INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 16 (2004) L421–L428

PII: S0953-8984(04)83466-0

LETTER TO THE EDITOR

Adhesion induced buckling of spherical shells
K Tamura, S Komura and T Kato
Department of Chemistry, Faculty of Science, Tokyo Metropolitan University, Tokyo 192-0397,
Japan

Received 12 July 2004, in final form 31 August 2004
Published 17 September 2004
Online at stacks.iop.org/JPhysCM/16/L421
doi:10.1088/0953-8984/16/39/L01

Abstract
Deformation of a spherical shell adhering onto a rigid substrate due to van der
Waals attractive interaction is investigated by means of numerical minimizationof the sum of the elastic and adhesion energies. The conformation of thedeformed shell is governed by two dimensionless parameters Cs / and Cb /,where Cs and Cb are respectively the stretching and the bending constants, and 
is the depth of the van der Waals potential. As a function of Cb /, we find both
continuous and discontinuous buckling transitions for small and large Cs /,respectively, which is analogous to van der Waals fluids or gels. Some scalingarguments are employed to explain the adhesion induced buckling transition.

Deformation of thin elastic sheets such as graphitic oxide [1], polymerized Langmuirmonolayers [2], or cytoskeletons of biological cell membranes [3] has attracted considerable
interest in recent years. From the theoretical viewpoint, scaling properties of stretching ridges
in a crumpled elastic sheet were investigated by analysing the nonlinear Föppl–von Kármán(FvK) equations [4]. The FvK equations for plates have been the subject of renewed interestin the context of developable cone (d-cone) singularities [5], pattern formations induced bybuckling [6], or formations of plastic tears [7]. However, most of these works are concernedwith flat elastic sheets, while less attention has been paid to the properties of initially curved
elastic sheets, i.e., shells.
Quite generally, curved shells exhibit a peculiar elastic feature because the stretching is a
first-order effect and they cannot be bent without being stretched [8]. This interplay between
bending and stretching, leading to a reduced flexibility of the shell, was studied for thermal
fluctuations of polymerized vesicles [9] or the asymptotic shape of fullerenes [10]. In theprevious paper, the deformation and mechanical stability of fullerene-like hollow nanoparticles
were investigated within the shell theory [11]. It was shown that van der Waals (vdW)
interaction between a rigid substrate and nanoparticles can cause considerable deformationof the latter. A similar problem was studied for elastic tubes by means of a numerical approachand a scaling theory [12]. As for the experiments on shells, a buckling instability of selfassembled actin-coated vesicles was observed under the application of a localized force using
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Letter to the Editor

z

nα

nβ

y
x
Figure 1. Bead and spring model of an elastic spherical shell adhering onto a substrate. n̂α(β) is
the unit normal vector of the triangle α(β).

optical tweezers [13]. Recently, the elastic properties of polyelectrolyte microcapsules werestudied by using AFM [14]. At the macroscopic level, the contact and compression problemof ping-pong and tennis balls was investigated in [15].
In this letter, we report a new type of instability that is found for spherical shells interactingwith a rigid substrate: namely the buckling transition induced by adhesion. We investigate
both numerically and theoretically the deformation of a spherical shell adhering onto a rigid
substrate due to vdW attractive interaction. Our work can be regarded in part as a contact
problem of spherical shells. Although the solution by Hertz for the contact problem of elastic
bodies is well known [8], there have been only a few works which deal with the corresponding
problem of shells. On the other hand, the present study is important for such as tribologicalapplications of fullerene-like balls since there is little control over the shape of adsorbed
nanoparticles [11]. We find both continuous and discontinuous buckling transitions for weak
and strong adhesion cases, respectively. Moreover, creation of a polygonal structure is observed
when the adhesion is strong enough. These results are compared with the experimentallyobserved buckling instability of a ping-pong ball which is pressed against a rigid plate [15].
Based on the continuum elastic theory, we also discuss the scaling theory of the observed
buckling transition.
Consider an elastic spherical shell interacting with a rigid substrate as shown in figure 1.
The normal direction to the substrate is taken as the z-axis, whereas the substrate spans the x yplane. The initial configuration of the shell consists of a spherically closed triangular mesh as asimplest approximation for two-dimensional elastic materials. To generate such a structure, westart from an icosahedron as the original network, and add new points on each triangle followed
by a subsequent rescaling of all bonds to the desired length [16]. This procedure ensures thatmost of the grid points have six neighbours and each bond has approximately the same length.
We then associate all the grid points and bonds with beads and springs, respectively. In thepresent work, we used a spherical shell consisting of N = 812 beads among which there are
12 beads having five neighbours1.
To describe the deformations of an elastic shell, both the stretching and the bending
energies should be taken into account [8]. Following the model of membranes with crystalline
order [17], or crushed elastic manifolds [18], the discretized stretching energy is given by the1

The numbers of triangles and springs are 1620 and 2430, respectively.

