of 10
L. C. Montemayor
Division of Engineering and Applied Science,
California Institute of Technology,
Pasadena, CA 91125
J. R. Greer
Division of Engineering and Applied Science,
California Institute of Technology,
Pasadena, CA 91125
e-mail: jrgreer@caltech.edu
Mechanical Response
of Hollow Metallic Nanolattices:
Combining Structural
and Material Size Effects
Ordered cellular solids have higher compressive yield strength and stiffness compared to
stochastic foams. The mechanical properties of cellular solids depend on their relative
density and follow structural scaling laws. These scaling laws assume the mechanical
properties of the constituent materials, like modulus and yield strength, to be constant
and dictate that equivalent-density cellular solids made from the same material should
have identical mechanical properties. We present the fabrication and mechanical proper-
ties of three-dimensional hollow gold nanolattices whose compressive responses demon-
strate that strength and stiffness vary as a function of geometry and tube wall thickness.
All nanolattices had octahedron geometry, a constant relative density,
q

5
%
, a unit cell
size of 5–20
l
m, and a constant grain size in the Au film of 25–50 nm. Structural effects
were explored by increasing the unit cell angle from 30 deg to 60 deg while keeping all
other parameters constant; material size effects were probed by varying the tube wall
thickness,
t
, from 200 nm to 635 nm, at a constant relative density and grain size. In situ
uniaxial compression experiments revealed an order of magnitude increase in yield stress
and modulus in nanolattices with greater lattice angles, and a 150
%
increase in the yield
strength without a concomitant change in modulus in thicker-walled nanolattices for fixed
lattice angles. These results imply that independent control of structural and material
size effects enables tunability of mechanical properties of three-dimensional architected
metamaterials and highlight the importance of material, geometric, and microstructural
effects in small-scale mechanics.
[DOI: 10.1115/1.4030361]
Keywords: cellular solids, hierarchical materials, size effect, metamaterials
1 Introduction
Hierarchically designed cellular materials have been used for
many decades as a method to create mechanically robust engi-
neered structures—like the Eiffel Tower and the Garabit Viaduct.
Introducing architectural elements enables creating structures that
are lightweight, because they use a fraction of monolithic material
with the same dimensions, and simultaneously strong since the
architecture provides a way to more efficiently carry load in a
structure. The mechanical performance of such architected solids
on the macroscale is a function of the deformation mechanism
and relative density of the structure, as well as of the constituent
material properties, and this has been studied for both materials
with cellular solid cores as well as bulk cellular solids [
1
16
]. Cel-
lular solids can deform by either bending or stretching of the ele-
ments, which is dictated by the geometry of the lattice and its
nodal connectivity, and this bending or stretching behavior defines
the deformation mechanism of the cellular solid [
1
,
2
,
5
,
17
,
18
]. A
three-dimensional structure must have a connectivity of
Z
¼
6at
the nodes to be rigid and a connectivity of
Z
¼
12 to be stretching-
dominated; structures with 6

Z
<
12 are bending-dominated
[
18
]. The structural deformation mechanism, determined by the
nodal connectivity, directly impacts the modulus and yield
strength of the overall structure [
1
,
2
]. The modulus and yield
strength are also related to the structure’s relative density,

q
,
which is defined as the volume of the lattice contained within a
unit cell divided by the volume of the unit cell [
2
]. The yield
strength and modulus of 3D open-cell
bending-dominated
struc-
tures, such as honeycombs or octahedron lattices, scale as
r
y
¼
0
:
3

q
1
:
5
r
ys
and
E
¼

q
2
E
s
, where
r
ys
and
E
s
are the yield
strength and modulus of the constituent material [
1
,
2
]. For 3D
stretching-dominated
structures, such as the octet-truss lattice, the
yield strength and modulus scale as
r
y
¼
0
:
3

