Hygromorphs: From pine cones to biomimetic bilayers

Abstract

We consider natural and artificial hygromorphs, objects that respond to environmental humidity by changing their shape. Using the pine cone as an example that opens when dried and closes when wet, we quantify the geometry, mechanics and dynamics of closure and opening at the cell, tissue and organ levels, building on our prior structural knowledge. A simple scaling theory allows us to quantify the hysteretic dynamics of opening and closing. We also show how simple bilayer hygromorphs of paper and polymer show similar behaviour that can be quantified via a theory which couples fluid transport in a porous medium and evaporative flux to mechanics and geometry. Our work unifies varied observations of natural hygromorphs and suggests interesting biomimetic analogues, which we illustrate using an artificial flower with a controllable blooming and closing response.

Full text

REPORT
Hygromorphs: from
pine cones to
biomimetic bilayers
E. Reyssat1and L. Mahadevan1,2,*
1
School of Engineering and Applied Sciences, and
2
Organismic and Evolutionary Biology, Harvard
University, Cambridge, MA 02138, USA
We consider natural and artificial hygromorphs, objects
that respond to environmental humidity by changing
their shape. Using the pine cone as an example that
opens when dried and closes when wet, we quantify
the geometry, mechanics and dynamics of closure and
opening at the cell, tissue and organ levels, building
on our prior structural knowledge. A simple scaling
theory allows us to quantify the hysteretic dynamics
of opening and closing. We also show how simple bilayer
hygromorphs of paper and polymer show similar behav-
iour that can be quantified via a theory which couples
fluid transport in a porous medium and evaporative
flux to mechanics and geometry. Our work unifies
varied observations of natural hygromorphs and
suggests interesting biomimetic analogues, which we
illustrate using an artificial flower with a controllable
blooming and closing response.
Keywords: pine cone; swelling; wicking;
biomimetics
1. INTRODUCTION
Many natural and artificial materials respond to
changes in environmental humidity by shrinking or
swelling. The implications of this mundane observation
can be seen in situations as varied as the formation of
cracks in drying mud (Shorlin et al. 2000), the shrinking
and swelling of wood in a variety of situations from its
role in sealing boat hulls to the warping of wooden
floors, the rippling of moistened paper, the curling of
fallen drying leaves, the wrinkling of wet skin, the wilt-
ing of flowers, the movement of wheat awns (Elbaum
et al. 2007), the self-burial of certain seeds (Erodium
cicutarium;Stamp 1984) and the shrinkage of certain
mosses (Funaria hygrometrica;Arnold 1899).
A common example of a robust natural hygromorph
is the pine cone, famous for the static phyllotactic pat-
terns of its scales. While tree-bound cones are closed,
fallen cones are invariably open. But when dead fallen
dry cones are moved into a humid environment, they
close and open again when dried, an experiment that
may be repeated many times. Here, we use this example
to first explore the dynamics of shape change at the
organ, tissue and cellular levels and use a simple
model inspired by the thermomechanics of the bimetal-
lic strip to quantify our observations and compare the
behaviour of cones that range in size from a few milli-
metres to 30 cm. Inspired by pine cones, we also build
bilayer hygromorphs of a thin layer of plastic attached
to paper which curl upon drying and straighten on wet-
ting. We quantify the dynamics of these biomimetic
hygromorphs by considering the mechanics of flow
through a porous medium with an evaporative flux
and compare the theory with experimental obser-
vations. We conclude with a discussion of the large geo-
metrical amplification provided by slender bodies in
hygromorphs.
2. PINE CONES
Pine cones offer a common example of how a structured
tissue responds to an environmental stimulus. A change
in the relative humidity causes a closed, tightly packed
cone to open gradually (electronic supplementary
material, movies S1 and S2), as shown in figure 1a.
