ARTICLE
Received 20 Mar 2016
| Accepted 15 Jul 2016 | Published 25 Aug 2016
Universal shape and pressure inside bubbles
appearing in van der Waals heterostructures
E. Khestanova
1
, F. Guinea
1,2
, L. Fumagalli
1
, A.K. Geim
1
& I.V. Grigorieva
1
Trapped substances between a two-dimensional (2D) crystal and an atomically flat substrate
lead to the formation of bubbles. Their size, shape and internal pressure are determined by
the competition between van der Waals attraction of the crystal to the substrate and the
elastic energy needed to deform it, allowing to use bubbles to study elastic properties of 2D
crystals and conditions of confinement. Using atomic force microscopy, we analysed a variety
of bubbles formed by monolayers of graphene, boron nitride and MoS
2
. Their shapes are
found to exhibit universal scaling, in agreement with our analysis based on the theory of
elasticity of membranes. We also measured the hydrostatic pressure induced by the
confinement, which was found to reach tens of MPa inside submicron bubbles. This agrees
with our theory estimates and suggests that for even smaller, sub-10 nm bubbles the pressure
can be close to 1 GPa and may modify properties of a trapped material.
DOI: 10.1038/ncomms12587
OPEN
1
School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK.
2
IMDEA Nanociencia, Faraday, 9, Cantoblanco,
28049 Madrid, Spain. Correspondence and requests for materials should be addressed to F.G. (email: francisco.guinea@manchester.ac.uk) or to
I.V.G. (email: irina.grigorieva@manchester.ac.uk).
NATURE COMMUNICATIONS | 7:12587 | DOI: 10.1038/ncomms12587 | www.nature.com/naturecommunications 1
V
an der Waals (vdW) heterostructures
1
—stacks of
atomically thin layers of different materials assembled
layer by layer—are making possible the design of new
devices with tailored properties. An essential feature of such
heterostructures is atomically clean interfaces that form due to
strong adhesion between the constituent layers
2
. Even though the
contamination (adsorbed water and hydrocarbons) is inevitably
present on individual layers before assembly, the vdW forces that
attract adjacent two-dimensional (2D) crystals squeeze out
trapped contaminants, usually pushing them into submicron-
size ‘bubbles’ and leaving large interfacial areas atomically sharp
and free of contamination
2
.
So far such bubbles have been used simply as signatures of
good adhesion between constituents of vdW heterostructures and
as indicators that the interfacial areas between the bubbles are
perfectly clean
3
. Now, we show that the bubbles can be employed
as a tool to study the elastic properties of the 2D crystals involved
and, also, to evaluate the conditions that nanoscale confinement
exerts on the enclosed material (for example, hydrostatic
pressure). This information is important in many situations,
where confinement can modify materials properties, with water
inside graphene nanocapillaries
4–6
, nanocrystals or biological
molecules confined in graphene liquid cells
7–10
, room-
temperature ice in a 2D nanochannel
11,12
and a hydrothermal
anvil made of graphene on diamond
13
being a few examples.
Furthermore, highly strained graphene nano-bubbles have been
shown to possess enormous pseudo-magnetic fields
14
, 4300 T.
The detailed knowledge of strain for commonly occurring
bubbles should facilitate studies of the electronic properties of
graphene under conditions inaccessible in high-field magnet
laboratories
15
.
Here we study bubbles formed between a 2D crystal
(monolayer graphene, monolayer hexagonal boron nitride
(hBN) or monolayer MoS
2
) and an atomically smooth flat
substrate (hBN, graphite and MoS
2
). By analysing shapes and
dimensions of the bubbles, and comparing them with the
corresponding predictions of the elasticity theory, we find that
the bubbles for all three materials are fully described by the
combination of a 2D crystal’s elastic properties and its vdW
attraction to a substrate. We find excellent agreement between the
experiment and theory, both for smoothly deformed bubbles, and
for bubbles with shape and dimensions modified by a residual
strain. Furthermore, using indentation of bubbles with an atomic
force microscope (AFM) tip, we extracted the hydrostatic (vdW)
pressure inside them, and Young’s moduli for graphene and
MoS
2
membranes. Through the experiments and analysis below,
we found that in-plane stiffness of 2D crystals plays a major role
in determining characteristic shapes and density of the bubbles
one can expect to find when such a crystal is part of a vdW
heterostructure. Stiffer 2D crystals, such as graphene or
monolayer hBN on an hBN substrate, form smaller, more
sparsely distributed bubbles, so that large (up to 100 mm
2
) areas of
the structure present a perfect vdW interface. This has been
exploited in fabrication of high-quality electronic devices. On the
other hand, stronger adhesion between a 2D crystal and the
substrate (monolayer MoS
2
on an MoS
2
substrate being an
example) can be exploited to achieve a higher vdW pressure
inside the bubbles, which is desirable if one wants to modify the
properties of a material through nanoscale confinement.