qr
ys
and
E
¼
0
:
3

q
E
s
,
which causes strength to decrease less rapidly than that of
bending-dominated structures as relative density decreases [
1
,
5
].
A theoretical upper limit for strength and modulus of cellular
solids exists as a function of relative density because the mechani-
cal properties of the lattice depend on the constituent material
properties. All existing engineering materials with densities lower
than 100 kg/m
3
fall below this theoretical maximum by an order
of magnitude or more, leaving a prominent “white space” in the
strength versus density material property space, shown in Fig.
1
[
1
]. When calculating this material property space, the strength
and stiffness of the constituent materials are assumed to be con-
stant so the deformation mechanism of the structure represents the
only variable parameter. For macroscale cellular solids, this
assumption is reasonable because yield strength is independent of
size for materials at this length scale. Below the micron scale,
many different classes of materials exhibit size effects, such as
“smaller is stronger” in single crystalline metals [
19
23
], “smaller
is weaker” in nanocrystalline metals [
20
,
24
27
], and “smaller is
ductile” in metallic glasses and ceramics [
20
,
28
31
]. This renders
the mechanical properties of materials at these dimensions differ-
ent from bulk, and these properties are no longer constant. Creat-
ing hierarchical structural geometries with submicron dimensions,
such as nanolattices, has been shown to provide a pathway to con-
trol and tune the strength and stiffness of cellular solids.
2 Background
2.1 Background: Cellular Solids.
Relative densities of cel-
lular solids can be modulated by several approaches, for example,
Contributed by the Applied Mechanics Division of ASME for publication in the
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OURNAL OF
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PPLIED
M
ECHANICS
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A. Amine Benzerga.
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by using hollow tubes instead of solid rods within the same archi-
tecture. When hollow tubes are utilized in low-density cellular
solids, structural effects can be activated by changes in the various
ratios of geometric parameters that define the lattice tubes. While
these structural effects do not fundamentally change the deforma-
tion mechanism of the entire lattice, deformation in the individual
lattice tubes can be tuned by changing the geometry. For example,
the compressive response of hollow thin-walled Ni-based micro-
lattices demonstrated that its deformation behavior and recover-
ability depend on the geometric parameters of the lattice tubes,
like the tube diameter-to-length ratio,
D
/
l
, and wall thickness-to-
diameter ratio,
t
/
l
[
9
11
,
28
,
32
34
]. Valdevit et al. proposed a
model in which the deformation mechanism of the individual
microlattice tubes transitions from Euler buckling to yielding of
the constituent material at the nodes at a critical relative density,
which is directly proportional to (
D
/
l
) and (
t
/
l
)[
9
]. This model
assumes that all members of the unit cell carry equal force and
that the maximum bending moment occurs at the nodes; the stress
at the node is not taken into account [
9
]. The results of this ana-
lytic model compared favorably with those of the finite element
method (FEM) simulations that account for the complex nodal
geometry, where the latter also exhibited a critical relative density
at which the deformation mode of the tubes transitioned from
Euler buckling to yielding, albeit at lower overall stresses, and the
transition occurred at a lower relative density [
9
]. Torrents et al.
proposed a simple analytic model that predicts a critical (
t
/
D
) ratio
above which plastic deformation begins. This critical (
t
/
D
)
crit
is a
function of the constituent material properties, tube angle, and
maximum global strain of the microlattice [
10
]. Compression
experiments showed that the microlattices fully recovered after
50% compressive strain for (
t
/
D
)
<
(
t
/
D
)
crit
, which has been dem-
onstrated for a variety of materials [
10
,
34
]. While the geometric
ratios of the microlattices impact the deformation mechanism of
the individual tubes, macroscopic parameters of the overall lattice,
such as the lattice angle, also affect the mechanical response of
the structure. Previous work defined the lattice angle to be the
angle between the horizontal midplane of the unit cell and the lat-
tice tubes, as shown in Fig.
2
[
35
,
36
]. Jacobsen et al. showed that
strength and stiffness of solid polymer microlattices increased
with higher lattice angle, as predicted by classical mechanics [
35
].
The calculated stiffness of the microlattices was found to be
within 15% of the model proposed by Deshpande et al., which
predicts that for a pyramidal core (stretching-dominated), the stiff-
ness scales as
E
¼
E
s

q
ð
sin
h
Þ
4
[
4
,
35
]. The microlattices presented
in the work by Jacobsen et al. have a connectivity of
Z
¼
8, which
does not rigorously satisfy the condition for a stretching-
dominated structure and might explain the observed discrepancy
Fig. 1 Hierarchically structured materials combine structural and material size effects to
enhance material properties and provide opportunities to create new materials that outper-
form current low-density materials. Material properties chart generated using CES SELEC-
TOR (image courtesy of S. Das). (Figures reprinted with permission from Greer and De
Hosson [
20
], Gu et al. [
26
], and Montemayor et al. [
49
]. Copyright 2011 by Elsevier Ltd. and
2012 by American Chemical Society, respectively.)
Fig. 2 (
a
) Schematic of relevant geometric parameters on a
nanolattice unit cell and (
b
) TEM image showing columnar grain
structure of tubes with grain size on the order of 50 nm (TEM
courtesy of Z. Aitken, scale bar 100 nm)
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between the model and the experiments [
35
]. Previous work on
microlattices demonstrates the importance of understanding struc-
tural effects that may potentially couple with material size effects
in complex hollow-tube architectures and provides insight to uti-
lizing structural effects in three-dimensional architected materials
[
4
,
9
11
,
28
,
32
35
]. Valdevit et al. have shown in simulation that
the analytical predictions for strength and stiffness where
r
y
/

q
m
and
E
/

q
n
overpredict the strength of a hollow cellular solid by
as much as an order of magnitude and the experimental results on
hollow microlattices agree with the finite element simulations for
a periodic cellular solid with hollow nodes [
2
,
9
]. Torrents et al.
have shown for the microlattices that the strength follows the ana-
lytically predicted scaling relations, however, again a knock-down
in strength is observed and this is attributed to the complex stress
state at the hollow nodes under uniaxial loading [
10
]. Recently,
Zheng et al. demonstrated the ability to create ceramic and
ceramic–polymer composite structures with multiple geometries,
on the same length scales as the microlattices, using a microster-
eolithographic process [
6
]. For stretching-dominated structures in
the work by Zheng et al., the strength was found to scale as
r
y
/

q
and the modulus as
E
/

q
, for 10

4
<

q
<
10

1
, which
agrees with the classical theory proposed by Gibson and Ashby
for stretching-dominated cellular solids [
2
,
6
]. Meza et al. have
proposed a model to describe the local deformation behavior of
the nanolattices by considering the compression of a single half
unit cell; they found that the deformation of the tubes was gov-
erned by lateral torsional bending which is caused by minor geo-
metric imperfections at the nodes and these results agree with the
experimental data [
8
,
37
]. Meza et al. have also reported the
mechanical behavior of stretching-dominated alumina nanolatti-
ces that were fabricated using two-photon lithography followed
by deposition of a rigid coating and etching out of the polymer
scaffold and found that the strength scales as
r
y
/

