of 11
Supplementary Materials
Hierarchical Nanolattice Design
The hierarchical nanolattices fabricated in this work
are
designed
using a recursive meth
od
that
combines different unit cells into hierarchical geometries
. The design process
takes place
as
follows
:
1)
two (unique or identical) unit cell geometries
are
prescribed
,
2) one unit cell
is
designated 1
st
order
and
the other 2
nd
order, and
3) the 1
st
order unit cell
is
patterned along the length of the 2
nd
order unit cell
with
N
repeating units
,
resulting in
a fractal
-
like geometry
(Fig. S1)
. Th
ese steps can be repeated
iteratively to create a
fractal
of any order, and the
method
is sufficiently general that it
can be repeated
for
a wide range of unit cell geometries
(Fig.
S2
)
. For the
samples
tested
in this
work, 1
st
order
axial
support beams
were
added along the length of the 2
nd
order beam to ensure that the hierarchical beams
form
ed
a stretching
-
dominated geometry.
Figure S1
: Hierarchical nanolattice design project showing the construction of an octahed
ron
-
of
-
octahedra unit cell.
Figure S2
:
H
ierarchical nanolattice geometries showing the versatility of the design process.
The different
geometries shown are
:
a
)
a cage of crosses, b) a cube of reinforced
-
BCC unit cells, c) an octet
-
truss
of octahedra,
an
d d) a
cub
octahedron of embedded
-
octahedra.
Fabrication Details
Hierarchical
nanolattice
s
were fabricated
out of
solid polymer, polymer
-
ceramic
core
-
shell
composite,
and
hollow ceramic.
There are three unique steps involved in the fabrication of these samp
les: 1) writing
of a polymer sample, 2) coating of that polymer sample with a ceramic thin film, and 3) removal of the
polymer
. E
ach material system tested in this work represents one of the steps in
volved in
the fabrication
process.
Fabrication of all sam
ples begins with the writing of a polymer sample out of photoresist
(IP
-
Dip
780)
using
two
-
photon lithography
(TPL)
direct laser writing
(DLW)
in
a
Photonic Professional
DLW
system (Nanoscribe GmbH).
Samples
are written using l
aser powers in a range from 6
-
12
mW and a
writing speed of ~50μm/s.
The minimum writing spot size (voxel) is an ellipsoid with a width between
150
-
600nm and approximately a 4:1 height to width aspect ratio.
The laser power is used to control the
diameter of the tubes, and the speed var
ies slightly during the writing process to control the quality of
the structure.
In the samples
,
individual 1
st
order beams
are written with three lines
one central line
and one on either side with a 150μm offset
perpendicular to the axis of the beam
t
o reduce the
ellipticity of the beam.
A
fter a polymer scaffold is
written
,
samples
are conformally coated
with
20nm of aluminum
oxide (alumina)
using atomic layer deposition (ALD). Deposition is done at 150°C in a Cambridge
Nanotech S200 ALD system using
the following steps: H
2
O is pulsed for 15ms, the system is purged for
20s, trimethyl aluminum (TMA) is pulsed for 15ms, the system is purged for 20s, and the process is
repeated. The carrier gas is nitrogen, which is used at a flow rate of 20sccm (standard
cubic centimeters
per minute). The process was
repeated
for
2
00 cycles to obtain the desired thickness coating. The
thickness of the coatings was verified using spectroscopic ellipsometry with an alpha
-
SE Ellips
ometer
(J.A. Wollam Co., Inc.).
After deposi
tion,
a
focused ion beam (FIB)
(Versa 3D DualBeam, FEI) is used to mill away
selected regions of the sample
to expose the polymer to air. O
nce the polymer is exposed,
samples are
placed into an O
2
plasma
etcher
(Zepto, Diener GmbH) for between 30
-
60
hours
at a pressure of 0.5
mbar and
at
100W of power in order to fully remove the polyme
r.
It is possible to
determine whether
the
polymer has been
fully
etched away by looking
for any contrast change in the beams using a
scanning e
lectron microscope
(1)
.
Polymer Constituent Properties
Mechanical characterization of the
IP
-
Dip
polymer was
performed throu
gh
micropillar
compression experiments
in a nanomechanical testing
device
(
TI 950 Triboindenter
, Hysitron Inc.
).
