Supplementary material
Structural origins of morphing in plant tissues
S.1. Structural analysis of the seed pods
S.1.1. The Leucaena seed pod valve
Figure S1: (a) The cross-section of the bi-layer structure of the Leucaena seed pod valve,
as viewed by SEM. (b) Higher magnification of the outer layer, showing its internal
architecture as a stack of sheets. (c) Reconstructed CT image of the inner layer showing
cellulosic tubules. The dashed and full lines represent the longitudinal and the local
1
tubule directions, respectively; 𝜃 is the tubule angle with respect to the longitudinal
direction. The image height is 0.4 mm. (d) Cross-section of the inner layer, viewed by
SEM, revealing its cellulosic tubular architecture. (e) Higher magnification of the tubules
forming the inner layer, showing their fine fibrous structure. The dashed and full lines
represent the tubule and the fibril directions, respectively; 𝜑 is the fibril angle with
respect to the tubule axis.
2
S.1.2. The Jacaranda seed pod valve
Figure S2: (a) Cross-section of the tri-layer structure of a Jacaranda seed pod valve, as
viewed by SEM. (b-c) Cross section (b) and top view (c) of the outer Jacaranda layer as
viewed by SEM, showing a rod-like pattern oriented (dashed lines) toward the valve
surface. (d) Reconstructed CT image (ortho-slice view) of the upper part of the
Jacaranda seed pod valve (comprising the outer layer and part of the middle layer). At
this scale the upper region in the picture (outer layer) appears to be quasi-homogeneous
whereas the lower region of the picture (middle layer) is foamy. The image height is 1.5
mm. (e) SEM view of the middle Jacaranda layer, showing that its foamy appearance
consists of hollow cells (f-g) Reconstructed CT image (f) and SEM view (g) of the
Jacaranda inner layer, showing randomly oriented domains made each of unidirectional
tubules (the lines represent the domain orientations). CT image height is 1.4 mm. (h-i)
Higher magnification views of the inner Jacaranda layer, revealing its cellulosic tubular
3
architecture. The dashed and full lines represent the tubule axis and the fibril direction,
respectively; 𝜑 is the fibril angle with respect to the tubule axis.
S.1.3. The Erythrina seed pod valve
Figure S3: (a) SEM view (longitudinal section) of the tri-layer structure of the Erythrina
seed pod valve. (b) SEM view of the outer and middle layers. (c) Higher magnification of
the green dashed rectangular in (b) (outer layer), showing thin channels embedded in the
bulk. (d) The middle and inner layers of the Erythrina valve: the tubules in the inner layer
4
appear to be perpendicular to the tubular pattern that appears in the middle layer (close to
the inner layer), see also figure 1i. (e) SEM image of the middle layer showing the
transition between the stack-of-sheet and the tubular patterns, shown by the dashed lines,
see also figure 1i in the main text. (f) Reconstructed CT image showing the inner layer.
The dashed and full lines represent the pod longitudinal and tubule directions,
respectively. The tubule angle with respect to the longitudinal axis is 𝜃 ≈ 90𝑜 . Image
height is 0.45 mm. (g) The cellulosic tubular architecture of the inner layer (SEM). The
dashed and full lines represent the tubule axis and the fibril direction, respectively; 𝜑 is
the fibril angle with respect to the tubule axis.
5
6
S.1.4. Methods
S.1.4.1. Scanning Electron Microscopy
Five-millimeter-long dry pod samples were carefully cleaned and then fractured either in the
longitudinal or in the lateral direction (precautions were taken not to deform the cross sections
during fracturing).The samples were sputtered with a gold-palladium alloy at 12 mA for 3
minutes using a Edwards S150 sputter coater instrument. This was followed by Scanning Electron
Microscopy imaging (SUPRA-55 VP ZEISS and ULTRA-55 ZEISS SEM instruments using InLens and secondary electron detectors). Images were collected at an acceleration voltage of 3 kV
and a working distance of 3-4 mm.
S.1.4.2. Cellulose analysis
Quantitative determination of the cellulose content was performed according to a procedure
adapted from [S1]. The layers of each pod were separated by hand (note that for the Jacaranda
and Erythrina the middle layer was inseparable from the outer layer). A sample of ~50 mg of each
layer was analyzed. All samples were weighted and put into a 15 ml glass test tube; 3 ml of nitricacetic reagent solution was added to each tube. The nitric-acetic reagent was prepared by mixing
9 parts of 80% acetic acid and 1 part of 60% nitric acid. The test tubes were shaken thoroughly
and kept in a boiling water bath for 1 hour without stirring.