q
1
:
76
, the mod-
ulus scales as
E
/

q
1
:
61
, and that the structures recovered after
compression in excess of 50% [
7
]. The decrease in scaling for
both strength and stiffnes
s is attributed primarily to the complex stress
state at the hollow nodes and the minor geometric imperfections of
the nanolattices [
7
].
2.2 Background: Material Size Effects
2.2.1 Uniaxial Deformation.
A variety of material size effects
have been observed in nanoscale samples with different micro-
structures [
19
27
]. Single crystalline nanopillars exhibit a smaller
is stronger size effect demonstrated through experiments, theory,
and computations for face-centered-cubic (FCC), body-centered-
cubic, hexagonal-close-packed metals [
19
24
,
38
40
]. For exam-
ple, the yield strength of single crystalline FCC metals can be
described by a power law dependence between strength and size,
r
/
D

n
, where
D
is the pillar diameter and
n
ranges between 0.5
and 0.7 [
19
,
20
,
23
,
24
]. Nanocrystalline metals with similar exter-
nal dimensions exhibit the opposite effect on strength, with the
emergence of a smaller is weaker size effect at a critical sample
size-to-grain size ratio, (
D
/
d
), unique to each material. For exam-
ple, Pt nanopillars with 12 nm grains had a critical (
D
/
d
) of 5–10;
in Ni nanopillars with 60 nm grains, the weakening occurred for
(
D
/
d
) between 15 and 30, and in polycrystalline Cu samples weak-
ening was observed in experiments and simulations for (
D
/
d
)
<
10
[
25
27
]. Gu et al. proposed a model that describes how the yield
strength of a nanocrystalline metal changes as a function of the
diameter of a cylindrical sample relative to the material grain size
[
26
]. This model assumes that only the grains in the outer shell
interact with the free surface and the grains in the inner core of
the sample are not affected by it [
26
]. The model predicts that for
large
D
/
d
, the strength approaches the bulk strength, and for small
D
/
d
, the deformation is dominated by grain boundary sliding,
which weakens the sample [
26
]. The following equation shows
the relationship between the yield strength and the sample size-to-
grain size ratio, where
d
is the grain size,
D
is the sample size,
r
s
Y
is the stress required to initiate grain boundary sliding, and
r
y
ð
bulk
Þ
is the strength of the bulk material with the same microstructure
[
26
]
r
n
ð
Au
Þ
r
y
ð
bulk
Þ
¼
1

d
D

2
þ
r
s
Y
r
y
ð
bulk
Þ
1

1

d
D

2
"#
(1)
2.2.2 Beyond Nanopillars: Deformation Under Complex
Stress States.
Hierarchical architected structures allow opportuni-
ties for material size effects to be utilized because the novel mate-
rial properties present at the submicron length scales can be
proliferated onto a larger structure. Hodge et al. showed that a sto-
chastic nanocrystalline Au nanoporous open-cell foam exhibits a
smaller is stronger size effect as the ligament size decreases from
900 nm to 10 nm; these foams outperform the expected strength
of a cellular solid with a relative density of

0.24–0.32 according
to the scaling law, where
r
y
¼
0
:
3

q
1
:
5
r
ys
[
16
]. Larger ligament
sizes lead to a reduction in strength, with the precise mechanism
still being a matter of ongoing discussion due to the complexity of
the interplay between microstructure and free surfaces in the
nanoporous foams [
16
].
In contrast to the stochastic nature of the Au foams, the microlatti-
ces have an ordered and controllable geometry, which allows the ex-
ploitation of the smaller is ductile material size effect for some
material systems [
28
]. For example, Rys et al. showed that the com-
pressive strength of NiP metallic glass microlattices with relative
densities of 10

4
is an order of magnitude higher than that of nano-
crystalline NiP microlattices with the same geometry [
28
]. That
work also demonstrated that these metallic glass hollow-tube micro-
trusses with 150 nm-thick tube wa
lls deformed in a ductile fashion,
in contrast to their thicker-walled counterparts, which undergo cata-
strophic failure upon loading. Such a transition in deformation mode
from brittle to ductile has been reported for nanosized metallic
glasses and has been attributed to the energetic balance between
shear band propagation and homogeneous deformation [
20
,
28
31
].
The authors also showed that the scaling of yield strength with den-
sity decreases from
q
2.3
to
q
1.4
when the wall thickness falls below
150 nm [
28
]. Gu et al. have shown that material size effects can also
be exploited to enhance the strength and stiffness of solid tube latti-
ces with relative densities between 40% and 80% with single crys-
talline regions within the ligaments [
41
]. For lattices with relative
densities above 50%, Gu et al. observed a strength 1.8 times that of
the bulk Cu as a result of the smaller is stronger size effect [
41
].
This work aims to show that nanola
ttices can exploit both structural
and material size effects that ha
ve been observed in previous work
independently to tune the strength and stiffness of an ordered cellu-
lar solid of a given relative density.
3 Experimental Methods
Nanolattices in this work were fabricated using two-photon
lithography with a 780 nm femtosecond pulsed laser (Nanoscribe
GmbH, Eggenstein-Leopoldshafen, Germany) [
42
48
]. This tech-
nique involves the constructive interference of two photons within
a three-dimensional voxel, which provides a sufficient amount of
energy to cross-link the photoresist within the voxel [
42
]. The
photoresist is placed on a substrate, which is then mounted on a
stage that can move in all directions with a resolution of
6
10nm
and a maximum displacement of 300
l
m. The cross-linked poly-
mer sample can be of any geometry with features down to 150 nm
[
44
,
48
]. The cross section of the tubes in the nanolattices is ellipti-
cal, which is an artifact of the two-photon fabrication process, and
the ellipticity of the individual tubes changes as a function of the
lattice angle. The samples were developed in propylene glycol
monomethyl ether acetate followed by an isopropyl alcohol rinse
to isolate the polymer nanolattices, which were then sputtered
with approximately 200–635 nm of columnar grained Au at 3
mTorr and 50 W. The internal polymer scaffold was then exposed
using a focused ion beam and removed using an O
2
plasma
etch [
49
].
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The transmission electron spectroscopy (TEM) image of the
tube walls shown in Fig.
2
confirms that the sputtered Au has a
columnar structure, with the grains oriented orthogonally to the
length of the tubes, which is consistent with the typical columnar
grains on flat substrates [
50
52
]. The size of the grains ranged
from 50 to 100 nm when measured using a cross section parallel
to the length of the columnar grains and 25–50 nm when meas-
ured from a cross section perpendicular to the length of the colum-
nar grains (Fig.
2(
b
)
). All samples were designed to have
octahedron geometry and a constant relative density, i.e., struc-
tures with thicker tube walls had larger unit cells. A geometric
model was developed in
SOLIDWORKS
and was used to predict the
geometric parameters needed such that the relative density would
be