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sum over Hooke’s law of each spring: 1  L n − L 0 2
Es =
Cs
.
(1)
L0
n 2
Here Cs is the stretching constant, L n is the length of spring n, and L 0 is the natural length of thespring. On the other hand, the discretized bending energy is calculated according to [17, 18]

2
1
Eb =
(2)
2 Cb |n̂α − n̂β | ,αβwhere Cb is the bending constant, n̂α(β) is the unit normal vector of triangle α(β), and the sum
is taken over each pair of triangles which share a common edge. We comment that both Cs
and Cb have the dimension of energy.
The adhesion energy of the shell is included through the vdW interaction between eachof the bead and the substrate [12]: 28/3  σ 12  σ 3
,
(3)

−
W =
3
zi
zi
i
where z i is the height of bead i from the substrate. When the adhesion energy of bead i is plottedagainst z i , the depth of the energy minimum is given by , and the distance corresponding tothis minimum is 22/9 σ . The first repulsive term in equation (3) is responsible for the excluded
volume interaction which prevents the beads from penetrating into the substrate. The second
term represents the long-ranged attractive interaction between the beads and the substrate [19].
Notice that the inverse cubic dependence results from the pairwise additivity of the vdW
interaction between two atoms.
The total energy E tot = E s + E b + W is numerically minimized using the conjugate
gradient method [20]. Hereafter all the energies and the lengths are respectively scaled by and σ which characterize the vdW interaction in equation (3). There are three independent
dimensionless parameters in the model, i.e., Cs /, Cb /, and L 0 /σ . In the present study, we
have mainly varied Cs / and Cb /. The natural length of each bond L 0 /σ is chosen suchthat the initial configuration of the shell does not store any stretching energy, and its value isroughly L 0 /σ ≈ 0.1. For N = 812, the scaled radius of the undeformed shell is R/σ ≈ 0.75.
We note that the bending energy is inherent even in the undeformed shell since the spontaneous
curvature is not included in the present calculation. This assumption is justified such as forfullerene balls.
Figure 2 shows various typical final configurations of the adhering shell as the combinationof Cs / and Cb / is varied. There are four qualitatively distinct patterns of deformation asclassified from (a) to (d) in figure 2. For case (a) when both Cs / and Cb / are large enough,the shell hardly deforms in spite of the adhesion, and keeps its spherical shape. As both values
are reduced, we observe case (b) where a flat contact disc develops at the bottom of the shell.
Further decrease of Cb / results in the buckling of the shell as illustrated in case (c). In such aconfiguration, both the stretching and the bending energies are localized at a narrow ‘bending
strip’ of contact. For case (d) with smaller Cb /, it becomes energetically favourable to createa polygonal structure composed of a number of ridges joined by the d-cones. The number ofridges (four in the case of (d)) depends on the strength of the vdW attraction, which will bereported separately. In contrast to spherical shells, the buckling phenomenon has never been
observed for elastic tubes [12].
In order to quantify the observed deformation, we employ the moment of inertia tensor
calculated by
1 (ri, p − r j, p )(ri,q − r j,q ),(4)
I pq =
2N 2 i j

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Figure 2. Side and bottom views of the adhering spherical shells which have the minimized totalenergy E tot when the sets of the scaled elastic constants (Cs /, Cb /) are (a) (1000, 1000), (b)(150, 9), (c) (150, 2), and (d) (100, 1.1).

