APPLIED
SCIENCES
AND
ENGINEERING
Knots
are not for naught:
Design,
properties,
and
topology
of hierarchical
intertwined
microarchitected
materials
Widianto
P. Moes
topo
1,2
*, Sammy
Shak
er
3
, Weiting
Deng
1
, Julia
R. Greer
1,4
Lightw
eight
and tough
engineer
ed materials
are often
designed
with
three-dimensional
hierarchy and intercon-
nected
structur
al members
whose
junctions
are detrimental
to their
performance
because
they serve as stress
concentr
ations
for damage
accumula
tion
and lower mechanical
resilience.
We introduce
a previously
unex-
plored class
of architected
materials,
whose
components
are interwo
ven and contain
no junctions,
and incor-
porate micro-knots
as building
blocks
within
these
hierarchical
networks.
Tensile
experiments,
which
show close
quantita
tive agreements
with
an analytical
model
for overhand
knots,
reveal that knot
topology
allows a new
regime
of deforma
tion capable
of shape
retention,
leading
to a ~92%
increase
in absorbed
energy
and an up to
~107%
increase
in failur
e strain compar
ed to woven structur
es, along
with
an up to ~11%
increase
in specific
energy
density
compar
ed to topologically
similar
monolithic
lattices.
Our explor
ation unlocks
knotting
and fric-
tional
conta
ct to create highly
extensible
low-density
materials
with
tunable
shape
reconfigur
ation and energy
absorption
capabilities.
Copyright
© 2023
The Authors,
some
rights
reserved;
exclusive licensee
American
Associa
tion
for the Advancement
of Science.
No claim to
original
U.S. Government
Works. Distributed
under
a Creative
Commons
Attribution
License
4.0 (CC BY).
INTR
ODUCTION
Many
na
tur
ally
formed
composites
are able
to
attain
unique
me-
chanical
properties,
such
as
high
strength
and
fractur
e toughness,
tha
t surpass
the
performance
of
individual
components
by
using
intrica
te
hier
archical
ordering
(
1
).
Ar
chitectur
al
hier
archy
has
been
kno
wn
to
allo
w the
activa
tion
of
multiscale
toughening
mech-
anisms
in
bone
and
enhance
the
structur
al
stability
of
he
xactinellid
sponges
(
2
–
4
).
Attaining
full
hier
archical
ordering
of
na
tur
al
ma
te-
rials
remains
a challenge;
progr
ess
has
been
enabled
by
modern
fab-
rica
tion
methods
capable
of
manufa
cturing
synthetic
ma
terials
with
comple
x prescribed
geometries
and
multiple
orders
of
hier
archy
(
5
–
8
).
Pr
ecisely
architected
arr
angements
of
cons
tituent
ma
terials
ha
ve
led
to
unique
ma
terial
properties
including
acous
tic
and
photonic
bandgaps
(
9
,
10
),
tunable
thermal
response
(
11
),
and
impa
ct
resis-
tance
(
12
,
13
).
Incorpor
ating
hier
archy
into
synthetic
architected
ma
terials,
such
as
by
forming
structur
al
elements
out
of
beams
at
a dis
tinctly
smaller
length
scale,
has
also
enabled
other
desir
able
mechanical
properties,
namely
, high
energy
absorption
and
deter-
minis
tic
failur
e beha
vior
(
14
,
15
).
As
advancements
in
manufa
ctur-
ing
techniques
continue
to
expand
the
available
design
spa
ce,
hier
archical
architected
ma
terials
ha
ve mos
tly
drawn
from
inter
con-
nected
design
principles
wher
e structur
al
members
are fused
to-
gether
at their
junctions
(
14
,
16
–
19
),
such
as
beam-based
lattices
whose
members
are composed
of
periodic
unit
cells
(
7
,
15
).