q

0
:
02

0
:
04 for all samples. After the samples were fabri-
cated, the geometric parameters were measured using scanning
electron microscopy (SEM) and these values were used in the
SOLIDWORKS
model to calculate the relative densities of the fabri-
cated nanolattices, which was found to be

q
¼
0.05
6
0.01. The
mean and standard deviation for the relative density were calcu-
lated by averaging the relative densities of all fabricated samples,
which was determined using SEM images for each fabricated sam-
ple. Thicker tube walls caused greater nonuniformity of the coating,
especially near unit cells along the edge of the nanolattice, and this
is shown in Fig.
3
. For all samples, the wall thicknesses were meas-
ured from SEM images at multiple points along the height, width,
and depth of the compressed regions of the nanolattice.
To exploit structural effects, we fabricated and tested octahe-
dron nanolattices with angles of 30 deg, 45 deg, and 60 deg, and a
constant relative density of

q
¼
0.05
6
0.01. The lattice angle is
defined in this work to be the angle between the horizontal mid-
plane of the unit cell and the lattice tubes, consistent with previous
work, and is shown in Fig.
2
. The wall thicknesses in these sam-
ples were
t
¼
336
6
58 nm. The wall thicknesses presented in this
work are in the regime where material size effects have been
observed for nanopillar geometries, which are discussed in
Sec.
2.2
. Material size effects were probed by varying the wall
thickness for the octahedron geometry nanolattices with constant
relative density of

q
¼
0.05
6
0.01 and a fixed angle of 45 deg;
the same experiment was repeated for lattices with a fixed angle
of 60 deg. Mechanical properties of the nanolattices were obtained
by conducting uniaxial compression experiments in an in situ
nanomechanical instrument (InSEM, Nanomechanics, Inc., Oak
Ridge, TN). Samples were compressed to strains of 40–60% at a
constant prescribed strain rate of
_
e
¼
10

4
s

1
or lower and
load–displacement data were collected. A diamond flat punch
with a diameter of 170
l
m was used to compress the nanolattices.
The stress was calculated according to
r
¼
F
=
A
, where
F
is the
measured load, and
A
is the total cross-sectional area of the nano-
lattice, and the yield stress was calculated using the 0.2% offset
method. The modulus was calculated using the slope from the
elastic loading regime and was calculated using
D
H
=
H
,
where
H
is the height of the nanolattice, and
D
H
is the measured
displacement. The unloading slope of the stress–strain curve was
not used to determine the modulus of the nanolattice, since the
structures deform plastically and the unloading slope would give
the modulus of the densified structure; initial regions where the
sample may not have fully been in contact with the punch were
excluded from the modulus calculation, though these were mini-
mal across the data.
4 Results
4.1 Results: Structural Effects.
To determine the effect of
geometry on the mechanical properties of the nanolattices, experi-
ments were performed on octahedron unit cells with angles of
30 deg, 45 deg, and 60 deg. Figure
4
shows SEM images of the
nanolattices before and after compression, as well as stress–strain
data for a representative sample for each lattice angle. A circle on
the stress–strain curves denotes the yield stress for each of the
nanolattices shown in Fig.
4
. Four samples were fabricated for
each lattice angle and the 0.2% yield stress and modulus were
found by averaging over all samples for each angle, and the data
are shown in Fig.
5
. These plots show that higher angle in the unit
cells leads to higher yield strength and stiffness. As the angle
changes from 30 deg to 60 deg, the yield stress increases from
161.3
6
18.5 kPa to 858.2
6
68.4 kPa, and the stiffness increases
from 8.42
6
0.7 MPa to 85.2
6
15.0 MPa. The data also show that
the yield stress was reached at lower strains at higher lattice
angles.
4.2 Results: Material Size Effects.
To examine the ability of
nanolattices to exploit material size effects, uniaxial compression
experiments were performed on the octahedral nanolattices with
two different fixed lattice angles, 45 deg or 60 deg, and the wall
thickness varied from
t
¼
200–635 nm. Figure
6
shows the images
Fig. 3 Representative images of the Au coating in a 60 deg nanolattice with a wall thick-
ness of

