Soft Matter
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Pattern formation in plants via instability theory of
hydrogels
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Zishun Liu,*ab Somsak Swaddiwudhipongc and Wei Hongd
In this paper, we demonstrate how deformation patterns of leaves and fruits in growing and drying
processes can be described via the inhomogeneous field theory. The distorted deformation of ribbed
leaves and the ridge formation on fruit surfaces can be understood as the energy-minimizing
mechanical buckling patterns. The swelling and de-swelling induced instabilities of various membrane
structures or elastic sheets on elastic or gel-like substrates are simulated using the inhomogeneous field
theory of a polymeric network in equilibrium with solvent and mechanical constraints. The article
describes briefly the inhomogeneous field theory of hydrogel deformation and the buckling patterns of
thin hydrogel films on thick substrates. The theory is then adopted to simulate the growth and drying
Received 16th July 2012
Accepted 11th October 2012
processes of leaves and fruits through the buckling phenomena observed in the film gel of various
shapes, geometric proportions, chemical potentials and mechanical constraints. The key idea is to show
that the hydrogel deformation theory can capture the deformation process and various states of plant
growth or drying. The study has been made in an attempt to mimic the shapes of fruits and leaves from
DOI: 10.1039/c2sm26642c
the swelling/deswelling patterns of hydrogel films. The study provides the possibility of exploring the
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origin of the intriguing natural phenomena of leaves and fruits.
Introduction
Natural growth is full of fascinating complex patterns and
shapes. The growth processes in plants have been observed to
produce various interesting complex three-dimensional shapes,
e.g. ribs on a saguaro cactus, parallelograms on cacti, and ridges
on a pumpkin surface. The geometrical features of plants or
phyllotaxis (i.e. the arrangement of leaves or other botanical
elements around a stem)1 has drawn much attention since
antiquity. It has long been recognized that plant leaves and
orets are organized in patterns consisting of whorls or spirals,
symmetric or antisymmetric. Pattern formations in plants and
animals are obviously varied and many. Thus, the following
interesting questions normally arise: how to explain these
patterns? Why does a squash have 10 equidistant longitudinal
ridges but a large pumpkin about 20 ridges? Why does a
cantaloupe show a reticular morphology which mixes ridge and
latitude patterns? What are the physical mechanisms of these
natural structural formations and deformations? How to
explain the morphogenesis and the natural growth of these
a
International Centre for Applied Mechanics, State Key Laboratory for Mechanical
Structure Strength and Vibration, Xi’an Jiaotong University, Xi’an, 710049, China.
E-mail: zishunliu@mail.xjtu.edu.cn
b
Institute of High Performance Computing, A-STAR, Singapore 138632. E-mail:
zishunliu@nus.edu.sg; Fax: +65 6467 4350; Tel: +65 64191289
c
Department of Civil and Environmental Engineering, National University of
Singapore, Singapore 119260
d
Department of Aerospace Engineering, Iowa State University, Ames, IA 50014, USA
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plants? How can we connect the range of these phenotypes with
gene expressions as currently understood? For example, in the
pine cone and sunower head, the pattern displays a crossed
spiral. How to explain this crossed spiral? Francis Darwin once
said that “the fascinating plant patterns can drive the sanest
man mad”. A simple and universal answer may be that the
patterns are caused by buckling and wrinkle deformation when
plants and animals are evolving or growing.2,3
Although the answers are not yet clear, there exists experimental evidence that chemistry and biophysics play certain roles
in pattern formation of plants and animals, e.g., growth hormones
as chemical signals, and growth forces as physical signals,
affecting the apical materials.4,5 Despite the complex biochemical
and biological processes involved in the growth of plants,
mechanics does affect the growth and drying processes of plants.
It was Simon Schwendener, a Swiss botanist, who rst studied the
role of biophysics in the growth of plants in the late 1800s.
However, he did not realize the connection between pattern
formation and the material properties of plants. From the 1990s,
Green, Steele and their co-workers6–8 have demonstrated through
experiments, analyses and simulations that the mechanical stress
and deformation play a key role in the pattern formation of plants.
Various forces including those from the environment acting on
the plant surface inuence the choice of plant patterns.4 There are
correlations between the regions of compressive stress on a plant
shoot and the regions where phyllotactic patterns are formed.
Most patterns of plants show that the growth processes minimize
the total potential energy of the plant surface.
