C A R B O N 5 0 ( 2 0 1 2 ) 1 7 9 3 – 1 8 0 6

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A theoretical evaluation of the effects of carbon nanotubeentanglement and bundling on the structural and mechanicalproperties of buckypaper

Ying Li a,*, Martin Kroger b

a Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-0311, USAb Department of Materials, Polymer Physics, ETH Zurich, CH-8093 Zurich, Switzerland

A R T I C L E I N F O

Article history:

Received 13 November 2011

Accepted 10 December 2011

Available online 19 December 2011

0008-6223/$ - see front matter � 2011 Elsevidoi:10.1016/j.carbon.2011.12.027

* Corresponding author: Fax: +1 847 4913915.E-mail address: yingli@u.northwestern.ed

A B S T R A C T

Structural formation mechanisms of carbon nanotube (CNT) buckypaper and their effects

on its mechanical properties are studied with numerical simulations. A bond swap algo-

rithm, resulting from coupling the molecular dynamics and Monte Carlo methods, has

been developed to equilibrate initial structures of buckypaper, generated by a random walk

approach. Entanglement and bundling mechanisms are found to affect major structural

features of buckypaper. Both mechanisms are evaluated quantitatively by calculating the

entanglement network and pore size of buckypaper. Compared with (8,8)-(12,12) double-

walled CNT, the structure of (5,5) single-walled CNT buckypaper is mainly dominated by

entanglement, due to its smaller adhesion energy. We show that the pore size of modeled

buckypaper, containing both types of CNTs, can be tuned from 7 nm to 50 nm by increasing

the double-walled CNT content from 0 wt% to 100 wt%, due to the transformation from

entanglement-dominated to bundling-dominated structures. Such an observation agrees

exceptionally well with experimental results. Both entanglement and bundling mecha-

nisms are also found to play important roles in the mechanical properties of buckypaper.

The findings open a way to tailor both structural and mechanical properties of buckypaper,

such as Young’s modulus or Poisson’s ratio, by using different CNTs and their mixtures.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The exceptional mechanical, thermal, electrical and optical

properties of carbon nanotube (CNT) make it to be one of

the most promising nanomaterials [1]. In order to utilize the

exceptional physical properties of CNTs, 2D and 3D well-

ordered CNT networks have been theoretically investigated

[2–4]. These well-ordered CNT networks also show the excep-

tional mechanical and thermal properties as those of a single

CNT [5–9]. At the same time, sensors and loudspeakers with

high sensitivities have been proposed based on these CNT

networks [10,11]. However, in the synthesis process of CNT

networks, it is not easy to obtain well-ordered CNT networks

er Ltd. All rights reservedu (Y. Li).

[12–14], due to entanglements during aggregation. The van

der Waals (vdW) interactions among CNTs enable them to

aggregate and form into close-packed bundles [15,16] (bun-

dling behavior), which, in turn, continue forming entangled

networks [17] (entanglement behavior). Therefore, the CNTs

are usually randomly distributed in the fabricated CNT thin

films and arranged into a nonwoven fibrous structure, as or-

dinary paper made from wood pulp fiber (see Fig. 1). These

CNT networks or thin films are called ‘buckypaper’. The pro-

duction of buckypaper is one of the simplest and most effi-

cient ways for large scale processing of CNTs. First, the

synthesized CNTs are dispersed in aqueous solutions. Then,

these suspensions can be membrane filtered under positive

.

Fig. 1 – Scanning electron microscopy (SEM) images for (a) SWCNT buckypaper and (b) MWCNT buckypaper with

corresponding representative volume elements (RVEs). The RVEs are initially built as random walks and relaxed through the

molecular dynamics coupled with the Monte Carlo method, which is so called ‘bond swap algorithm’ (Section 2.2). Different

CNTs are colored by different (random) colors in the snapshots. Two behaviors can be seen from these buckypaper RVEs:

entanglement and bundling. The structure of (5,5) SWCNT buckypaper is well entangled, but, not well bundled. In contrast,

the bundle size of (8,8)-(12,12) DWCNT buckypaper is very large. The two SEM images are reproduced from [17] with

permission.

1794 C A R B O N 5 0 ( 2 0 1 2 ) 1 7 9 3 – 1 8 0 6

or negative pressure to yield uniform films, which are the

buckypapers [18].

Buckypaper has undergone several mechanical tests and

attempted applications. Hall et al. [17] have found that its

Poisson’s ratio could be tuned from positive to negative values

by increasing the content of multi-walled carbon nanotubes

(MWCNTs). CNT buckypaper can thus exhibit auxetic behav-

ior, a property required for the fabrication of artificial mus-

cles. Chen et al. were able to fabricate highly oriented,

auxetic CNT networks [19]. This negative Poisson’s ratio can

be even maintained for CNT/polymer composites, if highly

oriented CNT networks were embedded [19], or if – as we will

see – an initial locally unordered system becomes locally or-

dered. Pham et al. studied the mechanical and electrical prop-

erties of polycarbonate/CNT buckypaper composites [20].

They found that the stiffness and toughness of buckypaper

could be enhanced by adding polycarbonate, at the expense

of a decrease of its electrical conductivity [20]. Xu et al. have

fabricated a random network of long interconnected CNTs

with a temperature insensitive viscosity from �196 to

1000 �C [21]. Meng et al. [22] have utilized the buckypaper as

a template to produce CNT/polyaniline composites with high-

er specific capacitance, lower internal resistivity, and higher

stability under different current loads, which have promising

applications for energy storage devices. In a recent experi-

ment by Jiang and co-workers [23], very thin CNT films emit-

ted a loud sound, once fed by sound electric currents. It thus

became possible to make flexible, stretchable and transparent

loudspeakers from the CNT film with ultra-low heat capacity,

or specific heat per unit area. Based on field-effect transistors

made by CNT networks, Star et al. [24] detected the DNA

immobilization and hybridization without any labels. In

terms of the high porosity of buckypaper, Brady-Estevez

et al. [25] developed a single-walled carbon nanotube

(SWCNT) based network filter for the effective removal of

bacterial and viral pathogens from water at lower pressure.

Dumee et al. [26] fabricated self-supported CNT membranes,

which were held together only by vdW forces. These mem-

branes were highly hydrophobic (contact angle of 113�), highly

porous (90%) and can be used for direct contact membrane

distillation [26].

Although the molecular level interactions between adja-

cent CNTs have been extensively studied [27–30], the complex

structure of buckypaper makes it computationally expensive

to understand its properties starting from the atomistic level.

The large scale of CNT networks requires mesoscale model-

ing, which is beyond the capabilities of traditional atomistic

and continuum simulations. Buehler and co-workers [31,32]

have developed a ‘bead-spring’ model to describe the

mechanical properties of the ultra-long CNTs and their self-

assembling process. Within the ‘bead-spring’ model, the

CNT is simplified to be composed of a number of uniformly

distributed beads, which are connected by springs. The

stretching and bending stiffness of these springs are adjusted

to describe selected mechanical behavior of CNTs. Recently,

the same authors proposed an in silico assembly process for

CNT buckypaper based on their results from bead-spring

models [33]. The porosity of the assembled buckypaper could

be changed from 0.3 to 0.9 [33]. The Young’s modulus of these

buckypapers can also be tuned from 0.2 to 3.1 GPa, according

to different CNT diameter and spacing [33]. Zhigilei et al. [34]

developed another mesoscopic model for static and dynamic

Fig. 2 – Bead-spring model for CNT [31]. The continuum,

fully atomistic CNT is uniformly discretized into a multibead

chain, connected through springs. The stretching and

bending stiffness of these springs are parameterized to

reproduce mechanical properties of CNTs. The vdW

interaction between pairs of beads, directly coupled with the

interaction between two chains, is determined through the

adhesion energy between two CNTs.