The resulting solutions were centrifuged at 2000 rpm for 2 minutes, and the supernatant was
decanted. 10 ml of DI water were added to each test tube, shaken thoroughly and centrifuged
under the same conditions. This was repeated three times, after which the samples were dried
overnight under reduced pressure. Dry cellulose samples were weighted, and cellulose weight %
was calculated accordingly.
S.1.4.3. µCT scanning
Specimens were fixed in a holder and scanned using a microXCT-400 (XRadia, Pleasanton, CA,
USA). High resolution scans (voxel size ≈ 1.5 µm) were carried out at 40 kV and 200 µA. Unless
otherwise indicated, 1000 projections were acquired over an angular range of 180º. Raw data
were reconstructed with the XRadia software which uses a filtered back projection algorithm.
Three-dimensional ortho-slice visualization was carried out with Avizo software (VSG).
7
S.2. Theoretical analysis
Here we summarize the analytical methods used to predict and simulate the opening
patterns of the seed pods considered in this work.
S.2.1. Hierarchical modeling of the seed pods via composite mechanics
S.2.1.1. The material level (nanoscale)
S.2.1.1.1 Elastic properties
We consider a composite material made of stiff fibrils with Young’s modulus 𝐸𝑓 and
a more compliant matrix material with Young’s
volume fraction 𝜙 , embedded in
modulus 𝐸𝑚 . Poisson ratios of both materials are assumed to be equal 𝜈𝑓 = 𝜈𝑚 .
Firstly, the elastic moduli of a unidirectional, parallel fibrillar, composite are evaluated
via the classical Halpin-Tsai (H-T) model [S2], as follows:
1+𝐴∙𝐵∙𝜙
𝑃𝑈𝑛𝑖 # = 𝑃𝑚 (
1−𝐵∙𝜙
)
;
𝑃 /𝑃𝑚 −1
𝐵 = 𝑃 𝑓/𝑃
𝑓
(S.2.1)
𝑚 +𝐴#
where 𝑃 = 𝐸 or 𝐺 designates Young’s or shear modulus respectively, and # indicates the
modulus direction of the composite: 𝐸𝑈𝑛𝑖 ∥ and 𝐸𝑈𝑛𝑖 ⊥ are the axial moduli along and
perpendicular to the fibril direction and 𝐺𝑈𝑛𝑖 ⊥∥ is the shear modulus. 𝐴# is a the
corresponding shape factor for the different moduli. The Poisson ratio of the composite is
taken as 𝜈∥⊥ = 𝜈𝑓 = 𝜈𝑚 .
Next, the quasi-isotropic Young’s and shear moduli of a planar composite are evaluated
via the common approximations [S2]:
3
5
1
1
𝐸𝑃𝑙𝑎𝑛𝑎𝑟 = 8 𝐸𝑈𝑛𝑖 ∥ + 8 𝐸𝑈𝑛𝑖 ⊥
(S.2.2)
𝐺𝑃𝑙𝑎𝑛𝑎𝑟 = 8 𝐸𝑈𝑛𝑖 ∥ + 4 𝐸𝑈𝑛𝑖 ⊥
(S.2.3)
where 𝐸𝑈𝑛𝑖 ∥ and 𝐸𝑈𝑛𝑖 ⊥ are the corresponding moduli of the unidirectional composite
eq.(S.2.1).
8
Lastly, the quasi-isotropic bulk (3D) composite material is viewed as spaghetti-like
randomly oriented fibrils embedded in a matrix. The isotropic Young’s and shear moduli
of such a bulk composite are approximately [S2]:
1
4
𝐸𝐵𝑢𝑙𝑘 ≈ 5 𝐸𝑈𝑛𝑖 ∥ + 5 𝐸𝑈𝑛𝑖 ⊥
(S.2.4)
1
𝐺𝐵𝑢𝑙𝑘 ≈ 15 𝐸𝑈𝑛𝑖 ∥
(S.2.5)
The Poisson ratios for the planar and the bulk composites are taken as 𝜈𝑃𝑙𝑎𝑛𝑎𝑟 = 𝜈𝐵𝑢𝑙𝑘 =
𝜈𝑓 = 𝜈𝑚 .