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Soft Matter
Since the 1990s, many investigators have adopted the principles of mechanics to explain natural phenomena such as
deforming patterns of plants.9–14 Besides the research work on
plant morphogenesis and phyllotaxis, which were studied from
botanical and biological points of view,15,16 some pioneering
work on pattern formation of plants with mechanical explanations will be reviewed herein. Green and Steele explained the
phyllotactic patterns of plants from a mechanics point of view
in their series of work.1,6–8,17 They adopted the beam, plate and
shell buckling theories to characterize and explain the patterns
of plant organs. They observed through the study on the mode
shapes and buckling patterns of annular rings that the patterns
are similar to the undulating patterns seen in plants and owers
originating in an annular region. They also showed that the
buckling mechanism is workable and robust for the initiation of
whorls of plant growth. Shipman, Newell, and co-workers
investigated the phyllotactic patterns of plants using a mathematical description.1,12,13,18–20 Through the theory of energy
minimization of buckling patterns of structures, they demonstrated how phyllotaxis and the ribbed, hexagonal, or parallelogram planforms on plants can be explained. They adopted the
Föppl–von Kármán–Donnell (FvKD) equations, the linear
stability analysis and the weakly nonlinear analysis to describe
the competition among the planforms on plant surfaces. Thus
they explained the kinetics of pattern transmission of plants via
a mathematical description for phyllotaxis.12
The morphogenesis of complex growth patterns involves
biological, chemical and physical factors which are multiscale
phenomena. The morphogenesis of plant growth was explained
and investigated by Goriely’s group21,22 from a mathematical
point of view. For example, Goriely et al.23 demonstrated the
possibility of spontaneous growth-induced cavitation in elastic
materials and considered the implications of this phenomenon
for biological tissues and in particular for the problem of
schizogenous aerenchyma formation. Another intriguing
problem in their analysis had shown that vertical growth is
achieved by discrete contact points and regions with continuous
contact, that the contact pressure creates tension in the stem as
observed experimentally, and that there is a maximal radius of
the pole around which a twiner can climb. Dervaux and
Amar’s24–26 mathematical description incorporated biochemical
details with a continuum mechanical framework to demonstrate the amplitude and direction of growth. In their work, the
equilibrium shapes of growing bodies were evaluated through
the minimization of appropriate energy. Their model was used
to explain the morphologies of geometry relevant to nuts, pollen
grains, leaves, petals and algae. Furthermore, they applied
Föppl–von Kármán theory to study the growth-induced pattern
of cockling of paper and the rippling of a grass blade. The major
difference between the gel model and the growth model of
earlier studies21,22,24–26 is that the growth deformation (volume
expansion) in the latter is set to be independent of the stress
state, while in the present gel model, the swelling ratio is highly
dependent on the local hydrostatic stress.
Mahadevan’s research group adopted a simple mathematical model and a combination of scaling concepts, stability
analysis, and numerical simulations to study the mechanisms
578 | Soft Matter, 2013, 9, 577–587
Paper
of various plant growth processes and deformations.27–30 For
example, Forterre et al.27 used high-speed video imaging, noninvasive microscopy techniques and a very simple theoretical
model to demonstrate that the fast closure of the trap results
from a snap-buckling instability, the onset of which is
controlled actively by the plant. Their study provided a general
framework for understanding nasty motion in plants. In the
study, to model the growth dynamics of snapping of plants they
treated the leaf tissue as a poroelastic material. They demonstrated that when the leaf snaps shut, the stored elastic energy is
dissipated via the viscous ow within the leaf tissue. The
evolution of vascular plants28 was investigated by using microuidic devices that allow biomimetic studies of plants. From
their study, it is shown that the same optimization criterion can
be used to describe the placement of veins in leaves. The origin
of long leaf morphologies was studied by using a combination
of scaling concepts, stability analysis, and numerical simulations.29 They found that as the relative growth strain is
increased, a long at leaf deforms to a saddle shape and/or
develops undulations. More recently, Gerbode et al.30 used
physical models of prestrained rubber strips and mathematical
models of elastic laments to explain the behavior of helical
coiling of cucumber tendrils. The work illuminates the origin of
tendril coiling.
Recently, Chen and his co-workers studied the deformation
patterns of fruits from the buckling theory of shell structures.31,32 They used the combination of theoretical and
numerical analyses to establish a quantitative mechanics
framework of elastic buckling of spheroidal-thin-lm-substrate
systems, to explain the morphologies of natural and biological
systems. A possible interaction pathway was postulated based
on the simple mechanical principles and the morphogenesis of
some natural and biological systems. The materials used to
model the plants in the above-mentioned studies were all
assumed to be linearly elastic, while the volume (or mass)
changes were not considered. In fact, the growth and drying of
plants and their changes of patterns are related to volume
changes and hence more realistic materials should be considered. Recently, Liu et al.33 attempted to use the inhomogeneous
gel theory to investigate the growth patterns of leaves and
owers, and reported that the volume changes would affect the
buckling and wrinkle processes.
In this study, a nonlinear inhomogeneous polymeric gel
theory is used to investigate gel behavior, and the resulting
buckling and wrinkle patterns are used to describe various plant
congurations in growth and drying processes. The present
work differs from others as we adopt gel theory to explain plant
growth and drying, considering volume changes as observed in
nature. The remainder of the paper is divided into three main
sections. In the second section, the thermodynamics of a gel in
equilibrium and the constitutive equations on the basis of a
nonlinear eld theory of swelling or deswelling due to Gibbs
theory34,35 are presented to predict the values of compressive
stress generated by swelling and deswelling. The stability
analysis is performed in the third section on a swollen gel lm
under compression, while attached to a substrate of gel or other
so elastic material. We present the analytical solutions of
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Soft Matter
swelling-induced instability of various thin-lm gel structures
by applying the constitutive relations of inhomogeneous eld
theory with an incremental modulus. The expressions for
buckling and wrinkle conditions and the critical stress of thin
lm gel structures are derived from the nonlinear buckling
theory of swelling gels. Then we consider the thin lm gel on an
elastic foundation of substantially different stiffness. This
section identies the buckling or critical wrinkle conditions and
predicts the critical stress and the corresponding buckling
wavelength. The section provides a theoretical basis for both
single- and multilayered gel-like material buckling and wrinkle
conditions under swelling and/or deswelling. These buckling
and wrinkle patterns of various models can be used to explain
the forms and shapes developed during the growth and drying
processes of plants and fruits.
