lable at ScienceDirect

Carbon 109 (2016) 19e29

Contents lists avai

Carbon

journal homepage: www.elsevier .com/locate/carbon

Effect of CNT length and structural density on viscoelasticity ofbuckypaper: A coarse-grained molecular dynamics study

Heng Chen a, b, Liuyang Zhang b, Jinbao Chen a, Matthew Becton b, Xianqiao Wang b, *,Hong Nie a, **

a College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, Chinab College of Engineering, University of Georgia, Athens, GA 30602, USA

a r t i c l e i n f o

Article history:Received 19 May 2016Received in revised form23 July 2016Accepted 26 July 2016Available online 29 July 2016

* Corresponding author.** Corresponding author.

E-mail addresses: xqwang@uga.edu (X. Wang), hn

http://dx.doi.org/10.1016/j.carbon.2016.07.0550008-6223/© 2016 Elsevier Ltd. All rights reserved.

a b s t r a c t

Carbon nanotube (CNT) buckypapers, having exceptional mechanical and electrical properties, have beenreported to demonstrate frequency-invariant and temperature-invariant viscoelastic properties. In anattempt to provide an in-depth insight into the viscoelasticity of buckypapers with (5, 5) single-walledCNTs (SWCNTs), we perform coarse-grained non-equilibrium molecular dynamics simulations toinvestigate the effects of oscillatory shear strain amplitude, buckypaper's density, and length of indi-vidual SWCNTs on the viscoelastic properties. SWCNT buckypapers exhibit linear viscoelasticity overshear strain amplitude from 0.03 to 0.05. Higher density SWCNT buckypapers can result in larger dy-namic stiffness and a higher loss factor of up to ~0.29. In the frequency-independent regime (�1 GHz),increasing the length of individual SWCNTs causes a very slight decrease of elastic properties and hasminor influence on the viscous mode. Thus, this study provides deep insight into the viscoelasticity of (5,5) SWCNT buckypapers and demonstrates controllability of the excellent energy dissipation potential ofbuckypapers, and can thus help us design new energy dissipation devices from carbon nanomaterials.

© 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Carbon nanotubes (CNTs) have shown unprecedented thermal,electrical, and mechanical properties due to their exceptionallylight weight, large surface area, and strong CeC bonds whencompared with traditional materials [1e4]. Hence, CNT network-based materials have attracted widespread attention in theareas of technology research and manufacture. For example,assembled CNTs with well-ordered orientations and configura-tions, such as three-dimensional sponge-array architectures, CNTfoams, and CNT aerogels, have demonstrated unprecedentedmechanical and thermal properties compared with materials ofsimilar density [5e9]. However, it is not easy to fabricate a well-ordered CNT network because of the likelihood of entanglementduring the fabrication process [10,11]. Owing to the large van derWaals (vdW) interactions between carbon structures, CNTs areprone to aggregate and then form into bundles and

ie@nuaa.edu.cn (H. Nie).

entanglements, thus forming randomly distributed CNT networkswhich are collectively called ‘buckypaper’ [12e14]. Several me-chanical and thermal tests for buckypapers have been imple-mented which indicate that buckypaper may be one of the mostpromising new nanomaterials [15e17]. For example, experimentalresults demonstrate that the Young's modulus of buckypaper canbe tuned from 0.2 GPa to 3.1 GPa by varying the buckypaper'sdensity and average CNT diameter [18,19], while the Poisson'sratio can be tuned between positive and negative values bychanging the content of multi-walled CNTs (MWCNTs) [14].Remarkably, buckypapers normally possess extremely low den-sities of 0.05e0.4 g/cm3 and high porosities of 0.8e0.9 [20] whichmake buckypapers very flexible, and it has been shown thatbuckypaper composed of single-walled CNTs (SWCNTs) with di-ameters 0.8e1.2 nm and lengths 100e1000 nm has a pore size ofabout 10 nm [21].

In current research, buckypapers have been widely used invarious applications, such as actuators, filtration systems, sen-sors, distillation devices, energy dissipation devices, etc. [22e31]For example, buckypaper-films composed of thin-walled carbonnanostructures filled with Fe3C, FeCo, FeNi, Co, and Ni crystalscan obtain enhanced saturation magnetization compared with

Table 1Potential parameters for the coarse-grained model of (5,5) SWCNT.

Parameters (5,5) SWCNT

Equilibrium bead distance, r0 (Å) 10Stretching constant, kb (kcal/(mol$Å2)) 1000Equilibrium angle, q0 (�) 180Bending constant, ka (kcal/(mol$rad2)) 14,300vdW distance, s (Å) 9.35vdW energy, ε (kcal/mol) 15.10

H. Chen et al. / Carbon 109 (2016) 19e2920

pristine Co, CoNi and Ni [23]. Chatterjee et al. [28] reported thatbuckypaper electrodes display superior sensitivity towardstryptophan, L-carnitine, tyrosine, and myoglobin; when engi-neered with specific molecules the buckypaper shows up to a1000-fold increase in signal compared with electrodes based onglassy carbon. Rein et al. [31] found that buckypaper sensors areable to measure strains in polymer with different elastic prop-erties while Chen et al. [29] revealed that buckypapers haveoutstanding impact behavior and high energy dissipation effi-ciency up to 95% when subjected to impacts. Additionally, Liuet al. [32] found that the interlaminar fracture toughness of acomposite laminate can be greatly improved by incorporating thebuckypaper into the middle interface of the laminate. Basiuket al. [33] demonstrated the possibility of fast and efficientsolvent-free functionalization of buckypaper mats prefabricatedfrom oxidized MWCNTs by utilizing three typical amines ofdifferent structure. Han et al. [34] made use of buckypaper-basedcoatings consisting of conductive buckypaper and insulatingadhesives to protect carbon fiber reinforced polymer laminates.Ma et al. [35] put forward a flexible tactile and shear sensingarray incorporating novel patterned buckypaper which exhibitedfavorable sensitivity and repeatability. Miao [36] revealed thatalumina-filled hybrid buckypaper composites have strong inter-facial polarization, high conductivity, and enhanced microwaveabsorbing performance.

Beyond these applications, it is also worth noting that severalpapers indicate that buckypapers exhibit frequency- andtemperature-independent viscoelastic properties from �196� to1000 �C and that the zipping-unzipping mechanism during theloading-unloading process can result in strong energy dissipation[37,38]. Moreover, Li et al. [37] also investigated the effects ofdouble-walled CNT (DWCNT) content on viscoelasticity of bucky-paper. Viscoelasticity features attributes of both viscosity andelasticity simultaneously, indicating capability towards energydissipation and reversible deformation, which both play importantroles in many biological and engineering systems, such as vibra-tion abatement, injury prevention, spacecraft construction, satel-lite stability, gasket reliability, computer disk storage, tireperformance, earplug functionality, and many others [39]. How-ever, most current research pertinent to the mechanical propertiesof buckypaper focus on purely elastic rather than viscoelasticproperties. Therefore, this work performs coarse-grained non-equilibrium molecular dynamic simulations to give further insightinto the viscoelasticity of SWCNT buckypapers.Wemainly focus onthe effects of oscillatory shear strain amplitude, buckypaper'sdensity, and the length of incorporated SWCNTs on the viscoelasticproperties of buckypapers. Our work aims to provide an in-depthunderstanding of the viscoelasticity of buckypaper relevant to itsstructure, and facilitate the design of new energy dissipationdevices.