Results
Experiment. Samples for this study were made by mechanical
exfoliation of graphene, hBN and MoS
2
monolayers onto hBN,
graphite and MoS
2
substrates using the now standard dry-peel
technique
3,16
. To this end graphene/monolayer hBN/monolayer
MoS
2
were first mechanically exfoliated onto a poly(methyl
methacrylate) membrane. The latter was then loaded into a
micromanipulator, where it was placed face-down onto a
substrate (a B100 nm thick crystal of graphite, hBN or MoS
2
on a Si/SiO
x
wafer), after which the supporting polymer
membrane was mechanically peeled off, ensuring residue-free
surface of a 2D crystal. The resulting heterostructures were then
heated (annealed
3,16
) at 150 °C for 20–30 min, which resulted in
spontaneous formation of a large number of bubbles filled with
hydrocarbons
2
, with typical separations from B0.5 to tens of
microns. The annealing time and temperature were optimized to
ensure that the bubbles reached equilibrium conditions, that is,
no further changes in their shape, size or position could be
detected with further annealing. After that the dimensions and
topography of many bubbles (up to 100 for each heterostructure)
were analysed using AFM.
Figure 1 shows typical examples of bubbles formed by
monolayer graphene on bulk hBN. The majority of the bubbles
were o500 nm in radius, R, and had a round or nearly round
base (Fig. 1a). Larger bubbles typically exhibited pyramidal
shapes, with either triangular (Fig. 1b) or trapezoidal (Fig. 1c)
bases. Bubbles formed by monolayer hBN on bulk hBN were also
either round or approximately triangular in shape, but smaller in
size compared with graphene (o100 nm for round and o500 nm
for triangular bases). Bubbles formed by MoS
2
monolayers were
mostly round, similar to those shown in Fig. 1a for graphene, but
exhibited a broader size distribution, with 30oRo1,000 nm. We
measured the cross-sectional profiles of the observed bubbles and
analysed their maximum height, h
max
, and the aspect ratio of h
max
to the radius, R, or to the length of the side, L, as appropriate.
The results for round-type graphene bubbles are shown
in Fig. 2a. The aspect ratio, h
max
/R, is remarkably universal,
that is, independent of the bubbles’ radius, R, or volume, V:
h
max
/RE0.11, within B10%. Moreover, if we discount the
smallest bubbles with Ro50 nm, the accuracy reaches 4% for
sizes varying by an order of magnitude. Very similar behaviour
was found for monolayer hBN, with h
max
/RE0.11 for bubbles
450 nm and a somewhat increasing h
max
/R for Ro50 nm—see
Fig. 2a. Only a few sufficiently large bubbles were found in this
case, limiting our analysis.
Aspect ratios for round bubbles formed by MoS
2
monolayers
are shown in Fig. 2b. For comparison, we analysed the MoS
2
bubbles formed on two different substrates, MoS
2
and hBN.
Again, for the same 2D crystal–substrate combination we find a
constant h
max
/R, but its value depends on the substrate and is
notably larger compared with graphene and hBN monolayers.
This can be attributed to different elastic properties of monolayer
0
10
20
30
40
50
h (nm)
ab
c
Figure 1 | Graphene bubbles. (a–c) AFM images of graphene bubbles of
different shapes. Scale bars, 500 nm (a); 100 nm (b); 500 nm (c).
The vertical scale on the right indicates the height of the bubbles.
ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12587
2 NATURE COMMUNICATIONS | 7:12587 | DOI: 10.1038/ncomms12587 | www.nature.com/naturecommunications
MoS
2
compared with one-atom-thick crystals (graphene and
monolayer hBN). Furthermore, the different h
max
/R found for
different substrates point at the importance of vdW adhesion, as
discussed below.
