Supporting information for ‘Understanding pop-ins in spherical nanoindentation’
Data Analysis Protocols for calculating Indentation Stress-Strain (ISS) Curves
The data analysis protocols used in Fig 1 to convert the recorded loaddisplacement data to indentation stress-strain (ISS) curves can be summarized as a twostep procedure (see Ref. 1 for details). The first step in the analysis process is an accurate
estimation of the point of effective initial contact in the given data set, i.e. a clear
identification of a zero-point that makes the measurements in the initial elastic loading
segment consistent with the predictions of Hertz’s theory 2. As shown in Ref. 1, the zero
point can be conveniently determined using the following equation for the initial elastic
segment in a frictionless, spherical indentation:




~
3 P 3 P  P*
S
 ~
.
(S1)
2 he 2 he  h*
~ ~
where P , he , and S are the measured load signal, the measured displacement signal, and
the continuous stiffness measurement (CSM) signal in the initial elastic loading segment
from the machine, respectively, and P* and h * denote the values of the load and
displacement values at the point of effective initial contact. Rearrangement of Eq. (SS1)
~ 2 ~
reveals that a plot of P  She against S will produce a linear relationship whose slope is
3
2
equal to  h * and the y-intercept is equal to P * . Therefore, a linear regression analysis
3
can then be performed to identify the point of the effective initial contact ( P * and h * )
very accurately.
It is important to recognize that the effective zero-point defined here may not
necessarily be the actual point of initial contact. The concept of an effective point of
initial contact allows one to de-emphasize any artifacts created at the actual initial contact
due to the unavoidable surface conditions (e.g., surface roughness, presence of an oxide
layer) and imperfections in the indenter shape. It has to be interpreted as the point that
brings the initial elastic loading segment to as close an agreement as possible with Hertz
theory.
In the second step, the values of indentation stress and strain can be calculated by
recasting Hertz theory for frictionless, elastic, spherical indentation as
 ind  Eeff  ind ,
a
S
,
2 E eff
P
4 he
he
,