661 nm. In the center region, the nanolattice walls are conformal; however, the
coating is not conformal near the edges of the lattice as a result of the anisotropy of the
sputtering process.
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of 45 deg nanolattices before and after compression, as well as the
representative stress–strain data for each structure. For the 45 deg
nanolattices, the yield stress was calculated by taking the average
of four samples for each of the
t
¼
200 nm and 327 nm cases and
one sample for
t
¼
635 nm case. The data indicate that for the 45
deg nanolattices, the yield stress increased by a factor of 2.1, or
by

120%, as
t
increased from 200 nm to 327 nm and by

13%
as
t
increased from 327 nm to 635 nm. The 0.2% yield stress
increased with
t
but occurred at approximately the same strain for
all 45 deg samples. Yield stresses were also calculated for nano-
lattices with a lattice angle of 60 deg for two samples with
t
¼
325
nm and one sample with
t
¼
661 nm. The 60 deg nanolattices
appeared to be

58% stronger than the 45 deg nanolattice for all
wall thicknesses, as shown in Fig.
7
, which is consistent with the
results from literature [
35
]. The yield stress of the 60 deg nanolat-
tices increased by

14% as
t
varied from 325 nm to 661 nm.
Yielding occurred at approximately the same strain of 1.2% for all
60 deg samples, which is lower than 1.7% yield strain in the 45
deg samples. The large error bars for tests on samples with the
thickest walls, shown in Fig.
7
, are likely a result of the anisotropy
introduced by the sputtering process. All deformation initiates at
the base of the nanolattice structures, where the coatings are more
conformal compared to the top layer on unit cells, so the mean is
a more representative wall thickness for the area of the nanolattice
that deforms during the experiments.
5 Discussion
Building upon an analytic model proposed by Valdevit et al. to
calculate the stress required to initiate Euler buckling or yielding
along the tubes for a microlattice with hollow circular tubes, we
account for the hollow elliptical tubes in the octahedron lattice
geometry presented in this work [
9
,
53
]. Since changes in the geo-
metric parameters of the nanolattices, such as wall thickness or
tube diameter can lead to structural effects, the initial deformation
of the tubes must be consistent across all samples with varying
geometric parameters such that lattice angle and material size
effects can be isolated. For example, under an applied load the lat-
tice tube can either undergo Euler buckling or yielding of the con-
stituent material as the tubes are compressed; as the geometric
parameters of tubes, such as the length and wall thickness, are var-
ied it may be possible to initial buckling prior to yielding of the
constituent material. The critical values to initiate Euler buckling
and yielding of the elliptical tubes for the nanolattice geometry
are shown in the following equations (supplemental derivations
are available under the “
Supplemental Data
” tab for this paper on
the ASME Digital Collection):
r
buckling
¼
8
p
3
E
cos
2
h
sin
h
D
min
l

