of 18
Supporting Information for
Ultralow Thermal Conductivity and Mechanical
Resilience of Architected Nanolattices
Nicholas G. Dou,
Robert A. Jagt,
,
Carlos M. Portela,
Julia R. Greer,
and
Austin J. Minnich
,
Division of Engineering and Applied Science, California Institute of Technology,
Pasadena, California 91125, United States
Faculty of Science and Engineering, University of Groningen,
9747 AG Groningen, The Netherlands
E-mail: aminnich@caltech.edu
1
Sample fabrication
To create hollow alumina nanolattice samples suitable for the 3
ω
experiment, we augment
the general fabrication process outlined by Meza et al.
1
The procedure begins by creating
an octet-truss polymer scaffold using a two-photon lithography direct laser write process.
We use the Photonic Professional system (Nanoscribe GmbH) with the proprietary IP-Dip
photoresist. The Nanoscribe system incorporates a mirror galvanometer that enables rapid
in-plane rastering and reduces the write time for this millimeter-length structure to a few
hours. The structures are developed using propylene glycol methyl ether acetate and iso-
propyl alcohol.
A highly conformal layer of amorphous aluminum oxide is coated onto this polymer
scaffold using a Savannah atomic layer deposition (ALD) system (Cambridge Nanotech).
One ALD cycle consists of pulsing trimethylaluminum for 0
.
015 s, purging for 20 s, pulsing
water for 0
.
015 s, and purging again for 20 s. The deposition chamber is held at 150
C as
20 sccm of N
2
carrier gas is flowed through continuously. This ALD recipe produces self-
limited growth of Al
2
O
3
at a rate of 1
.
07
̊
A per cycle. Film thickness measurements taken
with an alpha-SE ellipsometer (J. A. Woollam Co.) confirm the accuracy of the recipe
deposition rate to within 1 percent. The resulting nanolattice has beams with a polymer
core encapsulated by an alumina shell.
We then remove the polymer inside the beams by milling away sacrificial beams with a
focused ion beam (FIB) and etching away the polymer through these openings with oxygen
plasma. The FIBs are part of larger Nova 200/600 NanoLab systems (FEI Co.) and the
plasma asher is a Zepto unit (Diener electronic GmbH). To expedite complete removal of the
polymer, we write sacrificial beams at the vertical center of the structure so that the longest
diffusion paths to the top and bottom are minimized. All samples are etched for at least
36 hours with 100 W of power and the structures are inspected afterwards under a scanning
electron microscope to check for a contrast change indicating that the beams are hollow.
Finally, we deposit gold through an aligned shadow mask to create a continuous heater
2
line on the structure connected to four contact pads on the substrate. The mask is laser cut
from 0
.
005 in stainless steel and has maximum edge waviness less than 5
μ
m. It is suspended
above the sample by a spacer to avoid damaging the nanolattice and aligned under an optical
microscope using micrometer-controlled translation stages. Thin film deposition is done in
a LAB Line E-Beam evaporator (Kurt J. Lesker Co.) at pressures in the 10
7
Torr range.
A 10 nm layer of titanium promotes adhesion of the 100 nm gold layer. The final position of
the pattern deviates from the desired location by up to 10
μ
m due to imperfect alignment
and non-normal evaporation through the mask, but precise positioning and edge sharpness
are not critical for the experiment. Energy-dispersive x-ray spectroscopy shows negligible
overspray of gold onto the sides of the nanolattices, confirming that evaporated gold does
not contaminate the nanolattices.
Thermal experiments
3
ω
experiments are performed in a ST-100 cryostat (Janis Research Co.) at temperatures
from 95 to 300 K and pressures in the 10
6
Torr range. A 6221 current source (Keithley
Instruments) supplies power and an SR830 lock-in amplifier (Stanford Research Systems
Inc.) measures voltages. In each experiment, we tune input power in the 4 to 40
μ
W range
to ensure that the temperature rise at the heater line is less than 1 K. The thermal frequency
response is measured from 1 to 1000 Hz by averaging 30 samples at each frequency.
