonon PDOS with no ksampling calculations. However, we still have a good correspondence between the calculated radial and tangential PDOS with RBM modes and G-band. It should be pointed out that in phonon dispersion relation those branches associated with RBM and G-band are flat and present a very high density of states. Furthermore, we can correlate Gz and Gt PDOS to G+ and G− of achiral SWNTs as we will explain further.

III. SWNTS AND DWNTS UNDER PRESSURE: STRUCTURAL AND VIBRATIONAL PROPERTIES III.A. Nanotube Collapse. We start by discussing the lowpressure behavior of SWNT bundles, considering four basic cross-sectional shapes, namely: circular, oval, polygonal (hexagonal), and peanut-like. Tubes of different diameters exhibit a different sequence of cross-sectional shapes as a function of pressure.11 In particular, the polygonal or hexagonal phase is typical of bundled tubes, and it is often obtained in calculations for large diameter nanotubes since plane-parallel facing between adjacent tubes tends to decrease the van der Waals interaction energy.6,34,35 Figure 1 shows the enthalpy as a function of pressure calculated for circular, oval, and polygonized phases for bundles of (8,8), (18,0), and (24,0) SWNTs. The crossing of two curves in this plot (top panels) by following the lowest enthalpy for a given pressure indicates a transition between two different cross-sectional shapes. We have rescaled enthalpy curves looking for the enthalpy difference between those SWNT phases and the hexagonal graphite (bottom panels). The derivative discontinuities in the ΔH(p) plots are associated with the volume variation at the 22639



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predicted to occur at 5.5 ± 0.1 GPa and change in bulk modulus takes places around 5.9 GPa close to that value expected for the (12,0) SWNT (5.4 GPa). It is interesting to compare the pressure effects on isolated and bundled SWNTs/DWNTs. Pressure effects on isolated nanotubes have been the subject of many previous studies. In particular, Capaz et al.10 and Tangney et al.11 have studied isolated SWNTs under pressure using a similar methodology. In those studies, phase transitions were observed for SWNTs following a cross-sectional evolution of circular−oval−peanut. In our study we find similar phase transitions for small diameter tubes even arranged in bundles. However, the main difference that we observed between bundled or debundled SWNTs/ DWNTs is that we no longer observe the oval phase for large diameter tubes which is replaced by the polygonal phase. Indeed, the polygonal phase in DWNTs plays a more important role in their cross-section evolution being supported by inner tubes up to 5 GPa. III.B. Phonon Density of States Evolution under Pressure. Figure 3 and Figure 4 show the phonon projected

SWNTs. We find a very similar behavior for the other DWNTs such as (12,0)@(20,0) and (11,0)@(19,0). In Figure 2a and Figure 2b, we show, respectively, the p−V and enthalpy curves for the (10,0)@(18,0) DWNT bundle. Upon compressing the original circular structure of a DWNT, we find four distinct structures as shown in Figure 2a. The A configuration is the nondeformed structure with a circular shape for both inner and outer tube. The B configuration is characterized by the polygonization of the outer tube, while the inner tube maintains its circular cross section. The C configuration corresponds to a polygonal outer tube and oval inner tube. Finally, the D configuration stands for the case where the outer and inner tubes display oval/peanut cross-sectional shapes. The polygonization of the outer tube (A−B phase transition) in the (10,0)@(18,0) DWNT bundle is observed at 0.8 GPa which is lower than the value (1.2 GPa) that was calculated for the same transition in the (18,0) SWNT bundle. This result suggests that the presence of the inner tube enhances the outer−outer tube interactions within the bundle, thus reducing the critical pressure value for polygonization. Upon increasing the pressure, a B−C transformation occurs at 6.2 ± 0.2 GPa since the inner tube ovalizes and the outer tube reaches a higher degree of polygonization. We are able to increase the pressure in the C phase up to 13 GPa without inducing any strong deformation. However, from the enthalpy analysis (Figure 2b and Figure 2c), we conclude that the D phase is more stable than the C phase for the whole range of investigated pressures. Thus, the C configuration is actually metastable. By further increasing the pressure in the D phase, it reaches the same enthalpy as the B phase around 5.7 ± 0.2 GPa. The D phase is also metastable even at lower pressures, down to 5.5 GPa. In summary, the actual sequence of phase transitions for the (10,0)@(18,0) DWNT bundle is A → B at 0.8 GPa and B → D at 5.7 ± 0.2 GPa. This value contrasts with the 1.5 GPa calculated collapse pressure of the (18,0) SWNT outer tube, clearly pointing out the structural support given by the inner tube, as already proposed by some authors.30,31 We conclude that the C configuration (polygonized outer tube and ovalized inner tube) is unstable when we compare with both collapsed tubes which let us propose that the ovalization of the inner tube is the reason for the overall collapse of the DWNT. For isolated DWNTs, Ye et al.29 suggest that the DWNT transition could be essentially determined by the inner tube transition. They also found that collapse of DWNTs follows almost the same d−3 t law when the inner tube is used, which means that pc for DWNTs is the same as expected for the SWNT inner tube.29 In such a way, our results agree with the fact that the inner tube ovalization determines the critical pressure for collapsing of DWNTs. However, there is a clear pressure screening effect from the outer tube in the behavior of the inner tube because the critical pressure for the expected circular−oval transition for the inner tube was increased to 5.7 ± 0.2 GPa when it was encapsulated in the DWNT. This value is considerably higher than that observed for the (10,0) SWNT bundle, which was 1.5 GPa. Second, the full collapsing of the inner tube (oval−peanut transition) is continuous in the DWNT case, differently from the SWNT case, in which there is a discontinuous change in the volume (around 9.6 GPa). For the DWNT case, this transition is marked by a change in compressibility (which is proportional to the slope of the p−V plot) at around 10 GPa, clearly seen from Figure 2a. We found a similar behavior for the (12,0) @(20,0) DWNT bundle, where the transition B → D is