2. Model and computational methods

2.1. Coarse-grained model for SWCNTs

Since computational costs limit the implementation of fullyatomistic molecular dynamics (MD) simulations for large CNTnetworks with a large range of vdW interactions, a coarse-grainedmolecular dynamics (CGMD) model for CNT networks is emergingas a complementary tool to access time and length scales theatomistic model inevitably cannot [40,41]. To this end, a fullyatomistic CNT is discretized into multiple beads connected byelastic springs to form a chain model. The discrete beads interactthrough bonds between two adjacent beads, angles among threesuccessive beads, and a pairwise potential between non-

successive beads; this corresponds to stretching, bending, andintertubular adhesion energy respectively. The twisting behaviorwhich may make a slight effect on the mechanical properties ofbuckypaper is often neglected, as this CG model with relevantforce field has been proved by many publications to successfullydescribe the mechanical properties of the buckypaper, regardlessof the incorporation of twisting behavior [42e44]. Therefore, thetotal energy of our model of the buckypaper can be expressed as:

Etot ¼ Ebond þ Eangle þ EvdW (1)

where the stretching energy is calculated as

Ebond ¼ Pbonds

1=2kbðr � r0Þ2; kb being the stiffness constant, r is the

current bond length, and r0 is the equilibrium bond length. The

bending energy is given by Eangle ¼P

angles1=2kaðq� q0Þ2, where ka is

the bending constant, q is the current angle and q0 is the equilib-rium angle. The vdW interaction is given by

EvdW ¼ Ppairs

4ε½ðs=rÞ12 � ðs=rÞ6�, where ε denotes the depth of the

potential well and s denotes the equilibrium distance where theinter-bead potential is zero. A relatively large cutoff of 28 nm isused for the vdW interactions, in order to more thoroughly capturethe effects of entanglement and bundling between CNTs. Based onthe equilibrium bond distance and the assumption of homoge-neous distribution of CNT mass, the mass of each bead is deter-mined to be 1953.23 g/mol. The parameters kb, r0, ka, q0, ε, and s

used in our CGMD simulations can be derived from the full atom-istic MD simulations. Parameters r0, q0, and s can be obtained byequilibrium conditions while kb, ka, and ε can be determined byenergy conservation conditions based on tensile, bending, andadhesion tests. The details can be found in Refs. [19,45]. In oursimulations, the CGMD model of buckypaper is formed by (5, 5)SWCNTs (hereafter referred to as merely CNTs for short) and thepotential parameters are presented in Table 1. This CGMD model ofbuckypaper using these parameters has been adopted in manypublications, such as research into the behavior of entanglementand bundling, tensile tests, self-assembly, and so on[19,40,42e44,46]. The results of these computational studies havebeen proven to agree well with experimental results, which pro-vides a firm foundation for our work by using the same CGMDmodel. It has to be noted that for any buckypaper in this work, allindividual SWCNTs have the same length, avoiding the poly-dispersity effect. In order to investigate the effect of individualSWCNTs' length on the viscoelasticity of buckypapers, six models ofSWCNTs with different lengths (100 nm, 237 nm, 410 nm, 800 nm,1896 nm and 6400 nm) are used to construct six types of bucky-papers respectively while for each buckypaper the sum length of allSWCNTs is kept at 51,200 nm by adjusting the number of SWCNTs.In addition, for the fixed-density tests, the density of each specimenis set at 107.26 kg/m3 which closely matches those reported byexperiments [14,22]. We will now discuss how these SWCNTmodels are used to construct the buckypaper models.

H. Chen et al. / Carbon 109 (2016) 19e29 21

2.2. Configurations of buckypaper

A random walk method combined with the excluded volumealgorithm is adopted to generate the CGMD buckypaper model inorder to mimic the randomness and isotropy of buckypaper. In themodel generation process, a series of points are put into anorthogonal cell in a spatially uniform manner, serving as the posi-tions for the initial bead of each SWCNT. By changing the cell size,buckypaper models with different densities can be obtained. It isobvious that the number of these initial points is equal to thenumber of SWCNTs in the specimen. Each SWCNTgrows to the nextposition for the next bead by a random bond vector with themagnitude equal to the equilibrium bond length (10 Å). Anexcluded volume check is then performed to avoid the overlappingof beads. The critical check distance is set to be 2 Å, that is to say, ifthe newly generated position is less than 2 Å away from any otherbead, it is considered that they are overlapped. On this occasion, theSWCNT returns back to the last position and finds a new positionbased on a new random bond vector, followed by a new overlapcheck. This finding-checking-returning process is repeated untilthe new position passes the overlap check. A SWCNT's walk ter-minates when the total number of steps is equal to the desiredlength of the SWCNT. It has to be noted that a partial overlap of thebeads is likely to take place, which can be acceptable because thesepartially-overlap beads can reach their well-equilibrium positionsafter the subsequent energy minimization. The representativeorthogonal simulation box is set to be periodic, to simulate a large,connected mass of buckypaper.

Following themodel generation by randomwalk, an isothermal-isobaric (NPT) ensemble is used to relax the initial configuration ofthe buckypaper until the pressure and temperature of the systemreach 1 atm and 300 K, respectively. During the equilibrium pro-cess, the behavior of entanglement and aggregation of SWCNTs isobserved due to the large range of vdW interactions, which isconsistent with the conclusion reported by Li et al. [46] thatentanglement behavior dominates the structure of SWCNT bucky-paper while DWCNT buckypaper is mainly influenced by bundlingbehavior. The total energy of the buckypaper decreases rapidly andthen approaches an equilibrium value slowly. In Fig. 1(a) and (b)depict snapshots of buckypaper models made with SWCNTs of in-dividual lengths 100 nm and 6,400 nm, respectively. All coarse-grained molecular figures are represented with the visualizationsoftware OVITO [47].

2.3. Non-equilibrium simulations of buckypaper

The viscoelastic properties are investigated by applying an

Fig. 1. Computational cells of buckypapers made of CNTs with individual lengths (a) 100 nmdensity of the buckypaper are kept the same. (A color version of this figure can be viewed

oscillatory shear deformation and recording the resulting stress.This is a non-equilibrium molecular dynamics (NEMD) problemdue to an external field driving the buckypaper away from theequilibrium state. The SLLOD (so called because of its associationwith the DOLLS tensor algorithm [48]) equations of motion, one ofthe most widely used methods for NEMD problems, have beenproven to be equivalent to Newton's equations of motion for shearflow [49] and are able to generate the desired velocity gradient andthe correct production of work by stresses for all forms of homo-geneous flow [50]. The models are coupled to a Nose-Hooverthermostat [51,52] in a velocity Verlet formulation at a constanttemperature of 300 K in our simulations. According to Tuckermanet al. [53], the SLLOD equations of motion can be expressed as:

_qi ¼pimi

þ qi$Vu; (2)

_pi ¼ F i � pi$Vu� phQpi; (3)

_h ¼ phQ; (4)