S.2.1 1.2. Hygroscopic properties
Change in the water content of the composite material (Δ𝑤, in percent of the original
weight) leads to isotropic deformations of its matrix, which can be expressed as ~𝛽𝑚 ×
Δ𝑤 (𝛽𝑚 is the matrix hygroscopic coefficient). These matrix deformations are restricted
by the presence and architecture of the stiff fibrils. The overall hygroscopic behavior of
an anisotropic layered composite is characterized by the vector 𝛽 = [𝛽 ∥ , 𝛽⊥ , 𝛽 ∥⊥ ]𝑡 , where
𝛽 ∥ and 𝛽⊥ are axial hygroscopic deformation coefficients and 𝛽 ∥⊥ is the hygroscopic
shear deformation coefficient. The superscript t denotes a transpose operation
As a first case, the hygroscopic deformation coefficients of unidirectional parallel fibrillar
composite are:
𝛽𝑈𝑛𝑖 ∥ ≈ 0
(S.2.6)
𝛽𝑈𝑛𝑖 ⊥ ≈ (1 − 𝜙)𝛽𝑚
(S.2.7)
𝛽𝑈𝑛𝑖 ∥⊥ ≈ 0
(S.2.8)
where 𝛽𝑈𝑛𝑖 ∥ and 𝛽𝑈𝑛𝑖 ⊥ are the axial hygroscopic coefficients along and perpendicular to
the fibrils and 𝛽𝑈𝑛𝑖 ∥⊥ is the in-plane shear hygroscopic coefficient. As mentioned before,
𝛽𝑚 is the hygroscopic coefficient of the matrix and (1 − 𝜙) is the matrix content.
Second, the hygroscopic behavior of a random planar composite is generally a complex
function of the explicit fibril architectures. To simplify, this architecture is now
approximated as a stack of pairs of perpendicular unidirectional composites - randomly
9
oriented in-plane. By incorporating the hygroscopic models for a pair of perpendicular
unidirectional composites [S3] with the biologically relevant elastic properties of the
cellulose fibrils and the hemi-cellulose-lignin matrix, the isotropic hygroscopic behavior
of the random planar composite is given by the following (with no hygroscopic shear
deformations):
1
𝛽 𝑃𝑙𝑎𝑛𝑎𝑟 ≈ 5 (1 − 𝜙)𝛽𝑚
(S.2.9)
Lastly, the hygroscopic behavior of a quasi-isotropic bulk (3D) composite is mainly
dominated by the matrix (with no hygroscopic shear deformations):
𝛽 𝐵𝑢𝑙𝑘 ≈ (1 − 𝜙)𝛽𝑚
(S.2.10)
S.2 1.2 The layer level (microscale)
S.2.1.2.1. Tubule arrays with helical fibrils
Exact analytical expressions for the elastic and hygroscopic properties of arrays of
tubules made of wound fibrils are difficult to find due to coupling between the various
possible deformation modes (e.g.[S4,S5]). Cases where the tubules are also not aligned
with the longitudinal direction are even more complicated, and as far as we know have
never been considered in plant tissues. To obtain analytical approximations for the