Inhomogeneous field theory of a gel
In this section, we briey introduce the gel inhomogeneous
deformation theory. For more details, readers may refer to ref.
34–36. Consider a network of polymers in contact with a solvent,
subjected to mechanical loads and geometric constraints, and
held at a constant temperature. If we take the stress-free dry
network as the reference state, the deformation gradient of the
network is dened as35
FiK ¼
vxi ðXÞ
vXK
(1)
where XK and xi are the network coordinates of the gel system at
reference and deformed states, respectively. To describe the
solvent in the gel system, a new quantity C(X)dV(X) which
represents the number of solvent molecules in the element of
volume dV is introduced. Thus, the state of the gel system can
be described by two elds, namely (i) the deformation of the
network, xi, and (ii) the distribution of the solvent molecules in
the gel system, C. The free-energy density of the gel system, W, is
thus W ¼ W(F,C). When the gel equilibrates with the solvent and
the mechanical load (external load or geometrical constraint),
the chemical potential of the solvent molecules is homogeneous
in the gel system and in the external solvent, and thus
m¼
vW ðF; CÞ
:
vC
(2)
The thermodynamics principle requires that the change in
the free energy of the gel should equal the sum of the work done
by the external mechanical force and by the external solvent,
namely,
ð
ð
ð
ð
dW dV ¼ Bi dxi dV þ Ti dxi dA þ m dCdV :
(3)
This equation is the primary equation of the two-eld gel
theory. The rst and second terms on the right hand side of eqn
(3) represent the mechanical work done by the body force BidV
and the surface force TidA, as normally encountered in solid
mechanics. The main difference between eqn (3) and that of
classical solid mechanics is the presence of the third term in the
former, representing the work done by the external solvent in
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gel deformation. Eqn (3) holds for any small changes dx and dC
from the state of equilibrium. When the gel is in a state of
equilibrium, the chemical potential of the solvent molecules
inside the gel is homogeneous and is equal to the chemical
potential of the external solvent, m. At the same time, all
molecules in a gel are taken to be incompressible, so that the
volume of the gel is the sum of the volume of the dry network
and the volume of the pure liquid solvent. We denote v as the
volume per solvent molecule and hence the concentration of
the solvent in the gel relates to the deformation gradient of the
network as34,35
1 + vC ¼ detF.
(4)
To adopt the standard solid mechanics approach, a new freeenergy function Ŵ is introduced via Legendre transformation,
Ŵ ¼ W mC, where Ŵ is a function of the deformation gradient
of the network and the chemical potential of the solvent
molecules, i.e. Ŵ (F,m). Adopting the new free-energy function
Ŵ , the primary eqn (3) becomes
ð
ð
ð
^ dV ¼ Bi dxi dV þ Ti dxi dA:
dW
(5)
The equilibrium condition (5) takes the same form as that
observed in classical solid mechanics. Once the function Ŵ (F,m)
is prescribed, we can solve eqn (5) via either analytical or
numerical method as normally done in solid mechanics.
The behavior of a gel is mainly entropic in the gel system. As
the solvent molecules mix with the long-chained polymers, the
network swells to reduce the entropy of the network but increase
the entropy of the mixture. The balance of the two contributions
to entropy equilibrates the network and the solvent. According to
the well known Gibbs and Flory and Rehner37 theories and the gel
incompressible eqn (4), the modied free-energy function for the
polymeric gel system can be expressed as35
^ ðF; mÞ ¼ 1 NkTðI 3 2log JÞ
W
2
kT
J
c
m
ðJ 1Þ log
þ
ðJ 1Þ;
v
J 1 J
v
(6)
where I ¼ FiKFiK and J ¼ detF are invariants of the deformation
gradient, N is the number of polymeric chains per reference
volume, kT is the absolute temperature in the unit of energy,
and c is a dimensionless measure of the enthalpy of mixing.
When c > 0, the solvent molecules interact among themselves
more than the long-chained polymers. In this free-energy
function of gel, the deformation gradient of the network, F, and
the chemical potential of the solvent, m, are two dependent
variables. A representative value of the volume per molecule is
v ¼ 1028 m3.38 At room temperature, kT ¼ 4 1021 J and
kT/v ¼ 4 107 Pa. In the absence of solvent molecules, the dry
network under the small-strain condition has a typical shear
modulus value of NkT ¼ 104 to 107 N m2, and hence the range
of Nv ¼ 104 to 101.