The mechanism leading to cone opening when dried
(and closing when wetted) relies on the bilayered struc-
ture of the individual scales that change conformation
when the environmental humidity is changed (elec-
tronic supplementary material, movie S3). In
figure 1b, we show that the deformation is localized to
a small region close to where the scale is attached to
the midrib of the cone while the rest of the scale
simply amplifies this motion geometrically. In this
active outer layer of tissue, closely packed long
parallel thick-walled cells respond by expanding
longitudinally when exposed to humidity (Harlow
et al. 1964;Dawson et al. 1997), and shrinking when
dried, while the inner passive layer does not respond
as strongly. Consequently, the tissue behaves like a
thermally actuated bimetallic strip that curves in
response to temperature changes because of the differen-
tial expansion of the constituent strips that are glued
together.
In figure 1d, we show that a scale from a cone of
Pinus coulteri can rotate by angles as large as 1008
when the relative humidity of the active tissue
(measured by weighing the scale with an Ohaus Adven-
turer AR1530 balance) drops from saturation to almost
zero. This macroscopic change is related to the micro-
scopic humidity-induced strain in the tissue cells; our
measurements at the individual cell level in an environ-
mental scanning electron microscope (figure 1c) show
that the hygrometric expansion coefficient of the cells
that characterizes the strain per relative humidity
a
¼
0.001 +0.0001 (figure 1e), in agreement with measure-
ments at the macroscopic scale (Harlow et al. 1964). We
note that
a
measures the expansion of the cells along their
axis; no significant strain was measured in the radial
direction. This is consistent with the observation of stif-
fening fibres wound helically around the cell wall that
prevent the cell from increasing its girth. Furthermore,
*Author for correspondence (lm@deas.harvard.edu).
Electronic supplementary material is available at http://dx.doi.org/
10.1098/rsif.2009.0184 or via http://rsif.royalsocietypublishing.org.
J. R. Soc. Interface (2009) 6, 951–957
doi:10.1098/rsif.2009.0184
Published online 1 July 2009
Received 11 May 2009
Accepted 4 June 2009 951 This journal is q2009 The Royal Society

0.5 1.0 1.5 2.00
100
200
300
ha (mm)
m (mg)
τ (min)
φ (relative humidity at 20°C)
strain, ε
(b)
(e)
dry
wet
dry
wet
dry
wet
(a)
(d )
(g)( f )
200 µm
(c)
θ
100 200 300 400 5000
10
20
30
40
50
150
100
50 0.05
0.10
00 200 400 600 800 20 40 60 80 100
t (min)
θ (°) θ (°)
(i) (i)
(ii)
(ii)
Figure 1. (a) Cone from Picea abies in its wet (closed) and dry (open) states. (b) Cone scale in its wet (i) and dry (ii) states. We
call
u
the angular position of the scale, the dry state being chosen as a reference (
u
¼08). (c) Environmental scanning electron
microscopy (ESEM) images of a few cells from the responsive tissue of a scale of Pinus coulteri. The cells are about 20 per cent
longer in the wet state (i) than in a dry environment (ii). (d) Opening angle of a scale of P. coulteri as a function of its water
content. The reference of angle is chosen for a dry sample. (e) Single-cell ESEM measurements of strain– humidity relationship
in the active tissue of a scale of P. coulteri. Strains are measured along the axis of the cells, the radial expansion being negligible.
The line is a linear fit to the data with equation 1¼0.0011
f
þ0.0033. ( f) Plot of the opening angle
u
of a scale versus time,
when immersed in water and then drying in ambient air. The opened (dry) position was chosen to be the zero angle reference.
(g) Opening and closing times of scales of different sizes as a function of the thickness h
a
of external (i.e. responsive) layer of cells.
Filled circles are for opening times (in a 208C and 40% humidity environment) and open circles for closing times.