2
D
min
þ
3
D
max
l

t
l

(2)
r
yield
¼
4
p
sin
h
t
l

D
min
þ
D
max
l

r
y
ð
Au
Þ
(3)
The geometric parameters (
D
min
,
D
max
,
t
, and
l
) used to calcu-
late the critical values to initiate Euler buckling or yielding of the
nanolattice tubes are schematically defined in Fig.
2(
a
)
and tabu-
lated for each sample in Table
1
. The yield stress of the Au,
r
y
(Au)
, in Eq.
(3)
was measured via nano-indentation into 2.17
l
m
thick film on a glass substrate to a depth of 100–200 nm to be
583
6
41 MPa, and the indentation modulus to be
E
¼
95
6
4 GPa,
both calculated using the Oliver–Pharr method [
54
]. The stresses
required to initiate either yielding or Euler buckling in the Au
nanolattices were calculated using Eqs.
(2)
and
(3)
and are shown
in Table
1
. The experiments presented in this work consider struc-
tures of constant relative density and lattice type, therefore the
geometric parameters (
D
min
,
D
max
,
t
, and
l
) of the hollow tubes
must change with the wall thickness,
t
. As a result, all samples
presented in this work have varying values of
D
min
,
D
max
,
t
, and
l
,
which are shown in Table
1
along with the critical stress to initiate
Fig. 4 SEM image of octahedron nanolattices with lattice angles from 30 deg to 60 deg, as well as representative
stress–strain curve for each sample with an open circle showing the 0.2% yield stress of the structure. For the 30 deg lattice,
it should be noted that the error bars are included but are small enough to be obscured by the data point itself (scale bar
denotes 30
l
m).
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Euler buckling and yielding. We found that for all nanolattices in
this work, the threshold stress for yielding was in the range
between 164 and 290 MPa, which is more than an order of magni-
tude lower than that required for Euler buckling (3.0–12.5 GPa).
Potential errors in the magnitude of the critical stress values may
be introduced if size effects cause the value of
r
y
(Au)
to vary as
the wall thickness increases, however, based on previous work an
order of magnitude change in the yield strength of the Au is not
expected due to size effects [
24
,
26
]. This implies that all samples
will fail by yielding upon compression and that any geometric
changes required to keep relative density constant do not change
the fundamental deformation behavior of the nanolattice tubes.
We observed that the initiation of yielding always occurred at the
hollow nodes likely because of the substantial local stress. The
dominance of nodal deformation within the structure renders
minor nonuniformities within the wall thickness relatively incon-
sequential for mechanical behavior.
We found that the stresses predicted for the onset of yielding by
Eq.
(3)
, within the range of 164–290 MPa, did not match the
experimental data; which ranged from 161 to 976 kPa (see
Table
2
). This discrepancy is likely caused by the fundamental
limitations in the simple mechanics model in its inability to cap-
ture the complex stress state within the hollow structure. For
example, the model accounts for buckling or yielding along the
thin-shell hollow tubes with elliptical cross sections but does not
account for the complex geometry and stress state at the nodes
where the tubes connect. The nanolattices experience higher stress
concentrations at the nodes because of the small radius of
curvature where tubes converge induces the onset of yielding at
the node instead of along the length of the tube. Our experimental
observations and the analytic calculations confirmed that the
deformation was consistently initiated by yielding at the nodes for
all samples in this study. Valdevit et al. found that the analytic
calculations for microlattices were approximately 1–2 orders of
magnitude higher than those predicted by the FEM and than was
observed experimentally, which is consistent with the observa-
tions in this work [
9
]. A deviation of 1–2 orders of magnitude in
strength has also been observed experimentally in micro- and
nanolattices, and the deviation from the analytically predicted
strength and stiffness in this work is consistent with the existing
literature [
7
,
10
]. The tight distribution in the yield stresses in our
experiments, shown in Table
2
, implies that structural deformation
in all samples likely occurred via yielding rather than buckling
because this type of deformation is generally robust against
defects. Buckling is an example of instability phenomena where
minor defects in the nanolattice geometry could significantly
affect the critical stress required to initiate buckling and produce a
wider spread in the data, which is not observed in these experi-
ments [
53
]. The low standard deviation observed across all data in
this work is consistent with the prediction that failure of the tubes
initiates by yielding based on the model proposed by Valdevit
[
53
]. Experimental observations revealed that the sample first
began to rotate and deform at the nodes prior to bending of the
tubes, which may be induced by imperfections in the nanolattices.
5.1 Discussion: Structural Effects.
We found that the yield
stress and modulus of the structures increase with the lattice angle
of the unit cell (Fig.
5
). For the 30 deg case, the horizontal mem-
bers of the unit cell carry more load in tension than would be
experienced by the horizontal members in the 60 deg samples
when a uniaxial force is applied to a sample based on a simple force
balance. Alternatively, the vertical members of the 60 deg unit cell
carry more load in compression than those of the 30 deg unit cell
under the same uniaxial loading conditions. For beams loaded in
tension, a slight prebend or misalignment has a much less signifi-
cant effect on the mechanical behavior compared to a beam loaded
in compression with an equivalent prebend or misalignment. As a
result of the distribution of load across the unit cell as the lattice
angle varies, it is expected that larger standard deviation would be
observed in the 60 deg samples and this is observed in the data.
The model proposed by Deshpande et al., which relates the
modulus of a stretching-dominated pyramidal core, can be
extended to this work since the pyramidal core is half of the octa-
hedron unit cell [
4
]. Though the octahedron unit cell is a
stretching-dominated structure, the octahedron nanolattices
behave as bending-dominated structures due to rotation between
and within unit cells as the lattice is compressed. This rotation is
initiated by the initial imperfections in the unit cell layer closest
to the substrate, which is a result of the polymer contraction that
occurs during the development process. Minor imperfections and
misalignments within the nodes of the octahedron nanolattices
have been reported to induce lateral torsional bending in the nano-
lattice struts, which also contributes to the bending-dominated
behavior observed in this work [
8
,
37
]. The minor imperfections in
the nanolattices, which are caused by the position accuracy of the
two-photon fabrication system as well as the contraction of the
polymer upon cross-linking, are an order of magnitude smaller
than the unit cell size and account for the deformation mode of
lateral torsional bending, as shown in existing literature [
8
,
37
].
The trend in the data follows a modified form of the modulus rela-
tion proposed by Deshpande et al., shown in Eq.
(4)
and Fig.
8(
a
)
,
to include the correct scaling for a bending-dominated structure
E
¼
E
s

q
m
ð
sin
h
Þ
4
(4)
with
m
¼
2 which is consistent with the theory classical theory for
bending-dominated structures, where
E
a

q
2
[
1
,
2
]. Material size
Fig. 5 Calculated yield stress and modulus values for lattices
with angles ranging from 30 deg to 60 deg. The error bars were
calculated by using the standard deviation of the data for yield
stress and modulus (note:
t
5
352
6
87 nm for all samples).
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effects do not affect the modulus at these length scales, so
E
s