The nominal stress as the work conjugates to the deformation gradient can be dened as
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siK ¼
^ ðF; mÞ
vW
:
vFiK
(7)
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Inserting eqn (6) into (7), we can obtain the equation of state
for the gel:
siK
1
c
m
þ1þ J HiK :
¼ NvðFiK HiK Þ þ J log 1 J
J kT
kT=v
(8)
This is the constitutive equation of a gel system. In the
implementation of the inhomogeneous eld theory of a polymeric gel, the free energy in eqn (6) is singular when the
network is solvent-free, vC ¼ 0. To avoid this singularity, we
choose a reference state such that the network, under no
mechanical load, equilibrates with a solvent of chemical
potential m0, vC > 0. Relative to the dry network, the network in
this free-swelling state swells with isotropic stretches. We
denote this stretch by l0, which relates to the chemical potential
m0 by setting stress in eqn (8) to zero. This can be implemented
in the nite element package, ABAQUS, by coding eqn (6) into a
user-dened subroutine for a hyperelastic material (UHYPER)
or UMAT.35 Once ready, any complex boundary value problems
can be solved by using the FEM method.
The buckling and wrinkle analyses of a film
gel on a substrate
In the gel swelling or deswelling process, the buckling or
wrinkle of the gel system may take place. Many researchers have
studied the phenomena via either experimental or analytical
approaches.39–46 In this section, we will use some key results
from the existing literature36,47–50 to derive the expression for the
buckling condition of a thin lm gel layer bonded to another gel
layer or an elastic substrate. When the gel is exposed to a
solvent, the gel layer can swell to its equilibrium state, and if no
constraint, it is stress-free. However, when the gel layer is
conned to a rigid or stiff substrate, it can swell freely only in
the direction perpendicular to the substrate, but not in the
plane of the substrate (Fig. 1). The in-plane constraints of
deformations generate the in-plane compressive stresses.
According to the inhomogeneous gel theory, the in-plane
compressive stresses can be expressed as36
vs1
1
¼ Nv l1 kT
l1
þ l1 l2 l3 log 1 1
l1 l2 l3
þ1þ
c
m
l1 l2 l3
l1 l2 l3 kT
1
;
l1
(9)
vs2
1
¼ Nv l2 l2
kT
þ l1 l2 l3 log 1 1
l1 l2 l3
þ1þ
c
m
l1 l2 l3
l1 l2 l3 kT
1
;
l2
(10)
where indices 1 and 2 represent the in-plane quantities and 3
implies those in the perpendicular direction to the substrate.
The gel lm is stress free in the normal direction:
1
1
Nv l3 þ l1 l2 l3 log 1 l3
l1 l2 l3
þ1 þ
c
m
l1 l2 l3
l1 l2 l3 kT
1
¼ 0:
l3
(11)
Eqn (11) provides the relationship between the in-plane
stretches l1, l2 and the transverse stretch l3. As the gel swells with
the changing solvent environment, the stretches in the three
directions are l1 ¼ l2 ¼ l0, l3 ¼ l0l, where l is the stretch in the
lm normal, z, direction and the value is relative to the initial free
swelling state. When the generated in-plane compressive stresses
reach the critical values, the gel lm will buckle or wrinkle. These
phenomena have been observed in many experiments.47,51 For
simplication purpose, we assume that the in-plane stretches are
constant l1 ¼ l2 ¼ l0 and eqn (9) and (11) give
vs1
l3 2
¼ Nv l1 :
(12)
kT
l1
The buckling of a compressed elastic layer bonded to
another elastic layer has been studied by several researchers
recently.49,50,52 Allen analyzed the buckling problems by
approximating the deformation of the top layer by Euler’s linear
plate theory, with the Hookean constitutive equations for the
bottom layer.53 For a thin elastic lm bonded on the elastic
substrate, the buckling stress and critical wavelength are given
by the classical formulae,53,54
2=3
scr
1
Ef
¼
(13)
31=3
Ef
Es
Fig. 1 Schematic diagrams of swelling a thin gel film atop an elastic or gel
substrate before and after buckling.
580 | Soft Matter, 2013, 9, 577–587
1=3
lcr
2p
Ef
¼
h
Es
31=3
(14)
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where Ef is the elastic modulus of a thin lm with thickness h,
and Es is the elastic modulus of a thick substrate. lcr is the
critical wavelength at post-buckling.
While a gel lm is swelling, the effective elastic modulus
varies with the gel swelling deformation and chemical potential,
and the above equations cannot be directly applied. Introducing
the incremental modulus of a gel under swelling and using gel
constitutive equations, the buckling stress and critical wavelength for the swelling gel lm on an elastic substrate system
are,36 respectively,
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s1cr ¼
2=3
1=3
4p2 D 16p4 D
A 16p4 D
þ
h3
h3 A
2p2 h3 A
1=3
lcr
16p4 D
¼
h3 A
h cr
(15)
(16)
The current rigidity of a thin lm gel, D, can be expressed as
D ¼ h3NkT(1+l2)l3/12. The constant A has the form A ¼ 2pEs/((3
ms)(1+ms)), where ms is Poisson’s ratio of the substrate
material.