952 Report. Hygromorphs E. Reyssat and L. Mahadevan
J. R. Soc. Interface (2009)

over this humidity range, the expansion of the passive
layer in the tissue was also found to be negligible. To
understand the equilibrium bending of the tissue
bilayer, we adapt the theory of bimetallic thermostats
to hygrometry sensors (Timoshenko 1925), as the
actuation mechanism of pine cone scales relies on the
same principle, with humidity replacing temperature.
We assume that the thicknesses of the constituent pas-
sive and active layers are h
p
and h
a
(with a total thick-
ness h¼h
p
þh
a
), their Young’s moduli are E
p
and E
a
and the linear hygrometric expansion coefficients are
a
p
and
a
a
, assumed to be independent of the humidity
f
. Under a change D
f
of the relative humidity
f
, the
strain of an unconstrained layer of material iis then
given by
a
i
D
f
.If
a
¼
a
a
2
a
p
=0, the differential
expansion of both layers results in bending of the strip.
In the absence of external forces, all forces acting on
any cross section of the bilayer must be in equilibrium
Fp¼Fa¼Fð2:1Þ
Fh
2¼MpþMa;ð2:2Þ
where F
p
and F
a
are the axial forces. If M
p
and M
a
are
the bending moments in each of the layers, and the
bending rigidity (per unit width) of both layers is
E
p
h
p
3
/12 and E
a
h
a
3
/12, torque balance requires that
Fh
2¼D
k
Eph3
p
12 þEah3
a
12
!
;ð2:3Þ
where D
k
is the curvature change of the bilayer. Finally,
the displacement of both layers must be equal at the
contact surface of both layers so that
a
pD
f
þFp
EphpþD
k
hp
2¼
a
aD
f
þFa
EahaD
k
ha
2:ð2:4Þ
Combining equations (2.2)– (2.4) yields the change in
curvature due to a change D
f
in relative humidity
D
k
¼
a
D
f
fðm;nÞ
h;ð2:5Þ
where
fðm;nÞ¼ 6ð1þmÞ2
3ð1þmÞ2þð1þmnÞðm2þ1=mnÞð2:6Þ
and m¼h
p
/h
a
,n¼E
p
/E
a
.Infigure 2, we plot f(m,n)
as a function of the thickness ratio mfor different
values of the moduli ratio n. We note that for strips
of comparable thicknesses (m¼1), f(m,n) changes by
less than 10 per cent when nvaries from 0.3 to 3.5. In
the limit where nis 0 or þ1, one of the layers is infi-
nitely more rigid than the other, and f(m,n)¼0: no
bending occurs.
To compare this analytical result with our obser-
vations of pine cones, we measured the thicknesses of
the layers of pine cone tissues and the length of the
active zone for a range of cones. In figure 3, we show
the results of these measurements and note that the
thickness ratio m¼h
p
/h
a
’1, while the length of the
active zone L
a
’10h
a
. Prior experimental measure-
ments of the moduli of the bilayer tissues (Dawson
123450
0.5
1.0
1.5
m = hp/ha
f (m,n)
Figure 2. The dimensionless factor f(m,n) characterizing the
curvature change (see equations (2.5) and (2.6)) as a function
of the ratio of thickness of the passive layer to the active layer
m¼h
p
/h
a
and the ratio of the Young’s moduli n¼E
p
/E
a
¼
0.3 (solid line), 1 (dashed-dotted line) and 5 (dotted line).
For strips of comparable thickness (m¼1), the prefactor is
not very sensitive to changes in n:f(m,n) changes by less
than 20 per cent when nvaries from 0.3 to 5.
20 40 60 800
5
10
15
L
La / ha
0.5
1.0
1.5
2.0(a)
(b)
m = hp/ha
Figure 3. (a) Ratio mof thicknesses and (b) aspect ratio L
a
/h
a
as a function of the length Lof scales from different pine
cones. Although the data are strongly scattered, both mand
L
a
/h
a
are found to be weakly dependent on the scale’s size L.
