## Abstract

Order reduction methods are widely used to reduce computational effort when calculating the impact of defects on the vibrational properties of nearly periodic structures in engineering applications, such as a gas-turbine bladed disc. However, despite obvious similarities these techniques have not yet been adapted for use in analysing atomic structures with inevitable defects. Two order reduction techniques, modal domain analysis and modified modal domain analysis, are successfully used in this paper to examine the changes in vibrational frequencies, mode shapes and mode localization caused by defects in carbon nanotubes. The defects considered are isotope defects and Stone–Wales defects, though the methods described can be extended to other defects.

## 1. Introduction

Defects in carbon nanotubes (CNTs) are both inevitable and mathematically problematic. Not only have point defects been observed *in situ* by Fan *et al.* [1], but the presence of isotope defects—specifically a mixture of ^{12}C and ^{13}C—can be inferred by the ubiquitous presence of both isotopes [2]. Furthermore, any crystal lattice that is of finite size must at some point be terminated, and this termination is itself a defect [3]. These defects are a significant impediment to the analysis of CNTs, as they rob the nanotubes of their periodicity.

Like its planar analogue, graphene, a pristine CNT (one which is without defects) can be thought of as an unending sequence of identical, repeating units [4]. This allows the free vibration characteristics of the nanotube to be calculated by considering the behaviour of only one of these units and invoking the Floquet–Bloch theorem [2,5]. Breaking the nanotube's periodicity by introducing a defect renders this approach invalid, and results in greatly increased computational expense.

The presence of defects in an otherwise periodic structure has significant consequences for its vibrational properties. It has been shown that disorder in an atomic system leads to fundamental changes in the nature of wave propagation in that system, a result that has a commensurately large impact on the vibrational modes—that is to say, the phonon dispersion—of the system [6]. Furthermore, the phonon dispersion of a CNT has profound implications for its mechanical and transport properties [7]. Understanding and, perhaps, harnessing the emergent properties due to these defects require accurate and efficient methods for determining the vibrational characteristics of almost-periodic atomic systems. Presented in this paper are two such methods, each applied to two dissimilar types of defect.

Of the many defects that can affect CNTs, those under consideration here are isotope defects and Stone–Wales defects (SWDs) in single-walled CNTs. As previously mentioned, isotope defects are simply the presence of multiple isotopes of carbon distributed randomly throughout the structure [2]. This has the effect of perturbing the mass matrix of the system, but it does not influence interatomic interactions or the stiffness matrix. An SWD, which will be described in more detail later, is the rearrangement of certain bonds in the nanotube. While a pristine nanotube consists exclusively of hexagonal rings of carbon, an SWD is the presence of adjacent pentagonal and heptagonal rings [8]. This alters the manner in which the carbon atoms interact without changing their masses. In other words, the stiffness matrix changes, but the mass matrix remains unchanged.

Prior attempts have been made, notably by Chen *et al.* [9] and Georgantzinos *et al.* [10], to account for the impact of defects on the vibration of CNTs using numerical methods. These studies report changes in the natural frequencies of CNTs due to the presence of defects but do not consider changes in the corresponding mode shapes. Both of these investigations rely on the molecular structural mechanics model developed by Li & Chou [11]. They require a full-order model of the perturbed CNT—restricting their use to relatively short CNTs. Furthermore, the molecular structural mechanics model does not permit for the relaxation of an atomic structure after a defect is created and for the subsequent alterations to the interatomic stiffnesses.

Periodic systems with inevitable defects are not unique to the atomic scale. Rotor blades, the circular arrangement of which causes them to be periodic, have proved to be a fertile area of investigation not only for the behaviour of such systems with defects but also for the development of order reduction methods to handle the increase in complexity that these defects bring [12–14]. Two such methods, modal domain analysis (MDA) and modified modal domain analysis (MMDA), are here applied to CNTs with isotope defects and SWDs.