Exten-
siv
e experimental,
computa
tional,
and
analytical
studies
ha
ve been
conducted
on
periodic
architected
ma
terials
(e.g.,
beam-based,
pla
te-based,
and
triply
periodic
minimal
surfa
ce
lattices)
and
non-
periodic
ones
(e.g.,
foams
and
spinodal
architectur
es),
mos
t of
which
ha
ve inter
connected
designs
(
20
–
28
).
These
studies
reveal
tha
t the
unique
mechanical
attributes,
such
as
multis
table
reconfi-
gur
ability
and
high
stiffness-
and
strength-to-w
eight
ratios,
arise
from
the
combina
tion
of
stretching,
bending,
and
buckling
among
the
structur
al
members
(
29
–
32
),
as
well
as
from
energy
dis-
sipa
tion
within
the
cons
tituent
ma
terials
(
33
).
These
types
of
inter-
connected
architected
ma
terials
suffer
from
the
dev
elopment
of
inevitable
stress
concentr
ations
at the
junctions
upon
global
me-
chanical
loading,
which
deterior
ates
their
strength
and
stiffness
at
greater-than-theor
etical
rate
and
accumula
tes
damage
(i.e.,
micr
o-
cracks
and
localized
deforma
tions).
Interpenetr
ating
lattice
designs
ha
ve recently
been
explor
ed
as
alterna
tiv
es
to
inter
connected
design
(
34
,
35
),
sho
wing
the
potential
to
achiev
e multifunctionality
while
being
composed
of
mos
tly
two
inter
connected
lattices.
Exploiting
friction
betw
een
structur
al
members
has
also
been
sho
wn
as
a method
to
absorb
energy
without
accumula
ting
subs
tantial
damage
(
36
–
39
),
but
mos
t
designs
lack
hier
archy
to
further
augment
their
properties.
A differ-
ent
hier
archical
design
frame
work
has
been
intr
oduced,
in
which
multiple
interw
eaving
fibers
are arr
anged
into
effectiv
e beams
within
a micr
ola
ttice
tha
t contains
no
junctions
(
40
).
These
inter-
wo
ven
lattices
outperform
classical
monolithic,
inter
connected
lat-
tices
with
equivalent
unit
cell
designs
by
offering
two
to
thr
ee
times
higher
absorbed
energy
per
cycle
when
normalized
to
the
firs
t cycle,
>70%
greater
deformability
upon
tension,
>50%
compr
essiv
e strain
without
catas
trophic
failur
e,
and
dir
ectional
compliance
una
chiev-
able
in
their
monolithic
counterparts.
The
separ
ation
among
the
fibers
within
each
effectiv
e beam
opens
the
possibility
to
implement
ne
w
kinema
tics
be
yond
beam-joint
and
pla
te-hinge
mechanisms
(
30
,
41
,
42
).
For
example,
sev
eral
lessons
can
be
tak
en
from
knots,
which
can
be
found
in
a wide
range
of
length
scales:
from
sailing,
climbing,
and
sutur
es
to
the
entanglement
of
DNA,
protein,
and
polymer
strands
(
43
–
46
).
The
topology
of
knots
has
long
been
a
topic
of
ma
thema
tical
inter
est because
it uniquely
incorpor
ates
ge-
ometry
and
noncommuta
tiv
e algebr
a (
47
),
and
resear
chers
ha
ve dis-
covered,
for
example,
tha
t even
in
two
similarly
configur
ed
knots,
a
slightly
differ
ent
twis
t can
lead
to
diametrically
opposite
stabilities
1
Division
of Engineering
and Applied
Science,
California
Institute of Technology
,
Pasadena,
CA 91125,
USA.
2
Materials
Engineering
Division,
Lawrence Livermore
National Laboratory, Livermore, CA 94550,
USA.
3
Division
of Biology
and Biological
Engineering,
California
Institute of Technology
, Pasadena,
CA 91125,
USA.
4
Kavli
Nanoscience
Institute,
California
Institute
of Technology
, Pasadena,
CA 91125,
USA.
*Corresponding
author.