C A R B O N 5 0 ( 2 0 1 2 ) 1 7 9 3 – 1 8 0 6 1795

simulations of nanomechanics of CNTs. The model is based

on a coarse-grained representation of CNTs as ‘breathing flex-

ible cylinders’ consisting of a variable number of segments

[34]. A mesoscopic tabulated interaction potential for CNTs

of arbitrary length and orientation had been developed by

Zhigilei and Volkov [35] describing the vdW interactions be-

tween CNTs or other graphitic structures. These simulations

suggest that the structural stability of CNT networks is con-

trolled by bending buckling, which could reduce the bending

energy of interconnections between CNT bundles and stabi-

lize the interconnecting by creating the energy barriers for

CNT sliding [36]. However, all these aforementioned works

do not give a detailed link between the microstructure of

buckypaper and its mechanical properties.

We address this issue and focus on the structural formation

mechanisms of buckypaper and how these mechanisms will

influence its mechanical properties. By using a multibead

chain model representing a single CNT, an initial structure of

buckypaper has been generated by applying the random walk

theory. By combing molecular dynamics (MD) and Monte Carlo

(MC) methods, the initial configuration quickly equilibrates to

reach the free energy minimum state and then serves as a rep-

resentative volume element (RVE) for a buckypaper. Subse-

quently, the mechanical and structural properties of the

buckypaper are analyzed, shedding light on how the micro-

structure of buckypaper influences its mechanical properties.

This paper is organized as follows: Section 2 introduces the

computational methodology used to build the RVE and analyze

its entanglement behavior, pore size and local orientational or-

der. Section 3 encompasses the pore size, entanglement, den-

sity, porosity and tube-packing of buckypaper with different

SWCNT lengths and double-walled carbon nanotube (DWCNT)

contents. The Young’s modulus and Poisson’s ratio of buckypa-

per are obtained through the uniaxial compression in Section

4. Results and the relationship between microstructure of

buckypaper and its mechanical properties will be discussed

and conclusions are drawn in Section 5.

2. Model and methods

2.1. Bead-spring model for CNT

As CNTs in buckypaper are long and exhibit a large range of

vdW interactions, it is impossible to do the full atomistic sim-

ulation. The complex geometry of the buckypaper also hin-

ders the continuum modeling. Therefore, we adopt the

‘bead-spring’ model for CNTs developed by Buehler and co-

workers [31–33]. As shown in Fig. 2, the continuum, full atom-

istic CNT is discretized into a multibead chain. Adjacent

beads are connected through springs. The stretching and

bending stiffness of these springs are trained to reproduce

mechanical properties of CNTs. The behaviors of a fully atom-

istic single CNT could be obtained through highly accurate

reactive force field (ReaxFF) calculations [37,38] and the po-

tential parameters for the ‘bead-spring’ model could be ob-

tained through energetic comparisons between the full

atomistic model and the ‘bead-spring’ model. The energy of

a multibead-spring system is given by:

Uall ¼ UbondðbÞ þ UangleðhÞ þ UvdWðrÞ ð1Þ

here, UbondðbÞ ¼ kbðb� b0Þ2=2 is the intramolecular stretching

energy of springs, kb being the stretching constant. b and b0

represent the bond length and equilibrium bond length,

respectively. UangleðhÞ ¼ kaðh� h0Þ2=2 denotes the bending

energy of a trimer, ka is the bending constant related to the an-

gle formed by three successive connected beads, h. h0 is the

equilibrium angle and h0 = 180�. UvdWðrÞ ¼ 4e½ r=rð Þ12 � r=rð Þ6�describes the vdW interactions between all non-bonded

beads, separated by distance r. e and r represent the energy

depth and equilibrium distance at vdW equilibrium, respec-

tively. The stretching, bending and vdW energies of the

‘multibead-spring’ model could be determined through uniax-

ial tension experiments, bending of a single CNT, and the

adhesion energy measurements between two CNTs from the

fully atomistic ReaxFF [37,38]. For example, kb = EA/b0 and

ka = 3EI/(2b0), where E is the Young’s modulus, A and I denote

the cross-section area and bending moment of inertia for

the CNT, respectively. Similarly, we have e = cb0 and

r ¼ deq=ffiffiffi26p

, with c and deq for the interfacial binding energy

and equilibrium spacing between two CNTs, respectively.

Details for deriving the potential parameters of the

‘multibead-spring’ model from full atomistic simulations are

available from [31,32]. The CNTs considered in the current

work are (5,5) SWCNT and (8,8)-(12,12) DWCNT. Potential

parameters are collected in Table 1.In this work, both the

SWCNT length and DWCNT content effects will be explored.

To this end, we fix the total number of beads in our simula-

tions. All results to be presented are obtained with 50,000

beads. Five (5,5) SWCNTs with different lengths, i.e. 50 nm,

100 nm, 200 nm, 500 nm, 1000 nm and 2000 nm, are used to

construct the RVEs of buckypaper to consider the tube length

effect. In terms of the DWCNT content effect, the tube

length is fixed to be 500 nm. Then, the buckypapers

with 7.55 wt%, 17.39 wt%, 30.77 wt%, 50 wt%, 72.73 wt%,

Fig. 3 – Illustration of the BSA for fast equilibration of

buckypaper RVE. The different colors represent different

CNTs. See the details in Section 2.2.

Table 1 – Summary of the potential parameters used for the ‘multibead-spring’ models of (5,5) SWCNT [31] and (8,8)-(12,12)DWCNT [31,32] (see Section 2.1 for details).

Parameter Units (5,5) SWCNT (8,8)-(12,12) DWCNT

Equilibrium bead distance, b0 A 10 10Stretching constant, kb kcal mol�1 A�2 1000 3760Equilibrium angle, h0 Degree 180 180Bending constant, ka kcal mol�1 rad�2 14,300 180,000vdW energy, e kcal mol�1 15.10 21.6vdW distance, r A 9.35 19.70

1796 C A R B O N 5 0 ( 2 0 1 2 ) 1 7 9 3 – 1 8 0 6

85.71 wt%, 94.12 wt% and 100 wt% DWCNTs will be studied as

well, by mixing with a corresponding content of SWCNTs. A

larger system for 400 (5,5) CNTs (the length of each CNT is

1 lm) is also considered for exploring possible finite system

size effect. We find that both the entanglement length and

pore size of buckypaper remain unchanged upon increasing

system size. All the dynamic simulations are done via LAM-

MPS [39] software with a time step of 10 fs. Visualizations have

been done by using the VMD visualization package [40].