For a bi-layer system of a hard lm and a so substrate, the
buckling stress and critical wavelength can be obtained by
replacing Es with the so gel elastic modulus. Considering the
incompressibility of the gel, eqn (15) and (16) become
1=3
lcr
30p4 D
¼
h3 Esgel
h cr
s1cr ¼
2=3
1=3
4p2 D 30p3 D
4Esgel 30p3 D
þ
h3
h3 Esgel
15p h3 Esgel
(17)
(18)
Esgel represents the tangential modulus of the substrate gel
material. The above equations can be used to predict the critical
buckling and wrinkle values of the two-plied gel system.
The problem of wrinkle of an elastic gel top layer bonded to
an elastic bottom layer was also investigated by Basu et al.48
Although Allen53 analyzed the buckling problems by approximating the deformation by linear theory, and did not consider
shear stresses between layers, his results were similar to those
obtained by Groenewold55 via minimizing the bending and the
compressive energy of both layers together. Thus, Allen’s results
are still valid. Huang49 studied extensively the problems of the
buckling of an elastic top layer of thickness h bonded to another
viscous bottom layer or elastic bottom layer, which in turn is
bonded to a rigid substrate. In his problem, the top layer is
biaxially compressed. For the elastic substrate, the equilibrium
amplitude of the wrinkles of the top layer was derived by
modeling the deformation of the top layer by von Karman’s
nonlinear plate theory. Huang49 noted that the equilibrium
amplitude of the wrinkle was a function of the in-plane
compressive stress, wave number, the ratio of moduli of the two
layers, and the ratio of substrate thickness to top lm thickness.
For the two layered gel system as presented in this study, we
assume the thickness of the second layer substrate to be
substantially larger than that of the top thin lm wave amplitude. Huang49 showed that the equilibrium wavelength of the
wrinkle a can be derived to be
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
lcr 1 nf 2
a¼
p
s1cr
ðphÞ2
1 Esgel lcr
Ef
3ð1 nf 2 Þlcr 2 g22 Ef ph0
!1=2
(19)
where g22 ¼ 2(1 ms). Considering that the gel is incompressible, the above equation can be simplied to
!1=2
pffiffiffi
lcr 3
s1cr 4p2 h2 Esgel lcr
a¼
(20)
2p
Ef
Ef ph0
9lcr 2
The buckling and wrinkle of a two-layered gel system under
swelling can thus be analyzed. The wrinkle of a swelled,
deswelled, elastic gel layer bonded to another gel layer or elastic
bottom layer can be used to explain the buckling phenomenon
of the fruit skin while growing. Although the fruit surface is not
a plane surface, it resembles a thin layer on the thick substrate
and hence the theory can be applied with reasonable accuracy.
Application of gel film buckling and wrinkle
To explain owering cabbage/leaf growth by using annular
membrane gel swelling
The study on corona geometry membrane gel swelling was
carried out through analytical, numerical and experimental
methods.33,56 In Mora and Boudaoud’s56 experimental work, a
disk of stiffer gel is clamped to a corona of soer swelling gel,
with no swelling for hard gel. They observed that when the
assembly of membrane gel is swelling, the at gel plate deforms
and nally reaches the unstable buckling state. In our previous
work,33 a similar problem was studied by considering corona
geometry membrane gel buckling while swelling. Liu et al.33
indicated that the shapes of some plants may be described
using inhomogeneous gel deformation theory. The annular
membrane gel shapes can be further studied with various ratios
of inner radii and thickness to model different layers of cabbage
ower/leaves.
In the present study, the inner edge of the annular
membrane gel is assumed to be very stiff and hence there is no
swelling deformation, suggesting that the gel is clamped at the
inner edges. The schematic simulation model of constraint on
the inner circumferential surface of the annular membrane gel
is shown in Fig. 2. To study the buckling deformation behavior
of the membrane gel structure, the ratio of the outer radius of
annular membrane gel and initial thickness, Ro/h0, is kept
constant and greater than 15. A systematic study has been
carried out for various values of the ratio of the inner radius of
annular membrane gel to thickness, namely, Ri/h0 ¼ 5–18. In
the simulation, we assume that the annular membrane gel has
Fig. 2 Schematic simulation model of constraint on the inner circumferential
surface of the annular membrane gel (no swelling is assumed for the hard gel,
while the soft gel can swell).
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Fig. 4 Mimic of flowering cabbage flower/leaves. The buckling shapes and the
deformation patterns of swelling corona membrane gels can be used to explain
the flower patterns. The different parts of flowering cabbage can be modeled by
using different ratios of inner radius and thickness.
Fig. 3 The buckling shapes and the deformation patterns of corona membrane
gels of various inner radii (the ratio of outer radius and thickness, Ro/h ¼ 20).
tissue growth along the leaf outer edges induces the wrinkling
and wavy patterns. We also observe in nature that owers or
leaves do not display the wavy pattern in their seedling stage.