Email:
moestopo1@llnl.go
v, widi@caltech.edu
Moestopo
et
al.
,
Sci.
Adv.
9
, eade6725
(2023)
8 March 2023
1 of 10
SCIENCE
ADVANCES
|
RESEARCH
ARTICLE
Downloaded from https://www.science.org at California Institute of Technology on March 10, 2023
(
48
–
50
).
Mechanics-based
studies
on
phy
sical
tight
knots
ha
ve re-
vealed
the
importance
of
accounting
for
cons
tituent
ma
terial
prop-
erties
in
knot
failur
e pr
edictions
(
51
–
53
),
and
analy
ses
of
loose
knots
sho
w
their
potential
to
incr
ease
energy
dissipa
tion
and
intr
o-
duce
stable
tightening
and
untying
mechanisms
thr
ough
car
eful
se-
lection
of
knot
geometry
and
cons
tituent
ma
terials
(
54
–
57
).
Her
e, we combine
two
previously
independent
concepts,
hier
ar-
chical
architected
ma
terials
and
fiber
knotting,
to
dev
elop
building
blocks
for
architected
ma
terials
with
simultaneous
high
deformabil-
ity
under
every
loading
mode,
energy
dissipa
tion,
fractur
e resis-
tance,
and
shape
reconfigur
ability
. W
e classify
the
fiber
topology
of
hier
archical
intertwined
ma
terials
into
two
fundamental
topolo-
gies:
knotted
and
wo
ven
(Fig.
1).
To elucida
te
the
influence
of
fiber
topology
on
the
mechanical
properties
of
hier
archical
intertwined
structur
es,
we designed,
fabrica
ted,
mechanically
probed,
and
ana-
lyzed
hier
archical
wo
ven
and
knotted
rhombus-shaped
frames
with
equal
diagonals
under
quasi-s
tatic
tension.
The
probing
of
a two-
dimensional
(2D)
frame
with
intertwined
beams
focuses
on
an
identical
subs
tructur
e inside
a 3D
unit
cell
in
the
lattice,
in
this
case,
a rhombus
inside
an
octahedr
on,
while
maintaining
identical
hier
archy
of
each
beam
being
composed
of
thr
ee
intertwined,
sep-
arate
fibers.
Our
experiments
and
theory
sho
w tha
t the
knotted
fiber
topology
enables
a ne
w
regime
of
deforma
tion
and
reconfigur
ation
in
architected
ma
terials
spa
ce,
i.e.,
knot
tightening.
W
e inv
estiga
ted
the
effect
of
interfiber
friction
on
the
tying
process
by
(i)
applying
differ
ent
surfa
ce
treatments
to
the
fibers
—
passiva
tion
with
a thin
(~5-nm-thick)
layer
of
alumina
(Al
2
O
3
),
systema
tic
irr
adia
tion
with
ultr
aviolet
(UV)
light
(254-nm
wavelength),
and
aging
—
and
by
(ii)
conducting
quasi-s
tatic
in
situ
tensile
experiments
on
all
of
these
samples
with
differ
ent
surfa
ce
treatments.
W
e compar
e the
mechanical
response
of
UV-irr
adia
ted
versus
aged
rhombuses,
as
well
as
rhombuses
with
differ
ent
linear
dimensions,
to
inv
estiga
te
aging
mechanisms
and
size
effects
in
the
intertwined
polymeric
structur
es.
Intertwined
rhombus
frames
were fabrica
ted
out
of
IP-Dip
pho-
tor
esis
t using
two-photon
lithogr
aphy
(Nanoscribe
GmbH)
with
an
intended
fiber
radius
r*
of
1.69
μm,
beam
radius
R*
of
3.5
μm,
and
rhombus
height
H
of
70
μm,
as
well
as
rhombuses
with
twice
the
linear
dimensions
(
H
= 140
μm).