ind
a 2
3 a 2.4a ,
2
2
1  s 1  i
1
1
1
 ind 
1

E eff
Es

Ei
,
Reff

Ri

Rs
(S2)
where  ind and  ind are the indentation stress and indentation strain, a is the radius of the
contact boundary at the indentation load P, he is the elastic indentation depth, S (=
dP/dhe) is the elastic stiffness described earlier, Reff and Eeff are the effective radius and
the effective stiffness of the indenter and the specimen system,  and E are the Poisson’s
ratio and the Young’s modulus, and the subscripts s and i refer to the specimen and the
indenter, respectively.
A salient feature of the protocols described above is the use of CSM 3 to obtain a
reliable estimate of the radius of contact, a, at every point on the load-displacement curve
(Eq. S2). The rigorous derivation of Eq. S2 directly from Hertz theory makes the
Supporting Information - 1
Supporting information for ‘Understanding pop-ins in spherical nanoindentation’
estimates of contact radius from the measured CSM signals highly trustworthy.
Nanoindentations were carried out using two different nanoindenter machines – the
Agilent G200® system maintained and operated by the Center for Integrated
Nanotechnologies (CINT) at Los Alamos National Laboratory (LANL), Los Alamos,
NM, USA, and a similar Agilent G200® system located at the Paul Scherrer Institut (PSI),
Villigen, Switzerland. Both systems were equipped with the CSM attachment.
Sample preparation
As discussed in the Letter, surface preparation of metal samples is also known to
influence the occurrence of pop-ins in spherical nanoindentation 4, since any increase in
dislocation density under the indenter decreases the propensity for pop-ins. Thus only
annealed and electropolished samples, which consistently showed pop-ins for a 1 µm
spherical indenter, were considered in this study. All samples were polished initially
using mechanical polishing techniques (final polishing step 1 μm diamond polish), and
then electro-polished. Al was electro-polished using a mixture of 100 ml Perchloric acid
and 900 ml methanol at 4-5oC with a voltage of 11.8-12V for 1-5 min 5, W was electropolished using a sodium hydroxide solution at -20oC with a graphite cathode at 10V for
10-15 min 5,6 and Fe-3%Si was electro-polished at room temperature using a mixture of
95% acetic acid and 5% perchloric acid at a voltage of 60-90 V and a current of 0.5-1
Amp.
Supporting Information - 2
Supporting information for ‘Understanding pop-ins in spherical nanoindentation’
b)Effect
of indenter size
20
25
25
Load, mN
a) Pop-ins during
indentation
100 150 200 250 300
Displacement, nm
c) Effect of pre-strain
Close spaced dislocation loops
Indentation Stress, GPa
3.5
3.5
3.5
3
33
2.5
2.5
2.5
2
22
1.5
1.5
1.5
1
11
0.5
0.5
0.5
0
00
Eeff=
205
GPa
Yind
1 µm indenter
No Pop-ins
Fe-3%Si
30% deformed
0
00
0.01
0.02
0.03
10
10
10
20
20
20
0.04
0.01 0.02
0.02 Strain
0.03 0.04
0.04
0.01
0.03
Indentation
30
30
30
13.5 µm
indenter
Indentation Stress, GPa
Pop-in
10 20 30
10 20
20 30
30
10
50
0
00
0
50
100
150
50
100nm 150
150
00 Displacement,
50
100
3.5
3.5
3.5
3 Eeff=
33
Pop-in
2.5 190
2.5 GPa
2.5
2
22
1.5
1.5
1.5
Test 1 (pop-in)
1
Test 2
11
Test 3
0.5
0.5
0.5
Fe-3%Si, As cast
0
00
0
0.01 0.02 0.03 0.04
0.01 0.02
0.02 Strain
0.03 0.04
0.04
00 Indentation
0.01
0.03
d) Effect of polishing
Indentation Stress, GPa
1 µm
indenter
W,
Annealed
0
Stochastic
pop-ins
10
10
5
55
Indentation Stress, GPa
Load, mN
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
00
0
00
15
15
10
3
33
2
22
1
11
0
00
0
00
Widely spaced
dislocation loops
18
16
14
12
10
8
6
4
2
0
15
20
20
5
55 Eeff=
4 310
44
GPa
W
3
33
Yind
13.5 µm radius
2
22
indenter
1
mech polish (no pop-in)
11
electro-polish (pop-in)
0
00
0
0.01
0.02
0.03
0.01
0.02
0.03
00
0.01
0.02
Indentation
Strain0.03
5
55
4
44
3
33
Eeff=
170
GPa
Fe-3%Si
as cast
2 Y
22 ind
1
11
0
00
0
00
electro-polish, Ri 10 µm
vibro-polish, Ri 13.5 µm
0.02
0.04
0.06
0.02
0.04
0.06
0.02
0.04
Indentation
Strain0.06
Figure S1. (a) Pop-ins in nanoindentation are generally revealed as sudden excursions in
depth and strain (in a load controlled experiment) and occur most readily in indentation
experiments on annealed samples with very small indenter tip radii, such as during
indentation on annealed W with a 1 µm radius indenter 4. (b) Their occurrence is more
stochastic when using a larger indenter (such as during indentation on as-cast Fe-3%Si
with a 13.5 µm radius indenter 4) and (c) pop-ins are almost always absent in tests on
cold-worked samples with high dislocation densities. Thus tests on 30% deformed Fe3%Si steel do not show any pop-ins even with a small sized indenter of 1 µm radius 4. (d)
