Helical or coiled nanostructures have been object of intense experimental and
theoretical studies due to their special electronic and mechanical properties.
Recently, it was experimentally reported that the dynamical response of
foamlike forest of coiled carbon nanotubes under mechanical impact exhibits a
nonlinear, non-Hertzian behavior, with no trace of plastic deformation. The
physical origin of this unusual behavior is not yet fully understood. In this
work, based on analytical models, we show that the entanglement among
neighboring coils in the superior part of the forest surface must be taken into
account for a full description of the strongly nonlinear behavior of the impact
response of a drop-ball onto a forest of coiled carbon nanotubes.Comment: 4 pages, 3 figure

A. F. Fonseca

C. Daraio

D. S. Galvão

K. T. Lau

L. D. Landau

V. R. Coluci

W. Wang

English

arXiv

Crossref

Entanglement and the Nonlinear Elastic Behavior of Forests of Coiled Carbon Nanotubes

A model for the elastic response of a foamlike forest of coiled carbon nanotubes under mechanical impact is proposed according to recent experiments reported in the literature. A force vs. displacement relation is derived for different geometries of the impacting object (sphere, cube and circular cone). In the model, entanglement among neighboring coils in the superior part of the forest is explicitly taken into account and we show that it allows the full description of the strongly nonlinear elastic behavior of the forest

Coluci, Vitor R.

Fonseca, Alexandre F.

Galvao, Douglas S.

Daraio, Chiara

Caltech Authors - Main

FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCAPES - COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIORCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOHelical or coiled nanostructures have been objects of intense experimental and theoretical studies due to their special electronic and mechanical properties. Recently, it was experimentally reported that the dynamical response of a foamlike forest of coiled carbon nanotubes under mechanical impact exhibits a nonlinear, non-Hertzian behavior, with no trace of plastic deformation. The physical origin of this unusual behavior is not yet fully understood. In this Letter, based on analytical models, we show that the entanglement among neighboring coils in the superior part of the forest surface must be taken into account for a full description of the strongly nonlinear behavior of the impact response of a drop ball onto a forest of coiled carbon nanotubes.Helical or coiled nanostructures have been objects of intense experimental and theoretical studies due to their special electronic and mechanical properties. Recently, it was experimentally reported that the dynamical response of a foamlike forest of coiled carbon nanotubes under mechanical impact exhibits a nonlinear, non-Hertzian behavior, with no trace of plastic deformation. The physical origin of this unusual behavior is not yet fully understood. In this Letter, based on analytical models, we show that the entanglement among neighboring coils in the superior part of the forest surface must be taken into account for a full description of the strongly nonlinear behavior of the impact response of a drop ball onto a forest of coiled carbon nanotubes.100814FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCAPES - COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIORCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCAPES - COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIORCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOSem informaçãoSem informaçãoSem informaçãoWe acknowledge the financial support from IMMP/MCT, IN/MCT, BNN/CNPq, THEO-NANO, and the Brazilian agencies FAPESP, CAPES, and CNPq. A. F. F. acknowledges additional support from the CNPq

Coluci, V. R.

Fonseca, A. F.

Galvão, D. S.

Daraio, C.

Repositorio da Producao Cientifica e Intelectual da Unicamp

Physical Review Letters

Entanglement and the nonlinear elastic behavior of forests of coiled carbon nanotubes

AbstractA model for the elastic response of a foamlike forest of coiled carbon nanotubes under mechanical impact is proposed according to recent experiments reported in the literature. A force vs. displacement relation is derived for different geometries of the impacting object (sphere, cube and circular cone). In the model, entanglement among neighboring coils in the superior part of the forest is explicitly taken into account and we show that it allows the full description of the strongly nonlinear elastic behavior of the forest.</jats:p

Vitor R. Coluci

Alexandre F. Fonseca

Douglas S. Galvao

Chiara Daraio

MRS Proceedings

Helical or coiled nanostructures have been objects of intense experimental and theoretical studies due to their special electronic and mechanical properties. Recently, it was experimentally reported that the dynamical response of a foamlike forest of coiled carbon nanotubes under mechanical impact exhibits a nonlinear, non-Hertzian behavior, with no trace of plastic deformation. The physical origin of this unusual behavior is not yet fully understood. In this Letter, based on analytical models, we show that the entanglement among neighboring coils in the superior part of the forest surface must be taken into account for a full description of the strongly nonlinear behavior of the impact response of a drop ball onto a forest of coiled carbon nanotubes