Figure 3. Phonon density of states (DOS) projected in RBM (a) and tangential (b) probing vectors for the (18,0) SWNT. Some snapshots of the bundle structure evolution can be followed in panel (c).

density of states (PDOS) calculated for (18,0) and (8,8) SWNT bundles, respectively. The PDOS for RBM, Gz, and Gt

Figure 4. Phonon density of states (DOS) projected in RBM (a) and tangential (b) probing vectors for the (8,8) SWNT. Some snapshots of the bundle structure evolution can be followed in panel (c). 22640



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potential has been widely used to evaluate and predict mechanical properties of most of the carbon materials,32,44,45 and this limitation of our methodology does not affect the qualitative or even the semiquantitative trends: we obtain pressure transition values very close to those found by other authors using different methodologies, and we can still qualitatively analyze the phonon frequency evolution of SWNTs and DWNTs under high pressure. We note that the overestimation, especially of Gz frequency, is observed for all small diameter tubes, regardless of whether they are inner tubes of DWNTs or isolated SWNTs. Interestingly, we note that this overestimation is not observed for RBM modes and for their pressure coefficients when compared with experimental results. Before any structure collapse, we should note that both Gt and Gz contributions shift to high-frequency values as the pressure increases as expected for SWNTs under pressure. Furthermore, the Gz component shifts faster than the Gt component, suggesting a direct association with the evolution of G+ and G− Raman components under pressure.2,3,28,46−48 Following the pressure evolution of the (18,0) SWNT, we clearly observe two distinct peaks of this Gz band when polygonization takes place after 1.2 GPa, while Gt is not significantly affected. The two-peak feature of the G z component due to polygonization is also observed for the (24,0) SWNT bundle and, evidently, not observed for the (8,8) or (10,0) SWNT bundles. For the (8,8) SWNT, after the ovalization, we observe a large broadening of the G z component, and Gt is less affected. A similar behavior was observed for the zigzag (10,0) tube. Actually, the dependence of chirality is not significant, and the phonon evolution of SWNT bundles was found to be only dependent on tube diameter which determines transitions to polygonized and ovalized states. After collapse of the (18,0) SWNTs (see Figure 3b), we clearly see a sudden jump of Gz component to lower frequencies and an intensity enhancement of the lowerfrequency contribution, while the higher-frequency one continues to upshift. For (8,8) (see Figure 4b), (24,0) and (10,0) SWNT (not shown) similar behavior was also found. The Gt component is significantly modified after collapsing of tubes where a large broadening is observed especially for the (18,0) SWNTs. It is interesting to note that after collapse there is an overlap of Gt and Gz contributions, thus suggesting that the flattening of nanotube cross-section reduces the Gt−Gz splitting. However, a notorious peak of Gz PDOS can be observed at higher-frequency values and does not overlap with any Gt PDOS contribution. The phonon evolution with pressure for the (10,0)@(18,0) DWNT bundle can be observed in Figure 5. From (a) panels, the presence in the low-frequency region of two distinct peaks around 150 and 325 cm−1 corresponding to RBM modes of outer and inner tubes, respectively, is clear. The peaks are slightly shifted from the corresponding SWNTs in the circular conformation, as expected due to intertube coupling.49 It is interesting to note the evolution of both RBM peaks with pressure, where we clearly observe that the RBM frequency of the outer tube shifts much faster than the RBM frequency of the inner tube before collapse. This is clear evidence of a screening effect on the inner tube by the outer tube. Pressure screening effects on RBM and the G band in DWNTs have been observed in several Raman experiments.13,17,18,50−52 Our calculations of RBM and Gz confirm that this pressure screening effect occurs well before any structural collapse.