2.2. Initial configurations

The starting point of the simulation is to generate the initial

equilibrated RVE of the buckypaper. The randomness of

buckypaper in the presence of excluded volume is a challenge

to modeling. Different approaches have proposed to generate

the RVE of buckypaper. Berhan et al. [41] have performed 2D

finite element analysis on buckypaper by using the beam ele-

ments to represent the CNTs. The inter-tube connections are

represented by torsion springs. Cranford and Buehler [33]

have developed an in silico deposition method to generate a

buckypaper. In the in silico assembly of the buckypaper, one

layer of CNTs, containing 10 individual nanotubes, is depos-

ited on a rigid substrate through applied body force [33]. After

the CNTs agglomeration equilibrated, the body force has been

removed. Another layer of CNTs will be deposited on the pre-

vious equilibrated one, with a rotation of 30� about an out-of-

plane central axis. Such a process is repeated for succeeding

layers, up to 12 layers in total [33]. However, it is not easy

for these methods to guarantee both the randomness and

an energetic minimum of the buckypaper RVE. To inherently

ensure randomness, isotropy and excluded volume, we adopt

a random walk model for the buckypaper generation. After

the initial structure of buckypaper is produced by random

walks of finite thickness, it will be further relaxed through

the MD coupled MC method, bond swap algorithm (BSA)

[42], to reach an energetic minimum. During this latter pro-

cess the system might become slightly anisotropic and break

orientational symmetry, but this is induced by the large bind-

ing energy of CNTs, which enable them to be bundled and

highly local organized. Our RVE does not seem to exhibit a

nematic phase, but it exhibits a vanishing local orientational

order, to be discussed in detail below.

In our random walk generating process, spatially uniformly

distributed points in the simulation cell are selected as the

positions of terminal beads for each CNT. The CNT growth

1 For interpretation of color in Figs. 3 and 4, the reader is referred t

takes place by generating series of random bond vectors, con-

necting adjacent beads. The position of a new bead is obtained

by adding a randomly oriented bond vector of fixed equilibrium

length (10 A) to the previously generated site. This makes the

initial structure of the buckypaper system stochastic. While

CNT growth takes place, an overlap check (distance r) is per-

formed returning the occupancy of the newly generated site.

If the newly generated position for a bead is already occupied

we go back one step and generate a new site position. If growth

gets stuck, a new position of a terminal bead is randomly as-

signed. A partial overlap (distance r) of the beads, and thus

incomplete bead volume, is allowed during the CNT growth

which is later removed by subjecting the buckypaper system

to a soft repulsive potential to ensure that all the beads are sep-

arated by a distance conforming the absence of overlap. In this

growth process, all the CNTs are also assumed to have the

same length to avoid the polydispersity effect.

After the initial random network has been obtained, the

BSA [42] is applied for fast relaxation of the buckypaper RVE

and enables it to reach a minimum free energy state at room

temperature (T = 300 K). As shown in Fig. 3, there are two

CNTs with different colors (blue1 and red). On the left, the

red and blue CNTs have two beads B and b close to each other,

which are currently bonded to beads C and c, respectively,

within their own CNTs. The BSA will attempt to delete the

B–C and b–c bonds and replace them with B–c and b–C bonds.

If the swap is energetically favorable according to the Metrop-

olis criterion, the two CNTs on the right are the result and

each CNT has undergone a dramatic conformational change.

In order to keep the CNT length to be fixed in this bond swap

process, all the beads in each CNT have a unique ID as shown

in Fig. 3 as ‘� � �–A–B–C–D–� � �’ or ‘� � �–a–b–c–d–� � �’. A bond swap

o the web version of this article.

C A R B O N 5 0 ( 2 0 1 2 ) 1 7 9 3 – 1 8 0 6 1797

only occurs between beads with same ID, i.e. B and b. Once

the bond swap is accepted, the underlying architecture of

these two CNTs will not be changed; as we still have the sim-

ilar structures of CNTs (see Fig. 3) as ‘� � �–A–B–c–d–� � �’ or ‘� � �–a–

b–C–D–� � �’. For the buckypaper which contains both SWCNTs

and DWCNTs, the bonds swap only between SWCNT and

SWCNT or DWCNT and DWCNT, i.e. they do not artificially

change the CNT structure. In short, both SWCNT and DWCNT

RVEs are initially generated by random walks and further re-

laxed through a combined MD/MC method, BSA. Such a

method can guarantee both the randomness and free energy

minimization of the buckypaper. After the buckypaper RVEs

have been fully relaxed under NVT ensemble (T = 300 K), all

these RVEs are further relaxed through NPT simulations at

300 K and 1 atm for 20 ns to reach an equilibrated state under

constant pressure. The total energy of the system is also mon-

itored in this relaxation process: the total energy rapidly

drops in the first 5 ns and approaches a constant value in

the last 10 ns. Therefore, we believe our RVEs have been fully

equilibrated and can be used for representing the real CNT

buckypaper structures.

2.3. Pore sizes

To extract a mean pore size and pore size distribution we fol-

low an approach based on the Euclidean distance map (EDM)

[43]. Such a map is constructed for a given configuration by

first discretizing the whole system into voxels. Each cubic

voxel has the same fixed size and carries the value 1 (1-phase)

if a bead is located within the voxel volume, otherwise it is

part of the 0-phase. The EDM defined on the 0-phase is the

smallest distance to a voxel within the 1-phase, thus defining

an EDM sphere for each voxel. Once the EDM map has been

obtained, we calculate for each voxel of the 0-phase the ra-

dius of the largest EDM sphere which is able to reach its loca-

tion. These radii are collected to calculate a distribution of

pore size radii, and the directly related mean pore radius.

The result becomes insensitive to voxel size if the voxel is

small compared to the mean distance between beads. For

the results reported below the voxel size is 3/10 of the mean

distance between beads.

2.4. Local orientational order

Using the same grid, we calculate for each voxel j an orienta-

tion tensor, Sj ¼ uu� I=3h i, where I denotes the unit tensor,

from all segment vectors {u}, normalized to unit length, that

pass through voxel j. With all orientation tensors at hand,

we define a local orientational (uniaxial) order parameter for

that voxel as a sum over eight neighboring voxels k,

S2j ¼ ð3=16Þ

P8k¼1Sj : Sk. This orientational order parameter de-

pends on voxel size, and reaches unity only in the case of a

dense, completely aligned, system. Irrespective of its precise

choice, it characterizes the degree of local orientational order-

ing of spatially close bond vectors, as it should be pronounced

in the presence of bundles, and vanish in the completely

unordered, initial state. Therefore, the larger CNT bundle size,

the larger local orientational order. As for the pore sizes, we

have access to a distribution of order parameters and the

whole system is characterized by the mean, S ¼ Sj

� �j.

