of 22
1
1
2
3
Supplementary
Information
for
4
5
Deformation Characteristics of Solid
-
state Benzene
as a
Step towards
6
Understanding Planetary Geology
7
8
Authors:
Wenxin Zhang
1
*
, Xuan Zhang
2
, Bryce W. Edwards
1
, Lei Zhong
3
,
Huajian Gao
4,5,3
,
9
Michael J.
Malaska
6
, Robert Hodyss
6
, Julia R. Greer
1,7
10
Affiliations:
11
1
Division of Engineering and Applied Sciences, California Institute of Technology, 1200 E.
12
California Blvd., Pasadena, CA 91125, USA.
13
2
INM
Leibniz Institute for New Materials, Campus D2 2, 66123 Saa
rbrücken, Germany.
14
3
School of Engineering, Brown University, Providence, Rhode Island 02912, USA.
15
4
School of Mechanical and Aerospace Engineering, College of Engineering, Nanyang
16
Technological University, 70 Nanyang Drive, 639798, Singapore.
17
5
Institute of High Performance Computing, A*STAR, 138632, Singapore.
18
6
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena,
19
California 91109, USA.
20
7
Kavli Nanoscience Institute, California Institute of Technology, 120
0 E. California Blvd.,
21
Pasadena, CA 91125, USA.
22
*Corresponding author. Email:
wzhang2@caltech.edu
23
†These authors contributed equally to this work.
24
25
2
26
27
28
29
30
31
32
This file includes:
33
34
Supplementary
Notes
1
to
8
35
Suppleme
ntary Table 1
36
Supplementary Figures
1 to 1
1
37
38
39
40
3
Supplementary
Note 1
.
Calculation of Contact Pressure in Experiments
.
The pyramidal
41
sample geometry lends itself to the calculation of contact pressure, as representation of the
42
compressive stress. Contact pressure
P
contact
is defined as the ratio between the real
-
time load
F
43
and the instantaneous contact area
A
contact
between the indenter tip and the sample, or
44
P
contact
=
F
/
A
contact
(1)
.
45
For an ideal cuboid
-
corner pyramidal geometry, at displacement
h
,
46
A
contact
= 3
3/2
h
2
/2
(2)
.
47
Given minimal bulging was observed in SEM during compression, the self
-
similar cross
-
section
48
assumption is valid. For the real experimental
data, the finite curvature at the apex of the crystal
49
is taken into account. The adjusted contact area is
50
A
contact
= 3
3/2
(
h
+
δ
)
2
/
2
(3),
51
W
ith
δ
< 50
nm based on specific SEM observation. The contact pressure is then
52
P
cont
act
= 2
F
/
[3
3/2
(
h
+
δ
)
2
]
(4)
.
53
54
4
Supplementary Note
2.
Calculation of Loading Stiffness for Virgin
-
compression in Higher
-
55
reload Experiments
.
For the first loading cycles, due to the linearity of the loading segment, the
56
stiffness was taken as the best
-
fitted slope of the line. Whereas, for the reloading cycles, the
57
virgin
-
compression is considered as the loading segment from where the current l
oad just
58
surpassed the historical maximum to unloading. And the loading stiffness is then found as the
59
best
-
fitted slope of this segment. Note that the displacement from the bursts is included in
60
calculating the loading stiffnesses, which could cause these
values to slightly underestimate the
61
stiffness of the pre
-
burst pristine benzene.
The results are shown in
Supplementary Fig.
4.
62
63
5
Supplementary Note
3.
Identification of Displacement Bursts for the Loading Segments in
64
Higher
-
reload Experiments
.