Entanglement and the Nonlinear Elastic Behavior of Forests of Coiled Carbon Nanotubes
V. R. Coluci,1,* A. F. Fonseca,2 D. S. Galva˜o,1 and C. Daraio3
1Instituto de Fı´sica ‘‘Gleb Wataghin,’’ Universidade Estadual de Campinas, C.P. 6165, 13083-970 Campinas SP, Brazil
2Alan G. MacDiarmid NanoTech Institute, University of Texas, Richardson, Texas 75083-0688, USA
3Aeronautics and Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
(Received 28 August 2007; published 29 February 2008)
Helical or coiled nanostructures have been objects of intense experimental and theoretical studies due to
their special electronic and mechanical properties. Recently, it was experimentally reported that the
dynamical response of a foamlike forest of coiled carbon nanotubes under mechanical impact exhibits a
nonlinear, non-Hertzian behavior, with no trace of plastic deformation. The physical origin of this unusual
behavior is not yet fully understood. In this Letter, based on analytical models, we show that the
entanglement among neighboring coils in the superior part of the forest surface must be taken into account
for a full description of the strongly nonlinear behavior of the impact response of a drop ball onto a forest
of coiled carbon nanotubes.
DOI: 10.1103/PhysRevLett.100.086807 PACS numbers: 81.05.Uw, 81.05.Zx, 81.07.De
The study of nanostructures in special carbon nanotubes
(CNTs) and nanowires has been the object of intense
experimental and theoretical investigations due to their
large range of possible applications and new physical
phenomena [1,2]. Among these nanostructures, helical
and coiled formations have a special place due to their
differentiated mechanical behavior [3]. Coiled carbon
nanotubes (CCNTs) were first predicted to exist in the
early 1990s by Dunlap [4] and Ihara et al. [5–7] and
experimentally observed in 1994 by Zhang et al. [8].
CCNTs have been receiving increasing interest because
of their additional capability to serve as nanoscale me-
chanical springs [9] and electrical inductors [10], and for
their potential applications in composites [11].
Recently, the dynamical response of a foamlike forest of
CCNTs [Fig. 1(a)] under impact of a drop ball has been
reported [12]. The experiment consisted of producing ar-
rays of bundles of CCNTs [13,14], letting a stainless steel
bead fall down on the forest of CCNTs, and measuring the
dynamic force at the wall below the forest during the stages
of penetration and restitution. The analysis of the forest’s
morphology after impact has shown no trace of plastic
deformation and a full recovery of the foamlike layer of
CCNTs under various impact velocities.
The contact force exhibits a strongly nonlinear depen-
dence on displacement and appears fundamentally differ-
ent from the response of a forest of CNTs [15–18]. The
obtained results in [12] have been compared to the Hertz
elastic model [19] of a solid sphere and a planar surface in
contact where, if F is the force of contact and  is the
displacement, F 1:5 [19]. The results reported in
Ref. [12] showed a nonlinear response of the CCNT forests
having a force-displacement relation of F 2:2, different
from the Hertzian case. It has been hypothesized that
sideways interactions of the compressed CCNTs can be
associated with the strong nonlinear behavior in CCNT
forests, but the physical mechanisms behind it are still
unclear.
In this Letter, we propose a model for the impact re-
sponse of a forest of CCNTs. The model takes into ac-
FIG. 1 (color online). (a) Scanning electronic microscopy pic-
ture of a CCNT forest [12]. (b) Schematic diagram of a CCNT
forest deformed by a drop ball. H is the forest thickness, R is the
ball radius, and  is the total depth displacement of the ball into
the forest.  is the axial deformation of the spring at a distance r
from the center of the contact between the ball and the forest.