_ph ¼XN

i¼1

pi2

mi� dNkBT (5)

where, qi, pi, Fi and mi are the position vector, the momentumvector, the force vector and the mass of the i�th bead, and u is thestreaming velocity. The dot denotes a derivative with respect totime and V is a spatial gradient. h and ph are the coordinate- andmomentum-like variables, respectively. d is the dimension of space,N is the total number of beads, kB is the Boltzmann constant and T isthe temperature of the system. Q is the thermostat mass parameterexpressed as dNkBTt2, where t is the characteristic relaxation timeand is equal to 1 ps in this work with a timestep of 10 fs. For theSLLOD method, Lees-Edwards periodic boundary conditions [54]for SLLOD equations are applied, which generate homogeneousisothermal flow. The orthogonal simulation box is changed to benon-orthogonal, in preparation for the shear tests. Coordinates andvelocities of atoms are adjusted to create a shear velocity profile, sothat each point in the simulation box has a streaming velocity. Thisposition-dependent streaming velocity is subtracted from eachatom's actual velocity to yield a thermal velocity for temperaturecomputation and thermostatting. Velocities of atoms crossing theperiodic boundaries are remapped, keeping consistent with thevelocity profile. In this work, the simulation box is being sheared

and (b) 6400 nm at the equilibrium state. Both the total length of all the CNTs and theonline.)

H. Chen et al. / Carbon 109 (2016) 19e2922

along the x direction, relative to y. The shear strain is defined asgxy¼ dx/Ly0, where dx is the difference of transverse displacementsbetween the upper and lower xz planes, and Ly0 is the box length inthe y direction and is a constant in this work. For the oscillatoryshear simulations, the shear strain is given by a sinusoidal function:

gxy ¼ g0 sinðutÞ (6)

where g0 and u are the strain amplitude and angular frequency ofthe oscillatory shear deformation, respectively, and t is the time.The angular velocity u ¼ 2pf, where f is the oscillatory shear fre-quency. In our simulations, in order to find the viscoelasticity as afunction of the strain amplitude, the shear strain amplitude isvaried from 0.01 to 0.5 while the frequency is kept at 500 MHz. Tofind the relationship between the shear frequency and the visco-elasticity, the shear frequency is changed from 50�2000 MHz atconstant strain amplitude 0.05. Thus, the effects due to the lengthof individual CNTs on the viscoelasticity can be explored based onthe variations of length and frequency at strain amplitude 0.05. Thesimulations last for 30 cycles of oscillatory shear deformation andthe averaged shear stress is derived by using the latter 15 cycles,with the first 15 cycles discounted in order to allow for settlingtime. According to the theory of linear viscoelasticity [55], theoutput shear stress for buckypaper also has a sinusoidal formwith aphase shift relative to the input shear strain, expressed as:

txy ¼ t0 sinðut þ dÞ (7)

where t0 is the stress amplitude and d is the phase angle. The stressduring the latter 15 cycles of the shear deformation is recorded andthen fitted using a sinusoidal function with the same frequency asthe input strain, as depicted in Fig. 2, where the R-square is around0.997, showing an exceedingly close fit. The storage shear modulusG

0, loss shearmodulus G

00, and loss tangent tandwhich represent the

linear viscoelastic properties of buckypaper are defined as:

-0.06

-0.03

0

0.03

0.06

Shea

r str

ain

0 50 100 150 200 250 300-1.5

-1

-0.5

0

0.5

1

1.5

Time (ns)

Shea

r str

ess

(MPa

)

Computational datay=1.072sin(0.314x+0.2115)

y=0.05sin0.314x

Fig. 2. Input shear strain and output shear stress for the buckypaper made of CNTswith individual length 100 nm. The empty dots denote the computational shear stressand the continuous lines are sinusoidal fits to the empty dots. (A color version of thisfigure can be viewed online.)

G0 ¼ ðt0=g0Þcos d (8)

G00 ¼ ðt0=g0Þsin d (9)

tan d ¼ G0.

G00

(10)

The loss angle d is a dimensionless measure of the viscoelasticdamping of the material. For a linearly elastic solid, d ¼ 0, and for aviscous fluid, d ¼ p/2. Thus for viscoelastic materials, 0 < d < p/2. G

0

is the component of the stress-strain ratio in phasewith the appliedstrain, while G

00is the component 90� out of phase. The single and

double primes do not represent derivatives, but denote the real andimaginary parts respectively. The complex dynamic modulus canbe obtained by G* ¼ G

0 þ iG00

with the magnitude��G*

�� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðG0 Þ2 þ ðG00 Þ2

q¼ t0=g0, where i¼�1. Taking into account

the stiffness and the energy dissipation simultaneously, a dampingfigure of merit D describing the dynamic damping performance isdefined as:

D ¼ ��G*��tan d ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ tan2d

pG

00(11)

Generally, for low damping materials the damping figure ofmerit D is approximately equal to G

00, ignoring the order of tan2d.

Here NEMD simulations of buckypaper are implemented to obtaint0 and d, thus calculating G

0, G

00, tand and D according to Eqs.

(8)e(11).

3. Results and discussion

3.1. Strain amplitude effect

In order to determine the range of the strain amplitude corre-sponding to the linear viscoelastic regime and its effect on theviscoelastic properties of buckypaper, NEMD simulations of bucky-paper made with SWCNTs of individual length 100 nm are per-formed. The strain amplitude is varied over a wide range of

Fig. 3. Shear storage modulus, loss modulus, damping figure of merit, and loss factorfor buckypaper made of CNTs with individual length 100 nm as a function of strainamplitude (0.01e0.5). Shear frequency is 500 MHz. (A color version of this figure canbe viewed online.)

H. Chen et al. / Carbon 109 (2016) 19e29 23

0.01e0.5 at a frequency of 500 MHz, covering noise, elastic, andplastic deformation regimes. The viscoelastic properties of bucky-paper, including storage modulus G

0, loss modulus G

00, damping

figure of merit D, and loss factor tand, are presented in Fig. 3. Atextremely low strain amplitude (g0 < 0.03), the shear storagemodulus G

0related to the elastic energy decreases so slightly that it

can be considered to be strain amplitude insensitive. The reason isprobably that G

0is related to the elastic energy, and so during the

elastic regime it remains stable. Contrarily G00, related to the dissi-

pated energy, increases from 1.17 MPa to 5.33 MPa and tand in-creases from 0.04 to 0.24 when strain amplitude grows from 0.01 to0.03, which means that the viscosity increases quickly during thisregime. This is probably related to the noise effect resulting fromthermal fluctuations. As the strain amplitude grows, the noise effectbecomes softer. In addition, since the length of the simulation box isabout 1100 nm, when the strain varies from 0.01 to 0.03 thedisplacement of the upper surface (Fig. 4) ranges from 11 nm to33 nm, which is just close to the distance 10 nm between adjacentbonded atoms. Thus, the deformation during the oscillatory sheartests is rather tiny, especially for strain amplitude 0.01, and so thezipping-unzipping behavior occurs in few entanglement sites,which results in very little viscosity (small tand and G

00). As the strain

amplitude grows, more and more entanglement sites are able todisplay the zipping-unzipping mechanism, enhancing the viscosity(larger tand and G