Micropillars were compressed
using a 20μm diamond flat punch tip
to 10
-
15% strain at a
rate of
10
-
3
s
-
1
then
held at their peak displacement fo
r 50
s
before unloading
.
Samples
were fabricated out of IP
-
Dip
photoresist using an identical
DLW
method
to that described above for hierarchical nanolattices
.
Samples were fabricated
and tested
with
diameters between 2
-
10μm,
and
length
-
to
-
diameter (L/D)
r
atios
between 2 and 4.
For each
micropillar
,
stress
-
strain data was obtained and used
to determine the
Young’s
modulus (
)
and compressive yield strength
(
)
. The
Young’s
modulus is
calculated using the slope of
the linear regime
of the
stress
-
stra
in
curve.
The compressive y
ield strength is
calculated by finding the
intersection
of the stress
-
strain data with
a 0.2% strain offset
curve
from the linear
regime
.
Three
representative data sets
along with their corresponding yield strength and stiffnesse
s
are shown in
Figure S3.
The stress
-
strain data has an initial toe region followed by a linear regime and then a plastic
flow region. The
toe region
is likely due
to
improper
alignment or
contact of the indenter tip with the
sample
, and was consequently i
gnored in the calculation of the Young’s modulus
.
Based on
the stress
-
strain data,
we
found that the IP
-
Dip polymer had an average modulus of
=
.
±
.
푮푷풂
and an
average yield strength of
=
ퟔퟕ
.
±
.
푴푷풂
(Fig.
S3
)
.
Figure S3
:
Re
presentative stress
-
strain data
for each slenderness ratio tested (L/D = 2, 3, 4) with various radii
.
The
arithmetic mean of the Young’s
modulus (
E
) and compressive yield strength (
σ
y
) are
plotted
.
The inset
image
shows
a set of
pre
-
compression
mi
cropillar
samples
(scale bar: 50
μm
).
Composite Constituent Properties
The properties of the composite were calculated using a Voigt model rule of mixtures
, with the
properties of the ALD Al
2
O
3
taken from (
Refs.
2
6
).
In the 2
nd
order half cells, the polymer beams have
dimensions of
=
753
푛푚
and
=
317
푛푚
, where
and
are the major and minor radii of the ellipse
,
respectively. The ceramic shell has a thickness of
=
20
푛푚
;
the volume fraction of polymer in the
beams can be calculated to be
=
푝표푙푦푚푒푟
/
(
푝표푙푦
푒푟
+
푐푒푟푎푚푖푐
)
=
91
.
6%
. Given the polymer
properties of
=
2
.
1
±
0
.
3
퐺푃푎
and
푦푝
=
67
.
2
±
4
.
7
푀푃푎
, and the ceramic properties of
=
165
퐺푃푎
and
=
5
.
2
퐺푃푎
,
we can obtain the core shell composite properties to be
+
(
1
)
=
=
ퟏퟓ
.
푮푷풂
and
푦푝
+
(
1
)
=
풚풄
=
ퟓퟎퟗ
푴푷풂
.
Relative Density Calculations
The relative density of each hierarchical sample was determined using a representative CAD
model (
S
OLIDWORKS,
Dassault Systè
mes
).
The model enabled quick calculations for solid and hollow
samples. Figure S4 illustrate
s
the
design
process
of the CAD model.
Figure S4
:
Illustration of the design approach used to construct a
CAD model
. The above images show the
construction of
a
2
nd
order octahedron
-
of
-
octets unit cell. (1)
U
nique no
des that make up the unit cell.
(2)
Nodes are
minimally patterned
to create a unit cell beam. (3
-
4) Assembled nodes are patterned to create the full geometry
.
(5) Boundaries are cut to form
the unit cell
.
(6) Finished unit cell whose volume properties can be measured.
Failure Mode Formulation
Failure in hierarchical nanolattices primarily initiate
d
in 1
st
order beams
. In polymer samples, failure
occurs via beam
(Euler)
buckling, while in composite samples f
ailure occurs via brittle fracture.