## 2. Methods

While the equations of motion that govern the behaviour of a system of atoms are far from being simple [2,15], this study pertains only to the vibrational properties of such a system. Specifically, the properties of interest are the vibrational modes and natural frequencies of CNTs. The concepts of vibrational modes and frequencies are defined in a rigorous sense only for linear systems, that is to say, systems in which interactive force is proportional to displacement [16]. The equations of motion that will be used here are derived by first finding a static equilibrium for the system in question—effectively reducing its temperature to 0 K—and assuming that during vibration atoms are only displaced a small distance from this equilibrium. This ‘frozen phonon’ approach [17] allows the system to be linearized, and so permits the equations of motion for free vibration to be written as
**x**(*t*) contains the positions of the atoms in the system and **M**, **C** and **K** are the constant mass, damping and stiffness matrices, respectively. Vibrational modes and frequencies are properties of the undamped system; therefore, the damping matrix **C** can be set to zero [16]. Note that, for the remainder of this paper, the time dependence of **x** will be implied.

While the stiffness matrix in equation (2.1) could be found using *ab initio* methods [18], such calculations for large-scale aperiodic systems are unfeasibly expensive and time-consuming. Therefore, the stiffness matrix is here built using molecular dynamics techniques. It is typical to express the interactions among atoms in terms of an interaction potential, *Φ* [2]. Of the many functions proposed for the interaction potential of a CNT, the one chosen for this study is the Tersoff potential using the optimized parameters found by Lindsay and Broido [19–21]. While modifications and improvements have been made to this potential [22,23], which increase its ability to accurately model physical systems, the purpose of the interaction potential in this paper is primarily for demonstrating the validity of order reduction techniques. The Tersoff potential combines ease of implementation with sufficient accuracy to reflect the impact of structural changes in the CNT on the stiffness matrix. The methods described here are applicable to any potential function that can be linearized in the manner that will be described.

The first step in the linearization process is to ensure that the system is at a stable equilibrium. A standard procedure in molecular dynamics, this is achieved by altering the positions of the atoms until the total potential energy reaches a local minimum [15]. This computation has been performed using the built-in energy minimization feature of the software package LAMMPS, with energy tolerance of 10^{−9} eV and force tolerance of 10^{−12} eV Å^{−1} [24]. While the minimum potential energy condition does not ensure stability, its stability can be checked by ensuring that all the predicted natural frequencies of the linearized system are real [25].

The potential energy of the system is
*Φ _{ab}* is the interaction potential between atoms

*a*and

*b*. The relative stiffness of the

*α*th and

*β*th degrees of freedom is

*x*and

_{α}*x*indicate the

_{β}*α*th and

*β*th components of displacement vector

**x**[15].

At this point, it is convenient to explain the nomenclature that will be used in the remainder of this paper, as regards the positions of individual atoms. Consider a system of *n* atoms. Each atom is treated as a point mass, and has three degrees of freedom. It follows, therefore, that **x**, *x _{α}*, indicates the position of a single degree of freedom of the system. Of course, the order in which the degrees of freedom are written in the position vector does not alter the way it functions, but for simplicity the convention used here is

**u**

_{a}is the position vector of the

*a*th atom and

**x**is a function of time.

The formulation for *k _{αβ}*, the stiffness of one degree of freedom of the system with respect to another, allows a global stiffness matrix

*γ*= 3

*n*is the number of degrees of freedom in the system, to be constructed

The mass matrix **M** is defined as
*m _{a}* is the mass of the

*a*th atom, and

**I**

_{3}is the 3

*×*3 identity matrix. Using these definitions, the linearized equation of motion for this system of atoms can be written as

The ability of the linearized Tersoff potential to predict the phonon dispersion and thermal conductivity of pristine graphene sheets and single-walled CNTs is validated against experimental data in Lindsay & Broido's [19] derivation of the improved Tersoff parameters. In that work, the components of the interatomic stiffness matrix **K** are referred to as the harmonic interatomic force constants.

### (a) Order reduction techniques

For the purposes of this study, there are two systems to consider: a pristine (fully periodic) system, with mass matrix **M*** _{p}* and stiffness matrix

**K**

*, and a perturbed system, with mass matrix*

_{p}**M**

*=*

_{d}**M**

_{p}**+ δM**and stiffness matrix

**K**

*=*

_{d}**K**

_{p}**+ δK**. The vibrational frequencies and modes for each system can be found from the following eigenproblems:

*ω*and

**v**represent a natural frequency and the corresponding modal vector or mode shape, respectively.

The periodicity of the pristine system makes it unnecessary to solve the full-order eigenproblem (equation (2.8)) for that case. Instead, the Floquet–Bloch theorem allows the order of the problem to be reduced to that of one periodic unit (referred to as a unit cell) with no loss of accuracy [2,5,26]. A unit cell of a CNT is shown in figure 1.