2.5. Entanglement

Both the inter-tube entanglements and inter-tube vdW inter-

actions can greatly affect the load transfer efficiency among

the discontinuous CNTs [44]. The inter-tube vdW interactions

are mainly determined by the types/diameters of CNTs and

their inter-tube packing behaviors, which have been already

discussed by Cranford et al. [32]. However, the inter-tube

entanglements are not well understood, since it is hard to di-

rectly measure them through experimental techniques. Here,

we adopt a purely geometric algorithm, the Z1 code [45,46], to

extract the entanglement network and the number of inter-

tube entanglements present in our RVEs. In the Z1 code

[45,46], each multibead-spring CNT is mapped on a connected

path of infinitesimally thin, impenetrable and tensionless

straight lines. The total length of the multiple disconnected

lines for a system of CNTs is monotonically reduced, while

subjected to line-uncrossability (the so called ‘primitive path’

maintains the underlying entanglement structure), by intro-

ducing a smaller number of nodes. Within the Z1 code

[45,46], to obtain a physical path which carries information

about entanglement points, one removes non-physical infor-

mation from the mathematical version, by disregarding nodes

which do not change the direction of the path, and by disre-

garding (‘removing’) segments of vanishing length (it is an

algorithmic detail, that vanishing means small and finite

due to number precision). The remaining internal nodes are

called ‘interior kinks’; their number denoted as Z, which rep-

resents the number of entanglements in the system. Such a

method has been already successfully applied to polymeric

systems for obtaining their entanglement networks and

entanglement molecular weights [47,48]. More detailed infor-

mation about the Z1 code is available in [45,46].

2.6. Elastic property characterization

The uniaxial compression test has been performed to study the

effective Young’s modulus and Poisson’s ratio of buckypaper.

The deformation was simulated by controlling displacement

of the x dimension of the RVE and allowing the y and z dimen-

sions of the RVE to fluctuate according to the barostat [49], cor-

responding to the NLxryrzT ensemble. The compression rate

was about 108/s with time step 10 fs. The compression stress

of RVE in x dimension, rx, was calculated based on virial stress

formulation [50]. The compression strain was defined as

ex ¼ 1� Lx=Lx0, where Lx and Lx0 are the current and initial

RVE length in x dimension. Similarly, the lateral expansion

strain could be defined as ey ¼ Ly=Ly0 � 1 and ez ¼ Lz=Lz0 � 1,

where subscripts ‘y’ and ‘z’ denote the y and z directions,

respectively. From initial elastic deformation of buckypaper

(compression strain is smaller than 0.01), its Young’s modulus

and Poisson’s ratio were obtained by E ¼ �rx=ex, vy ¼ ey=ex and

vz ¼ ez=ex. Here, E, vy and vz are the Young’s modulus, Poisson’s

ratio in y and z directions, respectively.

3. Structure of buckypaper

Fig. 4 shows the snapshots of dynamically equilibrated bucky-

paper RVEs with different (8,8)-(12,12) DWCNT contents. All

Fig. 4 – Snapshots of RVEs for buckypaper models with different contents of (8,8)-(12,12) DWCNTs (a) 0 wt% (b) 17.39 wt% (c)

50 wt% (d) 85.71 wt% (e) 94.12 wt% and (f) 100 wt%. All the CNTs have the same length, 500 nm. The (5,5) SWCNTs and (8,8)-

(12,12) DWCNTs are colored by cyan and pink, respectively.

1798 C A R B O N 5 0 ( 2 0 1 2 ) 1 7 9 3 – 1 8 0 6

these CNTs are entangled and aggregated together. For the

buckypaper with pure (5,5) SWCNTs, the tubes are well entan-

gled (entanglement-dominated) and their bundle size is also

rather small (Fig. 1a). However, with increasing DWCNT con-

tent the pore size (further discussed below) of buckypaper

dramatically increases, from 7 nm to 50 nm. Such an observa-

tion on the pore size agrees reasonably well with the experi-

mental results for buckypaper [17] (Fig. 1). Interestingly, the

Table 2 – Summary of the density, porosity, mean pore size and mSWCNT and (8,8)-(12,12) DWCNT. The calculation methods for poand 2.4, respectively. The error bars for mean pore size and loc

CNT length(nm)

DWCNTcontent (wt%)

Densitya

(g/cm3)

50 0 0.229100 0 0.184200 0 0.154500 0 0.148

1000 0 0.1282000 0 0.112500 7.55 0.153500 17.40 0.164500 30.77 0.185500 50 0.216500 72.73 0.227500 85.71 0.214500 94.12 0.220500 100 0.191

a Experimental result for the density of buckypaper is around 0.05–0.4 g/cb Experimental result for the porosity of buckypaper is around 0.8–0.9 [56]c Wu et al. reported the pore size of SWCNT buckypaper (diameter 0.8–1.2d Muramatsu et al. reported the pore size of DWCNT buckypaper (outer d

experimental result on the pore size of SWCNT (diameter

0.8–1.2 nm with length 100–1000 nm) buckypaper is around

10 nm [51], which agrees reasonably well with the pore size

of (5,5) SWCNT (diameter 0.68 nm, length 1000 nm) buckypa-

per in the current work (pore size 8.07 ± 2.61 nm, Table 2).

Importantly, the bundle size of SWCNT buckypaper is much

smaller than that of DWCNT buckypaper counterpart (RVEs

in Fig. 1). In these thick bundles, the DWCNTs arrange them-

ean local orientational order of buckypaper containing (5,5)re size and orientational local order are given in Sections 2.3

al orientational order indicate their standard deviations.

Porosityb Poresize (nm)

Local orientationalorder

0.86 6.40 ± 2.38 0.273 ± 0.1580.92 6.97 ± 2.26 0.300 ± 0.1730.95 7.42 ± 2.44 0.298 ± 0.1710.96 7.55 ± 2.54 0.298 ± 0.1720.97 8.07 ± 2.61c 0.308 ± 0.1750.97 8.42 ± 2.57 0.302 ± 0.1760.96 7.74 ± 2.47 0.312 ± 0.1760.96 8.08 ± 2.62 0.318 ± 0.1810.95 8.31 ± 3.07 0.321 ± 0.1750.95 10.60 ± 5.62 0.325 ± 0.1770.94 14.83 ± 8.15 0.357 ± 0.1790.95 19.48 ± 9.00 0.406 ± 0.1860.95 25.08 ± 9.38 0.437 ± 0.1780.95 37.52 ± 12.8d 0.478 ± 0.175

m3 [17,36,55].

.

nm with length 100–1000 nm) was around 10 nm [51].

iameter 1.46–1.60 nm) was around 30–40 nm [53].

C A R B O N 5 0 ( 2 0 1 2 ) 1 7 9 3 – 1 8 0 6 1799

selves to increase the degree of close-packed hexagonal

ordering corresponding to the minimum of the potential

energy of the inter-tube interactions, which is also observed

in the experiments [15,16]. Moreover, such a bundling behav-

ior has been confirmed in DWCNT buckypaper (outer diame-

ter 1.43–1.60 nm, close to the outer diameter of (8,8)-(12,12)

DWCNT, 1.63 nm) by transmission electron microscopy

(TEM) (Fig. 1 in [52,53]). The bundle size of DWCNTs is esti-

mated to be around 10–30 nm [52,53], which is well in accor-

dance with our RVE on (8,8)-(12,12) buckypaper (bundle size

is around 10–25 nm as shown in Fig. 1). Also, the pore size

of the DWCNT buckypaper is measured by using N2 adsorp-

tion isotherms at 77 K and found to be around 30–40 nm

[53], which shows exceptional agreement with our estimation

(pore size 37.52 ± 12.8 nm, Table 2). The good agreement be-

tween experimental observations and our simulation results

on the pore size of buckypaper indicates the validities of mod-

el and methodology adopted in the current work. The adhe-

sion strength of CNTs has been studied by Buehler and co-

workers [31,32]. They found that the contact length of CNT

was linearly proportional to its bending stiffness [31,32]. As

shown in Table 1, the bending stiffness of (8,8)-(12,12) DWCNT

in this work is one order of magnitude above that of (5,5)

SWCNT. Therefore, it is not unexpected that the DWCNTs

are more likely to arrange themselves into thick bundles,

where they reduce both the inter-tube and bending energies.