A
3D
octahedr
on
cube
unit
cell
with
a rela
tiv
e density
(i.e.,
fill
fraction)
ρ of
5%,
calcula
ted
as
the
volume
of
ma
terial
in
the
unit
cell
compar
ed
to
the
total
volume
of
the
unit
cell,
and
width
equal
to
H
= 70
μm
was
formed
by
as-
sembling
thr
ee
rhombuses
together,
each
aligned
with
one
of
the
thr
ee
Cartesian
principal
axes
(see
Fig.
1A
for
repr
esenta
tion
in
a
lattice).
Ea
ch
beam
in
the
rhombus
was
composed
of
thr
ee
interwo-
ven
fibers,
and
a cus
tom
grip
was
fabrica
ted
on
top
of
each
rhombus
(see
Fig.
1D)
to
enable
in
situ
tensile
experiments
using
a nano-
indenter
inside
a scanning
electr
on
micr
oscope
(SEM).
Fig.
1. Knotted
and
woven fiber
topologies
in hier
archical
intertwined
materials.
(
A
) Computer-aided
design
(CAD) rendering
of a hierarchical
octahedr
on lattice
where each unit cell is composed
of three rhombuses.
The rhombus
in the rectangular
box is formed
by two knotted
fibers,
one being
highlighted
in purple
and tes-
sellated vertically
. (
B
) SEM images
with color shading
overlaid onto a single
fiber in the knotted
(left) and woven (right)
hierarchical
rhombuses,
each with two vertically
connected
fibers.
Anoverhand
knotisformed
byeachofthetwofibers(onecoloredpurple
andtheotheruncolor
ed)intheknotted
rhombus.
Vertical
fibersinthewoven
rhombus
arecoloredredandpink,respectiv
ely.(
C
) Photogr
aphofanoverhand
knotthatresembles
thepurple
knotin(A).(
D
) Aninsituexperimental
setupinside
anSEM
on a representat
ive intertwined
rhombus
frame. Scale bars, 10
μ
m (B) and 20
μ
m (D).
Moestopo
et
al.
,
Sci.
Adv.
9
, eade6725
(2023)
8 March 2023
2 of 10
SCIENCE
ADVANCES
|
RESEARCH
ARTICLE
Downloaded from https://www.science.org at California Institute of Technology on March 10, 2023
RESUL
TS
The
influence
of
fiber
topology
on
the
mechanical
response
of
in-
tertwined
structur
es
is
highlighted
in
Fig.
2 (A
to
C),
which
sho
ws
applied
uniaxial
load
F
versus
strain
ε and
the
corr
esponding
time-
lapse
images
during
the
in
situ
uniaxial
tension
experiments
on
knotted
versus
wo
ven
rhombus
frames
with
a designed
height
H
of
70
μm
pulled
to
failur
e (see
also
mo
vie
S1).
Tensile
experiments
demons
trate
dis
tinct
regimes
of
deforma
tion
to
failur
e,
with
the
wo
ven
topology
(Fig.
2A)
firs
t undergoing
fiber
alignment,
char
ac-
terized
by
a nearly
linear
region
with
a slope
of
~0.9
mN
(regime
1)
up
to
a strain
of
~40%,
follo
wed
by
fiber
stretching
(regime
2)
at a
fiv
e times
higher
slope
in
the
da
ta
up
to
failur
e at 73.4%
strain.
The
knotted
frame
(Fig.
2B)
also
firs
t underw
ent
fiber
alignment
(regime
1)
with
a similar
signa
tur
e up
to
a strain
of
~40%,
follo
wed
by
knot
tightening
(regime
2)
char
acterized
by
smoother
deforma
tion
at a
steady
stiffness
of
~0.5
mN,
and
the
combina
tion
of
knot
tightening
and
engaged
fiber
stretching
from
a strain
of
~115%
up
to
failur
e at
146.9%
(regime
3).
Figur
e 2C
contains
da
ta
for
fiv
e wo
ven
and
fiv
e
knotted
samples
fabrica
ted
in
thr
ee
separ
ate
ba
tches,
and
it demon-
strates
tha
t in
the
fiber
alignment
regime,
rhombuses
of
both
topol-
ogies
sho
w
similar
mechanical
signa
tur
e up
to
ε ~40%.