(c) Parameters of a helical spring: a is the coil radius, P is the
spring pitch, and D is the spring diameter. (d) Schematic repre-
sentation of the entanglement between two adjacent coil bundles
forming inactive, or ‘‘solid,’’ and active turns.
PRL 100, 086807 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending29 FEBRUARY 2008
0031-9007=08=100(8)=086807(4) 086807-1 © 2008 The American Physical Society
count: (i) the individual elastic contribution from each
CCNT in contact with the drop ball; (ii) the geometry of
the surface of contact between the forest and the drop ball;
and (iii) sideways interactions of the compressed CCNTs
through an entanglement process. As discussed below, we
show that this model can describe the strongly nonlinear
behavior observed in the recent experiments with CCNT
forests [12].
In the model, the CCNT forest of thickness H is con-
sidered as a set of identical springs that individually inter-
act with the drop ball and that may laterally interact with
other springs. The drop ball is considered to have a rigid
spherical surface of radius R. Figure 1(b) illustrates the
penetration of the ball into the CCNT forest and the
geometric features of the model. The total depth displace-
ment of the ball into the CCNTs forest is . Each bundle of
CCNTs is modeled as a helical spring of pitch P with a
Hooke’s constant k  GD4=64a3N [20], where G is the
shear modulus, N is the number of active turns, a is the coil
radius, and D is the bundle diameter [Fig. 1(c)]. Typical
values from experiment are H  100 m, R  1000 m,
2a  0:45 m, D  0:1 m,   3 m, and P 
0:9 m [12,14]. We also considered the situation where
the bundles of CCNTs are very close to each other. From
Ref. [12], the total density of CCNTs in the forest is
100=m2. Considering an ordered square grid and ac-
counting for each bundle to be composed by 25 tubes
[12], this roughly gives an estimative of 0:56 m as the
separation between two coiled bundles axis (and a 0:1 m
outer distance between them). Therefore, it is reasonable to
assume the existence of a certain degree of entanglement
between some parts of the CCNTs throughout the forest
which would contribute to sideway interactions. Further-
more, it is probable that changes in the entanglement occur
during the dynamic contact between the ball and the forest
surface. This mechanism is illustrated in Fig. 1(d), where it
is shown that the impact of the ball on the forest top surface
would cause a bending of the tips of each CCNT, leading to
contacts among the superior turns of the adjacent coils. The
entanglement mechanism can be very complicated and a
long-range process. In the present model, the effect of this
mechanism in the forest response is translated into the
reduction of the number of active turns of each CCNT, i.e.,
 N  NT  NS; (1)
where NT is the total number of turns of each CCNT (NT 
H=P  110), and NS is the number of turns that become
inactive due to the entanglement process [Fig. 1(d)]. The
inactive turns form a ‘‘solid’’ phase, deriving from a com-
plex entangled network of tubes, which is assumed not to
contribute to the elastic response of the CCNT. This ap-
proach has been used by Rodrigues et al. [21] to obtain a
nonlinear relation between the force and displacement of a
conic spring. Based on the small displacement (3 m) of
the top forest surface compared to the forest thickness
(100 m), we also assume a short-range effect of the
entanglement, i.e., only nearest neighboring coils interact.
Initial entanglement prior to the impact can be incorpo-
rated in the initial number of active turns. The short-range
interaction is included into the model by making the rate of
increase of the number of inactive turns during the contact
between the drop ball and the forest surface proportional to
the ball velocity:
 