The wrinkling and wavy patterns appear only aer growth reaches a certain level. From a mechanics point of view, the
compressive stress level has not yet reached its critical value and
hence no buckling is observed during the seedling stage. With
the growth of plants, the internal stress increases and the
buckling phenomenon arises from the energy minimization of
the plant under growth. This wrinkling that creates the wavy
patterns, such as those in owering cabbage leaves as shown in
Fig. 4, allows relaxation of the resulting strain in the leaf growth.
The different parts of owering cabbage can be modeled by
using different ratios of inner radius and thickness. In the head
bud part of owering cabbage, a less wavy pattern is observed
and this part resembles the buckling pattern of a membrane gel
with a smaller ratio of inner radius and thickness. The outer
part can be reected via the buckling pattern of a membrane gel
with a larger ratio of inner radius and thickness.
an initial chemical potential and then starts swelling. When the
chemical potential rises, the membrane gel swells further,
causing an increase in compressive stress due to the constraint
along the inner edge of the membrane gel. When the stress level
reaches its critical value, the deformed gel lm system becomes
unstable aer reaching the nal buckling state.
The nal deformed buckling patterns of the membrane gel
during swelling are illustrated in Fig. 3(a)–(h). As expected, it is
observed that the larger number of the buckling wave is associated with the increase in the values of the Ri/h ratio. The
results are consistent with Mora and Boudaoud’s56 experimental observations and buckling theory. When the inner
radius is small or at a lower value of Ri/h ¼ 6 or at large values of
Ro/Ri, the nal deformed pattern is the same as that for a
circular plate where a hyperbolic paraboloid or saddle shape is
normally generated.57 As Ri/h approaches 16–18, the deformation pattern is similar to a strip gel lm.33
Since many plant tissues can be regarded as polymeric gels
to a certain extent, the deformation patterns of these corona
membrane gels can be used to explain some ower/leaf
patterns. In nature, the wavy patterns or shapes can be observed
along the outer edges of the owering cabbage leaves. During
the growth of the corona shaped leaves, the inner part of the
circular leaves is normally stiffer than the outer surface and the
Fig. 5 Leaves of an episcia plant and the simulated model: (a) A typical leaf of
episcia plant. (b) A simulated model comprising a thin membrane gel and elastic
frames. The thin membrane is used to represent leafage and the elastic frames are
used to represent the leaf stem, leaf midrib and leaf vein.
582 | Soft Matter, 2013, 9, 577–587
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Membrane gel swelling and deswelling to mimic leaf growth
and drying
In nature, leaves of various plants display a fascinating variety of
complex patterns and shapes. These natural phenomena have
long drawn attention from botanical scientists. They normally
explain these via genetic effects of different species of plants.
Other approaches including physical mechanisms can also be
adopted to explain the morphogenesis and the natural growth
of leaves. Earlier studies demonstrated that mechanical effects
also play a signicant role in the growth and drying deformations of leaves.8,17,33 As many plant tissues consist of polymeric
gel like materials, the buckling patterns of a gel membrane
under swelling and deswelling may be used to explain this
interesting deformation pattern. Strictly speaking, gels are
different from living tissues of plants, but the mechanism of
leaf growth is similar to the swelling/deswelling phenomena of
a thin lm gel.15,16,33 In our previous work,33 we have presented
an analogy of plant leaf growth and drying processes by
observing the swelling and deswelling of gel materials. In the
current study, we use the polymeric gel inhomogeneous deformation theory to mimic the drying of episcia leaves.
A model with an initial shape and the same geometry of
episcia plant leaf is constructed as shown in Fig. 5. The leafage
Fig. 6 Deswelling process of a membrane gel leaf model. No buckling is
observed for stages (a) and (b). Further deswelling causes the out of plane
deformation due to buckling in stages (c), (d) and (e). The color in (a) to (e)
indicates the deformation level.
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Fig. 7 Comparison of the final drying shape of a leaf with the simulated
deformation pattern: (a) drying pattern, and (b) simulation pattern.
is modeled by a polymeric gel material with certain initial
chemical potential and initial shape of an actual leaf. The leaf
stem, midrib and leaf vein are modeled as stiffener frames with
different stiffness and elastic modulus. To simplify the
problem, the initial conguration of the leaf is assumed to be
at. In the simulation, we decrease the chemical potential of the
gel gradually to mimic the leaf drying process.
The various deswelling patterns of the gel leaf are illustrated
in Fig. 6(a)–(e). The buckling of the gel leaf membrane is
observed during the deswelling process and the at surface
deforms to a 3-dimesional pattern. During the early stages
shown in Fig. 6(a) and (b), the compressive stress in the leaf has
not yet reached the critical value and the membrane gel deforms
in plane. However, when the chemical potential is reduced
further, the leaf at surface becomes curved due to the buckling
and wrinkle of the leaf. The stiffer stem and vein of the leaf are
also bent to minimize the free energy of the system. Fig. 7(a)
shows the actual leaf drying status aer a few days of drying.
The simulated drying pattern is also shown in Fig. 7(b) for
comparison.
Similarly, to simulate the leaf growth process, we can
increase the gel chemical potential from an initial stage to the
nal status so as to make the gel membrane swell. The evolved
shapes of ower leaves under growth and a schematic view of
Fig. 8 (a) The evolved pattern shape of flower leaves under growth. (b) Schematic view of the simulated model and stiffener frames of a membrane gel leaf.