When the periodicity of the pristine system is broken by the presence of a defect, the Floquet–Bloch theorem is no longer applicable. Therefore, finding the modes and frequencies of the perturbed system is a much more computationally expensive process. To mitigate this, order reduction techniques are used to limit the size of the system while maintaining approximately accurate mode shapes and natural frequencies within a certain frequency range. The two methods of order reduction considered here are MDA and MMDA. While both have been described in detail elsewhere [12–14], their application to almost-periodic atomic systems is novel.

Both techniques follow the same basic steps: an order reduction matrix (*γ* is the number of degrees of freedom in the full-order system and *ζ* is the number of degrees of freedom in the reduced-order system) is constructed. The manner in which the matrix **T** is constructed is the only difference between the two methods. A reduced-order position vector **y** is defined such that

Equation (2.10) is substituted into the equations of motion (equation (2.7)), and the equations of motion are left-multiplied by **T**^{T}, resulting in

The corresponding eigenproblem that results in the mode shapes and modal frequencies is then

The prediction of natural frequencies from the reduced-order model is represented by *ω _{r}*. The modal vector

**v**

_{r}from the reduced-order model in physical coordinates is computed by the following relationship:

### (b) Modal domain analysis

MDA seeks to reduce the order of the system by first transforming it from physical coordinates—each of which corresponds directly to the position of a single atom in a certain direction—to modal coordinates, in which the position vectors of atoms are given as a linear combination of vibrational modes of the pristine nanotube. Order reduction is then achieved by eliminating certain of these modal coordinates [13].

As the pristine system is periodic, its vibrational modes and frequencies can be found efficiently using the Floquet–Bloch theorem. These modes, denoted **v**_{1}, …, **v*** _{γ}*, together form a

*γ*

*×*

*γ*matrix

Note that the modes of this system are orthogonal; therefore, the matrix **V** is of full rank. This matrix allows for position vectors to be transformed from the modal domain as follows:
**y** is a position vector in modal coordinates. As the matrix **V** is of full rank, this transformation is isomorphic [27]. Order reduction can be achieved by removing certain modes from matrix **V** to form the order reduction matrix, **T**, in equation (2.10)

It should be noted that the transformation from physical coordinates to reduced-order coordinates is no longer isomorphic, as the dimension of the modal coordinate space is lower than that of the physical coordinate space [27].

### (c) Modified modal domain analysis

MMDA applies the same rationale as MDA, with one significant difference: the order reduction matrix **T** is built from the modes of the pristine system and the modes of other periodic systems, chosen to represent the differences in mode shapes due to the presence of defects [12]. In this paper, only one such system will be used, and its modes are designated **v**_{A1 }… **v**_{Aγ}. Note that this system must be of the same order as the full system. Just as with MDA, the number of modes used to construct the transformation matrix must be less than the order of the full system, or else no order reduction is achieved. The order reduction matrix can, therefore, be written

Once again,

### (d) Defects

Two types of CNT defect are considered here: isotope defects and SWDs. Isotope defects refer simply to the presence of various carbon isotopes distributed through the structure that result in a perturbed mass matrix, while the stiffness matrix remains unchanged. As structural defects, SWDs influence how the atoms in a nanotube interact and so result in a perturbed stiffness matrix, but do not affect the mass matrix.

#### (i) Isotope defect

As the nanotubes comprise carbon atoms, the masses used in the presence of isotope defects correspond to those of ^{12}C and ^{13}C. The resulting mass matrix is referred to as **M*** _{d}*. The stiffness matrix is not affected—i.e.

**K**

*=*

_{d}**K**

*.*

_{p}For MDA, the only mode shapes that are required are those of the pristine system, which are found using the Floquet–Bloch theorem. MMDA attempts to improve on the accuracy of MDA by supplementing the order reduction matrix with mode shapes that reflect the characteristics of the defects, as well as the pristine system. To that end, a separate system is required that exhibits relevant behaviours of the perturbed system, but is itself periodic (and so can be readily analysed). Following the approach used in [12,28], this system is found by employing proper orthogonal decomposition (POD) [29,30] to identify the dominant features in the mass matrix due to the defects.