To interpret the experimental data, we developed a custom thermal model described in
the Supplementary Text. Our model accounts for lateral heat conduction through the gold
heater line, 1-dimensional heat conduction through the alumina plate, and 2-dimensional
heat conduction through the nanolattice and into the silicon substrate. The back side of
the silicon chip is assumed to be isothermal. We use handbook values for density, specific
heat, and thermal conductivity of silicon.
2
For amorphous alumina, we take the density to be
2900 kg m
3
,
3
and the specific heat as the handbook value. The temperature-dependent ther-
3
mal conductivity of bulk amorphous alumina is taken from measurements of RF-sputtered
alumina.
4
The heat capacities and effective thermal conductivities of the plate and nano-
lattice are further scaled by the calculated relative densities of the respective structures. For
the gold heater line, we use a density of 18 884 kg m
3
,
5
and calculate thermal conductivity
from electrical resistance via the Wiedemann-Franz law.
We extract thermal conductivity by performing a nonlinear least-squares fit on both the
in-phase and out-of-phase components of the temperature response from 1 to 100 Hz. In
this low frequency range, the thermal penetration depth is much longer than the height of
the nanolattice. The fitting parameters are nanolattice thermal conductivity and a constant
normalization factor.
Mechanical experiments and simulations
The uniaxial compression experiments provided stress-strain curves from which the Young’s
modulus and yield strength of each nanolattice was calculated. Several identical samples were
compressed for each set of parameters that were used for the thermal samples. Specifically,
all samples had an
r/l
ratio of 0
.
108,
t/r
ratios from 0
.
013 to 0
.
10, and spanned a relative
density range from 0
.
78 to 4
.
1 %. The relative density values were calculated using a CAD
model (SolidWorks), using dimensions obtained from SEM micrographs. Figure S3 depicts
representative stress-strain curves for each geometry. Slight misalignments between the in-
denter tip and the top of the lattice commonly caused a non-linear regime upon first contact,
followed by a linear loading regime, which transitioned into sequential or catastrophic failure
(depending on the
t/r
ratio). For samples with wall thicknesses of 24 nm, several instabilities
occurred within 20% strain increments, corresponding to a displacement equal to the unit
cell height, indicating that shell buckling in struts was a significant deformation mechanism
in addition to microcracking at nodes. Shell buckling is an elastic process that enables the
recovery observed in Figure 3. Thicker-walled nanolattices exhibit fewer and more extensive
4
strain bursts that correspond to discrete fracture events, such as the collapse and densifica-
tion of unit cell layers. Samples with wall thicknesses above 81 nm failed catastrophically.
No recovery was observed for samples with wall thicknesses of 36 nm and above. In order
to minimize the effects of initial misalignments between the lattices and the indenter, the
maximum slope of the linear loading regime was taken as the measured Young’s modulus.
The effective strength was determined as the stress at the initial instability for recoverable
samples or the maximum stress for samples that failed catastrophically.
Figure S4 shows the stiffness values of all fabricated samples compared to finite element
simulations with identical parameters (Abaqus FEA). The simulations consisted of a dis-
cretized octet-truss unit cell composed of S3R shell elements and a linear elastic material
model, where the Young’s modulus and Poisson’s ratio of ALD alumina were used.
6
Periodic
boundary conditions (PBC) were applied on all sides of the unit cell, and a uniaxial strain
was enforced as a linear perturbation. The simulation size ranged from 110 000 to 310 000
elements depending on the
r/l
and
t/r
parameters used.
The power-law fits of the form
E
ρ
m
in Figure S4 yielded scaling exponents of
m
= 1
.
41
and 1
.
91 for simulations and experiments, respectively. The vertical offset and the higher
scaling of the experimental samples in comparison to the defect-free simulations can be
attributed to imperfections such as wall waviness (on the order of a few nanometers), which
has been shown to significantly decrease the stiffness of thin-walled hollow beams.
7
The non-
linear scaling is consistent with the analysis done in that work, where the complex parameter
space of hollow nanolattices is explained.
Figure S5 presents the Young’s modulus and yield strength of the hollow alumina nano-
lattices in this work along with those reported for silica and carbon aerogels versus density.