dNS
dt
 v; (2)
where  is a measure of how many turns become inactive
per unit length of the displacement. Despite the complexity
of the interaction that forms the entanglement,  is con-
sidered here constant and determined from experimental
data a posteriori.
Since vdt  dz, where z is the direction perpendicular
to the forest surface (that is the direction of the movement
of the ball), Eq. (2) can be easily integrated to give the
following expression for the number of active turns:
 N  NT  ; (3)
where  is the axial deformation of each CCNT. Fig-
ure 1(b) shows how to determine the value of  as a func-
tion of  and the distance r from the center of the contact
between the ball and the CCNTs forest (hereafter called
‘‘center of contact’’ for short). From geometrical consid-
erations, we have
 r 

R2  r2
p
 R : (4)
The total force F consists of a summation of the forces
of each spring that interacts with the drop ball:
 F  kX
M
i0
ni
ri
1 =NTri ; (5)
where i represents the set of ni springs that are at the same
distance ri from the center of contact, and k is the effective
spring constant of a single bundle of CCNTs with all NT
turns active. At the edge of the contact area, M is such that
rM  0, i.e.,
 M 

2R 2
p
2a
: (6)
We assume a circular symmetry of the projection of the
surface of contact between the ball and the CCNTs forest,
on the plane of the forest. Therefore, ni is given by
 n0  1 and ni  2ri2a  2i; i  1; 2; 3; . . . ;M;
(7)
where we used that ri  2ai. The index i  0 is for the first
coil hit by the ball. Equation (7) is, of course, an approxi-
mation for the situation where the concentration of the
CCNTs is such that they are beside each other. Equa-
PRL 100, 086807 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending29 FEBRUARY 2008
086807-2
tion (7) will be a good estimative of the number of coils at
distance ri of the center of contact if the penetration of the
ball is enough to ensure that many coils are hit by the drop
ball. That is the case considered in the experiment, where a
total circular contact area has a radius of 77 m and the
CCNT bundle radius is 0:2 m [12]. For smaller depths,
Eq. (7) should be corrected for a better result.
Since =R  1, it is reasonable to use the following ap-
proximations: r   r2=2R and M  2Rp =2a.
Thus, Eq. (5) can be rewritten as
 F  k
1 =NT 2k
XM
i1
iR 2i2a2
	1 =NT
R 2i2a2
:
(8)
A simpler expression for F cannot be derived from Eq. (8).
However, by neglecting the term 2i2a2 from the denomi-
nator, the following expression can be obtained,
 F  k R
2
4a2	1 =NT