Some experimental results on SWCNT buckypaper find that

there exists bundling behavior in certain SWCNTs [54]. How-

ever, the diameters of these SWCNTs are very large and can

reach 1.6–2.6 nm [54]. The SWCNTs insides these bundles

are still highly entangled, and not as well aligned as the

DWCNTs (Fig. 2 in [54]). Since the bending stiffness of SWCNT

EI – D3 [34], with D for its diameter, the bundling behavior can

also occur in the SWCNT buckypaper when the diameter D is

very large. This regime is beyond the focus of this current

study. As the DWCNT content is increased, the bundle size,

pore size and local orientational order increase due to the

bundling behavior (see Fig. 4 and Table 2). We should empha-

size again that, although the initial structure of DWCNT

buckypaper is randomly generated (Section 2.2), the final

RVE of for DWCNT buckypaper is eventually highly locally

ordered (Table 2 for mean local orientational order), due to

the bundling behavior. The local orientational order will be

0 20 40 600.00

0.04

0.08

0.12

0.16

Dis

tribu

tion

Pore Size (nm)

0wt% 17.39wt% 30.77wt% 50wt% 94.12wt% 100wt%

(a)

Fig. 5 – (a) Pore size distribution and (b) mean pore size of buckyp

bars indicate the standard deviation of pore size. All the CNTs,

calculation method for pore size is given in Section 2.3.

seen to correlate with the auxetic behaviors of DWCNT

buckypaper.

Buckypaper has widespread applications, e.g. as a filter

membrane to trap microparticles in air or fluid, due to its

large porosity and pore size. To allow for designing, it is

important to characterize how the porosity and pore size of

buckypaper could be controlled through adjusting the CNT

length or DWCNT content. Fig. 5 shows the pore size distribu-

tion and averaged pore size (Section 2.3) of the buckypaper

with different (8,8)-(12,12) DWCNT contents. Obviously, the

pore size of buckypaper is below 10 nm, when the DWCNT

content is lower than 50 wt%. For 100 wt% DWCNT content,

the pore size can be as large as 50 nm, which is five-fold of

that of SWCNT buckypaper. Such an observation is in good

agreement with the SEM images given by Hall et al. [17],

which is also shown in Fig. 1. Sears et al. [55] also measured

the pore size of buckypaper from particle (polystyrene) rejec-

tion tests. There are two kinds of CNTs used in their work,

one has 9 nm diameter size and the other has 37 nm diame-

ter. They found that the pore size of buckypaper also could

be tuned from 25 nm to 50 nm, by changing the content of

CNTs with smaller diameter. As we observe in the RVEs of

buckypaper shown in Fig. 4, there are two important mecha-

nisms affecting the underlying structure of buckypaper:

entanglement and bundling. Both of them could reduce the

system energy of buckypaper. However, for the (5,5) SWCNT,

which has much smaller bending stiffness than that of

(8,8)-(12,12) DWCNT (Table 1), the buckypaper is mainly stabi-

lized by the entanglement as the bundle size is much smaller.

For the DWCNT buckypaper, the large bundle size greatly re-

duces the system energy due to its strong vdW interaction

as listed in Table 1. Therefore, the bundling behavior mainly

dominates the structure of DWCNT buckypaper. From Fig. 4

and Table 2, we clearly see that the greater amount of

DWCNT, the larger bundle size, the more well-ordered struc-

ture and the bigger pore size. For the SWCNT buckypaper,

the averaged pore size is slightly changed from 6.4 nm to

8.4 nm, with the tube length increasing from 50 nm to

2000 nm (Table 2).

The Z1 code [45,46] (Section 2.5) has been applied on our

RVEs of buckypaper to extract the number of entanglements.

Fig. 6 shows the number of entanglements per CNT, hZi, for

the buckypaper studied in this paper. The hZi value for (5,5)

0 20 40 60 80 1000

20

40

60(b)

Pore

Siz

e (n

m)

DWCNT Content (wt%)

aper with different DWCNT contents at 300 K. In (b), the error

(5,5) or (8,8)-(12,12), have the same length, 500 nm. The

0 500 1000 1500 20000

5

10

15

20

25

30

<Z>

SWCNT Length (nm)

slope 0.0128

SWCNT Buckypaper

(a)

0 20 40 60 80 1005

6

7

8

9(b)

slope -0.0588

<Z>

DWCNT Content (wt%)

slope -0.0313

SWCNT+DWCNT Buckypaper

Fig. 6 – Entanglement numbers per CNT, hZi, for (a) (5,5) SWCNT buckypaper with different tube lengths (b) buckypaper with

different (8-8)-(12,12) DWCNT contents at 300 K. In the second case, all the CNTs have the same length, 500 nm.

1800 C A R B O N 5 0 ( 2 0 1 2 ) 1 7 9 3 – 1 8 0 6

SWCNT buckypaper is found to be linearly increasing with

increasing tube length. However, for the buckypaper with dif-

ferent (8,8)-(12,12) DWCNT contents, hZi initially decreases

with the increment of DWCNT content (first stage), and

reaches a plateau around 50–72.73 wt% (second stage), then,

linearly decreases again (third stage). There are two different

slopes for characterizing the change of hZi. At the first stage,

the slope is around �0.03 and the slope for the third stage is

�0.06. From the snapshots in Fig. 4 and the discussion afore-

mentioned, at the first stage for buckypaper with lower

DWCNT content (<50 wt%), the underlying structure is entan-

glement-dominated. However, due to the extremely high

bending stiffness, the buckypaper with high DWCNT content

(>72.73 wt%) is bundling-dominated, implying a dramatic loss

of entanglements, compared with the first stage. Between

them, the entanglement and bundling effects will balance

each other, with hZi unchanged. Based on the relationship be-

tween hZi and CNT contour length, we can conveniently ob-

tain the entanglement length, defined as the mean length

between two adjacent entanglements: 57 nm and 93 nm for

(5,5) SWCNT and (8,8)-(12,12) DWCNT, respectively. Hall et al.

[17] also roughly estimated the inter-junction lengths of

54.3 nm and 39.5 nm for their MWCNT and SWCNT sheets,

respectively. The entanglement length of MWCNT is almost

two times of that for SWCNT, which we have observed from

our RVEs. However, they use 1.0 nm diameter SWCNT and

12 nm diameter MWCNT (nine walls) in their experiments

[17]. Our entanglement lengths thus do not exactly match

these results. Based on the TEM image of DWCNT buckypaper

(outer diameter 1.43–1.60 nm, close to the outer diameter of

(8,8)-(12,12) DWCNT, 1.63 nm) (see Fig. 1c in [53]), we roughly

measured the tube length between CNT junctions and esti-

mated it to be around 122 ± 26 nm, which is in good agree-

ment with our simulation result (93 nm for (8,8)-(12,12)

DWCNT) , given by Z1 code [45,46].