The
continuous and serrated flow behavior of the hollow nanolattices indicates the occurrence of Euler
buckling and shell buckling, the combination of which gives rise to a ductile
-
like deformation with a near
100% recov
ery. L
arge drops in the stress generally correspond to large Euler buckling events; smaller
drops are more likely to coincide with localized shell buckling events
. The failure criterion for
beam
buckling and shell buckling
in an elliptical beam with a semi
-
major axis
, semi
-
minor axis
, thickness
,
and length
are
푏푒푎푚
=
2
퐸퐼
(
푘퐿
)
2
2
(
3
4
2
)
2
(
휋푡
(
+
)
)
=
3
2
4
2
2
+
(S1)
푒푙푙
=
3
(
1
2
)
(
)
(S2)
Here,
is the
Young’s modulus and
is the Poisson’s ratio of the material. By equating the failure
criterion for beam buckling and shell buckling, i
t is possible to find a transition
between the two
. Here,
we quantify the transition using a critical dimensionless para
meter
푆퐵
as
3
2
4
2
2
+
=
3
(
1
2
)
(
)
(S3)
푆퐵
=
(
(
+
)
2
2
2
)
푐푟
=
3
2
4
3
(
1
2
)
12
.
2
(S4)
If the
critical buckling transition value
for a beam is
푆퐵
12
.
2
, shell buckling wil
l be the dominant
failure mechanism over beam buckling, and if
푆퐵
>
12
.
2
, beam buckling will be the dominant failure
mechanism. In the hollow samples tested in this work,
the
푆퐵
values range from 14.8 to 33.3, meaning
that all the samples tested here are
in a
beam buckling failure mode regime
.
Previous w
ork by Meza et. al. reported that hollow Al
2
O
3
nanolattices exhibit ductile
-
like behavior with
near complete recoverability when the wall
-
thickness
-
to
-
tube
-
radius ratio is below a critical transition
valu
e of
(
)
푐푟
=
푓푠
3
(
1
2
)
0
.
03
, where
푓푠
is the fracture strength of the constituent material
(1)
.
All samples tested in this work have
(
)
<
0
.
03
<
(
)
푐푟
, meaning that shell buckling will take place
preferentially over yielding. This analytical formulation shows that in the 1
st
order beams, beam buckling
will t
ake place preferentially over
shell buckl
ing, and after the occurrence of beam buckling, shell
buckling will take place over fracture, allowing samples to deform in a ductile
-
like manner and
accommodate large strains. This matches very closely to what is o
bserved experimentally
.
Full Hierarchical Nanol
attice Mechanical Behavior
2
nd
order full hierarchical nanolattices were fabricated in a 3x3x3 array using octahedra
-
of
-
octahedra
hierarchical unit cells. Samples were compressed using an in
-
situ nanoindenter
at a strain rate of 10
-
3
s
-
1
to 50% strain in a manner nearly identical to the half
-
cell samples.
Failure mechanisms in the
2
nd
order
full nanolattices
closely matched those in the 2
nd
order half
-
cells for each material type
:
p
olymer
samples underwent duc
tile failure
via beam buckling, composite samples underwent catastrophic brittle
collapse via fracture of the constituent Al
2
O
3
, and hollow samples underwent serrated ductile
-
like failure
via a combination of beam buckling and shell buckling.
The stiffnes
s and strength of
the
hierarchical
nanolattice samples
closely
matched that of the half
-
cell
samples, and a full list of results can be found in Figure S5.
The primary
distinction
between the failure
of the full samples and the half
-
cell samples
is th
e emergence of a layer
-
by
-
layer collapse deformation
mechanism, which
led to high local stresses
in the beams and
reduced the recoverability in the full
nanolattices compared with the half
-
cell samples. 2
nd
order polymer nanolattices recovered immediately
to 65
-
85% of their original height, with additional viscoelastic rec
overies of 5
-
20% after unloading
. 2
nd
order composite nanolattices had a catastrophic strain burst with failure localized to the top of the
lattice and exhibited little to no recovery. 2
n
d
order hollow nanolattices had layer
-
by
-
layer deformation
that limited their recovery to only 60
-
75% of the original height, compared with 85
-
98% in the
equi
valent hollow half
-
cell samples.
In hollow samples, the layer
-
by
-
layer collapse
generated local hig
hly strained regions where the buckled
beams partially fractured without complete failure
.
The bifurcated beams experienced marginal
recovery due to the
residual strain energy being
insufficient
to return the sample to its original
configuration. Preventin
g layer
-
by
-
layer collapse would improve recovery but requires a more efficient
hierarchical geometry to optimize load distribution.
Figure S
5
:
Comparison
of
mechanical data of
2
nd
order half
-
cells against full hierarchical nanolattices showing their
clos
e agreement for
(a)
Young’s modulus vs. relative density and
(b)
yield strength vs. relative density.