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Fig. 9 The deformation patterns of leaf growth via membrane gel swelling. (a)
to (c) The deformed configurations at the early stage are kept in the initial plane
of the gel leaf when growing. (d) to (f) The out of plane deformation patterns of a
gel leaf are observed due to buckling when it swells to a certain stage. The color in
(a) to (f) indicates deformation level.
the model and stiffener frames of a membrane gel leaf are
shown in Fig. 8 (a) and (b) respectively. Fig. 9 shows the simulated leaf growth process. The evolving patterns from the initial
to nal deformation stages of gel leaf swelling are depicted in
Fig. 9(a)–(f). It can be observed that the at membrane of the
leaf gel deforms into a hyperbolic paraboloid (saddle) shape,
demonstrating that the gel leaf membrane buckles while
swelling.
Though articial values are assumed for the material properties of a leaf gel membrane and the examples do not give the
actual values of deformation and the deformed shapes of gel
leaf growing or drying processes exactly, the study provides a
useful methodology for further studying the actual leaf growth
and senescence.
Paper
pumpkins, melons and tomatoes, not simply smooth, but
marked by ribs or ridges? A simple reason is that these patterns
are caused by buckling and wrinkle deformation during the
fruit growth process. As fruits typically consist of a so and
pulpy interior surrounded by a thin stiffer peel or skin, the
different mechanical properties of the skin and soer core can
cause buckling as they grow or dry up due to differential
swelling or shrinking of the two parts. From a mechanics point
of view, these buckling patterns are not arbitrary, but the
morphology of deformation is dependent on the material
properties of each fruit component and its original geometry.
According to Yin et al.31 using engineering materials for
spherical or ovoid (spheroid) objects, the deformation or
buckling depends on three key factors: (i) the ratio of the skin
thickness to the width of the spheroid, (ii) the difference in the
stiffness of the core and skin, and (iii) the shape of the spheroid.
From various values of these parameters, they found the buckling patterns of these types of fruits to be generally ribbed,
reticulated or banded around the circumference of the fruits.
As most fruits have shapes that look like spherical or
spheroid objects, we adopt these geometries to model fruits in
the present study. The simulated model consists of a shell
(spherical or spheroid) and a so pulpy interior. To simulate the
growing and drying processes, the thin shell of fruits is modeled
using a stiff gel material, while the interior part is modeled as a
so gel. We assume the gel materials to have different initial
chemical potential and different dimensionless measure of
enthalpy of mixing of gels. The geometric values of shell
thickness and the major and minor axes of the spheroid are
adjusted to represent various typical shapes of different breeds
of fruits. The schematic diagrams of a spheroid model of fruit
and its geometry are depicted in Fig. 10(a) and (b) respectively.
In the modeling, the material properties of the corpus and shell
of fruits are modeled using different initial chemical potential
to represent different stiffness of inner and outer materials of
plants. The ratio of initial incremental Young’s modulus of the
gel for corpus and shell is about 4 to 6.
Fig. 11 illustrates the simulated buckling pattern of a semispheroidal object in which the thin shell is made of a harder gel
material and the interior is of a soer gel material under
Swelling of a thin lm gel on an elastic or so gel substrate to
mimic fruit morphologies
It is interesting to observe various undulating surface
morphologies of fruits of various species. For example, pumpkins show ridged or ribbed shape on their surfaces, while many
other fruits display regular arrays of dimples or spiral arrangement patterns. Why are surfaces of some fruits, such as
584 | Soft Matter, 2013, 9, 577–587
Fig. 10 (a) Schematic representation of the geometry of a spheroid model; the
outer surface is made of a harder gel and the interior of a relatively softer gel.
Different gels can be achieved by assuming different values of initial chemical
potential and different dimensionless measure of enthalpy of mixing. (b) Spheroid
object which can represent various types of fruits.
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Fig. 11 Simulated buckling patterns of swelling of a semi-spheroidal body can
be used to mimic acorn squash or pumpkin or squash shape. (a) Simulated
buckling pattern of a spheroidal object, in which the thin shell is of a harder gel
and the interior is of a softer gel. (b) Patterns of some types of acorn squash.
Soft Matter
interior gel materials have to be carefully selected to represent
the actual fruit material values. As gel is not an elastic material,
the incremental modulus of the gel material can be regarded as
a reference to determine the initial chemical potential of the
material.33 This simulated swelling process produces a buckling
pattern that mimics the pumpkin shape, as shown in Fig. 11.
To model the growth of gourds, the model of a spheroidal
object with a stiff shell is constructed. In this model, the interior
part can be assumed to be hollow or with a very so gel material.
When the object is swelling, the spheroidal object can display
different buckling patterns at different swelling stages. Fig. 12
shows the initial model of a spheroidal object and two types of
different buckling patterns, as well as the pictures of bitter
gourds, which can be used to compare the patterns with the
simulated buckling model. From Fig. 12, it can be observed that
using a relatively long spheroid model, we can successfully
mimic the shape of a bitter gourd. In the swelling process of this
object, the transition of buckling status can be observed, and it
can be used to explain the different morphologies for fruit at
different stages.