A constant aspect ratio was also found for graphene bubbles
with triangular bases, such as those shown in Figs 1b and 3. In
this case, it is intuitive to use the length of the side, L,to
characterize their sizes. Similar to the round bubbles in Fig. 2a,
these bubbles usually had smooth round tops, but were larger in
size (typical L between 500 and 1,000 nm) and exhibited the
aspect ratio h
max
/L ¼0.07
±
0.01—see Fig. 3. We note that,
although this value appears to be lower than that for the round
bubbles, as if the triangular bubbles were somewhat thinner, this
is simply the effect of using a different measure to characterize the
lateral size (L versus R). Indeed, redefining the lateral size of
triangular-type bubbles as a distance L* from their centres to
corners, we find the same ratio h
max
/L* as for round bubbles,
within our experimental accuracy. As discussed below, the shapes
and dimensions of all smoothly deformed bubbles (round or
triangular) are expected to follow the same scaling.
The only class of bubbles that showed strong deviations from
the universal scaling behaviour were pyramidal-type bubbles with
sharp features. They exhibited sharp ridges that often extended
nearly to the full height of the bubbles. Two examples are shown
as insets in Fig. 4. The aspect ratio, h
max
/L, for such bubbles
showed relatively large variations (by a factor of 2), with most
values being higher than those for smoothly deformed bubbles—
c.f. Figs 3 and 4.
To summarize, all bubbles—formed by graphene, hBN and
MoS
2
monolayers—exhibited a small set of shapes (mostly, round
and triangular) with a universal aspect ratio. Different shapes
were found on each sample, but the frequency of occurrence was
different for different shapes, and there was a correlation between
the bubbles’ shapes and their sizes. For example, all possible
bubble shapes (round, triangular and pyramidal) were found on
the same sample of graphene on an hBN substrate. Of these,
bubbles with Ro400 nm were round or nearly round and most of
them were o200 nm; bubbles with 500oRo1,000 nm were
triangular with smooth tops, and triangular and pyramidal
bubbles with sharp features were very few, with a broad
distribution of sizes, from 400 to 1,400 nm. For monolayer
hBN, bubbles of all shapes tended to be smaller; accordingly, the
size ranges were different (20–100 nm for round bubbles;
150–350 nm for triangular with smooth tops), but a correlation
between shape and size was found as well. These statistics are
summarized in Supplementary Fig. 1.
In terms of the aspect ratio, monolayers of graphene and hBN,
which have similar elastic properties, showed the same aspect
ratio. The aspect ratio for MoS
2
, which has a lower elastic
100 200
10
3
10
4
0.05
0.10
0.15
Monolayer graphene
Monolayer hBN
R
300 400
10
5
10
6
10
7
0.15
0.20
0.25
0.30
1.9 µm
h
max
/ R
1.9 µm
Monolayer MoS
2
on MoS
2
substrate
0
0.05
0.10
0.15
0.20
0.25
R (nm)
h
max
/ R
a
100
0.00
0.05
0.10
Monolayer MoS
2
on hBN substrate
Monolayer MoS
2
on MoS
2
substrate
R (nm)
20
b
1,000
0 nm
100 nm
h
max
/ R
V (nm
3
)
Figure 2 | Universal shape of round-type bubbles. (a)Measuredaspect
ratios as a function of the base radius for graphene (blue symbols) and
monolayer hBN (red symbols). Dashed line shows the mean value. Top left
inset: sketch of a nearly round bubble and its effective radius R determined as
R¼
ffiffiffiffiffiffiffiffiffi
A=p
p
,whereA is the measured area of the base of the bubble. Right inset:
aspect ratio of the bubbles as a function of their volume. (b)Aspectratioof
MoS
2
bubbles on hBN and MoS
2
substrates. Dashed lines show the mean
values of h
max
/R ¼0.14 and 0.17, respectively. The logarithmic scale is used to
accommodate the large range of R. Inset: AFM image of a typical MoS
2
bubble.