The density and porosity of buckypaper are also calculated

based on our RVEs. The porosity, X, could be determined

through the volume of voids over the volume of buckypaper

[33],

X ¼ Vvoids

VRVE¼ 1� VCNT

VRVEð2Þ

where, VCNTand VRVE denotes the total volumes of the CNTs in-

side the RVE and volume of RVE, respectively. Then, all the

CNTs in the RVE are assumed to be solid cylinders with a diam-

eter that equals the sum of CNTwidth (dCNT, physical diameter

of CNT) and vdW spacing between adjacent tubes (dvdW, the

vdW spacing is taken to be 3.7 A [33]). The total volume of CNTs

is obtained by summation of their contour lengths, multiplied

by the cross-sectional area (pðdCNT þ dvdWÞ2=4). Fig. 7 shows the

density and porosity of buckypaper with different SWCNT

lengths and DWCNT contents. The density of SWCNT bucky-

paper dramatically decreases from 0.23 to 0.13 g/cm3, as the

tube length increases from 50 nm to 1000 nm. At the same

time, the porosity of SWCNT buckypaper increases from

0.86 to 0.97. However, the density for buckypaper is slightly

changed from 0.15 to 0.23 g/cm3, by increasing the DWCNT

content, without too much changing of porosity (almost

0.95). These obtained values conform very well with experi-

mental results as the density of buckypaper is around 0.05–

0.4 g/cm3 [17,36,55] and porosity is about 0.8–0.9 [56]. From

the above results and discussions, the pore size of buckypaper

can be greatly changed by increasing DWCNT content, due to

the competition between entanglement and bundling mecha-

nisms. The porosity of buckypaper is however mainly deter-

mined by CNT length. We notice that due to different

processing methods, the porosity and pore size of buckypaper

could also be changed by the surface tension effect [56],

which we do not consider in our current work.

To give a detail view on the structure of buckypaper, the in-

tra-tube gðrÞintra tube and inter-tube gðrÞinter tube pair distribu-

tions of both types of buckypaper are given in Fig. 8. Due to

the perfect structure of CNTs, the gðrÞintra tube of buckypaper

also shows a well-ordered behavior. All the peaks occur at

the 1 nm, 2 nm, 3 nm, . . ., 10 nm, which characterizes the or-

dered beads arrangement of each CNT, as the equilibrium

bond length of ‘bead-spring’ model is 1 nm shown in Table

1. Due to the high stretching and bending stiffness of DWCNT,

the peaks of its gðrÞintra tube are much higher than others. The

gðrÞinter tube of SWCNT buckypaper with different tube lengths

are quite similar to each other. At distance below 0.9 nm,

which is the equilibrium vdW distance between beads,

gðrÞinter tube ¼ 0 for all these buckypapers. The first peak of

gðrÞinter tube for SWCNT buckypaper occurs at 1.2 nm, then,

the gðrÞinter tube quickly decays to 1 after 3 nm. However, for

the DWCNT buckypaper, the first peak occurs at 2.2 nm, since

its vdW equilibrium distance is 1.97 nm. After that, the

0 500 1000 1500 20000.10

0.15

0.20

0.25

SWCNT Length (nm)

Den

sity

(g/

cm3 )

0.85

0.90

0.95

1.00

Porosity

(a)

0 20 40 60 80 1000.10

0.15

0.20

0.25

DWCNT Content (wt%)

Den

sity

(g/

cm3 )

0.85

0.90

0.95

1.00(b)

Porosity

Fig. 7 – Density and porosity of (a) (5,5) SWCNT buckypaper with different tube lengths and (b) buckypaper with different (8,8)-

(12,12) DWCNT contents at 300 K. In the second case, all the CNTs have the same length, 500 nm.

0 2 4 6 8 100

10

20

30

g(r)

Distance (nm)

50 nm 100 nm 200 nm 500 nm 1000 nm

(a) SWCNT length

0 2 4 6 8 100

10

20

30

40(b)

g(r)

Distance (nm)

0wt% 30.8wt% 50wt% 80.7wt% 100wt%

DWCNT content

Fig. 9 – Pair distribution function g(r) for (a) (5,5) SWCNT buckypaper with different tube lengths (b) buckypaper with different

(8,8)-(12,12) DWCNT contents at 300 K. In the second case, all the CNTs have the same contour length of 500 nm.

0 2 4 6 8 100

10

20

30

g(r)

intr

a_tu

be

Distance (nm)

(5,5) 50 nm (5,5) 100 nm (5,5) 200 nm (5,5) 500 nm (8,8)-(12,12)

500 nm

(a)

0 2 4 6 8 100

10

20

30

(b)

g(r)

inte

r_tu

be

Distance (nm)

(5,5) 50 nm (5,5) 100 nm (5,5) 200 nm (5,5) 500 nm (8,8)-(12,12)

500 nm

Fig. 8 – (a) Intra-tube gðrÞintra tube and (b) inter-tube gðrÞinter tube pair distributions of (5,5) SWCNT and (8,8)-(12,12) DWCNT

buckypaper at 300 K.

C A R B O N 5 0 ( 2 0 1 2 ) 1 7 9 3 – 1 8 0 6 1801

gðrÞinter tube is much higher than that of SWCNT buckypaper

(the area under the unweighted gðrÞinter tube curve of DWCNT

is also much larger than that of SWCNT), which indicates

the well-bundled behaviors of DWCNTs.

Fig. 9 shows the pair distribution function gðrÞ ¼gðrÞintra tubeþ gðrÞinter tube of SWCNT buckypaper and buckypaper

with different DWCNT contents. The SWCNT buckypaper with

different tube lengths exhibit a similar g(r). However, for bucky-

paper with different DWCNT contents, there are three different

behaviors. For DWCNT content below 50 wt%, the g(r) of bucky-

paper shows a behavior similar to that of SWCNT buckypaper,

since these systems belong to the entanglement-dominated

regime. For the buckypaper with 80.7 wt% DWCNT, there are

three peaks within the range between 0 nm and 2.5 nm. The

first peak at 1 nm characterizes the equilibrium bond length

between CNT beads. The third peak around 2 nm represents

the equilibrium distance between two beads bonded by two

successive bonds (see Fig. 8a) or the gðrÞinter tube of DWCNT

(Fig. 8b). Between them, the second peak denotes the equilib-

rium distance between SWCNT and DWCNT (see vdW param-

eters in Table 1), which also characterizes the mixing of these

two types of CNT. According to the obtained g(r) of SWCNT

buckypaper, DWCNT buckypaper is much more ordered than

SWCNT buckypaper (also see mean orientational order in Table

2), in accord with the formation of large CNT bundles shown in

Fig. 4f.

1802 C A R B O N 5 0 ( 2 0 1 2 ) 1 7 9 3 – 1 8 0 6

4. Elastic property of buckypaper

After the underlying structure of buckypaper is well under-

stood, it is important to characterize its mechanical proper-

ties, since it also has widely applications in the flexible/

stretchable electronics [23,57] and actuator devices [58,59].

