Eur. Phys. J. E 13, 73–77 (2004)
DOI: 10.1140/epje/e2004-00041-4

THE EUROPEAN
PHYSICAL JOURNAL E

Deformation of adhering elastic tubes
S. Komuraa, K. Tamura, and T. Kato
Department of Chemistry, Faculty of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japan
Received 11 September 2003
c EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2004
Published online 2 March 2004 – 
Abstract. Deformation of an elastic tube adhering onto a substrate due to van der Waals attractive force
is investigated by means of numerical minimization and scaling theory. The onset of the deformation isdetermined by the critical value of Cb /(N 2 ), where Cb is the bending constant,  is the depth of the vander Waals potential, and N is the size of the tube. For a signiﬁcantly deformed tube, we found a scaling
behavior of the bending energy, which is explained within the shell theory.
PACS. 68.35.-p Solid surfaces and solid-solid interfaces: Structure and energetics – 68.55.-a Thin ﬁlm
structure and morphology – 81.07.De Nanotubes

1 Introduction
Carbon nanotubes [1] have attracted great interests due
not only to their peculiar structure, but also to the electrical, chemical, and mechanical properties associated withthese structures. Examples of the possible applications are
such as nanowires or electronic devices. In practice, theelectric transport through nanotubes is studied after theirdeposition on a substrate with which they interact eachother. It is known, however, that the resistivity is aﬀectedby their elastic deformations. Since there is little control
over the alignment and the shape of adsorbed nanotubes,
it is crucial to know how they deform on the substrate.
There have been several works which report on the
deformations of nanotubes due to van der Waals (vdW)
interaction. For example, vdW interaction between adjacent nanotubes can deform them substantially, showing the ﬂattening along the contact region [2]. Multiwalled nanotubes can even fully collapse along theirlength [3]. Later, the deformations of multiwalled nanotubes on a substrate was investigated using both atomic
force microscopy (AFM) [4] and molecular-mechanics simulations [5]. They also showed that the vdW interaction
results in the ﬂattening of the tube. More recently, the collapse of a nanotube section due to the surface interaction
was observed by using AFM [6].
It is widely known that nanotubes have unique elastic properties such as exceptionally high Young’s modulus [7,8]. Bending and buckling of nanotubes [9,10] as wellas their axial [11,12] or radial [13] deformations have been
experimentally investigated. In general, deformations of
nanotube are intimately related to the problem of elastic
shell theory [14]. The stretching deformation which accompanies the bending deformation is a ﬁrst-order eﬀect for aae-mail: komura@comp.metro-u.ac.jp

shell, whereas it is only a second-order eﬀect for a plate.
As a result, the shape deformation of a shell is eﬀectively
suppressed compared to that of a plate. Such a coupling
between the bending and stretching modes was explicitly
calculated for a cylindrical polymerized vesicles [15].
In this paper, we investigate the deformation of an elastic nanotube adhering onto a rigid substrate due to vdW
attractive interaction both numerically and theoretically.
In our model, the equilibrium conﬁguration of the tube
is determined according to the competition between thestretching, bending, and vdW energies. The total energy
is numerically minimized by using the conjugate gradientmethod. We have performed the scaling analysis of theminimized energy, and examined how the tube deformation depends on its size and bending constant. The main
result of the present work is to show that the combinationof the bending constant, the tube size, and the depth ofthe vdW potential gives a scaling variable both for the onset of the deformation and the total energy. We found two
distinct scaling regimes, i.e., “bending regime” and “adhesion regime” which are connected by a crossover region.
For a largely deformed tube, the scaling behavior of thebending energy is explained in terms of the shell theory.

2 ModelConsider a cross section of an elastic tube interacting witha rigid substrate as shown in Figure 1. We assume that theaxial deformation is uniform along the tube. The elastic
tube is modeled by a circular network of N beads connected by harmonic springs. The parallel and the normal
directions to the substrate are taken as x- and z-axes, respectively. For the deformations of elastic shells, both thestretching and the bending energies should be taken into

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pairwise additivity of the vdW interaction between two
atoms [19]. In our model, we have not included the vdW
interaction between the diﬀerent pairs of beads.
The total energy
Etot = Es + Eb + W

Fig. 1. Beads and springs model for an elastic tube adhering
onto a substrate. The tube is assumed to deform uniformly
along the axial direction. Li is the length of the i-th springwhose natural length is L0 . n̂i is the unit normal vector of thei-th spring. Each bead interacts with the substrate through thevan der Waals potential energy equation (3).

account [14]. Following the model by Kramer and Witten
to simulate crushed elastic manifolds [16], the discretized
stretching energy is given by the sum over Hooke’s law ofeach spring:
Es =

1
i

2


Cs

Li − L0
L0

2
.