All
wo
ven
rhombuses
then
enter
ed
the
fiber
stretching
regime
indica
ted
by
a
fiv
e times
incr
ease
in
the
load-s
train
slope,
up
to
incipient
failur
e at
the
ultima
te
failur
e load
F
f
of
1.22
to
1.72
mN
and
a corr
esponding
failur
e strain
ε
f
of
67.4
to
75.4%.
At
the
transition
strain
of
ε ~40%,
the
knotted
rhombuses
were also
aligned
along
the
loading
dir
ec-
tion
and
continued
to
deform
pas
t the
failur
e strain
ε
f
of
the
wo
ven
rhombuses
via
the
knot
tightening
mechanism
available
to
this
geometry
. Two
knotted
rhombuses
origina
ting
from
the
firs
t
ba
tch
sho
wed
a dis
tinct
transition
betw
een
the
knot
tightening
and
fiber
stretching
regimes
at a strain
of
ε ~115%
up
to
failur
e at
a strain
of
144.3
to
146.9%
and
an
applied
failur
e load
of
1.23
to
1.27
mN.
Knotted
rhombuses
from
differ
ent
ba
tches
contained
a less
defined
tightening-to-s
tretching
transition,
failing
at a
lower
strain
(90.4
to
96.3%
for
the
second
ba
tch
and
108.1%
for
the
third)
and
similar
failur
e loads.
The
firs
t (ultima
te)
failur
e events,
depicted
with
symbols
corr
esponding
to
the
samples
’
ba
tch
numbers,
did
not
alw
ays inv
ok
e the
breaking
of
all
vertically
orient-
ed
fibers,
with
some
samples
being
able
to
bear
load
be
yond
their
reported
ultima
te
failur
e strains
(see
example
in
Fig.
2B).
The
unique
reconfigur
ation
mechanism
in
hier
archical
knotted
topology
is
further
sho
wcased
in
Fig.
2 (D
and
E),
which
contains
the
tensile
response
of
knotted
and
wo
ven
rhombuses
from
the
same
ba
tch
subjected
to
sev
eral
loading/unloading
cycles
at ~30%
strain
incr
ements
per
cycle
and
under
monotonic
loading
to
failur
e at a strain
rate
of
1 × 10
−
3
s
−
1
(see
also
mo
vie
S2).
These
plots
indica
te
tha
t the
mechanical
response
of
cyclically
and
mono-
tonically
loaded
rhombuses
of
the
same
fiber
topology
ma
tched
one
another
closely
at strains
be
yond
the
maximum
strain
of
the
previ-
ous
loading
cycle.
In
the
firs
t two
loading
cycles,
both
the
knotted
and
wo
ven
rhombuses
elonga
ted
via
fiber
reorienta
tion
and
uncoil-
ing
befor
e returning
to
their
original
shapes
upon
load
remo
val,
with
slight
twis
ting
of
the
rhombus
and
a concomitant
viscoelas
tic
response
present
in
the
unloading
region
in
both
topologies.
In
the
third
cycle,
the
wo
ven
rhombus
failed
via
fiber
ruptur
e around
the
same
failur
e strain
ε
f
and
failur
e load
F
f
as
monotonically
loaded
samples.
The
knotted
rhombus
retained
its
knotted
shape
follo
wing
unloading
from
ε ~90%
in
the
third
cycle
without
any
evidence
of
failur
e.
W
e assembled
two
knotted
rhombus
frames,
each
with
their
knot
tightening
dir
ection
aligned
along
the
loading
pa
th,
into
a
reduced
unit
cell
(RUC)
and
tes
ted
tessella
tions
of
RUCs
(also
termed
as
reduced
lattice)
in
the
knot
tightening
dir
ection
to
high-
light
ho
w the
knot
tightening
mechanism
in
each
frame
is transla
ted
to
a 3D
architectur
e with
multiple
unit
cells
(see
Fig.