; (9)
where terms proportional to =R were also neglected.
The result of these approximations to the final behavior
of F is shown in Fig. 2 using typical experimental parame-
ters. We can see that this approximation is physically sound
for  & 10=m. We will see that this condition is satisfied
in the experiments [12].
Using Eq. (9) to fit the experimental data [12], we obtain
k  1:988 N=m and   7:066=m. The comparison
with experiment is presented in Fig. 3.
The importance of the entanglement for the nonlinear
behavior of the forest of CCNTs can be tested by recalcu-
lating the total force with   0 in Eq. (3), yielding to
 F  kR
2
4a2
: (10)
Equation (10) shows that the force F is proportional to 2.
Even being close to the relation found in [12], this result
still does not capture the full nonlinear behavior of the
impact response of a forest of CCNTs. However, when the
entanglement is turned on again, our model, with the fitted
parameters k and , recovers the previous experimental
fitting, F  Am, A  0:031, and m  2:2 [12], which
shows that the entanglement formed by lateral deforma-
tions of the CCNTs is necessary to explain the full strongly
nonlinear behavior of the impact response of a drop ball
onto a forest of CCNTs.
In a loading experiment, Cheng et al. have measured the
spring constant value of a single amorphous carbon nano-
coil in a low-strain regime (nanocoil elongation &3 m)
as being 0:12 N=m [9]. The number of turns of that nano-
coil is about 10 which leads to 1:2 N=m for the value of
the spring constant of a single turn. Using the derived k
value of the present model from experimental data [12], we
can estimate the spring constant of a single turn of an
individual CCNT. According to Ref. [12], the bundle is
formed by 25 nanocoils. Assuming that the bundle is
formed by a parallel association of CCNTs, our estimative
for the spring constant of a single CCNT is ks  k=25 
0:08 N=m. The spring constant of a single turn of the
CCNT in a compression experiment is, then, ksNT 
8:7 N=m, which is the same order of magnitude of the
value for a single amorphous carbon nanocoil measured in
a tension experiment.
As previously mentioned,  is the number of turns that
become inactive per unit length of displacement. Suppos-
ing that the entanglement is formed by inactive turns that
0 1 2 3
δ (µm)
0
0.05
0.1
0.15
F 
(N
)
0 1 2 3
δ (µm)
0
0.05
0.1
0.15
F 
(N
)
0 1 2 3
δ (µm)
0
0.05
0.1
0.15
F 
(N
)
(a) (b) (c)
FIG. 2. Force-displacement curves obtained from Eq. (8) when
the term 2i2a2 is present (solid lines, no approximation) or not
(dashed lines) for (a)   1=m, (b)   5=m, and
(c)   10=m. The following parameters were used: a 
0:225 m, R  1000 m, NT  110, and k  1 N=m.
0 1 2 3
δ (µm)
0
0.1
0.2
0.3
0.4
F 
(N
) 
Experiment
Model
0 20 40 60 80
r (µm)
0
1
2
t (µ
m
)
FIG. 3 (color online). Behavior of the force as function of the
displacement of the drop ball during contact with the CCNT
forest. The model curve was obtained using Eq. (9) with k 
1:988 N=m and   7:066=m. The inset graph shows the
behavior of the thickness of the entanglement as a function of
distance from the center of impact (r  0).
PRL 100, 086807 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending29 FEBRUARY 2008
086807-3
form a ‘‘solid’’ phase in the top of the CCNT forest, we can
estimate the thickness, t, of the entanglement at a given
distance r from the center of impact as
 tr  Dr; (11)
where r is given by Eq. (4). The inset graph of Fig. 3
displays t as a function of r. For the center of impact, the
thickness is 2:1 m, and the number of inactive turns of
the coil at the center of impact is 21, about 20% of the
total number of turns of the total forest thickness. These
results are compatible with what we expect for an entan-
glement formed by the superior parts of the CCNTs due to
the impact of a drop ball on the forest.
If instead of a ball we have an approximated perfect cube
or parallelepiped with a finite contact area A with the
forest, falling down on the forest of CCNTs, its impact
response can be estimated using Eq. (5). In this case, all
nanocoils feel the same axial deformation , and the sum in
Eq. (5) can easily be performed to give F nc=	1
=NT
, where nc is the number of nanocoils in contact
with the face of the cube or parallelepiped, and is simply
given by nc  A=a2. The response force of the forest,
then, will be approximately linear with , for small 
values, and start growing nonlinearly for larger  ones,
as a consequence of the entanglement.
It should be stressed the limit of validity of our model
[represented by Eq. (2)] takes into account the experimen-
tal condition [12] of small forest deformations. The con-
tribution of the entangled part of the forest to the elastic
response is neglected because any deformation of the en-
tanglement is expected to be significantly smaller than the
forest deformations. One of the predictions of the model
that can be experimentally tested is the thickness value of
the entanglement of the top forest surface. We hope the
present Letter will stimulate further experiments to test the
validity of the present model.
In conclusion, we have derived an analytical model for
the nonlinear behavior of the impact response of a forest of
CCNTs including geometrical and physical aspects during
the forest compression. We showed that the nonlinear
behavior is fully described when the entanglement of the
coiled carbon nanotubes in the superior part of the forest
surface is incorporated into the model. This entanglement
among neighbors is due to the bending of the coil tips
produced by the ball impact. Under the experimental con-
ditions of small deformations [12], the model predicts an
entanglement thickness of 2 m at the maximum forest
compression. The model results point out to the importance
of the coil entanglements for the elastic behavior of such
systems. The present model is able to provide, by matching
experimental values, estimates of the spring constant of a
single CCNT and the level of entanglement between
CCNTs. These aspects can play an essential role in the
future design of micro-electro-mechanical systems de-
vices, new shock protecting layers, and composites for
microelectronic packaging and vibration mitigating mate-
rials where CCNT structural entanglement could be
present.
We acknowledge the financial support from IMMP/
MCT, IN/MCT, BNN/CNPq, THEO-NANO, and the
Brazilian agencies FAPESP, CAPES, and CNPq. A. F. F.
acknowledges additional support from the CNPq.
*Author to whom correspondence should be addressed.
FAX: +55-19-35215376.
coluci@ifi.unicamp.br
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Entanglement and the nonlinear elastic behavior of forests of coiled
  carbon nanotubes

Entanglement and the nonlinear elastic behavior of forests of coiled carbon nanotubes

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