E
of Au measured using nano-indentation. For bending-dominated
structures where
E
a

q
m
with
m

2, the data also scale with sin
4
(
h
)
as predicted by literature;
m
of

2.15 appears to provide a better
fit to experimental data, however, to do a proper regression model
to determine the best fit value for
m
, a wider range of lattice
angles must be tested [
35
]. The discrepancy in the relative density
exponent is likely an artifact of the hollow nodes and it is
expected that the nanolattices in this work would be less stiff than
a cellular structure with solid nodes, which would scale with
m
¼
2.
A similar analysis was done for the yield strength and compared
to the predicted value for a stretching-dominated structure pre-
dicted by Deshpande and Fleck [
4
]. A stretching-dominated py-
ramidal structure was predicted to scale as
r
¼
r
s

q
ð
sin
h
Þ
2
,
however, a bending-dominated geometry, such as the octahedron
nanolattices in this work, should scale as
ra

q
1
:
5
[
1
,
2
,
4
]. Modify-
ing the model proposed by Deshpande et al. to account for varia-
tion in the relative density exponent is given in the following
equation:
Fig. 6 SEM image of 45 deg octahedron nanolattices with
t
ranging from 200 to 635 nm, as well as representative
stress–strain curve for each sample with an open circle showing the 0.2% yield stress of the structure (scale bar denotes
20
l
m)
Fig. 7 Calculated yield stress for both 45 deg and 60 deg octa-
hedron nanolattices. The yield strength increases as (
t
/
d
)
increases due to the material size effect.
Table 1 Relevant average geometric parameters, as well as threshold stress values for both yielding and Euler buckling, for all
samples
Geometry
Angle
(deg)
Thickness,
t
(nm)
Major axis,
D
max
(
l
m)
Minor axis,
D
min
(
l
m)
Tube length,
l
(
l
m)
Thickness-to-length
ratio,
t
/
l
r
yield
(MPa)
r
bucking
(GPa)
Octahedron,

q
¼
0.05
6
0.01
30
357
2.68
1.31
7.97
0.05
164
12.5
45
200
2.39
0.80
4.77
0.05
290
16.4
327
2.76
1.17
7.96
0.04
210
8.8
635
3.21
1.65
11.12
0.06
259
10.6
60
324
2.71
1.18
9.71
0.04
170
2.4
661
3.14
1.53
13.07
0.05
229
3.0
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r
¼
r
s

q
n
ð
sin
h
Þ
2
(5)
Equation
(5)
captures the trend seen in calculated yield stresses
for
n
¼
1.5, shown in Fig.
8(
b
)
. The calculated yield stress devi-
ates from that predicted using Eq.
(5)
by approximately an order
of magnitude or more as the lattice angle increases. This is likely
because of the hollow nodes and small imperfections, such as
minor misalignment at the nodes or prebend in the unit cells con-
nected to the substrate, all of which may emerge as a result of the
fabrication process. The behavior or the lattice is more sensitive
to imperfections as the lattice angle increases, as shown by the
larger spread in the data at higher angles, and we attribute to the
increasing deviation from the predicted value according to Eq.
(5)
to this imperfection sensitivity at higher lattice angles. Material
size effects alter the strength of the constituent material at the
length scale present in the wall thickness of the nanolattices, how-
ever, a model to describe this change in constituent properties for
this geometry and microstructure is outside of the scope of this
work. The
r
s

q
n
term acts as a constant in the yield stress

f
(sin(
h
)) relation therefore the value of
n
cannot be determined
until the microstructure and material size effects present in the
nanolattices are further characterized. The deviation in the
m
and
n
values from that which is expected of a bending-dominated
structure (
m
¼
2 and
n
¼
1.5) is likely due to the complex stress
state at the hollow nodes that initiates failure well below analyti-
cally predicted values. The scaling laws for cellular solids assume
pin-jointed structures, which is not the case for the fabricated sam-
ples presented in this work, therefore it is expected that the
m
and
n
values would deviate from those predicted using analytic theory
[
1
,
2
]. Additionally, deviations of up to an order of magnitude
between experiments, models, and analytic predictions for
strength have been observed in previous work [
9
].
5.2 Discussion: Material Size Effects.
The yield strength of
the Au tube walls is expected to be lower than that of the same
material in monolithic form because the nanolattices in this work
have wall thicknesses in the range where the reduced sample
dimensions can lead to weakening in metals with nanometer-sized
grains [
19
27
]. The applicability of the model proposed by Gu
et al. in Eq.
(1)
to the nanolattice system is limited because the
tube walls of the nanolattices are effectively a freestanding thin
film in which the grains are confined laterally, although multiple
grains may also span the wall thickness. The thickness of the
nanolattice walls,
t
, and the microstructure of constituent material
are relevant parameters for predicting the strength of the hollow
Au tubes. The nanolattices in this work represent an
Table 2 Average measured and predicted yield stresses and modulus for the fabricated samples
Lattice
angle (deg)
Wall thickness
(nm)
Measured yield
stress (kPa)
Analytic scaling
law:
r
¼
0.3
r
s
q
1.5
(MPa)
Measured modulus
(MPa)
Analytic scaling law:
E
¼
E
s
q
2
(GPa)
45
200
246
1.96
15.8
4.75
327
540
1.96
34.9
4.75
635
610
1.96
56.0
4.75
60
324
858
1.96
85.2
4.75
661
976
1.96
77.0
4.75
Fig. 8 (
a
) Modulus versus sine of lattice angle according to Eq.
(4)
and (
b
) yield stress versus sine of lattice angle according
to Eq.
(5)
Fig. 9 TEM bright field image showing pores between grains
as viewed through a section of a nanolattice wall, where the
wall thickness is