Report. Hygromorphs E. Reyssat and L. Mahadevan 953
J. R. Soc. Interface (2009)

et al. 1997) suggest that n’5, so that from equation
(2.6) it follows that f(m,n)’1.3. Then the expected
curvature change of the scale is D
k
1.3 cm
21
(for
h1 mm and D
f
¼100, i.e. relative humidity chan-
ging from zero to saturation), and the angular ampli-
tude D
u
is D
k
L
a
, where L
a
is the length of the
deforming region of the scales. As f(m,n)1.3 and
L
a
/h10 is roughly invariant across samples and
species as shown in figure 3,D
u
758under similar con-
ditions for scales from different pine cones. The
observed invariance of these geometrical parameters
implies that the amplitude of the movements of the
scales of the cone is invariant. This in turn implies
that the release mechanism of the seeds is equally effi-
cient for different cone species and sizes and suggests
a simple design principle based on isometry.
We now move from the constant-humidity equili-
brium response of the material at the cellular and
tissue levels to the kinetics of swelling and shrinking.
In a typical experiment, the pine scale is fully saturated
with water and then allowed to dry in ambient air at
228C and 40 per cent relative humidity, and figure 1f
shows the evolution of the opening angle of a scale
versus time. Repeating the experiment with cone
scales from the different species (see electronic sup-
plementary material) yields swelling and drying times
ranging from a few minutes to a few hours as shown
in figure 1g, increasing monotonically with the thick-
ness h
a
of the active hygroscopic tissue of the scale.
This is consistent with simple dimensional consider-
ations that dictate that the time required for water to
penetrate tissues increases with their thickness, in
qualitative agreement with a capillary imbibition mech-
anism for water transport in porous media. Gravita-
tional effects are negligible at the small scales we are
dealing with here, so that wicking arises from a balance
between surface tension forces that drive the process
and viscous processes that resist it (Washburn 1921):
the imbibition time scale
t
h
2
/D, where his the size
of the porous sample and D
g
‘
p
/
h
is the diffusion
constant constructed from the interfacial tension
of the liquid in the pores
g
, the pore size ‘
p
and
the viscosity of the pore fluid
h
. For molecular pores,
D10
29
m
2
s
21
, the thickness of the active layer
h
a
1 mm and
t
¼h
a
2
/D1000 s, in the range of
experimental results shown by the open symbols in
figure 1g. However, when the data are fitted by a
power law, the shrinking (drying) time varies as
t
d
¼
96h
a
1.5
min and the swelling (wetting) time varies as
t
d
¼29h
a
1.3
min (where h
a
is in mm), quite different
from the Washburn law. This large difference between
drying and wetting is to be expected due to the different
mechanisms responsible for these processes; liquid
water is driven by capillary forces while wetting, while
drying is dominated by the slow diffusion of vapour
through pores at late stages, which leads to large vari-
ations in the effective diffusion coefficient in drying
porous media (Pel et al. 1996;Lockington et al. 2003).
The coupling of large strains (up to 20%) to water
transport and localization of viscous dissipation at
a bottleneck in the material may account for this
discrepancy in the wetting stage; indeed, when a bottle-
neck of size l
b
dominates water transport, we expect
t
h
a
l
b
/D. As drying and wetting in heterogeneous
structures are crucially dependent on the distribution
of pore structure and connectivity at long times
(Delker et al. 1996), the large heterogeneity in the
drying response seen in our experiments is probably
due in part to the heterogeneities in tissue structure
from one species to another, combined with the multi-
plicity of water transport mechanisms, and currently
does not allow us to go easily much beyond a scaling
approach.