(1)

Here, Cs is the spring constant, Li is the length of the i-th
spring, and L0 is the natural length of the spring taken
here as a constant. The discretized bending energy, on theother hand, is calculated according to [16]
Eb =

1
ij

2

Cb |n̂i − n̂j |2 ,(2)

where Cb is the bending constant, n̂i is the unit normal
vector of the i-th spring (see Fig. 1), and the sum is takenover each pair of springs which share a common bead.
A spontaneous curvature term is omitted here, which isvalid as long as we consider carbon nanotubes composed ofgraphite sheets. We remark that the same bending energy
was used before in the Monte Carlo simulations of twodimensional vesicles [17].
The adhesion energy of the tube is taken into accountthrough the vdW interaction between each of the beadand the substrate: 28/3  σ 12  σ 3

−
,
(3)
W =
3
zi
zi
i
where zi is the height of the i-th bead from the substrate.
When the adhesion energy per bead is plotted against zi , gives the depth of the energy minimum, and the distancecorresponding to this minimum is 22/9 σ. The ﬁrst repulsive term in equation (3) is responsible for the excluded
volume interaction which accounts for the fact that thebeads cannot penetrate into the substrate. The second
term represents the long-ranged attractive interaction between the beads and the substrate [18]. Notice that theinverse cubic dependence of this term results from the(4)is numerically minimized using the conjugate gradientmethod [20]. As regards the initial condition of the calculation, each bead is located on a circle and each spring
length is set to L0 , for which the stretching energy equation (1) is fully relaxed. Since there is no spontaneous
curvature, even the undeformed tube costs certain curvature energy. In the present study, we have mainly changed
the size N and the bending constant Cb .

3 Results and discussions
In our numerical calculation, all the energies and thelengths are scaled by  and σ, respectively. These are themodel parameters which appear in the vdW interaction(see Eq. (3)). Then, in addition to the number of beads
N , there are three independent dimensionless parametersin the model; Cs /, Cb /, and L0 /σ.
We ﬁrst show in Figure 2 the sequence of the equilibrated tube conﬁgurations as the bending constant ischanged from Cb / = 0.01 to 1000. Here the other dimensionless parameters are ﬁxed to Cs / = 250, L0 /σ =200/9 ≈ 22.2, and N = 100. In this case, each spring relaxes almost at its natural length. Hence the stretchingenergy given by equation (1) eﬀectively acts as a constraint of area (circumference) conservation. For a large
bending constant such as Cb / = 1000, the tube hardly
deforms in spite of the adhesion, and keeps its circularshape. As the bending constant is reduced to Cb / = 10,
a considerable deformation takes place and the contact
area (line) increases signiﬁcantly. Further decrease of thebending constant results in a conﬁguration obtained suchas for Cb / = 0.01. In this limit, the large curvature islocalized at the regions close to the contact line (point),
and a signiﬁcant ﬂattening of the tube is observed.
In order to quantify the deformation of the tube, we
use the inertia tensor calculated by
Iαβ =

1 
(ri,α − rj,α )(ri,β − rj,β ),2N 2 i j(5)where r i is the position of the i-th bead, and α, β = x, z.
Then as a measure of the deformation, we deﬁne the socalled anisotropic factor γ:γ=

λ1 − λ2,
λ1 + λ2(6)where λ1 and λ2 (λ1 > λ2 ) are the two eigenvalues of
the inertia tensor equation (5). This quantity vanishes
when the cross section of the tube is isotropic, but it deviates from zero as soon as the tube starts to undergo any
anisotropic deformations. In Figure 3, we have plotted γ

S. Komura et al.: Deformation of adhering elastic tubes

75

Fig. 2. Equilibrium conﬁgurations of the deformed tubes forvarious values of the scaled bending constant Cb / when N =100, Cs / = 250, and L0 /σ = 200/9.

Fig. 4. The critical value of the deformation t∗ as a functionof the scaled natural length of the spring L0 /σ for N = 200
and Cs / = 250.