0 200 400 600 800 1,000
0.00
0.05
0.10
0.15
h
max
/ L
L (nm)
Monolayer hBN
Graphene
1.3
µm
69 nm
0
.86
µm
0.4
µm
0.4
µm
0 nm
0 nm
21 nm
L
Figure 3 | Aspect ratio of smooth triangular bubbles. Symbols show the
measured aspect ratios of graphene and hBN bubbles (closed and open
symbols, respectively), both on hBN substrates, as a function of L. The
dashed line shows the mean aspect ratio, h
max
/L ¼0.07. Bottom left inset:
sketch of a triangular bubble. Its side length L was experimentally
determined as L¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4A=
ffiffiffi
3
p
q
, where A is the measured area of the base of a
bubble. The other two insets show typical AFM images of smoothly
deformed triangular bubbles.
0 500 1,000 1,500
0.0
0.1
0.2
0.3
h
max
/ L
L (nm)
1.3 µm
1.8 µm
0 µm
0.27 µm
1.3 µm
0.24 µm
1.8 µm
0 µm
Figure 4 | Aspect ratio of pyramidal graphene bubbles with sharp
features. Symbols show the measured aspect ratios of triangular
(red-closed symbols) and trapezoidal (blue-open) graphene bubbles on hBN
substartes, as a function of their side length, L,determinedasL¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4A=
ffiffiffi
3
p
q
for triangular bubbles and as L¼
ffiffiffiffi
A
p
for trapezoidal ones. Here, A is the
measured area of the base of a bubble. Insets show typical AFM images of
such bubbles: left and right are top and 3D views, respectively. Scale bars,
500 nm.
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12587 ARTICLE
NATURE COMMUNICATIONS | 7:12 587 | DOI: 10.1038/ncomms12587 | www.nature.com/naturecommunications 3
stiffness
17,18
, was also constant, but its value was up to 50%
higher than for graphene and hBN monolayers. The universal
behaviour for different 2D crystals points to the definitive role
played by their elastic properties, as analysed in the following
sections.
Scaling analysis. To model the observed bubbles, we consider a
material trapped between a flat substrate and a 2D crystal
attracted to the substrate by vdW forces (Fig. 5). We note that a
related situation—circular gas-filled graphene bubbles on a
substrate—was analysed recently using nonlinear elastic plate and
membrane theory
19
, and numerical simulations
20
. However, the
results of refs 19,20 are not applicable to our experiments because
they considered bubbles under constant pressure with clamped
edges. In contrast, our theory corresponds to the problem studied
experimentally, that is, bubbles of a constant volume, where the
edges adapt to the competition between the vdW attraction and
the internal pressure, while the pressure itself is determined by
the adhesion between the 2D crystal and the substrate.
Furthermore, the 2D crystal is free to adapt to the substrate
and the bubble profiles are not assumed (as in ref. 19), but found
self-consistently.
For simplicity, below we refer to graphene only. Its rigidity is
determined by a combination of the in-plane stiffness, and the
energy associated with out-of-plane bending. The in-plane stiffness
is described by the theory of elasticity
21
, which requires
the specification of two parameters, Young’s modulus, Y,and
Poisson’s ratio, n, or, alternatively, Lame
´
coefficients, l and m.As
graphene is an ultimately thin 2D membrane, out-of-plane
deformations lead to in-plane stresses, making the system highly
anharmonic
22
. The out-of-plane bending is described by the
bending rigidity, k. Relative contributions of the in-plane stiffness
and the bending rigidity to the elastic energy of a 2D membrane
are determined by the scale of deformations: beyond a length scale
‘
anh
ffiffiffiffiffiffiffiffiffi
Y=k
p
the stiffness is dominated by in-plane stresses. For
graphene, this scale is ‘
anh
4 Å, so that in most situations the
bending rigidity can be neglected (however, see further). The
equivalent length for MoS
2
is somewhat larger, but still o1nm.
The vdW energy associated with separating of a graphene layer
from the substrate is given by
E
vdW
¼ pgR
2
g ¼ g
GS
g
Gb
g
Sb
ð1Þ
where g
GS
, g
Gb
and g
Sb
are the adhesion energies between
graphene and the substrate, graphene and the substance inside
the bubble, and the substrate and the substance, respectively.
If the bubble is filled with a substance having a finite
compressibility, b, it can be written as
b
1
¼ V
@
2
E
b
VðÞ
@V
2
¼V
@P
@V
ð2Þ
where E
b
(V) is the free energy of the substance inside the bubble
of volume V and P is the pressure.