3A).
The
rep-
resenta
tiv
e lattice
stress
σ versus
strain
ε response
is overlaid
on
top
of
the
predicted
stress
σ versus
strain
ε beha
vior
obtained
by
dou-
bling
the
tensile
responses
of
knotted
and
wo
ven
rhombus
frames
to
account
for
the
two
knotted
frames
assembled
vertically
in
each
unit
cell.
Figur
e 3A
sho
ws
tha
t the
reduced
lattice
stress
versus
strain
re-
sponse
follo
wed
the
predicted
trajectory
for
a knotted
lattice
and
tha
t it reached
an
ultima
te
tensile
strength
(UTS
or
σ
UTS
) of
369
to
541
kP
a with
a corr
esponding
failur
e strain
of
77.3
to
97.1%,
both
close
to
the
expected
UTS
of
~522
kP
a and
a failur
e strain
of
~108%
for
a knotted
lattice
(see
fig.
S1
for
mor
e informa
tion
on
RUC
and
lattice
tes
ting).
To elucida
te
the
effect
of
interfiber
friction
on
the
mechanical
beha
vior
of
intertwined
structur
es,
we firs
t passiva
ted
the
samples
identical
to
rhombuses
in
Fig.
2C
with
a 5-nm-thick
alumina
film
deposited
via
atomic
layer
deposition
(ALD)
at 200°C
and
com-
par
ed
their
F
versus
ε response
with
tha
t of
the
as-fabrica
ted
rhom-
buses
(Fig.
3B).
W
e observ
ed
nearly
identical
load-s
train
beha
vior
betw
een
the
two
geometries,
with
the
initial
slope
a factor
of
~2.5
to
2.6
times
higher
than
tha
t of
the
as-fabrica
ted
samples,
follo
wed
by
ultima
te
failur
e in
regime
1 (fiber
alignment)
at a strain
of
~25
to
30%.
Passiva
ted
knotted
rhombuses
failed
at ε
f
= 27.1
to
28.6%
and
F
f
= 0.55
to
0.58
mN,
and
wo
ven
ones
failed
at ε
f
= 25.1
to
29.2%
and
F
f
= 0.59
to
0.63
mN,
which
are about
two
to
thr
ee
times
lower
than
those
of
unpassiva
ted
wo
ven
frames.
W
e observ
ed
a similar
trend
in
the
tensile
response
of
cylindrical
pillars
made
of
the
same
IP-Dip
resis
t with
radii
of
1.69
μm
and
heights
of
10
μm
subjected
to
dif-
fer
ent
pos
tpr
ocessing
procedur
es:
(i)
as-fabrica
ted
samples
(termed
pris
tine)
and
(ii)
samples
passiva
ted
with
a
5-nm-thick
ALD
alumina
immedia
tely
after
fabrica
tion
(fig.
S2);
tensile
experiments
on
the
passiva
ted
pillars
sho
w a ~144
and
~43%
incr
ease
in
Young
’
s
modulus
E
and
yield
stress
σ
y
compar
ed
to
pris
tine
pillars,
respec-
tiv
ely
, as
well
as
a ~27%
decr
ease
in
σ
UTS
and
an
~88%
drop
in
strain
at UTS
(ε
UTS
).
W
e explor
ed
ho
w varying
fiber
bulk
and
surfa
ce
properties
affect
the
mechanical
beha
vior
of
intertwined
structur
es
be
yond
the
fiber
alignment
regime
by
pulling
to
failur
e knotted
and
wo
ven
rhom-
buses
of
H
= 70
μm
tha
t had
undergone
the
follo
wing
treatments:
(i)
pris
tine,
(ii)
irr
adia
ted
under
UV
for
5 hours,
and
(iii)
irr
adia
ted
under
UV
for
29
hours
(Fig.
3,
C
and
D).