Fig. 3. The anisotropic factor γ deﬁned by equation (6) as afunction of t ≡ Cb /(N 2 ) for N = 200, 500, 1000. The otherparameters are ﬁxed to Cs / = 250 and L0 /σ = 200/9. The
deformation starts to occur at the critical value t = t∗ ≈ 0.05
irrespective of N .

as a function of the newly deﬁned variable t ≡ Cb /(N 2 )for three diﬀerent tube sizes, i.e., N = 200, 500, 1000.
The physical meaning of this scaling variable will be discussed later. With this combination of variables, we see
a clear transition at t∗ ≈ 0.05 between the undeformed
and the deformed tubes. This transition looks a continuous second-order transition while the buckling phenomena
does not occur. We stress here that the critical value t∗
does not depend on the tube size N . For a ﬁxed spring
constant Cs , however, we found that t∗ is weakly dependent on the natural length of the spring L0 . This resultis shown in Figure 4 where we plotted t∗ as a functionof L0 /σ for N = 200. We see that t∗ decreases almost
linearly with increasing L0 , although it is a rather weak
dependence. These results imply that tubes with larger N
and/or smaller L0 are more ﬂexible against the deformation.
For an almost undeformed tube, its bending energy
can be readily estimated, since the curvature is uniformly

Fig. 5. The minimized scaled bending energy Eb / as afunction of the scaled bending constant Cb / for N =200, 500, 1000. The other parameters are ﬁxed to Cs / = 250
and L0 /σ = 200/9. There are two scaling behaviors; Eb / ∼(Cb /)/N and Eb / ∼ (Cb /)0.75 for large and small Cb /,respectively.

distributed on the circle. In such a case, the radius of curvature is proportional to N , and the total bending energy
equation (2) becomes simply


Cb
Cb
,
(7)
∼
Eb ∼ NN2
N
or(Cb /)
Eb
∼
.
(8)

N
This linear relation between Eb / and Cb / in the undeformed region can be checked in Figure 5. The fact thatthe data of Eb / splits into three diﬀerent lines reﬂects

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The European Physical Journal E

Fig. 6. Various notations characterizing the tube deformation
due to the adhesion.

a size dependence on 1/N in equation (8) for large Cb /.
Since t = Eb /(N ) holds in this regime, the critical valueof t in Figure 3 can be understood as the value at which
the scaled bending energy per bead Eb /(N ) becomest∗ ≈ 0.05. For small Cb /, on the other hand, all the datacollapse on a single straight line regardless of the tube size
N . In this region of strong deformation, we observe a nontrivial scaling behavior, i.e., Eb / ∼ (Cb /)0.75 , which canbe understood within the shell theory [14] as explained
below.
When a tube of radius R is largely deformed on a substrate (see Cb / = 1 in Fig. 2), a contact area of width
2r develops as shown in Figure 6. For such a deformation,
the elastic energy is localized in the two parallel straight
strips near the edge of the bulge. Let us denote the widthof the bending strip by d. Then the displacement of points
in the bending strip ζ can be geometrically determined asζ ∼ dr/R. The associated strain and the curvature dueto the displacement ζ are given by ζ/R and ζ/d2 , respectively [14]. One can then show that the relevant stretching
and bending energies per unit area scale as   2   2 2
Cs d r
ζ
Cs
∼
,
(9)
L20
R
L20
R4
and



2
ζ
r2
∼
C
,
(10)
Cb
b
d2
d2 R2
respectively. These energies per unit area are multipliedby the width of the bending strip d to obtain the following
energies per unit length of the tube:
  3 2
Cs d r
Es ∼
,
(11)
L20
R4
r2
.
(12)
dR2
Minimizing the sum of these energies with respect to d,
the width of the bending strip is obtained as1/4
Cb L20R1/2 .
(13)d∼Cs
Eb ∼ Cb

By putting back this relation into equation (12), we ﬁndthat the minimized total bending energy per unit lengthscales as 1/4 1/4
2
Cs
Cs
3/4 r
3/4 H
Cb
∼
Cb
. (14)
Eb ∼
L20
L20
R5/2
R3/2

Fig. 7. The absolute value of the total energy per bead
|Etot |/(N ) as a function of the scaling variable t = Cb /(N 2 ).
The parameters are ﬁxed to Cs / = 250 and L0 /σ = 200/9.
The scaling Etot /(N ) ∼ t holds in the “bending regime”,
whereas Etot /(N ) asymptotically approaches to −1 in the“adhesion regime”.

