Supporting Information
Aitken et al. 10.1073/pnas.1519790113
Stress Analysis of Three-Point Bending Experiments
Stresses and strains were calculated along the bottom edge of the
beam directly under the point of applied load, which corresponds
to the point of maximum tensile stress:
σ
=
−
My
I
[S1]
«
=
−
y
ρ
,
[S2]
where
M
is the bending moment,
y
is the perpendicular distance
away from the neutral axis of the beam,
I
is the cross-sectional
moment of inertia, and
ρ
is the radius of curvature (31). At the
point of maximum tensile stress, the moment
M
=
PL
=
2and
=
−
h
=
2,
P
is the applied load,
L
is half the span of the gauge
section, and
h
is the height of the beam. The contribution from
the cribrum and basal plate to
the moment of inertia are
I
cribrum
=
ð
1
=
12
Þ
h
3
c
w
c
and
I
basal
=
ð
1
=
12
Þ
h
3
b
w
b
, respectively, where
w
is the thickness of the layer and subscripts
c
and
b
refer to
the cribrum and basal plate. As shown in Fig. S4
D
, the gauge
section, 2
L
, and the beam height,
h
, are dimensions that lie
within the shell plane and the width
w
, is perpendicular to the
shell plane. Estimation of the contribution to the moment of
inertia from the areola lattice can be difficult due to the direc-
tionality of the hexagonal cell array. Under this loading configu-
ration, an individual areolae wall has a thickness,
h
aerola
=
300
nm
and a width,
w
aerola
=
2.5
μ
m, so that
h
aerola
=
h
cribcrum
∼
0.1 and
w
aerola
=
w
cribcrum
∼
10. The predicted contributions to the area
moment of inertia from the areolae walls with these dimen-
sions are at least 2 orders of magnitude less than the individ-
ual contribution from the cribrum and from the basal plate,
which makes it reasonable to exclude the contribution of are-
olae from the analysis. We observed a small,
<
3° biaxial cur-
vature in the as-fabricated frustule beam, possibly due to
relaxation of residual stresses after its extraction from the
frustule. To account for this curvature, the height of the beam
was calculated as the average of the two layers measured after
sample extraction,
ð
h
c
+
h
b
Þ
=
2. This results in the following
expression for the uniaxial tensile bending stress in the bot-
tom edge of the beam:
σ
beam
=
3
PL
ð
h
c
+
h
b
Þ
2
h
3
c
w
c
+
h
3
b
w
b
.
[S3]
The beams tested in this work were not fixed to the substrate to
avoid generating boundary constraints, so any initial misorienta-
tion of the beam or misorientation of the indenter tip with respect
to the beam resulted in
“
settling
”
events during testing mani-
fested by marginal tilting and translation of the beam before
attaining full contact. As a result, the measured indenter dis-
placement could not be used to calculate strain in the sample.
Using the acquired video frames, we analyzed 450 still images
taken between incipient sample loading up to failure, to track the
position of seven points along the bottom edge of the bending
sample. A schematic in Fig. 3
A
shows these seven points whose
positions were fit to a circle with radius
ρ
and used to determine
the tensile strain as
«
beam
=
ð
h
c
+
h
b
Þ
4
ρ
.
[S4]
SEM Imaging Contrast Within Basal Plate
Fig. S1 shows a top-down view of a representative frustule beam.
A sharp contrast difference is observed within the basal plate. Fig.
S1,
Inset
shows a zoomed-in view of a section of the basal plate to
highlight this contrast. This contrast difference was also observed
in TEM analysis of the frustule (Fig. 2
A
), suggesting that these
regions indicate layers of differing material or microstructure.
FEM Simulations
A computer-aided design model of one of the bending samples
was meshed using 729,000 tetragonal elements in Abaqus sim-
ulation package 6.14 (Simulia). To emulate three-point bending,
the beam was modeled in contact with steel supports on both ends
and displacement was applied via a diamond roller from the top.