These
experiments
re-
vealed
tha
t in
the
fiber
alignment
regime,
i.e.,
at strains
belo
w
40%,
the
loads
of
UV-irr
adia
ted
(ii
and
iii)
wo
ven
frames
are up
to
~80%
higher
than
those
of
pris
tine
(i)
samples
(fig.
S3A).
Incr
eas-
ing
the
UV
radia
tion
time
strengthened
the
wo
ven
frames
in
the
fiber
stretching
regime,
which
failed
around
the
same
ε
f
: 67.4
to
71.4%
strain
for
pris
tine
samples,
68.5
to
69.9%
for
the
5-hour
UV-irr
adia
ted
samples,
and
70.3
to
72.1%
for
the
29-hour
UV-irr
a-
dia
ted
ones,
with
corr
esponding
F
f
values
of
1.23
to
1.40
mN,
1.60
to
1.66
mN,
and
2.49
to
2.57
mN,
respectiv
ely
.
No
clear
trend
in
the
differ
ence
betw
een
pris
tine
and
5-hour
UV-irr
adia
ted
knotted
samples
was
observ
ed;
the
loads
for
29-
hour
UV-irr
adia
ted
knotted
samples
are up
to
~90%
higher
than
Moestopo
et
al.
,
Sci.
Adv.
9
, eade6725
(2023)
8 March 2023
3 of 10
SCIENCE
ADVANCES
|
RESEARCH
ARTICLE
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those
of
other
knotted
rhombuses
in
all
regimes
of
deforma
tion
for
a giv
en
strain
(Fig.
3D
and
fig.
S3,
B and
C).
All
knotted
frames
ex-
perienced
firs
t failur
e events
at higher
strains
compar
ed
to
all
wo
ven
rhombuses
and
at similar
or
lower
F
f
compar
ed
to
wo
ven
rhombus-
es
tha
t underw
ent
the
same
pos
tpr
ocessing
procedur
es:
ε
f
= 90.4
to
146.9%
and
F
f
= 1.12
to
1.27
mN
for
pris
tine
samples,
ε
f
= 81.7
to
126.2%
and
F
f
= 1.07
to
1.66
mN
for
5-hour
UV-irr
adia
ted
samples,
and
ε
f
= 74.4
to
111.4%
corr
esponding
to
F
f
= 1.23
to
2.12
mN
for
29-hour
UV-irr
adia
ted
ones.
W
e inv
estiga
ted
the
mechanical
and
surfa
ce
properties
of
(i)
pris
tine,
(ii)
5-hour
UV-irr
adia
ted,
and
(iii)
29-hour
UV-irr
adia
ted
IP-Dip
samples
via
in
situ
tensile
tes
ting
of
pillars
(fig.
S2)
and
x-r
ay
photoelectr
on
spectr
oscopy
(XPS)
char
acteriza
tion
(fig.
S4;
see
the
Supplementary
Ma
terials
for
details
on
the
XPS
char
acteriza
tion).
These
experiments
unco
vered
tha
t greater
irr
adia
tion
time
incr
eases
the
modulus
E
and
the
yield
strength
σ
y
with
a concomitant
reduc-
tion
in
tensile
strain
ε
UTS
. The
UTS
σ
UTS
of
pillars
irr
adia
ted
for
5
hours
decr
eased
by
~10%,
while
the
σ
UTS
of
29-hour
UV-
irr
adia
ted
pillars
on
another
chip
incr
eased
by
~5%
compar
ed
with
pris
tine
IP-Dip
pillars
on
their
corr
esponding
chips.
XPS
char
acteriza
tion
reveals
inconsis
tency
with
regard
to
surfa
ce
prop-
erties
of
similarly
treated
polymer
samples,
with
no
unambiguous
surfa
ce
transforma
tions,
such
as
chain
scission
(
58
),
occurring
on
a
giv
en
sample
(see
the
Supplementary
Ma
terials).