00). Since in this regime tand is small, the damping

figure D ¼ jG*jtand ¼ 1 þ tan2dG00is approximately equal to G

00. As is

observed in Fig. 3, for 0.03 � g0 � 0.05, all the values of the visco-elastic properties (G

0, G

00, D, and tand) remain nearly independent of

the shear strain amplitude g0, indicating a linear response forbuckypaper subjected to external oscillatory shear deformation.Likewise, experimental results showed that the maximum strainamplitude for the reversible deformation linear regime is 0.05 [38].Past the linear regime, the buckypaper shows nonlinear viscoelasticproperties due to plastic deformation of the buckypaper. Thus as thestrain amplitude increases, the phase angle increases in anapproximately linear fashion; however, the shear storage modulusG

0decreases due to a decrease in the elastic stiffness. The loss

modulus G00remains nearly constant for 0.05 < g0 < 0.2, indicating

invariance in energy dissipation. As the strain amplitude continuesto increase, the damping ratio still rises rather linearly while boththe storage modulus G

0and the loss modulus G

00decrease, exhibiting

a process of temporary shear thinning at moderate strains [56]. Thedifference between G

0and G

00becomes smaller until g0 ¼ 0.5 where

G0 z G00. After that, G

00begins to be larger than G'. The variation

tendency between G0and G

00agrees well with relevant experimental

results [38]. It also can be found that beyond the critical strainamplitude (g0 ¼ 0.05) the damping figure of merit D gradually be-comes noticeably different from the loss modulus, thanks to theincrement of the damping ratio.

Fig. 4. Snapshots of buckypaper made of 100 CNTs of individual length 100 nm a

Fig. 4 depicts snapshots of the buckypaper model at threerepresentative states: undeformed state (gxy ¼ 0), reversibledeformation (gxy ¼ 0.05), and highly irreversible deformation(gxy ¼ 0.5). At the undeformed state, the buckypaper exhibits a 3Dhighway network where each SWCNT is in contact with manyother SWCNTs. Unlike the bundle dominated buckypaper wherenumerous CNTs are nearly straight and contact with each otherover a long distance, the entanglement behavior of long CNTsmakes the buckypaper possess high porosity. This porous nature ofthe buckypaper is favorable for fast and efficient heat dissipation,avoiding considerable heat accumulation. In addition, the entan-glement of SWCNTs yields a large range of reversible deformation,and thus sustains large linear strain amplitudes up to 0.05. Theconnections between SWCNTs, also called entanglement sites, areusually over a short span. Attachment and detachment betweenSWCNTs occur reversibly during the cyclic oscillatory deformationby zipping and unzipping mechanisms respectively, which is alsoobserved by both experiment [38] and simulation [37]. As Fig. 5shows, during the loading process (a)e(c), zipping behavior be-tween two SWCNTs occurs, forming an entanglement. Unzippingbehavior then happens during the unloading process, reducing theinter-tube entanglements. Plenty of energy is consumed duringthe unzipping process in order to overcome the large vdWattraction between SWCNTs, yet the zipping process is sponta-neous. Thus due to the high density of entanglement sites thebuckypaper is believed to have excellent energy dissipationcapability. Beyond the linear regime, several SWCNTs tend to alignand become bundled, which begins to take shape in Fig. 4(c) andwill be more obvious under larger strain according to the experi-mental results [38]. Therefore the number of entanglement sites isreduced, resulting in a degeneration of the energy dissipationcapability.

In the linear regime, the output stress is a typical sinusoidalfunction versus time, seen in Fig. 6(a) where the strain amplitudeg0 ¼ 0.05 is small. Another characteristic of the linear viscoelas-ticity is the so called elliptical Lissajous plot (Fig. 6(c)), where theshear stress is plotted versus the input shear strain within a cycleof oscillation. This loop demonstrates a hysteresis phenomenonthat is a lag between cause and effect. The area of the loop denotesthe energy dissipation density (energy dissipation per unit vol-ume per cycle of the material) which is proportional to the vis-cosity modulus G

00. In the nonlinear regime, the time-dependent

stress deviates from the sinusoidal function resulting in a distor-tion of the elliptical loop, which can be seen in Fig. 6(b) and (d)where the strain amplitude g0 ¼ 0.5, leading to high nonlinearity.Thus, a Fourier-transform method can be employed in order todescribe the stress distortion by calculating high odd harmonicscontribution, which can be expressed in the sine and cosine form[57]:

t varying shear strains. (A color version of this figure can be viewed online.)

Fig. 5. Snapshots of buckypaper made of 512 CNTs of individual length 100 nm during a cyclic oscillatory shear deformation at a frequency of 500 MHz and strain amplitude of 0.05.Only several CNTs are selected for observation while others remain hidden. Different CNTs are colored differently. (a) demonstrates the original state; (a)e(c) depict the loadingprocess during which zipping behavior of CNTs is observed; (c)e(e) show the unloading process when unzipping behavior of CNTs happens. (A color version of this figure can beviewed online.)

-0.05 0 0.05-1.5

-1

-0.5

0

0.5

1

1.5

Shear strain

Shea

r str

ess

(MPa

)

-0.5 0 0.5-3

-2

-1

0

1

2

3

Shear strain

Shea

r str

ess

(MPa

)

0 1 2 3 4-1.5

-1

-0.5

0

0.5

1

1.5

Time (ns)

Shea

r str

ess

(MPa

)

0 1 2 3 4-3

-2

-1

0

1

2

3

Time (ns)

Shea

r str

ess

(MPa

)

(a) (b)

(c) (d)

Fig. 6. Shear stress response as a function of time for strain amplitude g0¼ 0.05 (a) and g0¼ 0.5 (b) respectively. Lissajous plots of the shear stress versus the shear strain are shownfor shear strain amplitudes g0¼ 0.05 (c) and g0¼ 0.5 (d) respectively. The dots denote the simulation data; the solid line in the left column represents the sinusoidal fit; the solidlines in the right column are fitted by a Fourier function. (A color version of this figure can be viewed online.)

H. Chen et al. / Carbon 109 (2016) 19e2924

txy�g0 ¼

Xk¼1;3;5;…

hG

0k sinðkutÞ þ G

00k cosðkutÞ

i(12)

where k is odd. It has been proven that the Fourier expansion cancontain only odd modes since clockwise and anticlockwise rota-tions are mechanically equivalent, and thus only odd values of k areused. Fig. 6(b) and (d) exhibit fitting by Eq. (12) where only the first

two odd modes (k¼ 1, 3) are necessary for a near-perfect fittingwith R-square up to 0.99. It is obvious that the Fourier-transformmethod is also applicable in the linear regime using only the firstmode, which can be expressed as:

txy�g0 ¼ G

0sinðutÞ þ G

00cosðutÞ (13)

The following discussions in this paper are based on the strain

H. Chen et al. / Carbon 109 (2016) 19e29 25

amplitude g0¼ 0.05 to ensure that the buckypaper remains withinthe linear regime while still having a significant oscillation.