These plots demonstrate that nanolattices have more than an order of magnitude higher
stiffness and strength at a given density. The nanolattices achieve similar stiffnesses and
strengths as aerogels at half the density. This improvement in mechanical properties be-
comes even more pronounced at low densities because nanolattices have more favorable
5
density scalings. Compared to the stiffness scaling exponent for the nanolattice measure-
ments,
m
= 1
.
91, the scaling exponents for silica and carbon aerogels are considerably higher,
m
= 2
.
42 and 3.20. Using a similar power-law model for strength,
σ
y
ρ
n
, the scaling ex-
ponent for our nanolattices is
n
= 1
.
75 while the density scaling for aerogels ranges from
n
= 2
.
02 to 2.97.
Prior to manufacturing the samples, PBC simulations with constant wall thickness and
unit cell size but varying beam radius
r
were used to find an optimal
r/l
value at which the
stiffness of circular-beam hollow octet-truss nanolattices was maximized. Figure S6 shows
the resulting stiffness as
r/l
was varied from 0
.
014 to 0
.
14 and
t/l
remained at 0
.
007. The
observed trend guided the design of the samples with
r/l
0
.
1.
3
ω
thermal model
3
ω
measurement samples for ultralow thermal conductivity materials typically consist of a
slab or uniform film on a substrate with a metal heater line patterned on top.
4,8
In those
cases, the heater line is thermally insulated from the heat sink. The nanolattice fabrication
process prohibits us from creating macroscale structures, so we only construct the nanolattice
directly underneath the heater line.
The nanolattice sample geometry leads to non-negligible heat loss through the metal
heater line. This occurs because the heater line is connected to contact pads that lie directly
on the substrate, which serves as the heat sink. The nanolattice thermal conductivity is
so low that despite the small cross-sectional area of the metal line (50
μ
m
×
100 nm), the
thermal resistance of the conduction path through the nanolattice is comparable to the
thermal resistance through the metal line. Hence, the usual method of extracting thermal
conductivity from a 3
ω
experiment does not work for our nanolattice samples, and we need
to develop a thermal model that incorporates heat conduction along the heater line.
The following equations describe heat conduction through our nanolattice sample during
6
a 3
ω
experiment.
2
T
i
∂y
2
i
+
λ
i
2
T
i
∂z
2
1
α
i
∂T
i
∂t
= 0
in
0
< y
i
< d
i
,
0
< z < L
κ
1
κ
0
d
0
∂T
1
∂y
1
+
λ
0
2
T
0
∂z
2
1
α
0
∂T
0
∂t
=
g
κ
0
e
jωt
on
y
1
= 0
κ
i
∂T
i
∂y
i
=
κ
i
+1
∂T
i
+1
∂y
i
+1
on
y
i
=
d
i
, i
= 1
,...,m
1
T
i
=
T
i
+1
on
y
i
=
d
i
, i
= 0
,...,m
1
T
m
= 0
on
y
m
=
d
m
T
i
= 0
on
z
= 0
, z
=
L
(1)
(2)
(3)
(4)
(5)
(6)
In the equations above,
z
is the in-plane coordinate parallel to the heater line and
y
i
is the
cross-plane coordinate, defined such that
y
i
=
d
i
coincides with
y
i
+1
= 0. Because the width
of the heater line is identical to that of the nanolattice, we expect no temperature variation
in the other in-plane direction
x
, perpendicular to the heater line.
T
i
,
κ
i
,
λ
i
,
α
i
, and
d
i
denote the temperature, cross-plane thermal conductivity, ratio of in-plane to cross-plane
thermal conductivity, cross-plane thermal diffusivity, and thickness of the
i
-th layer, where
i
= 0 corresponds to the heater line. Instead of a prescribed surface heat flux, we impose a
boundary condition that describes a thin skin of highly conductive material that generates
heat (Equation 2).