Effect of Slenderness on Deformation
Samples with a range of structural parameters and slenderness ratios were tested in this work.
W
e show
a number of re
presentative samples of different material systems demonstrating the deformation and
failure modes of samples with high and low slenderness
in Figure S6 below.
Figure S
6
:
Compression experiments performed on
2
nd
order
half
-
cells showing the samples in an
undeformed
configuration, at 50% strain, and unloaded, along with the corresponding load (mN) v displacement (μm) data.
All
‘high slenderness’ samples are octahedra
-
of
-
octahedra and all ‘low slenderness’ samples are octahedra
-
of
-
octets.
Samples
a
-
c
have
=
8
휇푚
and
=
10
, and samples
d
-
f
have
=
12
휇푚
and
=
20
.
Samples
A
and
D
are pure
polymer,
B
and
E
are polymer
-
ceramic core
-
shell composites, and
C
and
F
are hollow Al
2
O
3
.
FEM Modeling
Modeling hierarchical lattices is a computationally demanding probl
em
because
the problem size scales
exponentially. To reduce problem size one can use either a simplistic method or an adaptive method.
The first, simplistic method uses bar elements in the Truss Model. Bar elements are the simplest and
computationally least ex
pensive method of modeling lattices.
O
ur simple refined model is significantly
more complex but can adaptively scale with the level of hierarchy through repeated use of substructure
generation and utilization.
Two modeling approaches we
re used to predict lattice stiffness and stress distributions. The first
approach uses an in
-
house finite element code with small
-
strain bar elements having only axial stiffness.
A linear elastic material model is assumed to govern the bar response. Figure
S
7
shows an example truss
structure with the half
-
cell bottom fully constrained and the vertical displacements imposed on the top
nodes of the cell. Vertical reaction forces were calculated at the top nodes to determine the lattice
stiffness.
Figure S
7
: Truss model of half
-
cell octahedron of octahedrons with applied boundary condi
tions: red
all
translation DOF
constrained, blue
prescribed
vertical displacement.
In contrast to the first model approach, the second, referred to as the refined model, i
s a scalable
solution to modeling hierarchical lattices by using degree
-
of
-
freedom (DOF) condensation. The refined
model consists of two stages: substructure generation and substructure utilization.
Substructure
generation
begins with identifying the geome
trically uniq
ue nodes and beams (Fig
.
4).
For compatibility between nodes and beams generalized DOFs, referred to as compatibility DOF, are
introduced at the center of each node
-
and beam
-
connecting edge or face and the latter are assumed to
deform rigidl
y; e.g. the end faces of each beam deform rigidly so that their motion can be described
solely by its compatibility DOFs (i.e., its generalized translational and rotational DOFs). A substructure
condensation method is applied to each beam or node part such
that the only remaining DOFs are the
compatibility DOFs.
Substructure utilization
then extracts the effective stiffness matrices of all
substructure parts, assembles those accordingly into a hierarchical half
-
cell lattice, and couples the
compatibility DO
F of each substructure to the appropriate adjacent substructure (Fig
.
4
D and E
).
Substructure generation and utilization can be performed repeatedly on increasingly larger levels of
hierarchy to reduce the computational cost of evaluating the final lattic
e stiffness. Also, the extraction of
the effective beam and node response from full
-
scale finite element models of the latter can be
performed a priori so there is no need to recover full
-
resolution simulations during the final simulation
run.
Two general
trends are observed for solid and hollow lattices. First, solid lattice stiffness predicted by
the simple truss model is generally less than that of the refined model. Secondly, hollow latt
ice stiffness
predicted by the truss model is greater than that of the refined model. These differences arise
because
of the contributions of
bending and node junctions
to
degrading
the
overall lattice stiffness in the
refined model. Figure S
8
summarizes co
mputational and experimental stiffness data and
compares the
simulated stiffness
es
obtained from
each model
for geometrically perfect lattices.
T
he refined model
more
accurately predicts
properties of
solid polymer and hollow ceramic lattice stiffness by 5.0 and 1.4
times
,
respectivel
y
,
relative to
the
truss predictions.