3.2. Density effects

Density, as one of the primary parameters to determine thestructure of buckypaper, has been proven to strongly influencemechanical properties such as the Young's modulus, bulk modulus,impact behavior, etc. [20,29] Thus it is meaningful to discuss theeffects of density on the viscoelastic properties of buckypaper.Because the energy dissipation of buckypaper is dominated by thezipping-unzipping mechanism at entanglement sites, we adoptthe Z1 code [58e61], a purely geometric algorithm, to count thenumber of entanglement sites of the buckypaper system, aparameter which is difficult to measure in experiments. In the Z1code, each buckypaper system is mapped on a multiple discon-nected path of straight lines which is infinitesimally thin, impen-etrable, and tensionless. The primitive path made of a linear chainwith several kinks and segments connects the space-fixed ends ofa CNT and is minimized for a shortest path while maintaining theunderlying entanglement structure. Subjected to line-uncrossability, the total length of all the lines for a buckypapersystem is reduced by introducing fewer nodes. To acquire ashortest multiple disconnected path carrying entanglement in-formation, nodes which do not change the direction of the path areremoved; two segments without overlap with other obstacles

Fig. 7. Number of entanglements per CNT (a) and viscoelastic properties (b) ofbuckypaper made of CNTs with individual length 100 nm as a function of buckypaper'sdensity. g0¼ 0.05 and f¼ 500 MPa. (A color version of this figure can be viewedonline.)

become a single one, otherwise they will change their orientationsimultaneously; two segments may grow to three if more than oneobstacle hinders the straightening behavior. The number of finalinternal nodes (interior kinks) represents the number of entan-glements of the buckypaper system. This Z1 code has been appliedby researchers to networks such as polymeric [60,62,63] andbuckypaper-based [46] systems for acquiring entanglementstructure and entanglement analysis. Therefore, using this Z1 codethe number of inter-tube entanglements per CNT for a buckypaper,denoted as Ze, versus density of buckypaper is displayed inFig. 7(a).

Fig. 7(a) depicts that for lower densities (from 39.7 kg/m3 to49.4 kg/m3) Ze grows little with increasing density. Yet for largerdensities (from 61.6 kg/m3 to 107.3 kg/m3), Ze increases approxi-mately linearly and the trend climbs as the density of the bucky-paper rises. The viscoelastic properties of buckypaper as a functionof density can be found in Fig. 7(b). As the density grows, both theloss factor and lossmodulus keep increasing and the increment ratetends to increase as well, indicating higher energy dissipationcapability at higher densities, due to the larger Ze. It can be deducedthat the energy dissipation is not simply proportional to Ze, but theyhave a close relationship since Ze strongly dominates the energydissipation capability. The buckypaper storage modulus also in-creases at higher densities due to the stronger elastic stiffnesswhich has been demonstrated in our previous work [29]. Thedensity varying from 39.70 kg/m3 to 107.30 kg/m3 results in anincrease of 4.2, 6.8, and 1.6 times for the storage modulus, lossmodulus, and damping ratio, respectively. Similarly, experimentalresults also show that when the density of buckypaper increases 4times, the increments of storage and loss modulus and the dampingratio are 5, 10 and 2 times respectively [38].

3.3. Length effects

The length of individual CNTs is another important factor indetermining the structure of buckypaper. However, there are nocurrent studies in its effects on the viscoelastic properties ofbuckypaper. Experiments about frequency-independence showedconstant viscoelastic properties including storage modulus, loss

Fig. 8. Shear storage modulus, loss modulus, damping figure of merit, and loss factorfor buckypaper models made of CNTs with individual lengths from 100 nm to 6400 nmas a function of frequency (50 MHze2000 MHz). (A color version of this figure can beviewed online.)

H. Chen et al. / Carbon 109 (2016) 19e2926

modulus, and damping ratio of the buckypaper in the range of0.1e100 HZ [38]. Related simulations also proved the frequency-independence of the viscoelasticity of buckypaper for the rangebetween 15.7 MHz and 3 GHz [37], but for significantly higherfrequencies the viscoelasticity becomes frequency-dependent.Since most engineering applications rarely involve extremelyhigh frequencies, in this segment we focus on the frequency-independent regime and the effects of SWCNT length on visco-elastic properties within this frequency-independent regime. Ourresults in Fig. 8 exhibit frequency-independent viscoelasticity forthe range of 50MHze1 GHz. It is known that the energy dissipationcapability of buckypaper mainly depends on the zipping-unzippingmechanism between SWCNTs, which is a fast process. If the shearloading rate is below the zipping-unzipping rate, energy dissipa-tion during a cycle is believed to be rate- or frequency-independent. It has to be noted that the critical frequenciesending the frequency-independence regime are not the same for allsamples because of the variant internal structures andmanufactureprocesses of buckypaper. If the loading rate is beyond the zipping-unzipping rate, the hysteresis between the input strain and outputstress becomes more severe, resulting in a larger damping ratiotand and thus a higher loss modulus. The shear storage modulusdecreases, indicating smaller dynamic elastic stiffness.

It is also shown in Fig. 8 that the individual length of SWCNTshas noticeable influence on the viscoelastic properties. For

Fig. 9. Number of entanglements per CNT (a) and viscoelastic properties of bucky-paper (b) as a function of the length of individual CNTs from 100 nm to 6400 nm underdifferent shear frequencies. (A color version of this figure can be viewed online.)

oscillatory shear frequency f� 1 GHz, the damping ratio tand of abuckypaper remains independent of the length of individualSWCNTs over a wide range from 100 nm to 6400 nm, indicating aninvariance of the ratio of energy dissipated to energy stored indynamic loading. In this regime, as the length of individual SWCNTsrises, the loss modulus exhibits a slight decline and seems to reacha plateau corresponding to a critically large length of individualSWCNTs, which can be seen in Fig. 9(b). This can be further un-derstood by the Z1 code analysis shown in Fig. 9(a). The number ofinter-tube entanglements per SWCNT (Ze) increases approximatelylinearly with increasing length of individual SWCNTs, which can beexpressed as Ze¼ kLCNTþ b (k is the slope, b is a positive constantand LCNT is the length per CNT). The total number of inter-tubeentanglements in a buckypaper system can be obtained byZt¼N� Ze/2 (N is the number of chains), as each entanglementinvolves two SWCNTs. Since N¼Nt/LCNT (Nt¼ 51,200, is the totalnumber of beads in each buckypaper system), we can getZt(LCNT) ¼ (bNt/LCNTþ kNt)/2¼ 46,976/LCNTþ 31,155, and thusdZT=dLCNT ¼ �46;976=L2CNT . Therefore, for the length of individualSWCNTs (LCNT) varying from 100 nm to 6400 nm, the total numberof entanglements in a buckypaper system Zt(LCNT) slightly decreaseswith a reducing rate, and becomes stable with relatively longerSWCNTs, which is coincident with the variation trend of the lossmodulus. With the growth of the length of individual CNTs, thestorage modulus also decreases, and the decline gradually appearsto reach a plateau. The reason is probably that SWCNTs with largerlengths are more prevalent to bending and buckling whendeformed, resulting in a more flexible buckypaper and thus a lowerstiffness. This buckling phenomenon was clearly observed in earlyexperiments and MD simulations and it was reported that the CNTshave remarkable buckling capacity and that the buckled CNTs havethe ability to fully recover elastically [64e67] corresponding to thelinear regime of viscoelasticity. Cao and Chen [68] found that thecritical bending curvature grows with respect to the aspect ratio,and after the aspect ratio reaches a critical value the critical buck-ling curvature becomes insensitive to the aspect ratio. Similarly,after the length is beyond a critical value (400 nm), the storagemodulus associated with elastic energy becomes unchanged withrespect to SWCNT length, which is depicted in Fig. 9(b). It has to benoted that, strictly speaking, the loss modulus and damping figureof merit have slight variationwith respect to the SWCNT length, butthis dependence is so slight that they can be considered to beapproximately independent of SWCNT length.