As the aim of this analysis is to efficiently generate the basis vectors that form the matrix **T**, equation (2.10), using a periodic system that retains important characteristics of the perturbed system, the perturbed system is first broken down into cells, analogous to the repeated unit cell of the periodic system. Let *η* be the number of degrees of freedom in a unit cell. The masses of atoms in these cells will be compared, and the most prominent features will be used to form the unit cell for a new, ‘periodically perturbed’ system.

To divide the system into cells, the mass matrix **M*** _{d}* can be split into

*ν*

*=*

*γ/η*sub-matrices as follows:

Each block-diagonal matrix of dimension *η* is the mass matrix of an individual cell. Furthermore, the atoms in the system are modelled as point masses, so each block-diagonal matrix **δM*** _{a}* is itself diagonal. A new, more concise matrix can be assembled thus

**δM**

*.*

_{a}**Σ)**= [

*σ*

_{1}

*σ*

_{2}…

*σ*0 … 0]

_{p}^{T}, where

*p*= min(

*ν,η*), and

*σ*≥

_{i}*σ*

_{i}_{+}

_{1}. Equation (2.22) can also be written as

**u**

*is the*

_{i}*i*th column of

**U**and

**v**

*is the*

_{i}*i*th column of

**V**[14,30]. The columns of

**U**are the POD features of

**M**

*, and form an orthonormal basis for the columns of*

_{c}**M**

*. Furthermore, since*

_{c}*σ*≥

_{i}*σ*

_{i}_{+1,}the first POD component represents the most dominant feature of

**M**

*, the second POD component represents the next dominant feature, and so on [29].*

_{c}Using these POD features, new matrices, **δM**_{POD1} to **δM**_{PODp}, are formed as follows: let **W*** _{i}* be a diagonal matrix, such that diag(

**W**

*) =*

_{i}**u**

*. Then*

_{i}**δM**

_{PODi}can be defined

From each **δM**_{PODi}, the periodically perturbed mass matrices **M**_{PODi} = **M*** _{p}* +

**δM**

_{PODi}can be defined. As these mass matrices are periodic, the natural frequencies and mode shapes of a system with stiffness matrix

**K**

*and mass matrix*

_{p}**M**

_{PODi}can be efficiently calculated using the Floquet–Bloch theorem. A selection of the resulting mode shapes, denoted

**v**

_{PODij}to indicate the

*j*th mode of the system built using the

*i*th POD feature, can be combined with certain modes of the pristine system to form the order reduction matrix,

**T**(equation (2.19)).

#### (ii) Stone–Wales defect

Introducing an SWD into the system requires rearranging the atoms, and so alters the stiffness matrix but not the mass matrix. An SWD can be created simply by taking two neighbouring (i.e. bonded) atoms and rotating them 90° about the centre of their bond, in the plane tangent to the surface of the nanotube (figure 2) [8,31]. As the positions of some of the atoms will have been changed, there is no guarantee that the system is at equilibrium. It is necessary, therefore, to repeat the energy minimization procedure. Once that has been completed, the perturbed stiffness matrix **K*** _{p}* can be assembled in the same manner as for the pristine system. Note that the defect causes the atoms in its vicinity to be arranged into two heptagons and two pentagons—a defining characteristic of an SWD [8].

Analysing the system using MDA proceeds in the same manner as for the system with isotope defects, and therefore needs no further explanation. MMDA, on the other hand, requires a different approach. It is neither feasible nor meaningful to break down the perturbed stiffness matrix and perform POD on it. The size of the matrix makes these computations very difficult, and there is no guarantee that the resulting POD features will produce a stable system. A similar problem occurs when the positions of the atoms are analysed using POD. The features that are isolated must be energy minimized before their stiffness matrices can be calculated, which frequently results in the system either returning to its unperturbed state or becoming unstable and breaking apart.