and modes are shown in red (a), blue, and orange (b) colors, respectively. From the figure, we can follow the RBM, Gz, and Gt evolutions as pressure increases (see Figure 3a and Figure 4a), and the corresponding snapshots of the bundle structural evolution are shown in the right panels (c). First, we observe the RBM modes of (18,0) and (8,8) SWNTs in the lowfrequency region centered, respectively, at about 150 and 200 cm−1, which gradually shift to higher frequencies as the pressure is increased. We estimated the pressure coefficient for those RBM modes (before any structural transition) as 7.0 ± 2.5 and 5.8 ± 1.5 cm−1/GPa for (18,0) and (8,8) SWNT, respectively, being in good agreement with experimental observations.2,25,39 It is interesting to note that the RBM mode is still well-defined even after polygonization of the (18,0) tube. However, after the collapse of the (18,0) SWNT bundle and after the ovalization of the (8,8) SWNT (and of (10,0) SWNT bundle), the contributions to radial displacements spread in the lowfrequency region, which is equivalent to say that the RBM disappears as a single well-defined mode, in good correspondence with the experimental difficulty to observe RBM after some suggested pressure phase transitions.2,3,28,46−48 The Gz and Gt band behavior is shown in panel (b) of Figure 3 and Figure 4 for (18,0) and (8,8) nanotubes, respectively. For isolated achiral SWNTs (zigzag and armchair), Gz and Gt can be directly assigned to longitudinal optical (LO) where atomic vibrations are along the SWNT axis and the transversal optical (TO) modes where carbon atoms vibrate along the SWNT circumference. In the case of chiral SWNTs, G band components have a mixed TO/LO nature depending on chirality. The respective intensities of these modes are chirality dependent, while their frequencies are only dependent on diameter.40,41 In general, the G band Raman spectrum of the isolated SWNT is characterized by the presence of two main peaks. The higher-frequency G+ component which is diameter independent can be associated to LO modes, while the lowerfrequency G− component is associated to TO modes of semiconducting tubes. Due to the electron−phonon interaction, in metallic tubes, the lower frequency component G− is associated with LO modes, and the higher frequency component G+ is associated with TO modes. Our model does not take into account the electron−phonon renormalization, and Gz and Gt in metallic armchair nanotubes are similar to semiconducting tubes regarding the G+ and G− assignment. The intensity profile of the G− component is enhanced in the case of metallic tubes due to the electron−phonon coupling, and their frequency is strongly diameter dependent. It has been observed that for small diameter tubes the G+−G− splitting is larger than for large diameter tubes where G+ and G− frequencies are equal, as in graphene.42,43 Furthermore, we are not dealing in our study with Raman active modes but the phonon zone-center DOS. In Figure 3b and Figure 4b, we clearly observe that the Gz− Gt splitting for the (8,8) SWNT bundle is larger than for the (18,0) SWNT in the circular phase (−1.0 GPa). This is related to diameter difference between those tubes. For the (24,0) SWNT, for example, this Gz−Gt splitting can not be resolved (not shown here). Even if the Gz component seems to be well overestimated compared with Raman experiments, the diameter dependence of Gz−Gt splitting is in agreement with Raman experiments.42 That overestimation of Gz frequencies comes from the fact that the Tersoff−Brenner interatomic potential is not parametrized for reproducing exactly the experimental vibrational frequencies. Nevertheless, the Tersoff−Brenner 22641



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Figure 6. (a) v-PDOS projected in z direction for the (18,0) SWNT at −1.0, 1.4, and 2.2 GPa. Arrows mark the center of modes whose eigenvectors are plotted in (b) panel. (b) Amplitude of z-displacements of the (18,0) SWNT bundle as a function of coordinate angle before the collapse (−1.0 and 1.4 GPa) and after collapse (2.2 GPa). From bottom to top: Eigenvectors for A1g mode centered at 1721.4 cm−1 (black) at −1.0 GPa; for modes centered at 1771.3, 1780.7, and 1786.1 cm−1 (red) at 1.4 GPa; for modes centered at 1696.2, 1705.4, 1710.1, 1715.8, and 1741.0 cm−1 (green) at 2.2 GPa; and for modes centered 1800.6, 1803.2, and 1815.4 cm−1 (blue) at 2.2 GPa.