Although Cranford and Buehler [33] have performed nanoin-

dentation simulations on our modeled buckypaper to calcu-

late its Young’s modulus, we are faced with two main

reasons for studying further its mechanical properties. First,

the buckypaper in the current work is generated by starting

from random walks, which is different with the in silico

assembled buckypaper proposed by Cranford and Buehler

[33]. Since the mechanical properties of buckypaper are

known to be affected by the method of synthesis [60], we can-

not just rely on the applicability of known results for our

modeled system. Second, the buckypaper in [33] was made

of CNTs of fixed length, 50 nm, and the maximum DWCNT

content explored in [33] was limited to 50 wt%. To explore

the effects of (i) CNT length, (ii) DWCNT content (up to

100 wt%), and (iii) the method of sample generation on the

mechanical behaviors of buckypaper, and most importantly,

to be able to (iv) correlate these behaviors unambiguously

with the underlying structural quantities like the entangle-

ment network, we need to extend the existing studies.

Fig. 10a shows the stress–strain curve for the SWCNT

buckypaper under compression. The compression stress is

initially linearly increasing with compression strain, indicat-

ing the elastic deformation of buckypaper. However, after

compression strain ex > 0:02, the compression stress in-

creases very slowly with the increment of strain, as densifica-

tion occurs in buckypaper under compression (see Movie 1,

Supplementary material). At the initial stage of compression,

the CNTs insides buckypaper are deformed through bending

and rotation. At the same time, the thick CNT bundles are

also formed due to the applied pressure. That is the reason

why the compressive stress increases rapidly during the ini-

tial stage. However, the bending rigidity of these thick bun-

dles, which is proportional to L4 (L is the diameter of the

bundle), can be greatly enhanced if compared with a single

CNT. Thus, when the buckypaper with locally bundled

microstructures is applied with further loading, the inter-tube

sliding will be initiated due to the weak vdW interaction (low

shear strength). Therefore, it is not hard to see why the

0.0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

Str

ess

(MP

a)

Engineering Strain

50 nm 100 nm 200 nm 500 nm 1000 nm

(a)

SWCNT length

(

Fig. 10 – Stress–strain curve of buckypaper under compression

buckypaper with different (8,8)-(12,12) DWCNT contents at 300 K

the same length, 500 nm. See Section 2.6 for details.

compressive stress increases very slowly, compared with the

initial stage. Such a behavior is also quite similar to the profile

of an open-cell foam structure [61] and CNT foam [62] under-

going compression. From the elastic deformation stage

(ex < 0:01), we calculate the Young’s modulus and Poisson’s ra-

tio of SWCNT buckypaper with different tube lengths (Table

3). For the SWCNT buckypaper, it is obviously shown that

the buckypaper with shorter length has larger Young’s modu-

lus. It is known, the Young’s modulus of foam structure is lin-

early dependent on its density [61], the Young’s modulus of

SWCNT buckypaper should have similar relationship with

its density. From Fig. 7a, the density of SWCNT buckypaper

monotonically decreases with the increment of its tube

length. Therefore, it is reasonably to see that the Young’s

modulus of SWCNT buckypaper is also monotonically re-

duced by increasing its tube length. The Poisson’s ratio of

SWCNT buckypaper is around 0.18–0.30, and thus in well

accordance with the experimental results [17]. After elastic

deformation, the buckypaper is much more compressed and

SWCNTs are undergoing the buckling behaviors [62] (Movie

1, Supplementary materials). Since the critical buckling of

an individual CNT with constant elastic modulus and diame-

ter is inversely proportion to its length, the SWCNT buckypa-

per with longer length should have smaller compression

stress after its elastic deformation. Such a phenomenon is

also observed in our compression simulations (see Fig. 10a).

Fig. 10b shows the compressive stress–strain curve of

buckypaper with different (8,8)-(12,12) DWCNT contents with

all the tube length 500 nm. It is interesting to see that under

the same strain, the compressive stress is enlarged by

increasing the DWCNT content from 0 to 50 wt%, and then re-

duced by further increment of DWCNT content. The corre-

sponding Young’s modulus of buckypaper with different

DWCNT contents is also given in Table 3. We clearly see that

the stiffness of buckypaper is enhanced by increasing its

DWCNT content, as long as it does not exceed 50 wt%. How-

ever, above 50 wt%, the stiffness of buckypaper weakens with

the increment of DWCNT content. From the aforementioned

discussions on the structure of buckypaper, both entangle-

ment and bundling processes tend to lower the system’s en-

ergy. For DWCNT content below 50 wt%, the buckypaper

structure is entanglement-dominated and the pore size is

small (see the snapshots in Fig. 4). When compressive loading

is applied on the buckypaper, the elastic deformation is

0.0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

b)

Str

ess

(MP

a)

Engineering Strain

0wt% 17.39wt% 50wt% 72.73wt% 100wt%

DWCNT content

(a) (5,5) SWCNT buckypaper with different tube lengths, (b)

and strain rate 108/s. In the second case, all the CNTs have

Table 3 – Summary of the elastic properties (Young’s modulus, Poisson’s ratio in y and z directions) of buckypaper obtained byour uniaxial compression simulations. See Section 2.6 for details.

CNT length(nm)

DWCNTcontent (wt%)

Young’s modulusa,E (MPa)

Poisson’sratiob, vy

Poisson’sratiob, vz

50 0 308 ± 7 0.25 ± 0.01 0.24 ± 0.01100 0 194 ± 3 0.24 ± 0.01 0.18 ± 0.01200 0 174 ± 4 0.22 ± 0.01 0.22 ± 0.03500 0 158 ± 4 0.28 ± 0.01 0.30 ± 0.01

1000 0 94 ± 2 0.26 ± 0.01 0.23 ± 0.032000 0 94 ± 3 0.25 ± 0.01 0.31 ± 0.01500 7.55 146 ± 4 0.29 ± 0.01 0.26 ± 0.02500 17.40 150 ± 6 0.21 ± 0.01 0.20 ± 0.02500 30.77 159 ± 6 0.23 ± 0.01 0.27 ± 0.01500 50 168 ± 7 0.27 ± 0.03 0.25 ± 0.01500 72.73 97 ± 3 0.32 ± 0.03 0.29 ± 0.02500 85.71 126 ± 6 �0.19 ± 0.05 0.27 ± 0.02500 94.12 75 ± 5 0.73 ± 0.06c �0.19 ± 0.02c

500 100 23 ± 4 �0.04 ± 0.01 0.63 ± 0.01a Experimental results for Young’s modulus of buckypaper are reported around 0.2–12.2 GPa [60]. Cranford and Buehler reported Young’s

modulus of buckypaper was around 220–500 MPa with porosity 0.858–0.902 [33].b Hall et al. [17] reported experimental results for Poisson’s ratios of SWCNT buckypaper were around 0.17–0.3.c Hall et al. [17] reported experimental results for in-plane and thickness direction Poisson’s ratios of MWCNT buckypaper as �0.20 and 0.75,

respectively.