An alternative approach is to once again break down the system into cells, but then directly compare the structure of each cell, choose the cell with the largest perturbation and use that cell as the basis for a ‘periodically perturbed’ system. For the case that is examined here—a single SWD—finding the most perturbed cell is simply a matter of identifying which cell contains the defect. To determine the modes of a system constructed using the most perturbed cell (the cell containing the defect), the Floquet–Bloch theorem requires information pertaining not only to the stiffnesses of the atoms in the unit cell relative to the other atoms in the unit cell, but also to the stiffnesses of those atoms relative to atoms in neighbouring unit cells. If the cell containing an SWD is used as a unit cell, then every cell in this new system contains an SWD. This raises a problem: the defect in one cell can potentially interact with the defect in another—a situation that could only arise if the perturbed system contained a high density of defects. To counteract this, the stiffnesses resulting from the interactions between the cell containing an SWD and the unperturbed cells around it are also included in the definition of the unit cell of the periodically perturbed system.

The entire process can be summarized mathematically as follows: it can be assumed without any loss of generality that the most varied cell is the first cell, i.e. the degrees of freedom pertaining to it are numbered 1 to *η*, where *η* is the number of degrees of freedom in a cell. A transformation matrix can be defined
*k* < *υ*. The eigenproblem for the periodically perturbed system is then
**T*** _{c}*. Note that the eigenproblem must be solved

*υ*times, once with each possible value of wavenumber

*k*, to produce the full set of frequencies and mode shapes for basis vectors in equation (2.19).

## 3. Numerical results

To demonstrate the validity and relative effectiveness of these two techniques—MDA and MMDA—in approximating the vibrational modes and natural frequencies of a CNT containing defects, a series of test cases are constructed. Each test begins with a model of a pristine CNT. To avoid end effects, periodic boundary conditions are applied to the models. Several nanotubes of varying length, diameter and chirality have been considered. These results are quite similar; therefore, for the sake of brevity, only the results from the nanotube with chiral vector (4,4) and length 59.3 Å (aspect ratio ≈ 10) will be presented. This nanotube model contains a total of 384 atoms (subsequently the full-order system contains 1152 degrees of freedom). Note that the periodic boundary condition means that the nanotube's length is, in effect, infinite, so the length given above refers to the maximum vibrational wavelength of the full-order system.

Each of the two types of defects under consideration, isotope and Stone–Wales, is applied separately to the nanotube. For the isotope defect case, the isotope masses correspond to ^{12}C and ^{13}C atoms, with approximately 1.1% of the atoms in the system assigned the mass of ^{13}C in an effort to match the relative abundance of these isotopes in nature [33]. The distribution of isotopes in this model is illustrated in figure 3. Only the mass matrix, equation (2.24), generated from the first POD feature is used to generate basis vectors for MMDA. The Stone–Wales case involves a single SWD, located at a randomly chosen position in the nanotube. The position of this defect and its impact on the overall shape of the nanotube is illustrated in figure 4.

Both MDA and MMDA achieve order reduction by approximating only certain modes of the full-order system. These modes are generally selected by first choosing a frequency range of interest, then building the order reduction matrices, equations (2.18) and (2.19), from modes with frequencies in this range [13]. In this study, approximations are made for the first 100 modes of the perturbed system—corresponding to the frequency range from 0 to 12.5 THz. The MMDA order reduction matrix contains the first 100 modes of the pristine system and the first 100 modes of the ‘periodically perturbed’ system that is generated using the methods described earlier. The MDA order reduction matrix is constructed from the first 200 modes of the pristine system. The choice of size of the order reduction matrices ensures that both order reduction techniques require approximately the same computational effort. Note that, while the use of 200 modes in creating the MDA order reduction matrix allows for the estimation of the first 200 modes of the perturbed system (with the expectation that the first 100 predicted modes will be more accurate than the second 100), only the first 100 modes of the MMDA approximation are expected to be relevant. For this reason, only the results relating to the first 100 modes are shown here.

In keeping with previous work on almost periodic systems [12,28], the accuracy of the approximations found here is measured by comparing the frequency deviation of a given mode (the predicted frequency of the perturbed system minus the frequency of the corresponding mode of the pristine system) with the frequency deviation of the full-order model. Each of these values is presented as a percentage of the pristine frequency for the mode in question. In the cases that are considered here, the modal frequencies of the perturbed systems do not deviate drastically from those of the pristine system—in general, the frequency deviations are less than 5% of the frequency of the pristine system (figures 5–7). Therefore, a plot of the frequencies themselves does not provide much information concerning the accuracy of the order reduction techniques under consideration. One such plot (figure 5) is presented to illustrate this. Note that in all the results shown here, rigid body modes (modes that correspond to zero frequencies) are excluded, as both approximation methods are capable of predicting those modes exactly. Plots of the frequency deviations for the isotope defect and SWD cases are presented in figures 6 and 7, respectively. Each plot contains the results for the full-order system, in addition to the frequencies estimated using MDA and MMDA.