properties of such architectures, the tubule section is assumed to be a square [S6]. An
array of tubules is then approximately viewed as an alternating laminate (AL) composed
of unidirectional layers with directions alternating between +𝜑 and −𝜑 with respect to
the tubule axis, i.e. at 𝜃 ± 𝜑 relative to the tissue longitudinal axis (see figure 3 in the
main text).
First, the in-plane elastic moduli of the tubule array are evaluated via the classical
laminate analysis in composites [S2]:
1
𝐶 𝑇𝐴 = 2 [𝑅(𝜃 + 𝜑, 𝐴) + 𝑅(𝜃 − 𝜑, 𝐴)]
(S.2.11)
where 𝑅 is a matrix rotation function (given explicitly in [S3]) and 𝐴 is stiffness matrix of
unidirectional parallel fibrillar composite:
10
1
−
𝐸𝑈𝑛𝑖∥
𝜈𝑈𝑛𝑖∥⊥
𝐴= −
[
𝐸𝑈𝑛𝑖∥
𝜈𝑈𝑛𝑖∥⊥
𝐸𝑈𝑛𝑖∥
1
0
𝐸𝑈𝑛𝑖⊥
0
−1
0
(S.2.12)
1
0
𝐺𝑈𝑛𝑖∥⊥ ]
Second, the hygroscopic coefficient of the tubular array, viewed as a balanced laminate
structure, are related to the hygroscopic coefficient vector and stiffness matrix of the
unidirectional composite (𝛽𝑈𝑛𝑖 and 𝐴) via a vector function 𝐷 (see [S3] for more details):
𝛽 𝑇𝐴 = [𝛽𝑇𝐴 ∥ , 𝛽𝑇𝐴 ⊥ , 𝛽𝑇𝐴 ∥⊥ ]𝑡 = 𝐷 (𝛽𝑈𝑛𝑖 , 𝐴, 𝜃, 𝜑)
(S.2.13)
S.2.1.2.2. Random planar architectures
Planar architectures are viewed as structures with a planar isotropic motif at either the
material level or at the layer level. The stiffness matrix and hygroscopic coefficients of
are:
1
𝜈
− 𝐸𝑃𝑙𝑎𝑛𝑎𝑟
𝐸𝑃𝑙𝑎𝑛𝑎𝑟
𝜈𝑃𝑙𝑎𝑛𝑎𝑟
𝑃𝑙𝑎𝑛𝑎𝑟
1
𝐶 𝑃𝑙𝑎𝑛𝑎𝑟 = − 𝐸
0
[
0
𝐸𝑃𝑙𝑎𝑛𝑎𝑟
𝑃𝑙𝑎𝑛𝑎𝑟
−1
0
(S.2.14)
1
0
𝐺𝑃𝑙𝑎𝑛𝑎𝑟 ]
𝛽 𝑃𝑙𝑎𝑛𝑎𝑟 = [𝛽 𝑃𝑙𝑎𝑛𝑎𝑟 , 𝛽 𝑃𝑙𝑎𝑛𝑎𝑟 , 0]𝑡
(S.2.15)
S.2.1. 2.3. Bulk (3D) architectures
The stiffness matrix and hygroscopic coefficients of layers made of bulk material are
approximately evaluated here as:
1
𝐸𝐵𝑢𝑙𝑘
𝜈𝐵𝑢𝑙𝑘
𝐶 𝐵𝑢𝑙𝑘 = − 𝐸
𝐵𝑢𝑙𝑘
[
0
𝜈
− 𝐸𝐵𝑢𝑙𝑘
𝐵𝑢𝑙𝑘
1
𝐸𝐵𝑢𝑙𝑘
0
−1
0
0
(S.2.16)
1
𝐺𝐵𝑢𝑙𝑘 ]
𝛽 𝐵𝑢𝑙𝑘 = [𝛽 𝐵𝑢𝑙𝑘 , 𝛽𝐵𝑢𝑙𝑘 , 0]𝑡
(S.2.17)
11
A summary of the composite models used for different layers of the seed pods are
summarized in supplementary table I.
S.2.1.3 The laminate level (macroscale)
Upon variations in water content, the laminate composite pod valve can in principle
experience in-plane axial deformations 𝜀𝑥 and 𝜀𝑦 , in-plane shear deformation 𝜀𝑥𝑦 , out-ofplane bending curvatures, 𝜅𝑥 and 𝜅𝑦 , and an out-of-plane twist curvature 𝜅𝑥𝑦 . These
deformations are directly dependent on the composite architecture of the tissue and are
evaluated here by means of classical laminate analysis, using closed-form analytical
relations [S3]:
𝜀
𝐴
[𝜅 ] = [
𝐵
𝐵 −1 𝑁
] [ ]
𝐷
𝑀
(S.2.18)
𝑡
𝑡
where 𝜀 = [𝜀𝑥 , 𝜀𝑦 , 𝜀𝑥𝑦 ] and 𝜅 = [𝜅𝑥 , 𝜅𝑦 , 2𝜅𝑥𝑦 ] .