The beam was assumed to have an elastic modulus of 35 GPa,
closely matching the experimental value obtained here. The steel
supports and diamond roller were assumed to have elastic moduli
of 200 GPa and 1,000 GPa, respectively. Nodes on the bottom
face of the steel supports were specified to have no displacement
and hard, impenetrable contact via penalty enforcement was
assumed between the beam and steel supports and between the
beam and diamond roller. A friction coefficient of 0.1 was applied
between the steel supports and the beam to prevent lateral sliding
during bending. Displacement of the diamond roller was chosen
such that the resulting maximum tensile beam strain was similar to
experimental values calculated from the captured video. For
comparison, bending simulations were also performed on a solid
beam of similar dimensions and elastic modulus.
Fig. S3 shows the distribution of stress through a three-point
bending sample recreated for FEM analysis. The stresses within
the areolae region are up to an order of magnitude lower than
the stresses supported in either the cribrum or basal layers.
Bending Beam, Nanoindentation, and TEM Sample
Fabrication
Fig. S4 shows a graphical description of the procedure to fabricate
samples for three-point bending experiments, nanoindentation,
and TEM analysis from the diatom shell.
Samples for bending experiments, TEM analysis, and ex situ
nanoindentation were fabricated using the focused ion beam
(FIB) lift-out technique in a dual-beam SEM (FEI Versa 3D)
(32). Once an intact frustule was identified, a cantilever beam
with plane dimensions of 3.5
×
24
μ
m was milled out of the center
of the frustule at 16 kV and 0.25 nA. The depth of the beam was
set by the thickness of the frustule shell and was
∼
3.5
μ
m. A site-
specific, platinum deposition needle and tungsten micromanip-
ulator (FEI EZLift) were used to extract the sample. The mi-
cromanipulator was welded to the free end of the cantilever
beam using Pt deposition and the opposite end of the cantilever
was milled away from the parent frustule. Fig. S4
A
–
C
show SEM
images of this lift-out procedure. The beam was then transferred
onto a stainless steel substrate and placed over prefabricated 20-
×
20-
×
40-
μ
m wells. The same micromanipulator was then used to
position the beam such that its gauge section was parallel to the
short edge of the well and spanned it entirely, ensuring that the
shell plane normal was perpendicular to the loading direction, as
shown in Fig. S4
D
. This type of parallel loading allows us to de-
form all frustule layers simultaneously and removes the compli-
cations caused by the compliance differences between layers and
the possibility of localized deformation that emerges when the
frustule layers are loaded in series (10, 11). This orientation also
Aitken et al.
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limits the influence of the areolae walls and simplifies the calcu-
lation of stresses within the beam.
Fabrication of a basal plate for nan
oindentation was prepared using
the described lift-out technique to extract a 25-
×
25-
μ
m plate from
the basal diatom plate. The removed plate was then transferred to a
copper TEM grid for thinning. The plane of the frustule shell was
oriented parallel to the ion beam, and the cribellum, cribrum, and
areolae walls were removed using milling conditions of 16 kV and 50
pA, leaving only the basal plate (Fig. S4
E
). Using the tungsten mi-
cromanipulator, the basal plate was transferred from the TEM grid
to a stainless steel substrate and se
cured by depositing Pt along the
edges of the plate. Decomposition and fracture of the diatom shell
during cleaning resulted in several girdle bands suitably oriented for
nanoindentation on the substrate, which was performed with no
modification to the structure. Fig. 3
C
and
D
show SEM images of
the basal plate and the girdle band used in this study.
Samples for TEM analysis were fabricated using the described
lift-out procedure, with the extracted lamella transferred and at-
tached to the Cu TEM grid using the Pt deposition needle in the
FIB. The attached TEM samples were then thinned using FIB to a
thickness of less than 100 nm using a voltage of 16 kV and pro-
gressively decreasing currents from 50 to 11 to 4 pA to ensure
electron transparency. A thinned TEM lamella is shown in Fig. S4
F
.
Analysis of Relative Density
We define the relative density of the diatom frustule as the ratio of
material volume in the frustule,
V
frustule
, to volume in a solid shell
of equal dimension,
V
solid
:
ρ
relative
=
V
frustule
V
solid
=
V
cribrum
+
V
areolae
+
V
basal
+
V
rim
V
solid
,
[S5]
where
V
cribrum
,
V
areolae
, and
V
basal
are the volume of the cribrum,
areolae, and basal layers, respectively, and
V
rim
is the volume of
the raised foramen rim. Owing to symmetry of the frustule, we
start by considering a single hexagonal unit of the frustule as
shown in Fig. S2 and develop a general model for the relative
density. Fig. S2
B
–
D
show schematics of the projected area in
each layer of the frustule. By defining the relative projected area
as the ratio of projected area for a given layer
i
,
A
i
, and the solid
area,
A
solid
, Eq.