9
The general solution that satisfies the diffusion equation (Equation 1) and isothermal
boundary conditions in
z
(Equation 6) can be expressed as a Fourier sine series. In layer
i
,
the temperature and heat flux have the form
T
i
=
e
jωt
n
=1
(
a
in
cosh
q
in
y
i
+
b
in
sinh
q
in
y
i
) sin
nπz
L
=
e
jωt
n
=1
T
in
sin
nπz
L
(7)
Q
i
=
e
jωt
n
=1
κ
i
q
in
(
a
in
sinh
q
in
y
i
+
b
in
cosh
q
in
y
i
) sin
nπz
L
=
e
jωt
n
=1
Q
in
sin
nπz
L
(8)
q
2
in
=
λ
i
(
L
)
2
+
α
i
(9)
7
The constants
a
in
and
b
in
are chosen to satisfy the boundary conditions in
y
i
. These can be
solved using the transfer matrix method.
9
T
in
Q
in
y
i
d
i
=
cosh
q
in
y
i
1
κ
i
q
in
sinh
q
in
y
i
κ
i
q
in
sinh
q
in
y
i
cosh
q
in
y
i
T
in
Q
in
y
i
=0
(10)
By enforcing continuity of temperature and heat flux across layers (Equations 3 and 4), we
can compose
m
of these matrices to relate
T
0
n
and
Q
0
n
at the heater line to
T
mn
and
Q
mn
at the isothermal surface.
T
mn
Q
mn
y
m
=
d
m
=
A
mn
B
mn
C
mn
D
mn
...
A
1
n
B
1
n
C
1
n
D
1
n
T
1
n
Q
1
n
y
1
=0
=
A
n
B
n
C
n
D
n
T
1
n
Q
1
n
y
1
=0
(11)
Here, matrix elements are evaluated at
y
i
=
d
i
. The isothermal boundary condition
T
mn
= 0
(Equation 5) thus implies
T
1
n
=
B
n
A
n
Q
1
n
=
γ
n
κ
0
q
0
n
Q
1
n
(12)
where we define the dimensionless quantity
γ
n
=
κ
0
q
0
n
B
n
/A
n
. The ratio
B
n
/A
n
can
be calculated using the algorithm given by Borca-Tasciuc et al.
10
Finally, the heater line
boundary condition is used to obtain
a
1
n
and
b
1
n
, giving
T
1
=
gd
2
0
κ
0
e
jωt
n
odd
γ
n
cosh
q
1
n
y
1
κ
0
q
0
n
κ
1
q
1
n
sinh
q
1
n
y
1
(1 +
γ
n
q
0
n
d
0
)
q
0
n
d
0
(
4
sin
nπz
L
)
(13)
Q
1
=
gd
0
e
jωt
n
odd
cosh
q
1
n
y
1
γ
n
κ
1
q
1
n
κ
0
q
0
n
sinh
q
1
n
y
1
1 +
γ
n
q
0
n
d
0
(
4
sin
nπz
L
)
(14)
Note that the sum is evaluated only over odd natural numbers. Because hyperbolic cosine
and sine grow rapidly, it is more numerically stable to evaluate the temperature and heat
8
flux using exponentials. For example,
T
1
=
gd
2
0
κ
0
e
jωt
n
odd
c
1
n
e
q
1
n
y
1
+
d
1
n
e
q
1
n
y
1
(1 +
γ
n
q
0
n
d
0
)
q
0
n
d
0
(
4
sin
nπz
L
)
(15)
c
1
n
=
1
2
(
γ
n
κ
0
q
0
n
κ
1
q
1
n
)
(16)
d
1
n
=
1
2
(
γ
n
+
κ
0
q
0
n
κ
1
q
1
n
)
(17)
Figure S1 shows an example temperature and heat flux field for a single layer.
In the 3
ω
experiment, the measured voltage is directly related to the average heater line
temperature, given by evaluating Equation 13 at
y
1
= 0 and integrating with respect to
z
.
The resulting thermal frequency response is
T
ω
=
gd
2
0
κ
0
n
odd
8
γ
n
n
2
π
2
(1 +
γ
n
q
0
n
d
0
)
q
0
n
d
0
(18)
Critical
t/r
and
r/l
ratios
Following the analysis done by Meza et al.,
1
the dominating deformation modes for circular
cross-section hollow nanolattices can be estimated. The three modes to be considered are
material fracture, beam buckling, and shell buckling. The critical stress values for each mode
are
σ
frac
=
σ
f
,
(19)
σ
buckle
=
π
2
EI
L
2
e
A
tube
,
(20)
σ
shell
=
E
3(1
ν
2
)
(
t
r
c
)
,
(21)
respectively. Here,
σ
f
is the fracture strength,
E
is the Young’s modulus, and
ν
is the
Poisson’s ratio of the constituent material. The cross-sectional area of the tube is denoted as
A
tube
, its second area moment is
I
, and its effective length is
L
e
(depending on the boundary
9
conditions). In the case of the octet-truss architecture,
L
e
=
L/
2, where
L
is the actual
length of the tube. The quantities
t
and
r
c
refer to the wall thickness and the wall’s radius
of curvature, respectively. For a circular hollow tube,
r
c
=
r
, where
r
is the tube radius.