𝑇
𝑇
𝑁 = [𝑁𝑥 , 𝑁𝑦 , 𝑁𝑥𝑦 ] and 𝑀 = [𝑀𝑥 , 𝑀𝑦 , 𝑀𝑥𝑦 ] are pseudo-forces (axial, transverse and
shear forces) and pseudo-moments (axial bending, transverse bending and twist) resulting
from hygroscopic effects, as given below. 𝐴, 𝐵 and 𝐷 are the compliance elements of the
structure: 𝐴 represents the relation between the pseudo-forces and the in-plane
deformations, 𝐷 represents the relation between the pseudo-moments and the out-ofplane curvatures and 𝐵 is the coupling between these effects (coupling effects vanish
when 𝐵 = 0):
𝑁
𝑙𝑎𝑦𝑒𝑟𝑠
𝐴 = ∑𝑖=1
𝐶𝑖 (ℎ𝑖 − ℎ𝑖−1 )
1
(S.2.19a)
𝑁
𝑙𝑎𝑦𝑒𝑟𝑠
2
)
𝐵 = 2 ∑𝑖=1
𝐶𝑖 (ℎ𝑖2 − ℎ𝑖−1
(S.2.19b)
12
𝑁
1
𝑙𝑎𝑦𝑒𝑟𝑠
3
)
𝐷 = 3 ∑𝑖=1
𝐶𝑖 (ℎ𝑖3 − ℎ𝑖−1
(S.2.19c)
𝑁
𝑙𝑎𝑦𝑒𝑟𝑠
𝑁 = ∑𝑖=1
{𝐶𝑖 𝛽 𝑖 (ℎ𝑖 − ℎ𝑖−1 )} Δ𝑤
𝑁
ℎ𝑖 +ℎ𝑖−1
𝑙𝑎𝑦𝑒𝑟𝑠
𝑀 = ∑𝑖=1
{𝐶𝑖 𝛽 𝑖 (ℎ𝑖 − ℎ𝑖−1 ) (
2
(S.2.19d)
)} Δ𝑤
(S.2.19e)
where ℎ𝑖 is the distance between the center of the layered structure and the interface
between the ith and (i+1)th layers, as shown in Figure S4 for a tri-layer structure. Note that
the ℎ𝑖 between the centerline and the inner surface are taken with a negative sign for the
calculations whereas the ℎ𝑖 between the centerline and the outer surface are taken with
ℎ𝑖 +ℎ𝑖−1
positive sign. Also note that (ℎ𝑖 − ℎ𝑖−1 ) is the layer thickness and (
2
) is the
location of the middle of the layer with respect to the center of the layered structure.
The hygroscopic deformation curvatures are the dominant effect in the opening of the
seed pod. Following the above, the progressive evolution of these curvatures is
proportional to the variations in water content and can be analytically expressed by the
following abbreviated form:
𝜅 = 𝛾 Δ𝑤
;
𝜅𝑥 = 𝛾𝑥 Δ𝑤 , 𝜅𝑦 = 𝛾𝑦 Δ𝑤 , 𝜅𝑥𝑦 = 0.5𝛾𝑥𝑦 Δ𝑤
(S.2.20)
where 𝛾 is the hygroscopic curvature coefficient of the laminate composite pod structure,
as determined by eqs. (S.2.17-S.2.20). The resultant 𝛾𝑥 , 𝛾𝑦 and 𝛾𝑥𝑦 values for the
different seed pods are summarized in supplementary table II.
13
Figure S4: Schematic representation of a tri-layered valve structure, in which the
different layers and the interfaces are indicated.
14
Supplementary table I: Composite models for the seed pods valves
Seed pod
Layer
model
Leucaena
Inner
Alternating laminate
Outer
Random planar
Inner
Random planar
Middle
Random planar
Outer
Bulk
Inner
Alternating laminate
Middle
Random planar
Outer
Bulk
Jacaranda
Erythrina
15
S.2.2. The deformations patterns of the seed pods
Based on the above theoretical developments we now introduce approximate expressions
for the opening patterns of the seed pods. We start with the simplest case, the Jacaranda.
S.2.2.1. The Jacaranda
The valves of the Jacaranda seed pod are attached together at the stalk; valve faces are
considered to be free of other constraints. Before dehydration (our reference state), the
Jacaranda valves are approximately flat and can be described analytically as 𝑤0 (𝑥, 𝑦) =
𝑥𝜉̂ + 𝑦𝜂̂ , where 𝜉̂ and 𝜂̂ represent the longitudinal and lateral directions, respectively; the
point (𝑥, 𝑦) = (0,0) represents the connection point of the valves to the stalk. Since the
deformations of the Jacaranda pod are at least an order of magnitude smaller than the