S5
can be rewritten as
ρ
relative
=
A
cribrum
A
solid
w
cribrum
w
total
+
A
areolae
A
solid
w
areolae
w
total
+
A
basal
A
solid
w
basal
w
total
+
A
rim
A
solid
w
rim
w
total
,
[S6]
where
w
cribrum
,
w
areolae
,and
w
basal
are the depth of the cribrum,
areolae, and basal layers, respectively, and
w
rim
is the thickness of
the raised foramen rim.
w
total
=
w
cribrum
+
w
areolae
+
w
basal
+
ð
1
=
2
Þ
w
rim
is the total depth of the beam.
From the observed hexagonal arrangement of pores in the
cribrum, we assume seven cribrum pores per hexagonal unit. The
projected area of the cribrum is then given:
A
cribrum
=
A
solid
−
7
π
4
d
2
c
,
[S7]
where
d
c
is the diameter of a cribrum pore. The total area of the
walls that make up the areolae is
A
areolae
=
A
solid
−
3
ffiffiffi
3
p
2
a
−
t
ffiffiffi
3
p
2
,
[S8]
where
a
and
t
are the length and thickness of the areolae wall,
respectively. The area within the basal layer and area of the raised
rim are
A
basal
=
A
solid
−
π
4
d
2
b
,
inner
[S9]
A
rim
=
π
4
h
d
2
b
,
outer
−
d
2
b
,
inner
i
,
[S10]
where
d
b
,
inner
and
d
b
,
outer
are the inner and diameter of the fora-
men rim. Noting that the solid area is simply the area of the
hexagonal cell,
A
solid
=
ð
3
ffiffiffi
3
p
Þ
=
2
a
2
, Eq.
S6
then gives the final
expression for the relative density of the frustule:
ρ
relative
=
1
−
7
π
6
ffiffiffi
3
p
d
c
a
2
w
cribrum
w
total
−
1
−
1
ffiffiffi
3
p
t
a
2
w
areolae
w
total
−
π
6
ffiffiffi
3
p
d
b
,
inner
a
2
w
basal
w
total
+
π
6
ffiffiffi
3
p
"
d
b
,
outer
a
2
−
d
b
,
inner
a
2
#
w
rim
w
total
.
[S11]
Average measured values for
d
c
,
d
b
,
inner
,and
d
b
,
outer
as well as
measured values for
w
cribrum
,
w
areolae
,
w
basal
,
w
rim
,and
w
total
of each
frustule beam are given in Table S1. Average values of
t
and
a
taken from 50 measurements of the indentation sample are
0.17
μ
m and 1.19
μ
m, respectively. For these values, the average
relative density of each frustule beam is calculated and given in Table
S1. The average relative density as determined by Eq.
S11
is 36.4%.
By taking advantage of the periodicity of the frustule we have
been able to develop a general model, but the simplifying as-
sumptions used could lead to a misrepresentation of the relative
density of a beam sample. From SEM imaging it is seen that the
pores in the cribrum layer are typically elliptic and vary in number
per hexagonal cell and that foramen are not necessarily regularly
spaced. Similarly, hexagonal areolae cells are irregular and the
number of cell walls can vary from 5 to 6. To test the veracity of
this general model, we used direct measurement of SEM images
of each frustule beam to calculate
V
cribrum
,
V
areolae
,
V
basal
, and
V
rim
.
Ellipses were manually fit to each cribrum pore in the beam and
the area of each ellipse was summed to provide the pore area,
A
pore
. The inverse area is determined by subtracting the pore area
from the rectangular area,
A
beam
. Multiplying by the depth gives
the cribrum volume
V
cribrum
:
V
cribrum
=
w
cribrum
A
beam
−
A
pore
.
[S12]
A similar procedure was used to determine the volume of the
basal layer,
V
basal
, by fitting circles to the inner diameter of each
foramen:
V
basal
=
w
basal
A
beam
−
A
foramina
.