In the last expression, we have used the relation r ∼(HR)1/2 [14], where H is the depth of the bulge (seeFig. 6). For a ﬁxed size of tube, the obtained scaling
relation in equation (14) accounts for the behavior of
Eb / ∼ (Cb /)0.75 in Figure 5. We comment here thatthe corresponding expression of the bending energy for aspherical shell has similar dependences on Cs and Cb , but
diﬀerent dependences on H and R [14,21]. Moreover, thescaling relation in equation (14) is valid as long as H is
ﬁxed and given.
In Figure 7, we have plotted the scaled absolute valueof the total energy per bead |Etot |/(N ) as a function of
t = Cb /(N 2 ) for three diﬀerent tube sizes as before. Notice that Etot can take negative value due to the vdW
interaction when the tube is strongly adsorbed. It is remarkable that all the data collapse onto a single curve irrespective of the tube size N . When the tube is undeformed
for large t, the bending energy is the main contribution to
the total energy. As discussed before, we have the scaling
behavior
Eb
Cb
Etot
∼
∼
∼ t,
(15)
N
N
N 2
in this regime (see Eq. (8)). This relation is strictly satisﬁed in Figure 7 for t > t∗ where we call as the “bending regime”. For small t, on the other hand, |Etot |/(N )asymptotically approaches to a constant (unity). This behavior is consistent with the fact that most of the beads
are strongly adsorbed on the substrate when t is small.
We remind that the depth of the potential minimum inthe vdW adhesion energy per bead (see Eq. (3)) is −1 inunit of . Hence we call this small t limit as the “adhesion
regime”. These two regimes are connected by a crossover
region in which the bending and the adhesion energies
compete with each other.

S. Komura et al.: Deformation of adhering elastic tubes

Here we discuss the physical meaning of the scaling
variable t. Consider the case where the indentation H inequation (14) is caused by the adhesion. The corresponding energy gain per unit length of the tube is
Ea ∼ wr ∼ w(HR)1/2 ,L0 w 2 R 4
1/2 3/2
Cs Cb,

exhibit universal behaviors when Cb /(N 2 ) is chosen as
the scaling variable. Further studies such as a tube with aspontaneous curvature, or deformation of a spherical shell
are under progress.

(16)

where w is the vdW energy per unit area. In reference [21], we showed that w can be approximately given
by w ∼ A/(12πD2 ), where A is the Hamaker constantand D is an atomic cutoﬀ for the vdW interaction. If thedeformations are driven by vdW adhesion, the adhesionenergy Ea is expected to balance with the bending energy Eb in equation (14). By setting Ea ∼ Eb , we arriveat the estimate for indentation H:
H∼

77

(17)

for given w and R. Since the tube radius R is the only
length scale in the problem, we replace H by R and obtain
the relation:
2/3
Cb
L0 w4/3
∼
.
(18)
1/3
R2
Cs
This result suggests that Cb /(R2 ) ∼ Cb /(N 2 ) ∼ t canbe the dimensionless scaling variable for any deformation
because the right hand side does not depend on R.
We note here that the upper part of the tube is not apart of a circle with radius R especially when the tube isstrongly deformed as shown in Figure 2. Since the spontaneous curvature is not considered in our model, the ﬂattening of the tube reduces the bending energy. In such acase, the bending energy is localized at the regions close
to the contact line, and the above analysis becomes moreaccurate.
We ﬁnally comment that the present work is related
to the problem of vesicle adhesion which was consideredby Seifert and Lipowsky [22]. For spherical vesicles, various axisymmetric shapes were calculated by minimizing
the sum of the bending and the adhesion energies under the constraint of constant area or volume. Apart fromtheir shapes, one of the major diﬀerences between the nanotubes and the vesicles is that the vesicles do not support
any stretching energy. Moreover, the vesicles feel the adhesion potential only when they are in contact with thesubstrate, whereas the long-ranged interaction acts on allof the beads in our model.

4 Conclusion
We have investigated the deformation of the elastic
tube adhering onto the substrate by using the conjugate
gradient method and the scaling argument. We found thatthe onset of the deformation and the total energy per beadWe thank D. Andelman and S.A. Safran for useful discussions
and comments. This work is supported by the Ministry of Education, Culture, Sports, Science and Technology, Japan (Grantin-Aid for Scientiﬁc Research No. 15540395).

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