pod length, the curved shape (𝑤) of the valves can be approximated via [S3]:
1
𝑤 ≈ [𝑥]𝜉̂ + [𝑦]𝜂̂ − 2 (𝑘𝑥 𝑥 2 + 𝑘𝑦 𝑦 2 + 2𝑘𝑥𝑦 𝑥𝑦)𝜁̂
(S.2.22)
S.2.2.2. The Leucaena
The valves of the Leucaena seed pod are also attached together at the stalk and the valve
faces are considered to be free of other constraints. The Leucaena valves are
approximately flat rectangles (reference state), which again can be described by
𝑤0 (𝑥, 𝑦) = 𝑥𝜉̂ + 𝑦𝜂̂ . In contrast with the Jacaranda seed pod, the curvatures of the
Leucaena pod result in both bending and twisting deformations, with magnitude similar
to that of the pod length. In view of the elongated geometry of the Leucaena, the
deformation function 𝑤 is dominated by the longitudinal bending and twist curvatures
(𝑘𝑥 , 𝜅𝑥𝑦 ), and is evaluated via [S7-S8]:
1
2 )]𝜉̂ + [𝑘 𝑥 𝑘 2 + 𝑘 2 ]𝜂̂ + [𝑘 [𝑐𝑜𝑠(𝑥 𝑘 2 + 𝑘 2 ) −
𝑤 ≈ 𝑘 2 +𝑘 2 {[𝑘𝑥 𝑠𝑖𝑛(𝑥 √𝑘𝑥2 + 𝑘𝑥𝑦
√ 𝑥
𝑥𝑦 √ 𝑥
𝑥𝑦
𝑥
𝑥𝑦
𝑥
𝑥𝑦
1]] 𝜁̂} − 𝑦
1
2
√𝑘𝑥2 +𝑘𝑥𝑦
2 )]𝜉̂ + [−𝑘 ]𝜂̂ + [−𝑘 𝑠𝑖𝑛( 𝑘 2 + 𝑘 2 )]𝜁̂}
{[𝑘𝑥𝑦 𝑐𝑜𝑠(√𝑘𝑥2 + 𝑘𝑥𝑦
√ 𝑥
𝑥
𝑥𝑦
𝑥𝑦
16
(S.2.23)
Note that this function includes an additional pseudo rigid-body rotation around the 𝜁̂
𝑘𝑥𝑦
axis with respect to the reference configuration, ~ tan−1 ( 𝑘 ), but since this rotation is
𝑥
not physical in this case it was removed in the graphical representation (figure 2 in the
main text).
S.2.2.2. The Erythrina
The Erythrina seed pod includes two hemispherical valves of radius 𝑟0 , bonded along the
perimeter 0 ≤ 𝑥 ≤ 𝜋𝑟0 , 𝑦 = 0 (see Figure S5).
The reference configuration of the Erythrina valve is given by [S7] 𝑤0 (𝑥, 𝑦) =
1
𝜅0
[1 − 𝑐𝑜𝑠(𝜅0 𝑥)]𝜉̂ +
1
𝜅0
𝑠𝑖𝑛(𝜅0 𝑥)[−𝑐𝑜𝑠(𝜅0 𝑦)𝜂̂ + 𝑠𝑖𝑛(𝜅0 𝑦)𝜁̂] , where 𝜅0 = 1/𝑟0 is the
curvature of the valves and the 𝜉̂ direction represent the axis along the pod stalk. Both
biological samples and the theoretical calculations indicate that Erythrina valves
experience both 𝑘𝑥 and 𝑘𝑦 curvatures. However due to the bonding confinements from
𝑥 = 0 to 𝑥 = 𝜋𝑟0 , only 𝑘𝑦 affects the deformations (as a first approximation) and the
deformed shape is expressed via:
1
1
𝑤(𝑥, 𝑦) ≈ 𝜅 [1 − 𝑐𝑜𝑠(𝜅0 𝑥)]𝜉̂ + 𝜅 𝑠𝑖𝑛(𝜅0 𝑥)[−𝑐𝑜𝑠(𝜅𝑒𝑞 𝑦)𝜂̂ + 𝑠𝑖𝑛(𝜅𝑒𝑞 𝑦)𝜁̂]
0
𝑒𝑞
(S.2.24)
where 𝜅𝑒𝑞 = 𝜅0 + 𝑘𝑦 is the deformed curvature of the hemisphere in the cross section
of the pod.
17
Figure S5: Coordinate systems of the hemispherical Erythrina seed pod valves of radius
𝑟0 : (red) cartesian coordinates 𝜉̂, 𝜂̂ and 𝜁̂ where 𝜉̂ represents the axis along the pod stalk,
and (blue) longitudinal and lateral coordinates 𝑥 and 𝑦 (for the right valve: 0 ≤ 𝑥, 𝑦 ≤
𝜋𝑟0 ).
18
S.2.3. Calculated results for the hygroscopic curvature coefficients
Theoretical hygroscopic curvature coefficients of the different seed pods were calculated
by considering the geometrical parameters shown in figure 1 (main text) and typical plant
Young’s
tissues properties. The results are summarized in supplementary table II.
modulus of the cellulose fibrils is taken as 𝐸𝑓 = 120𝐺𝑃𝑎; Young’s modulus of the noncellulose (hemicellulose-lignin) matrix is taken as 𝐸𝑚 = 3𝐺𝑃𝑎; Poisson ratios of the
fibrils and matrix are given by 𝜈𝑓 = 𝜈𝑚 = 0.3; the matrix hygroscopic coefficient is
𝛽𝑚 = 0.3 [S9,S10].