Next we consider a spherical-like object which can be used to
represent the apple shape. In the modeling, we can reduce the
geometry of the long axis of a spheroidal object to spherical-like
shapes with two poles. The top pole end of the spheroidal object
can be constrained to mimic the apple stake part, while the
other pole at the bottom is modeled using a slightly harder
material than the interior to represent the stamen part of the
apple. When the spheroidal gel structure is swelling under these
boundary conditions, compressive stresses will be generated
near the two pole ends and once reaching the critical value, the
wrinkle or buckling takes place on the apple surface as depicted
in Fig. 13. This example demonstrates that the growing process
of apple-shape-like fruit can be mimicked by using a spheroidal
object of two gel materials under swelling.
At certain specic values of properties of gel materials and
geometries, the mechanical buckling of a spheroidal object can
Fig. 12 Growth of a bitter gourd can be simulated by spheroidal body gel
swelling. (a) Initial geometry of the spheroid model, (b) buckling patterns for the
swelling of gel material, (c) the shape of an actual bitter-gourd.
swelling. To mimic the fruit swelling by using inhomogeneous
gel theory, we assume that the shell and the interior parts of the
model have different initial chemical potential values and they
swell to different nal chemical potential values. In the implementation of object swelling, although the actual values for the
gel materials are not crucial, the relative values of the shell and
This journal is ª The Royal Society of Chemistry 2013
Fig. 13 Apple fruit growth can be simulated by spheroidal body gel swelling. (a)
Initial configuration of a spheroid object, (b) and (c) buckling patterns after
swelling, (d) the actual apples.
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indeed display a range of spectacular patterns which can be
used to mimic some fruit morphology and growing process. The
mechanical buckling is likely to play an important role in plant
and fruit morphology study. For the patterns formed in plants,
such as the spiral arrangement, combined mathematical regularity arrays of dimples are more difficult to describe and
explain by a simple mathematical method. The gel inhomogeneous deformations under swelling and deswelling provide an
alternative approach of great potential to explain the plant
complex morphology. The simulation shows that spheroid-like
objects may produce precisely the same number of ribs as
observed in certain fruits.
Further simulations have shown that the ridges on some
cacti and the spiral arrangements of budding stems at the head
of a plant shoot might result from the buckling of the stiff lm
that covers the surface. Combining gel material deformation
and buckling theory of lm gel and solid gel, certain plant
growth processes can be explained more rationally. Even
though the selected gel material properties are not those of real
fruit or plant materials, the ndings from the present study are
interesting and the approach as presented herein has great
potential in describing the morphology of fruits and other parts
of plants.
Concluding remarks
In this study, the inhomogeneous eld theory of a polymeric
network in equilibrium with solvent and mechanical load or
constraint is used to investigate deformations of gel swelling
and deswelling. The swelling- and deswelling-induced instabilities of thin lm gels, layered gels on either an elastic or
another gel layer substrate, are discussed. The critical stress and
wavelength as well as the equilibrium amplitude of wrinkle and
buckling of two layered gel–gel substrate and gel–elastic
substrate systems are derived based on the inhomogeneous
eld theory which has been implemented in the nite element
package, ABAQUS. The buckling phenomena of membrane gels
and a two layered gel system are simulated. It was demonstrated
that using the present eld theory and the developed user
subroutine of the nite element method, the membrane gel
deformation patterns in buckling can be simulated and predicted. The paper also presents examples demonstrating the
possibility of studying the natural plant tissue deformation in
growing or senescence processes via the application of
membrane gel and layered gel system swelling or deswelling.
The natural phenomena can thus be more rationally explained.
In this paper, we simulate the leaf drying and growing
processes by studying the deformation and buckling patterns
of membrane gel structures. The wavy owering cabbage leaves
are explained by investigating buckling models of annular thin
lm membrane gel structures of various geometries, while
growth patterns and morphologies of fruits, such as apples,
bitter gourds and pumpkins, are simulated by using a spheroidal body of a hard gel layer and an internal so gel layer
system under swelling. From a biological point of view, most
models which recognize the two layered fruit structure at the
apex assume that patterning of fruits arises in the deeper layer
586 | Soft Matter, 2013, 9, 577–587
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via chemical diffusion and that the outer layer is passive.
However, from a mechanics point of view, we suggest that the
origin of patterning of fruits may also be physical, not just
chemical, and involves both the hard outer layer and the so
pulpy interior. We hope that this eld theory and the developed FE subroutine will motivate more future research to
elucidate other complex natural phenomena. The study may
also be extended to assist the design of physical growing
tissues by varying the gel parameters. The examples demonstrate the physical mechanism of plant leaf growth and drying,
and the fascinating complex deformation patterns of the leaf
and ower via simulation through gel swelling and deswelling
processes. Once the values of the natural plant material
properties are available, the detailed deformation pattern of
the plants can be quantitatively predicted and simulated based
on the same procedure adopted herein. The methodology and
simulation processes also provide great potential for further
study on plant evolutions under various environmental
conditions.
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