For f beyond the critical value (1 GHz), the damping ratio ofbuckypaper seems to be more sensitive to the length of individualSWCNTs, as is seen in Fig. 8. The reason is probably that a bucky-paper made of longer SWCNTs has more bending and bucklingdeformation, resulting in a larger lag between the input strain andoutput stress subjected to the oscillatory shear deformation withsignificantly high frequency. That is to say, buckypaper made oflonger SWCNTs behaves more like a viscoelastic material thanbuckypaper made of shorter SWCNTs does. Figs. S1 and S2 in theSupporting Information illustrate the trend of viscoelasticity be-tween 1 GHz and 2.3 GHz in detail. It can be found that for CNTlengths from100 nm to 1896 nm, shear storagemodulus G0 declinesslightly while shear loss modulus G

00, loss factor tand, and damping

figure of merit D increase slightly with respect to CNT length. ForCNT length of 6400 nm, there appears something different: asfrequency varies from 1.9 GHz to 2.3 GHz, tand and D increasequickly. That is related to the increase of phase angle d. Afterd reaches ~1 rad, a slight increase of d can result in an extremeincrease of tand and thus a rapid increase of D, which may result inthe calculation error for the loss factor and damping figure of merit.Here, we mainly focus on the frequency-independent regime, and

Fig. 10. Stress versus strain of buckypapers made of different lengths of individual CNTs with shear frequencies 500 MPa and 2 GHz respectively. The notation in the left figurecorresponds to the model with CNTs of length 100 nm. (A color version of this figure can be viewed online.)

Fig. 11. Energy dissipation density of as a function of the individual length of CNTswith shear frequencies ranging from 50 MHz to 2 GHz. (A color version of this figurecan be viewed online.)

H. Chen et al. / Carbon 109 (2016) 19e29 27

thus the corresponding effects and properties of buckypaper athigh frequencies (>1 GHz) need to be further investigated in thefuture.

Fig. 10 depicts the elliptic Lissajous figures for different lengthsof individual SWCNTs under oscillatory shear frequencies 500 MHzand 2 GHz, representing frequency-independence and frequency-dependence respectively. The values of storage and loss modulican be revealed in the elliptic Lissajous figure directly and thewidthof the figure is a measure of the loss angle d. The enclosed area ofthe elliptic Lissajous figure represents the dissipated energy pervolume of the material during a period, which is presented inFig. 11. It is obvious that the dissipated energy per volume per cyclewith respect to the length of individual SWCNTs under differentoscillatory shear frequencies has a similar variation trend as theloss storage modulus does. In the regime of frequency-independence, with the increase of SWCNT length the dissipatedenergy declines slightly from around 0.04 J/cm3 to around 0.03 J/cm3. Therefore, the dissipated energy can be considered to be

approximately independent of SWCNT length. For f¼ 2 GHz, energydissipation is obviously higher at lower frequencies.

4. Conclusions

In this paper, a series of coarse-grained NEMD simulations ofbuckypaper with (5, 5) SWCNTs subjected to oscillatory sheardeformation are performed in order to investigate the effects ofoscillatory shear strain amplitude and frequency, buckypaperdensity, and length of the constituent SWCNTs on the viscoelasticproperties of buckypaper. The key findings revealed by our simu-lation results are as follows: (i) with the growth of the strainamplitude the viscoelasticity of buckypaper experiences noise,linear, and nonlinear regimes in turn, and the linear regime in thispaper corresponds to a strain amplitude range of 0.03 � g0� 0.05;the damping ratio for the linear regime is around 0.25 which ismuch higher than those of conventional materials (~10�3 or less formetals). (ii) Energy dissipation is mainly dominated by the zipping-unzipping mechanism at entanglement sites; the loss modulusrelated to energy dissipation is not simply proportional to thenumber of entanglements (which can be calculated by Z1 code),however they do have a close relationship. (iii) Due to an increase inentanglements and the resulting stronger stiffness of buckypaperswith higher density, the loss factor, loss modulus, storage modulus,and damping figure of merit all increase with increasing density,and the increment rate becomes larger for larger densities. (iv)Buckypapers exhibit frequency-independent viscoelasticity whenthe frequency is below a critical value (1 GHz); for frequenciesbeyond the critical value, both damping ratio and loss modulusincrease indicating larger energy dissipation while the storagemodulus decreases indicating smaller elastic energy. (v) In thefrequency-independent regime, the shear storage modulus de-creases slightly with respect to the length of individual SWCNTs;the damping ratio, the loss modulus and damping figure of meritare approximately independent of the length of SWCNTs, whichmay be attributed to the linear increasing of Ze with respect to thelength. It has to be noted that buckypaper used in this work is madeof pure (5, 5) SWCNTs; MWCNT buckypapers have potentiallydifferent properties. Overall, this work can provide deeper insightinto the viscoelasticity of buckypaper and thus benefit the design ofnovel energy-dissipation devices for protecting critical structuresor humans under extreme conditions by using fabricated carbonnanomaterials.

H. Chen et al. / Carbon 109 (2016) 19e2928

Competing financial interests

The authors declare no competing financial interests.

Acknowledgements

HC, JC, and HN appreciate the support from the Scholarship ofNanjing University of Aeronautics and Astronautics (China) (GrantNo: [2013]59). XW would like to acknowledge support from theUniversity of Georgia (UGA) Research Foundation and NationalScience Foundation (Grant No. CMMI-1306065). Calculations areperformed at the UGA Advanced Computing Resource Centre.

Appendix A. Supplementary data

Supplementary data related to this article can be found at http://dx.doi.org/10.1016/j.carbon.2016.07.055.

References

[1] X. Chen, Y. Huang, Nanomechanics modeling and simulation of carbonnanotubes, J. Eng. Mech. 134 (3) (2008) 211e216.

[2] M.R. Delfani, H.M. Shodja, F. Ojaghnezhad, Mechanics and morphology ofsingle-walled carbon nanotubes: from graphene to the elastica, Philos. Mag.93 (17) (2013) 2057e2088.

[3] J.P. Salvetat, J.M. Bonard, N.H. Thomson, A.J. Kulik, L. Forr�o, W. Benoit, et al.,Mechanical properties of carbon nanotubes, Appl. Phys. A 69 (3) (1999)255e260.

[4] H. Chen, L. Zhang, M.D. Becton, H. Nie, J. Chen, X. Wang, Molecular dynamicstudy of a CNT/buckyballs-enabled energy absorption system, Phys. Chem.Chem. Phys. 17 (2015) 17311e17321.