Based on the current loading rate, data acquisition frequency, and
65
cryogenic system’s vibration magnitude, the criteria for displacement burst identification are: (to
66
simplify notation, total cumulative displacement is called
h
, and the incremental displac
ement
67
per data point is called
Δ
)
68
(a)
Δ
>
δ
= 0.7 nm, i.e., approxi
ma
tely twice median
Δ
(median instead of mean is
69
considered because median is less sensitive to cryogenic vibrations) as well as the unit
70
cell size of benzene;
71
(b) the mean
Δ
from the adja
cent five data points should be greater than
δ
/3;
72
(c) for all data points identified by criteria a & b, the last identified point should be
73
separated by no more than one data point from the current identified point;
74
(d)
h
at the current data point identifi
ed by criteria a, b, & c should be greater than
h
at
75
the last data point identified by criterion a, b, & c;
76
(e) the mean
h
of the adjacent seven data points is less than the mean
h
of the next seven
77
data points.
78
The criteria b
-
e partially serves for removi
ng false idenfication caused by vibration. Still, very
79
occasionally the criteria falsely identified artificial displacement bursts or omits likely real
80
displacement bursts satisfying criteria a & b, in which case the relevant data points were visually
81
exam
ined and manually removed/reselected. The results are shown in
Supplementary Fig.
5.
82
83
6
Supplementary Note
4.
Estimation of the
Longitudinal
Modulus of the Densified Region
.
84
We estimated the longitudinal modulus of the densified region along the axial
compression
85
direction
from the unloading segment of the load
-
displacement data and the post
-
mortem SEM
86
images:
87
E
l
=
kH
(1
D
pl
/
H
)/
A = ckH
/2.6
h
(5)
,
88
where
D
pl
is the plastic displacement,
H
is total deformed height
,
k
is the unloading stiffness,
A
is
89
the cross
-
sectional area calculated at distance
h
from the initial apex location
, and
90
c =
1
D
pl
/
H
(6)
91
represents the densification ratio
.
A
ssuming linear elastic
unloading and approximating cross
-
92
sectional area o
f
the compressed pyramid as a constant
(i
n reality, it had
a slight variation over
93
the height due to the small taper of the side)
,
E
l
is estimated as ~0.7
-
1.8 GPa for
c
~ 0.4±0.04
94
(
Supplementary F
ig. 10).
95
96
7
Supplementary Note
5.
Influence of Temperature on the Mechanical Properties
.
As applied
97
in a previous room
-
temperature nanoindentation study of a selection of candidate Titan minerals
12
,
98
the Young’s modulus E scales as a function of temperature
T
as
99
E
(
T
)
E
0
(1
T
/2
T
m
)
(7)
,
100
where
E
0
is the Young’s modulus at 0 K, and
T
m
is the melting temperature. Applying this
101
relation to the benzene crystallites in this work, whose melting temperature is 278.7 K
27
, we find
102
that
E
(94 K)
1
.07
E
(123.39 K)
1.12
E
(144.40 K). This implies that the Young’s modulus at
103
Titan’s surface temperature is only ~12% below the one measured in our work. The agreement
104
should hold for the longitudinal modulus, as well because it is linearly related to the
Young’s
105
modulus through a geometric factor independent of temperature. Yu et al. also suggested the
106
empirical relation between hardness
H
and Young’s modulus
E
through a power
-
law scaling,
107
with a close
-
to
-
unity power coefficient
12
, further suggesting that
the hardness as well as the
108
strength have similarly weak temperature
-
dependence and validates our measurements.
109
110
8
Supplementary Note
6
.
Calculation of Density Distributions from MD results
.
We used the
111
following method to plot the density
distribution: we calculated the coordination number (CN) of
112
Carbon atoms around one Carbon atom within a cut
-
off of 8 Å and calculate the density of this
113
position by the equation:
114
퐴푡표푚
=
(
CN
+
1
)
×
13
×
4
/
3
×
3
(8)
115
Where
m
is the at
omic mass constant, which is
1
.
66
×
10
27
kg
,
R
is the cut
-
off radius. By
116
substituting the initial lattice constants of the solid benzene for MD simulations, we can calculate
117
the initial density of the sample as:
118
푖푛푖푡푖푎푙
=
24
×
13
×
푎푏푐
=
1
.