Supplementary table II: Summary of the hygroscopic curvature coefficients for the
different seed pod valves, calculated for the geometries presented in figure 1, with 𝐸𝑓 =
120𝐺𝑃,𝐸𝑚 = 3𝐺𝑃𝑎, 𝜈𝑓 = 𝜈𝑚 = 0.3, 𝛽𝑚 = 0.3.
Seed pod
Leucaena
Jacaranda
Erythrina
Hygroscopic curvature
𝜸𝒙 *
~0.3
~0.05
0#
coefficients
𝜸𝒚 **
~0.45
~0.05
~0.15
𝜸𝒙𝒚 ***
~2
0
0
[𝟏/(𝒎𝒎 ∙ 𝚫𝒘)]
* Bending curvature along the longitudinal direction.
** Bending curvature along the lateral direction.
*** Twist curvature.
# An approximation due to the confinement at the edges of the valves in the longitudinal
direction.
19
S.3. Bio-inspired synthetic seed pods
S.3.1. Materials:
Poly(vinyl alcohol) (PVA) with molecular weight of 78,000 was purchased from
Polysciences; hydrochloride acid (HCl, 37%) was purchased from BioLab LTD., Israel;
Glutaric dialdehyde (GDA, 25 wt% solution in water) was purchased from Sigma
Aldrich. Commercial mono-PAN fiber tows were obtained from Dralon, Germany.
HCl was diluted into 1M aqua solution. PVA was dissolved in water at 100 oC under
vigorous stirring for two hours to get a PVA solution (10 wt%).
To prepare a PVA gel, 2 g GDA crosslinker were added to 100 ml of PVA solution,
stirring vigorously to get a homogeneous solution. 0.86g of GDA was added to 100 ml
PVA solution. To accelerate crosslinking, 3 ml of the HCl solution used as catalyst was
added to the mixture under heat (about 40oC). The mixture was stirred for 2 minutes
before being poured into the mold/bath for the subsequent preparation of the synthetic
pods.
The Young’s moduli of PAN fiber and PVA gel are 4.8 GPa [S11] and 0.12 MPa,
respectively. The modulus of the PVA gel was determined by means of tensile test using
an Instron 4502 instrument (equipped with 10N load cell and with a crosshead speed of
10mm/min). The PVA gel samples were made by pouring the gel mixture into a 15 x 1.5
x 1 mm3 dogbone mold. 8 specimens were tested.
After drying in laboratory conditions, the PVA gel lost about 90% of its weight.
Compared to the hygroscopicity of PVA gel, that of PAN fiber was negligible.
20
S.3.2. Fabrication strategies
S.3.2.1. Synthetic Leucaena pod
The natural Leucaena pod has two valves, which are mirror symmetric of each other.
Each half is made of two layers as described in the main text. As to the synthetic
analogue, the PVA gel was used to get an isotropic structure for the outer layer. The inner
layer was made by first winding two layers of PAN fibers to approximately mimic the
microfibril helical structure (𝜃 ± 𝜑 relative to the longitudinal direction). Subsequently,
the PVA gel was added to these two fiber layers. The same strategy was used for the
opposite half but with reversed angles (-(𝜃 ± 𝜑)o relative to the longitudinal direction) so
as to conserve mirror symmetry.
For the PAN fiber winding, a special device made of Delrin was designed (shown in
Figure S6) which consists of two parts placed in a surrounding square bath. Part 1 is a
square plate placed in a second, bigger plate (Part 2) with 24 triangular teeth separated by
15o. Both plates have tiny notches (3 mm long, 1 mm wide, separated by 1 mm intervals)
used for fiber winding. With such a device laminates can be prepared with various
relative fiber orientations.
Figure S6. Winding device which has two plates sitting in a bath.
Since only multiples of 15o can be obtained for the fiber angle, we used 15o and 60o to
approximate the fibril orientations of the different layers (i.e. 𝜃 ≈ 37° and 𝜑 ≈ 23°) –
which are typical values for the natural configuration (table I, main text). After the
21
winding of the two layers of PAN fibers, we introduced PVA gel into the bath and let the
fibers soak in the PVA gel until complete cross-link. The same procedure was repeated
for the other half, with fiber angles of -15 o and -60 o.
Cross-linking of the PVA gel took about two hours. Thereafter the fibers at the edges of
the plate were carefully cut and the laminate was peeled off from the plates. The laminate
was then cut along the longitudinal direction to form 2 x 15 cm2 strips. This laminated
strip simulates a single natural valve. The resulting fiber-to-PVA volume ratio was about
3 %.