Figure 5. Phonon density of states (DOS) projected in RBM (a) and tangential (b) probing vectors for the (10,0)@(18,0) DWNT. Some snapshots of bundle evolution can be followed in (c).

After the collapse, the radial contribution spreads over a large frequency range similar to SWNTs. In Figure 5b, we can follow the Gz and Gt component for the outer/inner tube system. It should be noted here that the inner (10,0) tube Gz component is well overestimated, being at about 1850 cm−1 compared with the outer (18,0) one which is located at around 1650 cm−1. However, as explained before, qualitative analysis could still be performed. For the Gt component, no clear separation of inner/outer contribution is observed at the DWNT circular phase (−1.0 GPa), and one single peak for Gt PDOS is presented in Figure 5b. It is interesting to note that pressure screening effects are also observed in the longitudinal Gz component, as we can see that the outer tube Gz components shift faster than the inner ones. As observed for SWNTs, the transition of the outer tube to a polygonized phase at 0.85 GPa is clearly marked by a separation in two peaks of that Gz band. After collapse, the PDOS spreads over a large frequency range for both Gz and Gt tangential contributions, but it is possible to see that the lowest frequency components of Gz are suddenly shifted to lower frequencies for both inner and outer tubes, as in the case of SWNT bundles. In Figure 6, we plot the z-displacement (along the tube axis direction) of some eigenvectors as a function of the coordinate angle θ defined from the center of the (18,0) tube. First, we have identified the A1g mode centered around 1721.4 cm−1 for the structure at −1.0 GPa (Figure 6b). This is the mode which mainly contributes to v-PDOS as we can see in Figure 6a. As the pressure is increased up to 1.4 GPa, the (18,0) cross section is polygonized as we observe a separation of Gz PDOS contribution. Analysis of the atomic displacements (Figure 6b) shows that the higher frequency component is mainly due to modes which are localized on the regions with high curvature (vertices of the polygonized cross-section). Modes centered at 1771.3, 1780.7, and 1786.1 cm−1 have the same symmetry, and their maxima of displacement are localized for θ equal to ±30°, ±90°, and ±150° which are the vertices of polygonized shape. After the collapse of the (18,0) SWNT bundle, we have calculated phonon eigenvectors for a collapsed structure at 2.2

GPa. In Figure 6a, we show modes centered at 1696.2, 1705.4, 1710.1, 1715.8, and 1741.0 cm−1 which mainly contribute to the low-frequency peak in PDOS (Figure 6a), while the modes centered at 1800.6, 1803.2, and 1815.4 cm−1 mainly contribute to the higher frequency peak in PDOS. Then, we clearly see that the higher frequency contribution in PDOS at 2.2 GPa comes from modes that are localized in the high-curved regions of the peanut-shaped tube (θ ∼ ± 180° and θ ∼ ± 0°). In opposition, we can observe that the modes which mainly contribute to the low-frequency peak of PDOS at 2.2 GPa have their z-displacements far from the high-curved regions of the peanut-shaped tube. Therefore, the low-frequency and more intense peak of PDOS is due to vibration of modes which are localized in the flat regions of the collapsed nanotube. From an experimental point of view, it has been recently proposed that a saturation or even a negative pressure slope of the Raman G+ component for SWNTs and DWNTs could be associated to the collapse of the nanotubes.5,18,39 Furthermore, after the collapse this band follows the graphite pressure evolution as observed in Figure 7 with a smaller pressure coefficient.18,53 This was also observed in the PDOS of SWNT bundle calculations (cf. Figure 3b). Our calculations confirm this hypothesis and identify the atomistic origin of the lowfrequency bands. Collapsing of the nanotube cross-section leads to flattened regions where the reduced stress in C−C tangential bonds reduces the Gz frequency, which can be associated to Raman G+ component. However, as we observe in our calculations, a small high-frequency component of the Gz component, arising from the high-curvature regions, remains after the collapse. Since the flattened portion of the tubes is much higher than the curved portion, we expect that the lowfrequency shift of the G band is dominant in the experiments. The experimental red-shift observed of the G-band is then explained from our calculations as related to the dominance of the flat region of the collapse phase in the Raman signal. This is 22642