3. BIOMIMETIC BILAYERS
Inspired by these natural hygromorphs, to study the
dynamics of wetting and drying quantitatively, we
built active biomimetic bilayer structures with control-
lable characteristics. A simple model consists of a natu-
rally flat piece of paper about 5 cm long, 5 mm wide and
0.3 mm thick glued with epoxy onto a flat strip of poly-
mer. Here the active material is cellulosic paper that
swells while softening in a moist environment and
shrinks while stiffening in a dry atmosphere. In con-
trast, the passive polymer layer is insensitive to humid-
ity changes. In a dry atmosphere, the bilayer bends with
the papered side inwards (figure 4a), while in humid
environments, its curvature is reversed, albeit very
slightly, as wet paper is relatively soft compared with
the naturally flat plastic layer. The first wetting and
drying cycle ‘wears-in’ the system and leaves the dried
bilayer in a curved state. Subsequent cycles lead to
repeatable and reproducible behaviour (electronic sup-
plementary material, movies S4 and S5); when one
end of the bilayer is dipped in a water reservoir, the
curved bilayer straightens when it absorbs water and
swells, but dries and closes when removed from the
reservoir. Though the planar bilayer exhibits relatively
simple morphologies during wetting and drying, the
transient paths from one stationary state (when wet,
say) to the other (when dry, say) can be complex. For
example, during wetting, the end of the strip in the
water reservoir is almost straight, while the free dry
end remains curved. In contrast, during drying, homo-
geneous evaporation over the entire paper surface
leads to a uniform water content everywhere, so that
the time-dependent curvature is constant along the
strip. The temporal complexity of the shapes thus
arises from the inhomogeneous distribution of water
either along the strip or across its thickness.
To quantify the water movements during wetting
and drying, we place one end of the bilayer in contact
with a water reservoir, so that a wet front progresses
along the paper driven by capillary imbibition through
the porous structure. The driving force is of order
g
‘
p
per pore in the strip, ‘
p
being the typical pore size of
the absorbing medium, while the viscous resistance to
flow through the narrow pores at height zfrom the
reservoir on a segment dzof the strip is of order
h
v(z)dz, where v(z) is the mean velocity at location z.
Then force balance yields
g
‘p¼ðzm
0
h
vðzÞdz,ð3:1Þ
954 Report. Hygromorphs E. Reyssat and L. Mahadevan
J. R. Soc. Interface (2009)

where z
m
is the position of the wetting front. However,
evaporation from the exposed, wet paper surface implies
that the dynamics of wicking is altered. Assuming a con-
stant areal flux of water at a rate Jfrom the wetted paper
surface implies that the velocity of liquid is an affine
function of the distance to the wetting front: v(z)¼
v(z
m
)þJ(z
m
2z)/h
p
¼z
˙
m
þJ(z
m
2z)/h
p
,whereh
p
is
the thickness of the paper layer. Then equation (3.1)
reads D¼z
˙
m
z
m
þ(
d
/2)z
m
2
,withD¼
g
‘
p
/
h
and
d
¼J/h
p
. Integrating this last equation yields
1
zm¼ffiffiffiffiffiffiffi
2D
d
r1e
d
t

1=2ð3:2Þ
in good agreement with our experiments on the pro-
gression of the wetting front shown in figure 4c.At
0.10
(b)(c)
(d )0.08
0.06
0.04
0.02
0
80
60
40
20
0
0.05
(mm–1)
(mm–1)
zm (mm)
0
0 20 40 60 80 100 10 20 30 40 50
s (mm) t (min)
20 40 60 80 100
(relative humidit
y
at 22°C)
(a)
Figure 4. (a) Paper–plastic bilayers are sensitive to humidity. When one extremity is put in contact with a water bath, water
invades the texture by capillarity, inducing a change in the curvature (first three pictures in (a)). The drying process is homo-
geneous, making the curvature of the structure to be the same along the strip (last three pictures in (a)). The first three pictures
were taken 0, 9 and 100 min after contact with the water bath. The last ones were taken 8, 12 and 50 min after the beginning of the
drying process. (b) Curvature measured along the bilayer, as a function of the distance sto the water bath. s¼0 is the position of
the bath. Circle, 0 min; square, 1 min; plus, 2 min; triangle, 5 min; dot, 10 min; star, 40 min; inverted triangle, 119 min. (c)Impreg-
nated length z
m
of a strip as a function of the time after contact with a water bath. The wicking process saturates due to evaporation.