200 nm
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interconnected system of such tubes and demonstrate that the
material size effects observed in a simple nanopillar geometry can
be exploited to control and tune the mechanical properties of the
entire structure. Though a model to describe the material size
effect is outside the scope of this work, any geometric parameters
that potentially cause structural effects or changes in relative den-
sity are held constant and the structures are self-similar as wall
thickness increases, so we attribute the observed lower yield stress
to changes in the microstructure of the constituent material as the
wall thickness dimension decreases. Table
2
shows the predicted
yield stress and modulus of a structure with a fixed geometry and
relative density, assuming the constituent material properties are
constant, as well as the measured yield stress and modulus. The
discrepancy in the measured and analytically predicted values is
1–2 orders of magnitude, which may be attributed to the complex
stress state at the hollow node and imperfections in the lattice due
to the fabrication process. The scaling laws of Gibson and Ashby
predict a constant yield stress for all structures, yet this work dem-
onstrates that tuning the microstructure of the constituent material
and its dimensions at small length scales enables tuning the
strength in the fabricated nanolattice structures [
2
]. Increasing the
lattice angle from 45 deg to 60 deg, we observe an improvement
over the lower angle case by

1.6 times for multiple wall thick-
nesses and again show the tunable strength of the nanolattices as a
result of material size effects.
The 45 deg nanolattices were nominally conformally coated
with a

200 nm-thick Au. TEM analysis revealed that these
200 nm-thick films were more porous than samples with thicker
walls. This is likely due to the porosity of the polymer onto which
the Au is deposited; the 200 nm-thick walls were the thinnest con-
formal layer possible with the sputtering technique and the pores
within the thin walls are shown in Fig.
9
. We discovered that
thicker films had no observable pores, as revealed by TEM and
ion channeling of

2
l
m
n
-Au thin films deposited under identi-
cal conditions. This decrease in porosity as
t
increases is consist-
ent with the mechanisms for polycrystalline film growth on a
substrate where the nucleation sites grow to coalesce into a fully
dense thin film [
55
]. The greater porosity impacts the mechanical
response of the film because at low wall thicknesses the tubes of
the nanolattices act as a thin film with defects rather than a fully
dense constituent material. The films that comprise the nanolattice
tubes are oriented at an angle to the loading direction, as defined
by the lattice angle, so it is expected for the structures to be
weaker when loaded in this configuration compared to loading
applied along the length of the columnar grains. We propose this
to be the cause of the large change in both yield stress and modu-
lus as
t
increases from 200 nm to 327 nm. Ion channeling and
TEM confirm that as the wall thickness increases, more grains
span the thickness of the film, as shown in Fig.
10
, however, there
are single grains that may span the thickness of the nanolattice
wall. Additionally the average height of the grains is on the order
of

100 nm so it is suspected that for
t
¼
200 nm there are fewer
grains that span the thickness of the film; however, as
t
increases
more grains span the thickness and the behavior approaches that
of the bulk constituent Au. The data for the 45 deg nanolattices
show a significantly larger increase in yield strength as
t
increased
from 200 to 327 nm compared to the increase in yield strength as
t
increased from 327 to 635 nm where there is no porosity. This
suggests that the strength of the constituent material is approach-
ing the bulk value for
t
>
327 nm, which is also consistent with
the literature for nanocrystalline nanopillars with increasing ratios
of (
t
/
d
)[
24
26
]. Existing models for thin films or nanopillars geo-
metries are not directly applicable to this work since the stress
state of the hollow nanolattice structures is complex and the
microstructure varies as a function of wall thickness. Characteriz-
ing the deformation mechanism of the constituent material at
varying wall thicknesses is the subject of future work to further
understand the deformation mechanism of hollow metallic nano-
lattices at all length scales.
6 Conclusions
We fabricated hollow Au octahedral nanolattices with variable
wall thicknesses and lattice angles and discovered that upon uni-
axial compression, the structural and material size effects could
be utilized to tune strength and stiffness. Without inducing a
change in deformation mechanism of the lattice tubes, changes in
the lattice angle can increase both the yield strength and modulus
of the lattice by approximately an order of magnitude. Tuning the
wall thickness of the nanolattices makes it possible to exploit
material size effects of the small-scale constituent material, thus
increasing the yield strength by a factor of 2. Classical mechanics
of cellular solids predicts a constant strength and stiffness for all
nanolattice geometries explored in this study, however, by utiliz-
ing material size effects, the strength of a lattice with a constant
deformation mechanism and relative density can be increased by
up to a factor of 2. Exploiting structural effects can provide up to
an order of magnitude increase in strength and stiffness for a
nanolattice with a constant deformation mechanism and relative
density. We show that by exploiting the structural and material
size effect parameters of nanolattices, the strength and stiffness of
these cellular solids are no longer constant for a given deforma-
tion mechanism and relative density, as predicted by classical the-
ories. By combining an understanding of structural mechanics and
the behavior of small-scale materials, technology development is
no longer limited by the constraints of existing cellular solids and
new materials can be designed to meet the needs of emerging
technology by exploiting structural and material size effects to
tune mechanical properties.
Acknowledgment
The authors gratefully acknowledge the financial support from
the National Science Foundation through NSF Graduate Research
Fellowship of L.C.M. and Grant Nos. of J.R.G. (CMMI-1234364
and DMR-1204864). The authors would like to acknowledge Pro-
fessor Lorenzo Valdevit at the University of California Irvine for
the analytic model used to determine the deformation mechanism
of the fabricated samples. The authors also acknowledge the criti-
cal support and infrastructure provided by the Kavli Nanoscience
Fig. 10 (
a
) Ion channeling (sample at 52 deg tilt) and (
b
) TEM images of a

2
l
m thin
n
-Au film showing columnar grain structure with multiple grains spanning the film
thickness
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