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dependent. It is also evident that the transition from polygonized to a collapsed cross section for large diameter tubes has a first-order character with a large bundle volume discontinuity, whereas for small diameters the transition is continuous, with intermediate oval or racetrack cross sections. For DWNT bundles, screening pressure effects were observed on the inner tube, which keeps its circular cross section for higher pressures than would be expected from the SWNT behavior. Furthermore, the inner tube acts as a structural support to the outer tube. Phonon calculation of Gz and Gt PDOS also reveals screening effects in radial and tangential components as the pressure increases. Polygonization of the (18,0) SWNT bundle is characterized by a separation of Gz PDOS contribution which is mainly due to localized modes on the high-curved region of polygonal shape. After the collapse transition, the tangential modes associated with the flat regions of the tubes are suddenly shifted to low frequencies, and the high-curved contribution of the peanut shape is less pronounced but still observed in Gz PDOS. We also found that there is a G−-Gz increase of splitting as the pressure is increased independent of zigzag/armchair symmetry before the collapse. Such Gt−Gz splitting is reduced after the collapse of nanotubes due to flattening of the tube cross-section. The experimental observation of the G-band red-shift at nanotube collapse let us conclude, using z-displacements of modes related to the Gz band, on the dominance on the Raman signal from the flat regions of collapsed tubes. Moreover, this provides a coherent explanation of the lower pressure slope (∼4 cm−1/ GPa) of the G-band observed after nanotubes collapse. Consequently, the pressure slope could be a useful means for the identification of the geometry shape state of SWNTs and DWNTs.

Figure 7. Experimental pressure Raman shift evolution of the G+ component for SWNTs,39 DWNTs,18 and graphite.53,54 Horizontal dashed lines indicate experimental observations of nanotube collapse pressure for SWNTs (black) and DWNTs (red).

in addition perfectly coherent with the fact that in the collapsed state the pressure evolution of the G-band clearly matches (see Figure 7) the one of graphite or graphene under triaxial compression.53,54 Recently, Araujo et al. have studied the G band evolution of SWNTs under local pressure application using the AFM/ Raman coupled setup.55 For some cases, they observed a broadening and splitting of G+−G− modes, which was associated to chiral tubes which have no LO/TO modes purely defined. Furthermore, a symmetry breaking of the TO mode was observed from the G− splitting in two peaks TO1 and TO2, while the LO (G+) mode remained unaffected. The authors associated the effect with chiral tubes. Calculations of phonon DOS were used to suggest that such TO splitting could be associated with ovalization of the nanotube cross-section. However, no signal of RBM mode was found, and the pressure range used was not quantified.55 In our study, we observe, for some cases, a broadening of Gt PDOS, but no clear splitting of this contribution was found in circular−oval cross section modification. Furthermore, we found that the Gz PDOS contribution shifts to higher-frequency values before circular− oval and circular−polygonal phase transitions. We also found that there is a Gt−Gz increase of splitting as the pressure is increased independent of zigzag/armchair symmetry. We should note that we worked in the hydrostatic pressure regime where the pressure is transmitted by an interaction between the adjacent tubes in bundles. Another important point is that we worked in the high-pressure regime which is normally achieved for in situ Raman pressure experiments and too different from that found using AFM tip-induced deformations.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: agsf@fisica.ufc.br; [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the CAPES-COFECUB collaboration program (grant 608) grant for the partial support of this research. This is a contribution of NanoBiosimes Instituto Nacional de Ciências e Tecnologia (INCT). A.L.A. acknowledges the PNPD-CAPES program for partial funding. A.G.S.F. acknowledges funding from Fundaçaõ Cearense de Apoio ao ́ Desenvolvimento Cientifico e Tecnológ ico (FUNCAP) through PRONEX (grant PR2-0054-00022.01.00/11). R.B.C. acknowledges financial support from Brazilian agencies CNPq, FAPERJ, and INCT - Nanomateriais de Carbono.

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REFERENCES

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Structural and Phonon Properties of Bundled Single-and Double-Wall

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