Approximating the second area moment to
I
=
πr
3
and the cross-sectional area to
A
tube
=
πrt
, setting
σ
buckle
=
σ
frac
and
σ
shell
=
σ
frac
, and solving for
r/l
and
t/r
yields
(
r
l
)
crit
=
1
2
π
σ
f
E
,
(22)
(
t
r
)
crit
=
σ
f
E
3(1
ν
2
)
,
(23)
where (
r/l
)
crit
and (
t/r
)
crit
are the critical ratios below which elastic Euler buckling and
elastic shell buckling are expected to dominate over material fracture, respectively.
10
Supplementary figures
z
y
Figure S1: Temperature and heat flux fields calculated from our custom thermal model. This
simplified example includes only a single layer. The top boundary is a thin skin of highly
conductive material and the left, right, and bottom boundaries are isothermal.
11
75
100
150
200
300
400
Temperature (K)
0.5
1
2
5
10
20
50
Thermal conductivity (mW m
-1
K
-1
)
3
ω
FEM
36
24
81
182
121
54
0
25
50
75
100
125
Density (kg m
-3
)
0
5
10
15
20
25
Thermal conductivity (mW m
-1
K
-1
)
3
ω
FEM
Cellular
Resistor
b
a
Figure S2: (a) Room temperature thermal conductivity versus density—instead of relative
density in Figure 2b—depicting 3
ω
measurements (symbols), finite element simulations (solid
line), a thermal conductivity model developed for cellular solids (dotted line),
11
and our
previous thermal resistance model (dashed line).
12
(b) All measured thermal conductivities
(symbols) of the six nanolattice samples along with finite element simulation predictions
(lines). Data sets are labeled with the wall thickness (nm) above the simulation curve in
the same color as the data. Low temperature measurements of the 24 nm sample are not
available.
12
(a)
t
= 24 nm
(b)
t
= 36 nm
(c)
t
= 54 nm
(d)
t
= 81 nm
(e)
t
= 121 nm
(f)
t
= 182 nm
Figure S3: Representative stress-strain data for compression tests of each nanolattice geom-
etry with varying wall thickness,
t
.
13
1.41
1.91
Figure S4: Experimental and computational stiffness values. The experiments where
t/r <
(
t/r
)
crit
are marked by a thicker black outline. Power-law fits of the form
E
ρ
m
for both
experiments and simulations are shown as dotted lines.
14
10
1
10
2
10
3
Density (kg/m
3
)
10
-2
10
-1
10
0
10
1
10
2
Str
ength
(MPa)
silica
aer
ogels
carbon
aer
og els
alumina
nanolattices
Dou et al.
(th
is wo
rk)
Parmen
ter
& Milstein (1
998)
Woignier & Phal
ippo
u (198
8)
Pekala et
al. (1990
)
1.75
2.97
2.02
2.82
10
1
10
2
10
3
Density (kg/m
3
)
10
-1
10
0
10
1
10
2
10
3
10
4
Young's Mo
dulus (MPa)
silica
aer
ogels
carbon
aer
og els
alumina
nanolattices
Dou et al. (this work)
Parmen
ter
& Milstein (1
998)
Pekala et
al. (1990
)
1.91
3.20
2.42
b
a
Figure S5: (a) Young’s modulus and (b) yield strength versus density of alumina nanolattices
in this work compared to silica
13,14
and carbon
15
aerogels.
15
Figure S6: Periodic boundary condition simulations with a constant wall thickness but
varying
r/l
, showing optimal values of
r/l
to maximize stiffness.
16
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