05977
g
cm
3
(9)
119
which is consist
ent with the initial density distribution in MD videos (
Supplementary
Movies
2
120
to
4
).
121
122
9
Supplementary Note
7
.
Calculation of Radial Distribution Functions from MD results
.
123
Base
d on observations, we isolated the molecules of the densification regions by
selecting the
124
atoms in OVITO that satify non
-
affine square displacement
>100 and shear strain
<0.5 with cuf
-
125
off radius of 10 Å (see
Supplementary Fig
. 7
for example). And then the RDF of C
-
C pairs was
126
calculated in OVITO, setting the cut
-
off of 15 Å (
Fig.
5b
and
Supplementary Fig.
11
).
All curves
127
are
normalized by the
height of the
first peak.
128
129
10
Supplementary Note
8
.
Complexity of stress distribution due to “pyramid”
-
shaped solid
130
benzene
.
The stress tensor within the pyramid is non
-
uniform when
subjected to a uni
-
131
directionally applied external load because of the increasing cross
-
sectional area during
132
compression. The pressure distribtuion in the loading direction decreases with the rising cross
-
133
sectional area, with the largest local deformation
occurring at the maximum
-
pressure region, i.e.,
134
the apex. The non
-
negligible shear stress component perpendicular to the laoding direction arises
135
as a result of transverse expansion of adjacent atomic planes under pressure.
136
137
11
Supplementary
Table 1.
Elasti
c constants of solid benzene
57
.
138
c
11
c
22
c
33
c
44
c
55
c
66
c
12
c
13
c
23
170 K
8.01
9.26
7.88
3.18
5.53
1.95
3.85
4.80
5.08
138 K
8.61
10.01
8.63
3.56
6.13
2.10
4.15
5.10
5.38
Units: GPa
139
140
12
141
Supplementary Figure
1.
The loading profiles for (
A
) the
S
ame
-
R
eload
test and (
B
) the
H
igher
-
142
R
eload test.
143
144
13
145
Supplementary Figure
2. Same
-
R
eload tests.
Load
-
displacement raw data and calcualted
146
contact pressures.
147
148
14
149
Supplementary Figure 3
. Higher
-
R
eload tests.
Load
-
displacement raw data and calcualted
150
contact pressures.
151
152
15
153
Supplementary Figure
4.
Higher
-
R
eload
virgin
-
compression stiffness (outliers removed): (
A
)
154
fitting results for individual datasets; (
B
) comparison of the stiffening effect over loading cycles.
155
Error bars represent the standard deviation
of
each
measurement.
156
157
16
158
Supplementary Figure
5.
Identification of displacement bursts (red) for
H
igher
-
R
eload
159
experiments.
160
161
17
162
Supplementary Figure
6.
Collective re
-
orientation of benzene rings along the 45°
local shear
163
direction.
T
he re
-
orientation process
during compression
in
the local region in the black box
is
164
snapshotted
and
magnified
in the
lower
zoomed
-
in images
.
165
166
18
167
Supplementary Figure
7
.
Selection of the desification region using
pa
rameters of non
-
affine
168
square displacement >100 and shear strain
<
0
.5 set in software OVITO.
169
170
19
171
Supplementary Figure
8
.
The benzene molecule orientation in the lateral face of the pyramidal
172
sample for
simulations under different loading directions.
173
174
20
175
Supplementary Figure
9
.
Simulated contact pressure
-
normalized displacement relations using
176
different (
A
) unit cell orientations and (
B
) temperatures in the range of 10
-
30 K.
177
178
21
179
180
Supplementary Figure
1
0
.
Estimation of the
longitudinal
modulus of densified region: (
A
)
181
Extracting
D
pl
,
H
,
k
, and
c
from compression data (inset: post
-
mortem SEM image defining the
182
dimensions); (
B
) Bounds of modulus as a function of densification ratio.
183
184
A
B