The full line is a fit of the experimental data (filled circles) with equation (3.2). The dotted line corresponds to the Washburn law in
the absence of evaporation. (d) Curvature variation of a bilayer as a function of the relative humidity of the atmosphere.
1
After this work was done, J. Bico (2008, personal communication)
informed us of his independent derivation of this result.
Report. Hygromorphs E. Reyssat and L. Mahadevan 955
J. R. Soc. Interface (2009)

short times, i.e. t
d
,z
m
¼ffiffiffiffiffiffiffiffi
2Dt
p, which is the so-
called Washburn law (Bell & Cameron 1906;Lucas
1918;Washburn 1921), while at longer times, evapor-
ation limits the impregnated length to ffiffiffiffiffiffiffiffiffiffiffi
2D=
d
p. The
observed shapes of these artificial hygromorphs then
result from the local strain in the bilayer, which is
simply slaved to the water content in the structure; sol-
ving for the shape of the hygromorph simply requires
determining the local curvature in terms of the theory
of bimetallic strips (Timoshenko 1925), which we will
not pursue here in detail. Qualitatively, the hygrometric
expansion coefficient
a
’410
24
, while the thickness
of the polymeric part of the bilayer h
p
¼200 mm and
the thickness of the hygroscopically active paper is
h
a
¼100 mm so that m¼2 and h¼300 mm. The exper-
imentally measured stiffness of the materials of both
layers yields n’0.4. Knowing
a
,mand nfor these
devices yields an expected curvature change of about
0.19 mm
21
comparable to the experimental observations
in figure 4d. The slender geometry of the biomimetic
hygromorph (h
paper
0.1h
cone
) compensates for the
weak hygroscopic response of paper (
a
paper
0.1
a
cone
),
so that comparable curvatures are seen in both the
natural and artificial hygromorphs. While drying,
the papered region loses water uniformly and thus
shrinks uniformly, so that the entire strip now curls
with a constant curvature that varies in time, as here
again the shape of the hygromorph is slaved to the
water content in the structure.
4. DISCUSSION
The pine cone with a hygromorphic hinge and the con-
tinuously deformable slender bilayer are examples of
how geometric amplification of motion is achieved in a
long, thin sheet or filament by coupling deformation
to swelling or shrinkage in the transverse direction.
This is a design that is repeatedly used in nature in
seed dispersal and burial strategies, and elsewhere,
and our results suggest a simple design principle that
nature has converged on in the context of pine
cones—isometric scaling implies an efficient mechanism
for opening and thus seed dispersal. Our experiments on
pine cones also suggest designs for bioinspired sensors or
actuators that respond to environmental variations in
humidity using responsive materials. Here, we limit our-
selves to an amusing toy by creating an artificial flower
with a dynamically controllable blooming and closing
response. In figure 5, we show our system of a few
paper– plastic bilayer petals, with the paper on the
inside. When the flower is dipped in water it blooms,
while when left to dry it closes (electronic supplemen-
tary material, movies S6 and S7).
Natural pine cones and their biomimetic counter-
parts including the simple floral mimic are driven pas-
sively by environmental humidity variations; the rate
of change of shape in these systems is controlled by a
combination of system geometry and the mechanical
and hydraulic properties of the material. Although the
rate of evaporative or capillary-driven water movement
in tissues is a relatively slow process with a strong
asymmetry between drying and wetting, it is possible
to achieve a switch-like response rather than the gra-
dual transition that we have explored: this simply
requires bilayers that are naturally doubly curved, i.e.
they are shell-like. Nature has already evolved many
examples of such systems such as the Venus flytrap
(Forterre et al. 2005) and related examples that use
actively controlled water movements within the tissue
coupled to a primitive calcium-based nervous system
to generate fast snapping movements. Indeed, these
active hygromorphs and their passive counterparts
may simply be stages in a convergent evolutionary pro-
cess that uses water movements to control the geometry
and dynamics of plant organs, a topic worthy of further
exploration.