The bubble’s height profile is described by
hrðÞ¼h
max
~
h
r
R
ð3Þ
where h
max
is the maximum height of the bubble, so that
~
h 0ðÞ¼1;
~
h 1ðÞ¼0. The in-plane displacements are defined by the
function u
r
r
ðÞ¼
h
2
max
=R
~
u
r
R
ðÞ
. We assume radial symmetry, so
that the azimuthal displacements vanish, that is, u
y
¼0. Details of
calculating the in-plane displacements and the total energy as a
function of h(r) are given in Supplementary Note 1.
Neglecting the bending rigidity, the total energy can be written
as
E
tot
¼ E
el
þE
vdW
þE
b
VðÞ¼
¼ c
1
~
h
hi
Y
h
4
max
R
2
þc
2
~
h
hi
YEh
2
max
þpgR
2
þE
b
VðÞ
ð4Þ
where dimensionless coefficients c
1
and c
2
depend only on the
function
~
h, describing the height profile, and the volume V is
V ¼ c
V
~
h
hi
h
max
R
2
: ð5Þ
Below, we show that the function
~
hxðÞis generic, that is,
independent of the material parameters Y, g and E
b
(V).
By minimizing Equation (4) with respect to h
max
and R,we
obtain
c
1
Y
4h
3
max
R
2
þ2c
2
YEh
max
c
V
R
2
P ¼ 0
c
1
Y
2h
4
max
R
3
þ2pgR 2c
V
h
max
RP ¼ 0;
ð6Þ
where we have used P ¼qE
b
/qV. By eliminating P in
Equation (6), we obtain
5c
1
Y
h
max
R
4
þ2c
2
YE
h
max
R
2
pg ¼ 0 ð7Þ
This equation defines the aspect ratio of the bubble, h
max
/R,in
terms of the coefficients c
1
and c
2
, parameters Y and g, and an
external strain, E:
h
max
R
2
¼
c
2
E
5c
1
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c
2
E
5c
1
2
þ
pg
5c
1
Y
s
ð8Þ
In the absence of external strain, E ¼0, this expression reduces to
h
max
R
¼
pg
5c
1
Y
1=4
ð9Þ
that is, the value of h
max
/R is determined solely by the balance
between vdW and elastic energies of a 2D crystal, independent of
the properties of the substance captured within the bubble. This
result is in excellent agreement with the constant aspect ratios
observed experimentally—see Figs 2 and 3.
The presence of finite E (induced, for example, during
fabrication) should modify the bubbles’ shape, reducing the
aspect ratio h
max
/R for tensile strains and increasing it for
compressive strains—see Supplementary Notes 2 and 3.
The above analysis also shows that the fluid material inside the
bubble is under a constant hydrostatic pressure P, which is
described by Equation (6) and, following ref. 12, is referred to
below as vdW pressure. Accordingly, our case of bubbles formed
by the competition of vdW and elastic forces can be considered as
a particular case of the membrane deformed by applying a
constant external pressure.
h
2R
Figure 5 | Sketch of the bubble considered in our theoretical analysis.
The bubble is formed by material trapped between a substrate and a 2D
layer (graphene).
ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12587
4 NATURE COMMUNICATIONS | 7:12587 | DOI: 10.1038/ncomms12587 | www.nature.com/naturecommunications
Using the change of variables, x ¼r/R, we write the total energy
as
E
tot
¼ E
el
þE
bend
þPV
E
el
¼ c
1
~
hxðÞ
hi
Y
h
4
max
R
2
þc
2
~
hxðÞ
hi
YEh
2
max
E
bend
¼ c
3
~
hxðÞ
hi
k
h
2
max
R
2
E
P
¼ c
V
~
hxðÞ
hi
Ph
max
R
2
ð10Þ
We consider first the bubble’s profile,
~
hxðÞ, determined solely
by the competition between the pressure and the in-plane
stresses, E
el
and E
P
, and we set E ¼0. Minimization of E
tot
with
respect to h
max
gives
h
max
¼
c
V
~
h
ðÞ
c
1
~
h
ðÞ
1=3
PR
4
4Y
1=3
E
tot
~
h
hi
¼
3
4
c
4
V
~
h
ðÞ
c
1
~
h
ðÞ
1=3
P
4
R
10
4Y
1=3
ð11Þ
We can now calculate
~
h by minimizing E
tot
. This yields that
~
hxðÞis universal, that is, independent of Y , P and R. The function
~
hxðÞis shown in Fig. 6. The in-plane stresses associated with the
bubble formation can also be expressed in a scaled form,
~
s
rr
xðÞ¼R
4
= h
4
max
Y
s
rr
r=RðÞand
~
s
yy
xðÞ¼R
4
= h
4
max
Y
s
y
r=RðÞ. These functions are plotted in the inset of Fig. 6. It
is interesting to note that the hoop stress, s
yy
, becomes negative
(compressive) near the base of the bubble. In the absence of vdW
pressure, a compressive stress can lead to an instability with
respect to the formation of wrinkles
23
. The existence of in-plane
stresses outside the bubble (see inset in Fig. 6) implies that the
bubbles interact with each other—see Supplementary Note 4,
where this interaction is analysed. It is attractive and decays as
Y/d
2
, where d is the distance between bubbles.