[5] X. Gui, Z. Zeng, Y. Zhu, H. Li, Z. Lin, Q. Gan, et al., Three-dimensional carbonnanotube sponge-array architectures with high energy dissipation, Adv.Mater. 26 (8) (2014) 1248e1253.

[6] L. Lattanzi, L. De Nardo, J.R. Raney, C. Daraio, Geometry-induced mechanicalproperties of carbon nanotube foams, Adv. Eng. Mater. 16 (8) (2014)1026e1031.

[7] L.Q. Liu, W.J. Ma, Z. Zhang, Macroscopic carbon nanotube assemblies: prepa-ration, properties, and potential applications, Small 7 (11) (2011) 1504e1520.

[8] M.B. Bryning, D.E. Milkie, M.F. Islam, L.A. Hough, J.M. Kikkawa, A.G. Yodh,Carbon nanotube aerogels, Adv. Mater. 19 (5) (2007) 661e664.

[9] K.H. Kim, M. Vural, M.F. Islam, Single-walled carbon nanotube aerogel-basedelastic conductors, Adv. Mater. 23 (25) (2011) 2865e2869.

[10] W. Ma, L. Song, R. Yang, T. Zhang, Y. Zhao, L. Sun, et al., Directly synthesizedstrong, highly conducting, transparent single-walled carbon nanotube films,Nano Lett. 7 (8) (2007) 2307e2311.

[11] H. Dai, J. Kong, C. Zhou, N. Franklin, T. Tombler, A. Cassell, et al., Controlledchemical routes to nanotube architectures, physics, and devices, J. Phys.Chem. B 103 (51) (1999) 11246e11255.

[12] A. Rinzler, J. Liu, H. Dai, P. Nikolaev, C. Huffman, F. Rodriguez-Macias, et al.,Large-scale purification of single-wall carbon nanotubes: process, product,and characterization, Appl. Phys. A Mater. Sci. Process. 67 (1) (1998) 29e37.

[13] A. Thess, R. Lee, P. Nikolaev, H. Dai, P. Petit, J. Robert, et al., Crystalline ropes ofmetallic carbon nanotubes, Sci. AAAS Wkly. Pap. Ed. 273 (5274) (1996)483e487.

[14] L.J. Hall, V.R. Coluci, D.S. Galvao, M.E. Kozlov, M. Zhang, S.O. Dantas, et al., Signchange of Poisson's ratio for carbon nanotube sheets, Science 320 (5875)(2008) 504e507.

[15] J.M. Romo-Herrera, M. Terrones, H. Terrones, V. Meunier, Guiding electricalcurrent in nanotube circuits using structural defects: a step forward innanoelectronics, ACS Nano 2 (12) (2008) 2585e2591.

[16] Y. Li, X. Qiu, F. Yang, Y. Yin, Q. Fan, Stretching-dominated deformationmechanism in a super square carbon nanotube network, Carbon 47 (3) (2009)812e819.

[17] I. Zsoldos, I. Laszlo, Computation of the loading diagram and the tensilestrength of carbon nanotube networks, Carbon 47 (5) (2009) 1327e1334.

[18] L. Zhang, G. Zhang, C. Liu, S. Fan, High-density carbon nanotube buckypaperswith superior transport and mechanical properties, Nano Lett. 12 (9) (2012)4848e4852.

[19] S.W. Cranford, In silico assembly and nanomechanical characterization ofcarbon nanotube buckypaper, Nanotechnology 21 (26) (2010) 265706.

[20] R.L.D. Whitby, T. Fukuda, T. Maekawa, S.L. James, S.V. Mikhalovsky, Geometriccontrol and tuneable pore size distribution of buckypaper and buckydiscs,Carbon 46 (6) (2008) 949e956.

[21] Q. Wu, W. Zhu, C. Zhang, Z. Liang, B. Wang, Study of fire retardant behavior ofcarbon nanotube membranes and carbon nanofiber paper in carbon fiberreinforced epoxy composites, Carbon 48 (6) (2010) 1799e1806.

[22] K. Sears, L. Dumee, J. Schutz, M. She, C. Huynh, S. Hawkins, et al., Recent de-velopments in carbon nanotube membranes for water purification and gas

separation, Materials 3 (1) (2010) 127e149.[23] J. Guo, M. Lan, S.L. Wang, Y. He, S.J. Zhang, G. Xiang, et al., Enhanced saturation

magnetization in buckypaper-films of thin walled carbon nanostructures fil-led with Fe3C, FeCo, FeNi, CoNi, Co and Ni crystals: the key role of Cl, Phys.Chem. Chem. Phys. 17 (27) (2015) 18159e18166.

[24] Q. Cao, Kim H-s, N. Pimparkar, J.P. Kulkarni, C. Wang, M. Shim, et al., Medium-scale carbon nanotube thin-film integrated circuits on flexible plastic sub-strates, Nature 454 (7203) (2008) 495e500.

[25] Q. Cao, J.A. Rogers, Ultrathin films of single-walled carbon nanotubes forelectronics and sensors: a review of fundamental and applied aspects, Adv.Mater. 21 (1) (2009) 29e53.

[26] A.S. Brady-Est�evez, S. Kang, M. Elimelech, A single-walled-carbon-nanotubefilter for removal of viral and bacterial pathogens, Small 4 (4) (2008) 481e484.

[27] C. Meng, C. Liu, S. Fan, Flexible carbon nanotube/polyaniline paper-like filmsand their enhanced electrochemical properties, Electrochem. Commun. 11 (1)(2009) 186e189.

[28] J. Chatterjee, J. Cardenal, A. Shellikeri, Engineered carbon nanotube bucky-paper: a platform for electrochemical biosensors, J. Biomed. Nanotechnol. 11(1) (2015) 150e156.

[29] H. Chen, L. Zhang, J. Chen, M. Becton, X. Wang, H. Nie, Energy dissipationcapability and impact response of carbon nanotube buckypaper: a coarse-grained molecular dynamics study, Carbon 103 (2016) 242e254.

[30] I.W.P. Chen, Z. Liang, B. Wang, C. Zhang, Charge-induced asymmetricaldisplacement of an aligned carbon nanotube buckypaper actuator, Carbon 48(4) (2010) 1064e1069.

[31] M.D. Rein, O. Breuer, H.D. Wagner, Sensors and sensitivity: carbon nanotubebuckypaper films as strain sensing devices, Compos. Sci. Technol. 71 (3)(2011) 373e381.

[32] L. Liu, L.L. Shen, Y.X. Zhou, Improving the interlaminar fracture toughness ofcarbon/epoxy laminates by directly incorporating with porous carbon nano-tube buckypaper, J. Reinf. Plast. Compos. 35 (2) (2016) 165e176.

[33] E.V. Basiuk, I.J. Ramirez-Calera, V. Meza-Laguna, E. Abarca-Morales, L.A. Perez-Rey, M. Re, et al., Solvent-free functionalization of carbon nanotube bucky-paper with amines, Appl. Surf. Sci. 357 (2015) 1355e1368.