We thank J. Dumais and I. Kulic
´for fruitful discussions and
D. Goldman for providing pine cone specimens and for help in
identifying most samples. We acknowledge support from the
French Ministry of Defense, the DARPA Programmable
Matter project and the NSF-Harvard MRSEC.
REFERENCES
Arnold, W. 1899 The cord moss and its allies. Bryologist 2,
52–55.
Bell, J. M. & Cameron, F. K. 1906 The flow of liquids through
capillary spaces. J. Phys. Chem. 10, 658–674. (doi:10.
1021/j150080a005)
Dawson, C., Vincent, J. F. V. & Rocca, A.-M. 1997 How pine
cones open. Nature 390, 668. (doi:10.1038/37745)
Delker, T., Pengra, D. B. & Wong, P.-Z. 1996 Interface
pinning and the dynamics of capillary rise in porous
media. Phys. Rev. Lett. 76, 2902– 2905. (doi:10.1103/
PhysRevLett.76.2902)
Elbaum, R., Zaltzman, L., Burgert, I. & Fratzl, P. 2007. The
role of wheat awns in the seed dispersal unit. Science 316,
884–886. (doi:10.1126/science.1140097)
Forterre, Y., Skotheim, J., Dumais, J. & Mahadevan, L. 2005
How the Venus flytrap snaps. Nature 433, 421–425.
(doi:10.1038/nature03185)
Figure 5. A floral mimic made of paper– plastic bilayer petals shows a controllable blooming and wilting response. When dipped
in water, the petals open and the flower blooms, while when the water supply is exhausted, the bilayer dries out and the flower
closes (see electronic supplementary material, movies S6 and S7). The length of the petals is about 4 cm, and the typical time
scale of a cycle is an hour.
956 Report. Hygromorphs E. Reyssat and L. Mahadevan
J. R. Soc. Interface (2009)

Harlow, W. M., Cote, W. A. & Day, A. C. 1964 The opening
mechanism of pine cone scales. J. Forestry 62, 538– 540.
Lockington, D. A., Parlange, J.-Y., Barry, D. A. & Leech,
C. A. 2003 Drying of porous building materials:
hydraulic diffusivity and front propagation. Mater.
Struct. 36, 448–452. (doi:10.1007/BF02481524)
Lucas, V. R. 1918 Ueber das Zeitgesetz des kapillaren Auf-
stiegs von Flu
¨ssigkeiten Kolloid.Zeitschrift 23, 15– 22.
Pel, L., Brocken, H. & Kopinga, K. 1996 Determination of
moisture diffusivity in porous media using moisture con-
centration profiles. Int. J. Heat Mass Transf. 39, 1273–
1280. (doi:10.1016/0017-9310(95)00201-4)
Shorlin, K. A., de Bruyn, J. R., Graham, M. & Morris, S. W.
2000 Development and geometry of isotropic and
directional shrinkage-crack patterns. Phys. Rev. E 61,
6950–6957. (doi:10.1103/PhysRevE.61.6950)
Stamp, N. E. 1984 Self burial behavior of Erodium
cicutarium seeds. J. Ecol. 72, 611–620. (doi:10.2307/
2260070)
Timoshenko, S. 1925 Analysis of bi-metal thermostats.
J. Opt. Soc. Am. 11, 233– 255. (doi:10.1364/JOSA.11.
000233)
Washburn, E. W. 1921 The dynamics of capillary flow. Phys.
Rev. 17, 273– 283. (doi:10.1103/PhysRev.17.273)
Report. Hygromorphs E. Reyssat and L. Mahadevan 957
J. R. Soc. Interface (2009)