To probe
into
the
influence
of
fea
tur
e size
on
the
mechanical
be-
ha
vior
of
architected
ma
terials
composed
of
IP-Dip
fibers,
which
ha
ve been
reported
to
lack
size
effects
in
modulus
and
yield
strength
within
1 to
10
μm
(
59
),
we fabrica
ted
and
tes
ted
one
ba
tch
of
larger
self-similar
pris
tine
knotted
and
wo
ven
frames
with
designs
and
printing
par
ameters
identical
to
those
sho
wn
in
Fig.
2 and
with
double
the
linear
dimensions
(
H
= 140
μm).
W
e define
a normalized
load
F
as
the
applied
load
F
divided
by
the
product
of
the
Young
’
s
modulus
of
pris
tine
IP-Dip
E
pr
and
the
fiber
cross-sectional
area
Fig.
2. Micr
oscale
tensile
experiments
of hier
archical
knotted
and
woven
rhombus
frames.
(
A
and
B
) Mechanical
data and time-lapse
images
during
tension
of representa
tive woven (A) and knotted
(B) frames pulled
up to failure,
showing distinct deforma
tion regimes
in each fiber topology
. Throughout
this
figure, red data points
correspond
towoven geometry
and purple
data points
cor-
respond
to the knotted
one. In (B), mechanical
data beyond ~147%
strain corre-
sponds
to the unraveling of broken fiber(s)
after the first failure event. (
C
)
Combined
tensile
response
up to the first failure event of five knotted
and five
woven rhombus
frames from three separate rounds
of fabrica
tion. Bolder
datasets
come
from the same tests shown in (A) and (B). (
D
and
E
) Load versus strain for
woven (D) and knotted
(E) rhombuses
cyclically
loaded
in tension,
prestrained to
incrementally
higher
extents
in each subsequent
cycle. The still frames correspond
to sample
images
at the end of each cycle, showing failure in the woven rhombus
and stable reconfigur
ation via knot tightening
in the knotted
rhombus
during
the
third cycle. Tensile
responses
up to failure of rhombuses
with identical
fiber topol-
ogy origina
ting from the same batch are shown in gray dashed
lines. Scale bars,
15
μ
m.
Fig.
3. Tensile
response
of a knotted
lattice
and
effects
of passiva
tion
and
UV
irradia
tion
on the
mechanical
beha
vior
of intertwined
architectur
es.
(
A
) Rep-
resenta
tive tensile
response
of a tessella
tion of RUCs, where two knotted
rhombus
frames are assembled
in each unit cell with their knot tightening
direction
aligned
in the loading
direction.
The first failure event for each experiment
is marked with
“
x.
”
Predicted
responses
for a lattice consisting of woven and knotted
frames,
as
well as a woven lattice with a similar
relative density
from literature, are shown by
dashed
curves. (
B
) Representa
tive tensile
responses
up to first failure and corre-
sponding
still frames of knotted
and woven rhombuses
passiva
ted with 5-nm-
thickAl
2
O
3
film.Theresponse
ofequivalent
as-fabrica
ted(unpassiva
ted)rhombus-
es from Fig. 2 is also shown. First failure events from all experiments
are marked
with
“
x.
”
(
C
and
D
) Representative tensile
responses
up to first failure events of UV-
irradiated and pristine hierarchical woven (C) and knotted
(D) rhombuses
with cor-
responding
stillframesandtheirfirstfailureeventsmarkedwithindicat
edsymbols.
All woven samples
in (C) and some knotted
ones in (D) origina
ted from the same
batch (see fig. S3D). Mechanical
data for pristine hierarchical
rhombuses
are also
shown in Fig. 2C. Scale bars in all SEM images,
15
μ
m (A, B, and D).
Moestopo
et
al.
,
Sci.
Adv.
9
, eade6725
(2023)
8 March 2023
4 of 10
SCIENCE
ADVANCES
|
RESEARCH
ARTICLE
Downloaded from https://www.science.org at California Institute of Technology on March 10, 2023