[S13]
The difference in area between circles fit to the outer and inner
diameter of each foramen provides the total projected area of the
foramen rims,
A
rim
. Multiplying by the average thickness of the
foramen rims gives the total rim volume,
V
rim
:
V
rim
=
w
rim
ð
A
beam
−
A
rim
Þ
.
[S14]
Determination of
V
areolae
is difficult because the areolae walls are
obscured by the cribrum and basal layers. To provide an estima-
tion of the location of areolae walls, we generated a Voronoi
diagram using the centers of the foramen as the seeds of each
cell and bounded by the dimensions of the beam. This method
generates a remarkably good match when applied to the visible
areolae walls in the indentation plate shown in Fig. 2
E
.
V
areolae
could then be determined from the total length of the cell walls,
Aitken et al.
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l
areolae
multiplied by the average areolae wall thickness,
t
and
depth of the areolae layer:
V
areolae
=
w
areolae
p
l
areolae
p
t
.
[S15]
The solid volume was determined using the average heights and
depths of each beam:
V
solid
=
h
cribrum
+
h
basal
2
p
L
cribrum
+
L
basal
2
p
w
total
,
[S16]
where
L
cribrum
and
L
basal
are the lengths of the cribrum and basal
layers, respectively. These volumes were then directly substituted
into Eq.
S5
to provide the measured relative density. The vol-
umes
V
cribrum
,
V
areolae
,
V
basal
,
V
rim
, and
V
solid
and resulting relative
densities are shown in Table S2. The average relative density as
calculated from direct image measurements was 30.1%, showing
decent similarity with the generalized model.
For analysis and discussion in this paper we use the relative
density calculated from image measurements as it provides a
direct value for each unique beam sample.
Fig. S1.
SEM image giving a top-down view of a representative beam sample. (
Inset
) The sharp contrast difference within the basal plate.
Fig. S2.
Schematic showing the hexagonal unit of frustule. (
A
) A cross-section of the frustule shell. (
B
–
D
) Top-down views of each layer of the frustule shell.
Aitken et al.
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Fig. S3.
Displays the von Mises stress distribution throughout the bending beam in an (
A
) oblique view to highlight the stress distribution in the areolae walls
and (
B
) top-down view.
Fig. S4.
(
A
–
C
) SEM snapshots demonstrating the focused ion beam lift-out method. The FIB is used to mill out a rectangular sample and a Pt deposition needle
is used to attach to the tungsten micromanipulator for lift-out. This technique can be used to prepare (
D
) bending samples, (
E
) nanoindentation samples, and
(
F
) TEM lamella from the frustule shell. Dimensions of the beam sample are shown in
D
. The gauge length of the beam is
2L
and the height of beam is
h
.
Thickness of the cribrum layer,
w
c
, and basal layer,
w
b
, are also shown.
Aitken et al.
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Table S1. Pore dimensions and layer widths used in the general model for relative density
Sample
d
c
,
μ
m
d
b, inner
,
μ
m
d
b, outer
,
μ
m
w
cribrum
,
μ
m
w
areolae
,
μ
m
w
basal
,
μ
m
w
rim
,
μ
m
ρ
relative
1
0.34
0.80
1.47
0.42
2.66
0.62
0.29
0.401
2
0.34
0.82
1.37
0.26
2.97
0.41
0.19
0.314
3
0.35
0.89
1.49
0.36
2.67
0.48
0.28
0.369
4
0.27
0.90
1.43
0.31
1.94
0.32
0.20
0.373
5
0.30
0.70
1.33
0.38
2.49
0.35
0.21
0.360
Table S2. Layer and solid volumes for each beam sample and associated
relative density
Sample
V
cribrum
,
μ
m
3
V
areolae
,
μ
m
3
V
basal
,
μ
m
3
V
rim
,
μ
m
3
V
solid
,
μ
m
3
ρ
relative
1
25.05
20.38
44.80
4.20
280.65
0.336
2
17.98
28.46
35.84
1.38
322.33
0.260
3
38.40
39.75
57.39
5.01
455.15
0.309
4
20.52
16.30
21.80
1.80
196.42
0.308
5
27.84
22.17
28.83
2.51
277.52
0.293
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