The modal frequencies are only one part of the results produced by these order reduction methods; both MDA and MMDA provide approximate mode shapes that correspond to each frequency. These mode shapes are critical in predicting or matching the results of spectroscopic analyses [7]. A comparison with the mode shapes of the full-order system is achieved here using the modal assurance criterion (MAC) [35], which takes two mode shapes and returns a value between zero and one (inclusive), where one indicates identical modes and zero indicates orthogonal modes.

To give a full picture of the relationship between the modes of the full-order system and the modes of the reduced-order systems, the MAC of each mode of the reduced-order systems is calculated with respect to each mode of the full-order system. Should the values on the diagonal of the plot be close to 1, this will show not only that the modes of the full-order system are correctly predicted by the reduced-order system, but also that the associated modal frequencies are similar. Plots of the MAC for MDA and MMDA approximations of the system containing an isotope defect are shown in figures 8 and 9, while figure 10 shows the MAC of the modes of the pristine system with respect to the full-order system. Similar results for the SWD case are shown in figures 11–13.

The presence of defects in the otherwise periodic structure of a CNT has the potential to cause mode localization—the confinement of the majority of the energy of the mode to a small portion of the CNT [6,36]. To visualize localization in the defect-bearing CNT, the participation factor—after [36]—is used. Originally defined for a system in which each atom has one degree of freedom, the participation factor can be extended to three-dimensional systems as follows (note that this extension is necessary, as the application of the original form of the participation factor to a system with atoms having multiple degrees of freedom results in participation factor values that are dependent on the coordinate system used). For each mode **v** = [*v*_{1} *v*_{2} … *v _{γ}*], a new vector

**is defined such that**

*μ*In other words, ** μ** contains the magnitude of the displacement of each atom in that mode of vibration. The participation factor (

*p*

_{f}) is then calculated by first normalizing

**such that (**

*μ***)**

*μ**· μ*^{2}= 1, then

The value of the participation factor lies between zero and the number of atoms in the system, with smaller values indicating more localized modes.

The participation factors of the full-order perturbed system and the pristine system are shown for the CNT with isotope defects and with an SWD in figures 14 and 15, respectively. As localization requires some perturbation of a periodic linear system, there is no localization in the modes of the pristine system [6]. Subsequently, the modes of the perturbed system with participation factors much lower than those of the pristine system can be said to show localization.

To illustrate the relative efficiency of MDA and MMDA compared with solving the full-order eigenproblem, the Matlab implementations of MDA and MMDA developed for this article are timed as they estimate the eigenvalues and eigenvectors of the system. Note that the algorithm for finding the eigenvalues of the full-order system and the MDA algorithm do not change between the isotope defect case and the SWD case, while the MMDA algorithm does change. Running on a single core of an Intel Xenon E5-2650 v4 processor, with a clock speed of 2.2 GHz, the full-order system requires on average 19.251 s to solve. For the same system, the MDA algorithm requires only 0.652 s to run. For the isotope defect and SWD cases, MMDA requires 1.225 s and 1.395 s respectively. Each of these results is the average of 10 trials. Further increases in efficiency are possible as both the MDA and MMDA algorithms are highly parallelizable.

## 4. Discussions

The aim of this paper has been to determine whether either MDA or MMDA is a suitable technique for approximating the vibrational response of a CNT containing one or more defects. In this context, the vibrational response is summarized by the vibrational mode shapes and their associated frequencies. First, isotope defects are considered. From the frequency deviation of the full-order and reduced-order systems shown in figure 6, it is clear that both MDA and MMDA are successful in approximating the natural frequencies of the system. However, the MMDA results diverge slightly from the full-order results towards the end of the plotted frequency range—a result that is most likely due to the use of only 100 modes of the pristine system [12].