C A R B O N 5 0 ( 2 0 1 2 ) 1 7 9 3 – 1 8 0 6 1803

mainly induced by the axial compression and bending of

CNTs (Movie 2, Supplementary material). As the stretching

and bending of stiffness of DWCNT are much higher than that

of SWCNT, the stiffness of the buckypaper can be strength-

ened by adding DWCNT to buckypaper. Yet, above 50 wt%,

the underlying structure of buckypaper is mainly affected

by the bundling behavior, with a pore size up to 50 nm, and

a corresponding increase in CNT bundle size (see the snap-

shots in Fig. 4). The sliding of CNTs inside the bundle occurs

in the elastic deformation of buckypaper under compression

(Movie 3, Supplementary material) [36]. Since the inter-tube

vdW interaction is very weak compared with stretching and

bending stiffness, the Young’s modulus of buckypaper is re-

duced by increasing the DWCNT content. Experimental re-

sults on the mechanical properties of buckypaper also find

that the optimal content of MWCNT is around 50–60 wt%

[17], which enables the largest Young’s modulus and tensile

strength of buckypaper. This observation is in reasonable

agreement with our simulation results. The Poisson’s ratio

of buckypaper with DWCNT content below 73 wt% is around

0.2–0.32, in good agreement with the experimental results

[17,19]. However, for DWCNT content above 73 wt%, the Pois-

son’s ratio of buckypaper can be tuned from positive values to

negative values down to �0.2, due to its local ordered behav-

iors (Table 2). Such a sign changing behavior of Poisson’s ratio

of buckypaper has also be found experimentally to be located

at around 77–80 wt% MWCNT content by Hall et al. [17]. The

measured Poisson’s ratio of buckypaper with highly content

MWCNT is �0.2 [17], exactly coinciding with the value we

have obtained in the current work. Coluci et al. [63] have

developed a theoretical model, based on a well-ordered 3D

structure of buckypaper, to understand such a uncommon

behavior. In their 3D model [63], the in-plane Poisson’s ratio

of buckypaper can be evaluated by m ¼ ð1� bÞ=ð3þ bÞ, as

b ¼ 3kB=kSB, kB and kSB represent the force constant of bending

and stretching of CNTs/CNT bundles. By varying b value

(changing the diameter/length of CNT or the size of CNT bun-

dle), the in-plane Poisson’s ratio of buckypaper can be tuned

from 0.33 to �0.2, which has been well captured by our RVEs

of buckypaper. As aforementioned, the SWCNT buckypaper

shows a more isotropic behavior than its DWCNT counter-

part. The Poisson’s ratio of the (5,5) CNT buckypaper is around

0.3 in accord with its isotropic cellular structure [61]. On the

other hand is auxetic behavior had been suggested to appear

in locally well-ordered structures [64]. For the DWCNT bucky-

paper we observe a negative Poisson ratio, well supporting

previous arguments, since its DWCNTs are well-bundled to-

gether and forming locally ordered structures.

In order to explore the possibility of strain rate effect on

the mechanical properties of buckypaper, we have performed

the similar compression tests on (5,5) SWCNT buckypaper

with tube length 500 nm under three different strain rates,

108/s, 107/s and 106/s. The Young’s modulus of the buckypaper

is slightly reduced from 165 MPa to 145 MPa, with the decre-

ment of strain rate. However, the Poisson’s ratio of buckypa-

per is still in the range of 0.27–0.34. The persistence length

is used to characterize the thermal energy effect on the self-

bending of CNTs at the molecular level. The persistence

length, lp ¼ EI=kBT, where EI is the bending stiffness, kB and

T denote the Boltzmann constant and temperature, respec-

tively. Thus, the persistence length for (5,5) SWCNT is about

16 lm at 300 K, which is much longer than the tube length

studied in the current work. Therefore, we do not see the tre-

mendous strain rate effect on the mechanical properties of

buckypaper. Xu et al. [60] have compared the experimental re-

sults on Young’s modulus of buckypaper. They found that the

Young’s modulus of buckypaper can vary from 0.2 to 12.2 GPa,

due to the different types of CNTs used and different types of

processing methods adopted. Cranford and Buehler also cal-

culated the Young’s modulus of buckypaper via nano indenta-

tion method [33]. Their simulation results on the Young’s

modulus of buckypaper are around 220–500 MPa with porosity

1804 C A R B O N 5 0 ( 2 0 1 2 ) 1 7 9 3 – 1 8 0 6

0.858–0.902 [33]. We can see that our simulations results on

buckypaper (Table 3) are both in well accordance with these

experimental and numerical results.

5. Conclusions

By using ‘multibead spring’ models for both (5,5) SWCNT and

(8,8)-(12,12) DWCNT, the RVEs for buckypaper with different

SWCNT lengths and DWCNT contents are obtained from ran-

domly oriented and non-overlapping walks. All these RVEs

had been fully relaxed at given temperature and pressure by

the BSA, a combined MD/MC approach. Such a method has

been proved to be able to efficiently generate RVEs of bucky-

paper. There are two important mechanisms found to affect

the underlying structure of buckypaper: entanglement and

bundling. Both mechanisms have been evaluated quantita-

tively in this manuscript by calculating the entanglement net-

work and the pore size. For the (5,5) SWCNT buckypaper, the

entanglement behavior dominates its structure since the in-

ter-tube vdW interaction and bending stiffness of SWCNT

are smaller than that of DWCNT. However, the structure of

(8,8)-(12,12) DWCNT buckypaper is mainly influenced by the

bundling behavior due to its strong adhesion energy and large

bending rigidity. Therefore, by increasing the DWCNT content

from 0 to 100 wt%, the pore size of buckypaper can be tuned

from 7 nm to 50 nm, following the transformation from

entanglement-dominated behavior to bundling-dominated

behavior. Such an observation is perfectly consistent with

experimental results on the pore size of SWCNT and DWCNT

buckypaper. However, the SWCNT length effect on the pore

size is found to be negligible. The inter-tube entanglements

of buckypaper are calculated by using the Z1 code [45,46].

The entanglement lengths for (5,5) SWCNT and (8,8)-(12,12)

DWCNT are 57 nm and 93 nm, respectively, which are close

to the experimental results The elastic properties of the mod-

eled buckypaper are obtained through uniaxial compression

tests. The stiffness of SWCNT buckypaper decreases with

the increment of tube length, as the density of buckypaper

is reduced. Due to the competition between entanglement

and bundling behaviors, 50 wt% is found to be the optimal va-

lue for the (8,8)-(12,12) DWCNT content, which enables bucky-

paper to have the largest Young’s modulus. This optimal

value, 50 wt%, is also well in accordance with the reported va-

lue given by Hall et al. [17] The Poisson’s ratio of the modeled

buckypaper can be tuned from �0.2 to 0.3 by decreasing the

DWCNT content, which can be explained by the ratio of

CNT/CNT bundle bending stiffness over its stretching stiff-

ness. We have studied the structural formation mechanisms,

the pore size, Young’s modulus and Poisson’s ratio of bucky-

paper. The findings suggest tuning buckypaper properties by

using different CNTs and their mixtures, could be exploited

in the design of artificial muscles, stress/strain sensors or

actuators and filtration/distillation membranes.

Acknowledgements

Y.L. would like to thank Northwestern High Performance

Computing Center for generous allocation of computer time

on QUEST cluster and financial support from Ryan Fellowship.

M.K. acknowledges support by SNSF grant IZ73Z0-128169.

Appendix A. Supplementary data

Supplementary data associated with this article can be found,

in the online version, at doi:10.1016/j.carbon.2011.12.027.

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