[34] J.H. Han, H. Zhang, M.J. Chen, D. Wang, Q. Liu, Q.L. Wu, et al., The combinationof carbon nanotube buckypaper and insulating adhesive for lightning strikeprotection of the carbon fiber/epoxy laminates, Carbon 94 (2015) 101e113.

[35] C.-W. Ma, L.-S. Hsu, J.-C. Kuo, Y.-J. Yang, A flexible tactile and shear sensingarray fabricated using a novel buckypaper patterning technique, Sens. Actu-ators a Phys. 231 (2015) 21e27.

[36] H.Y. Miao, J.H. Liu, L. Saravanan, Fabrication of alumina filled hybrid bucky-paper composites and their enhancement of microwave absorbing perfor-mance, J. Phys. D Appl. Phys. 48 (21) (2015) 6.

[37] Y. Li, M. Kroeger, Viscoelasticity of carbon nanotube buckypaper: zipping-unzipping mechanism and entanglement effects, Soft Matter 8 (30) (2012)7822e7830.

[38] M. Xu, D.N. Futaba, T. Yamada, M. Yumura, K. Hata, Carbon nanotubes withtemperature-invariant viscoelasticity frome196 to 1000 C, Science 330 (6009)(2010) 1364e1368.

[39] R.S. Lakes, Viscoelastic Materials, Cambridge University Press, 2009.[40] M.J. Buehler, Mesoscale modeling of mechanics of carbon nanotubes: self-

assembly, self-folding, and fracture, J. Mater. Res. 21 (11) (2006) 2855e2869.[41] S. Cranford, H. Yao, C. Ortiz, M.J. Buehler, A single degree of freedom ‘lollipop’

model for carbon nanotube bundle formation, J. Mech. Phys. Solids 58 (3)(2010) 409e427.

[42] X. Yang, P. He, H. Gao, Modeling frequency-and temperature-invariant dissi-pative behaviors of randomly entangled carbon nanotube networks undercyclic loading, Nano Res. 4 (12) (2011) 1191e1198.

[43] C. Wang, L. Wang, Z. Xu, Enhanced mechanical properties of carbon nanotubenetworks by mobile and discrete binders, Carbon 64 (2013) 237e244.

[44] Y. Li, M. Kr€oger, Computational study on entanglement length and pore size ofcarbon nanotube buckypaper, Appl. Phys. Lett. 100 (2) (2012) 021907.

[45] S. Cranford, M.J. Buehler, Mechanomutable carbon nanotube arrays, Int. J.Mater. Struct. Integr. 3 (2e3) (2009) 161e178.

[46] Y. Li, M. Kroeger, A theoretical evaluation of the effects of carbon nanotubeentanglement and bundling on the structural and mechanical properties ofbuckypaper, Carbon 50 (5) (2012) 1793e1806.

[47] A. Stukowski, Visualization and analysis of atomistic simulation data withOVITOethe open visualization tool, Model. Simul. Mater. Sci. Eng. 18 (1)(2010) 015012.

[48] G.P. Morriss, D.J. Evans, Statistical Mechanics of Nonequilbrium Liquids, ANUPress, 2007.

[49] D.J. Evans, G. Morriss, Nonlinear-response theory for steady planar Couetteflow, Phys. Rev. A 30 (3) (1984) 1528.

[50] P.J. Daivis, B. Todd, A simple, direct derivation and proof of the validity of theSLLOD equations of motion for generalized homogeneous flows, J. Chem. Phys.124 (19) (2006) 194103.

[51] S. Nos�e, A unified formulation of the constant temperature molecular dy-namics methods, J. Chem. Phys. 81 (1) (1984) 511e519.

[52] W.G. Hoover, Canonical dynamics: equilibrium phase-space distributions,Phys. Rev. A 31 (3) (1985) 1695e1697.

[53] M.E. Tuckerman, C.J. Mundy, S. Balasubramanian, M.L. Klein, Modifiednonequilibrium molecular dynamics for fluid flows with energy conservation,J. Chem. Phys. 106 (13) (1997) 5615e5621.

[54] A. Lees, S. Edwards, The computer study of transport processes under extreme

H. Chen et al. / Carbon 109 (2016) 19e29 29

conditions, J. Phys. C Solid State Phys. 5 (15) (1972) 1921e1929.[55] J.D. Ferry, Viscoelastic Properties of Polymers, John Wiley & Sons, 1980.[56] F. Renou, J. Stellbrink, G. Petekidis, Yielding processes in a colloidal glass of

soft star-like micelles under large amplitude oscillatory shear (LAOS), J. Rheol.54 (6) (2010) 1219e1242.

[57] K.D. Danov, G.M. Radulova, P.A. Kralchevsky, K. Golemanov, S.D. Stoyanov,Surface shear rheology of hydrophobin adsorption layers: laws of viscoelasticbehaviour with applications to long-term foam stability, Faraday Discuss. 158(1) (2012) 195e221.

[58] M. Kr€oger, Shortest multiple disconnected path for the analysis of entangle-ments in two- and three-dimensional polymeric systems, Comput. Phys.Commun. 168 (3) (2005) 209e232.

[59] S. Shanbhag, M. Kr€oger, Primitive path networks generated by annealing andgeometrical methods: insights into differences, Macromolecules 40 (8) (2007)2897e2903.

[60] R.S. Hoy, K. Foteinopoulou, M. Kr€oger, Topological analysis of polymeric melts:chain-length effects and fast-converging estimators for entanglement length,Phys. Rev. E 80 (3) (2009) 031803.

[61] N.C. Karayiannis, M. Kr€oger, Combined molecular algorithms for the genera-tion, equilibration and topological analysis of entangled polymers:

methodology and performance, Int. J. Mol. Sci. 10 (11) (2009) 5054e5089.[62] J.M. Kim, D.J. Keffer, M. Kr€oger, B.J. Edwards, Rheological and entanglement

characteristics of linear-chain polyethylene liquids in planar Couette andplanar elongational flows, J. Newt. Fluid Mech. 152 (1e3) (2008) 168e183.

[63] K. Foteinopoulou, N.C. Karayiannis, M. Laso, M. Kr€oger, M.L. Mansfield, Uni-versal scaling, entanglements, and knots of model chain molecules, Phys. Rev.Lett. 101 (26) (2008) 265702.

[64] S. Iijima, C. Brabec, A. Maiti, J. Bernholc, Structural flexibility of carbonnanotubes, J. Chem. Phys. 104 (5) (1996) 2089e2092.

[65] M. Falvo, G. Clary, R. Taylor, V. Chi, F. Brooks, S. Washburn, et al., Bending andbuckling of carbon nanotubes under large strain, Nature 389 (6651) (1997)582e584.

[66] W.H. Knechtel, G.S. Düsberg, W.J. Blau, E. Hern�andez, A. Rubio, Reversiblebending of carbon nanotubes using a transmission electron microscope, Appl.Phys. Lett. 73 (14) (1998) 1961e1963.

[67] O. Lourie, D. Cox, H. Wagner, Buckling and collapse of embedded carbonnanotubes, Phys. Rev. Lett. 81 (8) (1998) 1638e1641.

[68] G. Cao, X. Chen, Buckling of single-walled carbon nanotubes upon bending:molecular dynamics simulations and finite element method, Phys. Rev. B 73(15) (2006) 155435.