The frequency results alone are hardly enough to justify the use of order reduction techniques—the frequency deviation is so small that an analysis of the pristine system produces frequencies that are within 0.15% of the frequencies of the system that contains defects. A comparison of the mode shape of the pristine system and the system with isotope defects tells a very different story, however. In figure 10, it is clearly shown that the mode shapes of the pristine system differ greatly from those of the perturbed system, with only about 10 modes of the two systems matching closely. Figures 8 and 9, which show the MAC results for the MDA and MMDA mode shapes, respectively, demonstrate that both techniques produce mode shapes that correspond almost exactly to the mode shapes of the full-order system.

The presence of an SWD—a much more notable defect than an isotope defect—results in relatively large frequency deviations. For the nanotube used here, that deviation peaks at roughly 4% of the frequency of the pristine nanotube. Therefore, the frequencies of the pristine nanotube alone are enough to motivate the development of reduced-order models. Of the two reduced-order models discussed here, it is clear from figure 7 that MMDA produces much more accurate frequency approximations than MDA. MMDA succeeds in matching almost every frequency of the full-order system much more closely than MDA.

The MAC results further indicate that MMDA produces a valid and useful approximation of the perturbed system. The modes of the pristine system deviate significantly with respect to those of the full-order perturbed system across the frequency range in question (figure 13), indicating that the presence of an SWD has a large impact on the system's mode shapes. The MDA modes only matched the modes of the full-order system closely in a handful of cases (figure 11). MMDA, by contrast, produced high MAC values for the vast majority of modes (figure 12).

It is of especial interest that the reduced-order methods described here, particularly MMDA, are capable of efficiently predicting the mode shapes of the perturbed systems, as those systems have the potential to exhibit mode localization. The participation factor for each of the first 100 modes of the pristine and perturbed system are plotted in figures 14 and 15. Figure 14, which presents the results for the isotope defect case, shows that the participation factor for the modes of the perturbed system is generally lower than those of the pristine system, but not enough to be considered as localized modes. The lowest participation factor for the isotope defect case is slightly below 200, which is similar to the lowest participation factors of the pristine system. In the SWD case (figure 15), several modes have notably low participation factors. In particular, mode 51 has a participation factor of roughly 125, far lower than any modes of the pristine system. This is a strong indication that mode 51 of the full-order system exhibits localization.

An illustration of the relative ability of the two order reduction techniques to predict mode localization is given in figure 16. Here, mode 51 of the system containing the SWD is examined. As in figure 3, the nanotube is shown ‘unrolled’, to allow each atom to be clearly seen. The shade of each atom indicates the maximum displacement of that atom, divided by the largest displacement in that mode. Note that, due to frequency errors in the MDA results, the mode displayed is numbered mode 53 in the previous figures. This mode most closely matches mode 51 of the full-order system, and was therefore picked for this comparison. No such step was necessary for the MMDA results—mode 51 of the MMDA results matched mode 51 of the full-order results closely. It can be seen in figure 16 that, in both the full-order and MMDA results, the vast majority of atomic movement is confined to a small band containing the SWD. The atoms with the greatest displacement in both these results are members of the five-atom rings formed by the defect. The MDA results, on the other hand, erroneously show many more atoms participating in this mode, and the greatest displacement is not confined to atoms immediately involved in the defect.

## 5. Conclusion

Overall, the results found in this study indicate that order reduction techniques—namely MDA and MMDA—allow the prediction of the vibrational properties of CNTs containing isotope defects and SWDs with significantly decreased computational effort. MDA produces accurate results for the isotope perturbation case, but is unable to closely match the frequencies or mode shapes when the stiffness matrix was perturbed by the presence of an SWD. For both types of defect, MMDA is able to accurately predict the first 100 natural frequencies and mode shapes based on a system with 200 degrees of freedom (the full-order system contained 1152 degrees of freedom). Furthermore, the mode shapes predicted using the MMDA method replicate the mode localization in the full-order system. The success of these methods demonstrates that they are suitable tools for future investigations into the impact of defect types and locations on the phonon dispersion of CNTs of varying size and chirality.

## Data accessibility

This article has no additional data.

## Authors' contributions

Both authors have contributed equally to this work.

## Competing interests

We declare we have no competing interests.

## Funding

This research was supported by a Seed Grant award from the Institute for CyberScience at the Pennsylvania State University.

- Received August 9, 2017.
- Accepted February 16, 2018.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.