where ri is the position of bead i , and p, q = x, y, z. The three eigenvalues of I pq are orderedaccording to magnitude λ1  λ2  λ3 . As a quantitative measure of the anisotropy of thedeformed shell, we have calculated [21]λ2 + λ22 + λ23 − (λ1 λ2 + λ2 λ3 + λ3 λ1 )
.
(5)
= 1
(λ1 + λ2 + λ3 )2This quantity vanishes for an isotropic configuration, but it deviates from zero when the shellundergoes anisotropic deformations. In figure 3 we have plotted  as a function of Cb / forvarious different values of Cs /. The buckling of the shell is manifested in the sharp increase of as Cb / is decreased. There are even jumps of  for larger values of Cs /, which indicatesthe occurrence of a discontinuous buckling transition. It is worthwhile to mention that 
attains its minimum at Cb / larger than its threshold value of the buckling.
For Cs / = 300, we have plotted in figure 4 all the minimized energies, equations (1)–(3),and the total energy E tot as a function of Cb /. In this case, the shell exhibits a discontinuous
buckling transition at around (Cb /)∗ ≈ 5 as indicated by the dashed line. When the valueof Cb / crosses this critical value from above, both the stretching energy E s and the bending
energy E b increase abruptly. These losses in the elastic energies are compensated by the gainin the vdW energy W which decreases discontinuously at the transition point. In other words,
the shell buckles at the expense of the elastic energy when the adhesive force is strong enough.
Interestingly, however, we do not see any apparent discontinuity in E tot around the transition
point. For different values of Cs /, the behaviours of each energy are qualitatively the same
when the discontinuous buckling occurs.

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10

Cs/ε=260
300
340
380

-4

∆

10

-2

-6

10

10

0

10

1

Cb/ε

10

2

10

3

Figure 3. The anisotropy factor  defined in equation (5) as a function of the scaled bending
constant Cb / for various values of Cs / ranging from 260 to 380.

500

E/ε

250

Eb
Es
Etot
W

0
-250
-500 0
10

10

1

Cb/ε

10

2

Figure 4. The minimized total energy E tot / as a function of Cb / when Cs / = 300. The threeenergies E s , E b , and W contributing to E tot are also shown. The discontinuous buckling transition
occurs at (Cb /)∗ ≈ 5 indicated by the dashed line.

To investigate the property of the buckling transition in more detail, we have measured
the indentation length H which is defined in the inset of figure 5. The same figure shows thescaled indentation length H /R as a function of Cb / for various Cs / ranging from 100 to
900. In accordance with the aforementioned discussion, H changes discontinuously at thetransition point for larger Cs /, revealing the first-order nature of the buckling transition. This
discontinuous buckling transition takes place between the configurations, figures 2(b) and (c),i.e., the contact region changes from a disc to a ring at the transition point. However, thediscontinuous jump in H becomes smaller as Cs / is decreased, and finally vanishes at aroundCb / ≈ 6.1 (the filled circle). The corresponding critical indentation length is H /R ≈ 0.18,
and that of the stretching constant is Cs / ≈ 210. Below this value of Cs /, the buckling
occurs continuously between the two configurations (figures 2(b) and (c)). We immediately

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10

0

R
w

H/ R

H
r
10

-1

10

0

10

1

Cb/ε

10

2

Figure 5. The relative indentation length H /R as a function of Cb / for various Cs / ranging from100 (top) to 900 (bottom) for every 200. The discontinuous buckling occurs in the shaded region.
The filled circle located roughly at (Cb /, H /R, Cs /) = (6.1, 0.18, 210) indicates the point at
which the discontinuity vanishes. The inset defines the geometry of the deformed spherical shell(see the text).

note that figure 5 is very reminiscent of the isotherms of vdW fluids. Analogous to the liquid–gas coexistence region, the region of discontinuous transition has been shaded in figure 5.
In the present model, the parameter Cs / plays a role similar to the temperature of vdW
fluids. Another similar phenomenon is the volume transition of gels which is induced either
by changing the temperature or the ionic strength.
When the size of the shell N is changed, the indentation length H behaves similarly tofigure 5 although the location of the critical point shifts systematically. Roughly speaking, thecritical values of the elastic constants become larger for bigger shells as long as the potential
range satisfies R/σ  1. When this ratio is much larger than unity, the buckling does notoccur. Details of the size effect will also be published elsewhere.
We now pay attention to cases (b) and (c) in figure 2, and interpret the corresponding
deformations within the scaling argument [8]. Notice again that the buckling transition
corresponds to the transition between these two configurations. In the following continuum
treatment, we use Cs /L 20 to be the two-dimensional Young’s modulus, Cb the bending
rigidity as our discretized model. Then the effective thickness of the shell is expressed as
h = (8Cb L 20 /Cs )1/2 [17]. As shown in figure 2(b), the shell deforms only slightly at thebottom when the adhesive force is weak. Let d be the dimension of the deformed region
which is caused by the contact with the substrate. Following the case of a shell subjected to asmall localized force [8], we balance the associated stretching and bending energies to obtaind ∼ (Cb L 20 /Cs )1/4 R 1/2 . Hence the area of the contact region S (b) scales as
S (b) ∼ d 2 ∼ (Cb L 20 /Cs )1/2 R.