Time lapse pictures were produced while hanging such synthetic pod on a frame in an
air-conditioned room for about two days. The time interval between each frame was 10
minutes, from which a movie was prepared.
S.3.2.2. Synthetic Jacaranda pod
The natural Jacaranda pod has two valves as well, each made of three layers (see main
text). For the synthetic pod, a transparent polymer sheet (about 0.2 mm thick) was used
for the isotropic inner layer. For the middle layer we used Styrofoam (about 5 mm thick),
and for the outer layer a sheet of paper (about 0.2 mm thick). Since the whole structure of
Jacaranda pod is isotropic, two identical synthetic valves were used.
The preparation of the synthetic Jacaranda pod was as follows. First, both the plastic
sheet and the paper sheet were cut to fit a 9 cm-diameter Petri dish. The plastic sheet was
roughened with sand paper for better adhesion with the PVA gel, then put at the bottom
of the Petri dish. A 5 cm-diameter Styrofoam layer was then attached to the plastic sheet
using double side scotch tape, and 25 ml pre-mixed PVA gel was poured onto the
Styrofoam and plastic sheet inside the Petri dish. The PVA gel was covered with the
paper sheet and the PVA gel cross-link took about two hours. The resulting fiber-to-PVA
volume ratio of the sample was about 8%.
22
Again, time lapse pictures were produced while hanging the synthetic Jacaranda pod on
a frame in an air-conditioned room for about two days. In this case the time interval
between each frame was 60 minutes, from which a movie was prepared.
S.3.2.3. Synthetic Erythrina pod
The Erythrina pod has three layers. For the synthetic pod we again used PVA gel for the
isotropy. To prepare a synthetic Erythrina pod, the shape of the pod was approximated by
a hollow sphere. First, four layers of PAN fibers were wound at angles of 30, 150, 30 and
150° on a silicon mold relative to the vertical axis of the hemisphere, then immersed in
100 ml of PVA gel held in another silicone semi-spherical bowl. Following PVA gel
crosslinking, the mold was carefully removed. The other half was made in exactly the
same way, with opposite fiber angles (150, 30, 150, 30°). The resultant fiber-to-PVA
volume ratio of the sample was about 0.9%.
As before, time lapse pictures were produced while hanging the synthetic Erythrina pod
on a frame in an air-conditioned room for about two days. In this case the time interval
between each frame was 160 minutes, from which a movie was prepared.
S.3.2.4. Synthetic Venus fly-trap
To mimic the Venus fly-trap, a spherical shape was adopted here as well, with thinner
wall thickness. One layer of fibers was wound circumferentially to provide anisotropic
deformations effects as in the natural configuration [S12]. 50 ml of PVA gel were used
for each half. The resulting fiber-to-PVA volume ratio of the sample was about 0.4%.
As before, time lapse pictures were produced while hanging the synthetic Venus fly-trap
pod on a frame in an air-conditioned room for about two days. In this case the time
interval between each frame was 60 minutes, from which a movie was prepared.
23
S.4. Multimedia files and movies
Movie S1: Opening of the Leucaena leucocephala seed pod (Multimedia View).
Movie S2: Opening of the Jacaranda mimosifolia seed pod (Multimedia View).
Movie S3: Opening of the Erythrina corallodendrum seed pod (Multimedia View).
24
Movie S4: The simulated opening pattern of the Leucaena leucocephala seed pod, as
calculated for water content decreasing from Δ𝑤 = 0 to Δ𝑤 = −5%. The blue and red
surfaces represent the pod valves and the white surface represents the initially closed
configuration (Multimedia View).
Movie S5: The simulated opening pattern of the Jacaranda mimosifolia seed pod, as
calculated for water content decreasing from Δ𝑤 = 0 to Δ𝑤 = −10%. The blue and red
surfaces represent the pod valves and the white surface represents the initially closed
configuration (Multimedia View).
25
Movie S6: The simulated opening pattern of the Erythrina corallodendrum seed pod, as
calculated for water content decreasing from Δ𝑤 = 0 to Δ𝑤 = −20%. The blue and red
hemi-spheres represent the pod valves and the white sphere represents the initially closed
configuration (Multimedia View).
Movie S7: Opening of the synthetic Leucaena leucocephala (Multimedia View).
26
Movie S8: Opening of the synthetic Jacaranda mimosifolia (Multimedia View).
Movie S9: Opening of the synthetic Erythrina corallodendrum (Multimedia View).
Movie S10: Snapping of the bio-inspired synthetic Venus flytrap (Multimedia View).
27
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