Rough mechanical polishing can cause the near-surface dislocation density to increase,
thus reducing pop-ins but artificially increasing the yield stress (Yind) in annealed W 4.
Vibro-polished samples show the ideal combination for measuring Yind: suppressing popins in as-cast Fe-3%Si but not adversely affecting the Yind value 7.
Supporting Information - 3
Supporting information for ‘Understanding pop-ins in spherical nanoindentation’
Indentation Stress, GPa
50
100 150 200 250 300
50
100
150
200
250
300
50
50Displacement
100
100 150
150 200
200
250
250 300
300
(h
), nm
t
375
375
375
375
325
325
325
325
275
275
275
a 275
225
225
225
225
175
175
175
175
Rapid
125
increase
125
125
125
Pop-in
in a
75
75
75
75
15
25
35
15
15
25
25
35
35
15
25
35
Displacement
(h
),
nm
t
Pop-in
8
88
Elastic
unloading
6
66
4
44
2
22
0
00
0
00
Es = 404 GPa
0.05
0.1
0.15
0.05
0.1
0.05
0.1
0.15
Indentation
Strain 0.15
Contact radius (a), nm
Load (P ), mN
Fe-3%Si
Contact radius (a), nm
Indenter radius 1 µm
3
33 3
430
430
430
430
2.5
a
2.5
2.5
2.5
330
330
330
330
2
Pop-in region
22 2
230
(expanded below)
1.5
230
230
230
1.5
1.5
1.5
130
1
130
130
130
11 1
30
0.5
30
30
30
Indenter radius
0.5
0.5
0.5
P
1 µm
0
-70
-70
-70
00 0
-70
0
50
100
150
50
100
150
50
100
150
00 0
50
100
150
Displacement
(h
),
nm
t
430
2.1
430
430
430
2.1
2.1
2.1
1.9
330
1.9
1.9
1.9
Rapid
330
330
1.7
a 330
increase
1.7
1.7
1.7
1.5
230
in a
1.5
1.5
1.5
230
230
230
1.3
1.3
1.3
1.3
130
1.1
130
130
130
1.1
1.1
1.1
P
0.9
30
0.9
0.9
0.9
30
3030
0.7
Pop-in
0.7
0.7
0.7
0.5
-70
0.5
-70
0.5
-70
0.5
-70
20
40
60
80
100
20
40
60
80
100
2020
40
60
80
100
40
60
80
100
Displacement (ht), nm
Load (P ), mN
a
Contact radius (a), nm
Load (P ), mN
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.6
0.6
0.6
0.6
0.5
0.5
0.5
0.5
0.4
0.4
0.4
0.4 P
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0
0 00
5
5 55
10
10
10
P
1400
1400
1400
1400
1200
1200
1200
1200
1000
1000
1000
1000
800
800
800
800
600
600
600
600
400
400
400
400
200
200
200
200
0
000
Contact radius (a), nm
Tungsten
Load (P ), mN
11
11
11
11
9
999
7
777
5
555
3
333
1
111
-1
-1-1-1
-3
-3-3-3
0
000
(b) Large (~52nm) pop-in
14
14
14
12 E =
Pop-in
s
12
12
198
10
10
10
Non-linear
8 GPa
88
unloading
6
66
4
44
2
22
0
00
0
0.05
0.1
0.15
0.2
00
0.05
0.1
0.15
0.2
0.05
0.1
0.15
0.2
Indentation
Strain
Indentation Stress, GPa
(a) Small (~20nm) pop-in
Figure S2. Effect of pop-in size on the unloading slope.
(a) For a smaller ~20 nm pop-in in W the unloading slope after pop-in in the ISS curve
(bottom graph) is very close to the slope in the initial elastic loading segment of the
curve. The top images show the pop-ins as a displacement burst in the load-displacement
(P-ht) and contact radius-displacement (a-ht) responses. The magnified pop-in region in
the middle images show that the contact radius remains constant during the pop-in but
increases rapidly immediately afterwards.
(b) A larger ~52 nm displacement burst in Fe-3%Si steel shows a much more rapid
increase in contact radius after the pop-in event (middle graph), and the unloading slope
after pop-in in the ISS curve (bottom graph) is also much lower than the elastic loading
slope. Note the data point marked in gold color, which signifies the end of the regime of
rapid increase in the contact radius, and its corresponding location in the ISS curve.
Supporting Information - 4
Supporting information for ‘Understanding pop-ins in spherical nanoindentation’
7
0.25
13.5 µm
indenter
Pop-in
0.2
6
0.15
Load, mN
5
0.1
0.05
4
0
0
10
20
30
40
3
Ag
2
1 µm indenter
1
0
(a)
5 µm
indenter
0
100
200
Displacement, nm
300
3
Indentation Stress, GPa
1.2
1
2.5
2
Eeff=100
GPa
0.8
0.6
0.4
0.2
1.5
0
0
0.05
0.1
1
1 µm indenter
5 µm indenter
13.5 µm indenter
0.5
0
(b) 0
0.1
0.2
0.3
Indentation Strain
0.4
Figure S3. Effect of indenter size on pop-ins in annealed electro-polished Ag. Pop-ins
are seen as displacement bursts in the (a) load-displacement curves and as strain bursts in
the (b) ISS curves. Note that the pop-ins are largest for the smaller 1 µm radius indenter,
and their size (and propensity) decreases with increase in indenter radius. The larger popins also lead to a non-linear unloading slope in the ISS curves, while the unloading (after
pop-in) slope for the ISS curves with relatively small pop-ins is very close to their
loading modulus (slope).
Supporting Information - 5
Supporting information for ‘Understanding pop-ins in spherical nanoindentation’
1
2
3
4
5
6
7
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Supporting Information - 6