(6)

For strong adhesion, on the other hand, the buckling takes place as in figure 2(c), and mostof the elastic energies are concentrated over a narrow ‘bending strip’ of width w and radius r(see the inset of figure 5). Applying the case of a large deformation also described in [8], we
obtain w ∼ (Cb L 20 /Cs )1/4 R 1/2 and r ∼ H 1/2 R 1/2 . Note that the scalings for d and w are thesame. The area of the bending strip that contacts with the substrate is then given as
S (c) ∼ wr ∼ (Cb L 20 /Cs )1/4 H 1/2 R.

(7)

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L427

Comparing equations (6) and (7), we see that the contact area of the bending strip becomeslarger when H > (Cb L 20 /Cs )1/2 ∼ h, namely, when the indentation length exceeds the effective
shell thickness. The increase in the contact area between the shell and the substrate results in
the gain in the vdW adhesion energy, and hence W decreases when the buckling takes place asshown figure 4. The ‘critical point’ in figure 5 corresponds to the elastic constants Cb / ≈ 6.1
and Cs / ≈ 210 from which the effective thickness is calculated as h/R ≈ 0.064. Since thenumerically obtained critical indentation length is H /R ≈ 0.18, we have H ≈ 2.8h at thispoint. This result indeed confirms the fact that the buckling transition takes place when the
indentation length exceeds the shell thickness.
Experimentally, Pauchard and Rica studied the deformation of a ping-pong ball which
is forced to be in contact with a rigid plate [15]. In their work, the boundary of the halfsphere was fixed in order to avoid non-axisymmetric deformations. For low applied forces, theshell flattens against the horizontal plate. For higher compression forces, the discontinuous
buckling transition occurs when a deformation is close to twice the thickness of the shell(H ≈ 2.4h). Although their experimental set-up is not identical to our model of adhesion,thesebehaviours are in excellent agreement with our simulation results. However, the occurrenceof the continuous buckling transition of the shell with small stretching constant is our new
finding.
For comparison with other adhesion experiments, we give below some numbers to themodel parameters. For a layered material made of carbon, the two-dimensional Young’s
modulus and the bending rigidity are roughly 1.3 × 105 erg cm−2 and 1.6 × 10−12 erg,
respectively [22]. Assuming that the adhesion energy  is of the order of thermal energy
kB T , we can deduce the model parameters as Cs / ≈ 480 and Cb / ≈ 40. According tofigure 5, the adhesion of a single-walled fullerene with radius R/σ ≈ 0.75 should correspondto a point well above the critical point. In this regime, the fullerene may deform as in figure 2(b),which is consistent with the previous prediction [11]. Another example is the adhesion of ahollow polyelectrolyte microcapsule onto a flat substrate [14]. The three-dimensional Young’s
modulus of this shell material was measured to be in the range of 1.5–2 GPa, but its thickness
h tends to be in the 10 nm range. Since this gives fairly large bending rigidity of the order of
Cb / ≈ 104 , we expect that a microcapsule would hardly deform due to vdW adhesion as infigure 2(a). However, other attractive interaction such as electrostatic interaction can lead toadhesion induced buckling of microcapsules [14].
We have investigated the deformation of an elastic shell adhering onto a rigid substrateby using the conjugate gradient method and the scaling argument. We find that both thediscontinuous and the continuous bucking transitions occur for large and small stretching
constants, respectively. Further studies such as the analysis of the ridges and d-cones in thestrongly deformed region, or a spherical shell with a spontaneous curvature, are under progress.

We thank S A Safran for useful discussions. This work is supported by the Ministryof Education, Culture, Sports, Science and Technology, Japan (Grant-in-Aid for Scientific
Research No. 15540395).
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