I
n simulating composites
nano
lattices
,
the truss
model is 1.4 times more accurate than the refined model. If one ta
kes into account
geometric
imperfections
of the nanolattices
, such as the
beam waviness, the predicted stiffness of both the refined
model and
the
truss model would decrease such that the refined model is always more accurate than
the truss model relative t
o experimental data.
Figure S
8
:
Comparison of model approximations and experimental data of second order half
-
cell lattices consisting
of (a) solid polymer, (b) solid polymer with ceramic coating,
(c) hollow ceramic
.
Complete List of Samples Tested
Co
mposition
Geometry
Unit Cell Size (μm)
Fractal Number
Relative Density
Solid Polymer
Octahedron
-
of
-
Octahedra
8
10
1.03E
-
2
8
15
4.49E
-
3
8
20
2.50E
-
3
12
10
4.80E
-
3
12
15
2.09E
-
3
12
20
1.16E
-
3
Octahedron
-
of
-
Octets
8
10
2.10E
-
2
8
15
9.46E
-
3
8
20
5.34E
-
3
12
10
9.82E
-
3
12
15
4.41E
-
3
12
20
2.49E
-
3
Composite
Octahedron
-
of
-
Octahedra
8
10
1.12E
-
2
8
15
4.88E
-
3
8
20
2.72E
-
3
12
10
5.23E
-
3
12
15
2.28E
-
3
12
20
1.27E
-
3
Octahedron
-
of
-
Octets
8
10
2.28E
-
2
8
15
1.03E
-
2
8
20
5.83E
-
3
12
10
1.07E
-
2
12
15
4.81E
-
3
12
20
2.73E
-
3
Hollow Al
2
O
3
Octahedron
-
of
-
Octahedra
8
10
8.95E
-
4
8
15
3.90E
-
4
8
20
2.17E
-
4
12
10
4.30E
-
4
12
15
1.87E
-
4
12
20
1.04E
-
4
Octahedron
-
of
-
Octets
8
10
1.82E
-
3
8
15
8.36E
-
4
8
2
0
4.89E
-
4
12
10
8.78E
-
4
12
15
3.95E
-
4
12
20
2.41E
-
4
Table S1
: Full list of fabricated 2
nd
order half
-
cell geometries with corresponding
relative densities
Composition
Unit Cell Size (μm)
Fractal Number
Relative Density
Solid Polymer
8
5
2.32
E
-
2
3
10
3.55
E
-
3
Composite
8
5
1.83E
-
2
8
5
2.51
E
-
2
3
10
2.15
E
-
3
Hollow Al
2
O
3
8
5
1.83
E
-
3
3
10
4.73
E
-
4
Table S2
: Full list of fabricated 3
rd
order octahedron half
-
cell geometries with corresponding
relative densities
Composition
Unit Cell Size (μm)
F
ractal Number
Relative Density
Solid Polymer
8
10
1.03
E
-
2
6
15
4.95
E
-
3
4
20
3.26
E
-
3
Composite
8
10
1.12
E
-
2
6
15
5.49
E
-
3
Hollow Al
2
O
3
8
10
8.95
E
-
4
6
15
5.35
E
-
4
4
20
4.71
E
-
4
Table S3
: Full list of fabricated 2
nd
order octahedron full
-
lattice ge
ometries with corresponding
relative densities
References
1.
Meza LR, Das S, Greer JR (2014) Strong, lightweight, and recoverable three
-
dimensional ceramic
nanolattices.
Science
345(6202):1322
1326.
2.
Ber
dova M, et al. (2014) Mechanical assessment of suspended ALD thin films by bulge and shaft
-
loading techniques.
Acta Mater
66:370
377.
3.
Ilic B, Krylov S, Craighead HG (2010) Young’s modulus and density measurements of thin atomic
layer deposited films us
ing resonant nanomechanics.
J Appl Phys
108:1
11.
4.
Jen S
-
H, Bertrand JA, George SM (2011) Critical tensile and compressive strains for cracking of
Al2O3 films grown by atomic layer deposition.
J Appl Phys
109(8):084305.
5.
Tripp MK, et al. (2006) The m
echanical properties of atomic layer deposited alumina for use in
micro
-
and nano
-
electromechanical systems.
Sensors Actuators A Phys
130
-
131:419
429.
6.
Bauer J, et al. (2015) Push
-
to
-
pull tensile testing of ultra
-
strong nanoscale ceramic
polymer
composi
tes made by additive manufacturing.
Extrem Mech Lett
.