A similar analysis can be carried out when the shape of the
bubble is determined by the bending rigidity, and E
tot
¼E
bend
þE
P
.
In this case, we find
h
max
¼
c
V
~
h
ðÞ
2c
3
~
h
ðÞ
PR
4
k
E
tot
~
h
hi
¼
c
V
~
h
ðÞ
4c
3
~
h
ðÞ
P
2
R
6
k
ð12Þ
The generic profiles in the two cases (elastic energy is dominated
by either in-plane stresses or bending) are given in Fig. 6.
In the following, we neglect the bending rigidity term, E
bend
,as
appropriate for 2D membranes with
ffiffiffiffiffiffiffiffiffi
k=Y
p
h
max
; R, and
corresponds to the case studied in our experiments. In principle,
the coefficients c
1
and c
2
, and the function
~
h can depend on strain
E. However, we have found numerically that this dependence is
negligible for Et0:1, that is, can be neglected in realistic
situations because even smaller strains (a few %) are likely to
cause slippage along the substrate due to limited adhesion. The
numerical parameters that relate h
max
, L, Y and P are found as
c
1
0:7
c
2
0:6
c
V
1:7:
ð13Þ
The above scaling analysis can also be applied to bubbles of
other shapes, such as the pyramidal bubbles found experimentally
(Fig. 3). For simplicity, we model smooth triangular bubbles as
having an equilateral triangle as their base. The bubbles are then
characterized by two length scales: height, h
max
, and the side
length, L. The scaled universal profile for a triangular bubble is
shown in Fig. 7. The numerical parameters in this case are
c
1
0:6
c
2
0:3
c
V
0:2:
ð14Þ
The corresponding average strain,
u
rr
, for graphene/hBN/MoS
2
monolayers enclosing a bubble is of order
u
rr
h
max
R
2
1 2 % ð15Þ
To gain further insight, we estimate the parameters in
Equation (9), corresponding to the experimentally observed
0.2 0.4 0.6 0.8 1.0
x
0.2
0.4
0.6
0.8
1.0
0.5 1.0 1.5 2.0
x
–0.2
0.2
0.4
a
0.2 0.4 0.6 0.8 1.0
x
0.2
0.4
0.6
0.8
1.0
b
h(x)
~
h(x)
~
rr
(x),
(x)
Figure 6 | Bubble profiles. (a) Scaling function,
~
hxðÞ, obtained by
minimizing numerically the elastic energy. It is well approximated by a
quartic function,
~
hxðÞ¼1 x
2
þcx
2
x
4
(red dots), with c ¼0.25. The
inset shows the scaled stresses, ~s
rr
xðÞand ~s
yy
xðÞ(blue and red curves,
respectively). (b) Comparison between the bubble profiles under a
hydrostatic pressure for the cases dominated by in-plane strains (blue) and
bending (red).
0.0
–1.0
–0.5
0.0
0.5
1.0
–0.5
0.0
0.5
1.0
y / L
x / L
h / h
max
Figure 7 | Universal shape of smooth triangular bubbles (theory). Shown
is the result of numerical calculations of the 3D shape of a triangular bubble
with an equilateral triangle as its base. The bubble dimensions are scaled to
its maximum height, h
max
, and the side length, L.
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12587 ARTICLE
NATURE COMMUNICATIONS | 7:12 587 | DOI: 10.1038/ncomms12587 | www.nature.com/naturecommunications 5