On the origin of indentation size effect and depth dependent mechanical properties of elastic polymers

Abstract

Indentation size effects have been observed in both polymers and metals but, unlike in metals, the origin of size effects in polymers is not well understood. To clarify the role of second order gradients of displacements, a model polymer is examined with spherical and Berkovich tips at probing depths between 5 and 25 μm. Applying different theories to determine the elastic modulus, it is found that with a pyramidal tip, the elastic modulus increases with decreasing indentation depth, while tests with the spherical tip yielded essentially constant values for the elastic modulus independent of indentation depth. The differences between these tips are attributed to second order displacement gradients, as they remain essentially constant with a spherical tip while they increase in magnitude with decreasing indentation depth applying a Berkovich tip.

Full text

J Polym Eng 2015; aop
*Corresponding author: Chung-Souk Han, Department of
Mechanical Engineering (Department 3295), University of Wyoming,
1000 East University Avenue, Laramie, WY 82071, USA,
e-mail: chan1@uwyo.edu
Seyed H.R. Sanei and Farid Alisafaei: Department of Mechanical
Engineering (Department 3295), University of Wyoming, 1000 East
University Avenue, Laramie, WY 82071, USA
Chung-Souk Han*, Seyed H.R. Sanei and Farid Alisafaei
On the origin of indentation size effects and
depth dependent mechanical properties
of elastic polymers
Abstract: Indentation size effects have been observed in
both polymers and metals but, unlike in metals, the origin
of size effects in polymers is not well understood. To clar-
ify the role of second order gradients of displacements,
a model polymer is examined with spherical and Berko-
vich tips at probing depths between 5 and 25 μm. Apply-
ing different theories to determine the elastic modulus,
it is found that with a pyramidal tip, the elastic modulus
increases with decreasing indentation depth, while tests
with the spherical tip yielded essentially constant values
for the elastic modulus independent of indentation depth.
The differences between these tips are attributed to sec-
ond order displacement gradients, as they remain essen-
tially constant with a spherical tip while they increase in
magnitude with decreasing indentation depth applying a
Berkovich tip.
Keywords: indenter tip geometry; length scale-dependent
deformation; nanoindentation; polymers; second order
displacement gradients.
DOI 10.1515/polyeng-2015-0030
Received January 30, 2015; accepted March 26, 2015
1 Introduction
Nanoindentation is widely applied for the determination
of mechanical properties [1–4] at small length scales.
While the testing of materials with indentation is rela-
tively simple, a reliable interpretation of the obtained
data can pose challenges, particularly at small indenta-
tion depths where the deformation mechanisms can be
quite different from those at macroscopic scales. At the
micron to submicron length scales for instance, inden-
tation size effects have been observed in metals [5–8] as
well as polymers [9–13]. However, unlike in metals where
length scale dependent deformation is attributed to geo-
metrically necessary dislocations and related gradients of
plastic strain (see, e.g., [14–17]), the origin of length scale
dependent deformation in polymers is not well under-
stood. Compared to metals, the deformation mechanisms
in polymers are arguably more complex because of their
complex molecular structure. Polymers are also known
to be more sensitive to deformation rates and tempera-
tures than metals. While in metals, length scale depend-
ent deformation is associated with plastic deformation,
bending experiments of epoxy micro-beams [18] and
indentation of the highly elastic polydimethylsiloxane
(PDMS) [19, 20] have shown that in polymers, size effects
are also present in elastic deformation. For both bending
and indentation experiments [18, 20] a non-local rotation
gradient material formulation could predict such behav-
ior [21–23], which will be examined here.
Indentation size effects have been observed in many
polymers where the hardness H increases with decreas-
ing indentation depth h [24]. A straightforward possible
explanation for the observed increase in H with decreasing
indentation depth would be changes in the material prop-
erties through the depth of the polymer. For PDMS, it has
been found that the elastic modulus E is strongly inden-
tation depth dependent, as reported in [1, 2], where con-
siderable increases in E have been found with decreasing
indentation depth. While E increases by a factor of about
two from 600 μm down to 100 μm in [1], an even stronger
rise has been found at lower h where E increases by a
factor of about 25 from 4000nm down to about 200nm [2].
Assuming that (i) the increases in H [9–13, 19, 20] and E [1,
2, 9, 13] are related and (ii) the determined depth depend-
ent E is factual, then a possible straightforward explana-
tion would be that the properties of the polymers change
with depth. It is, however, not clear why the PDMS material
properties should vary so strongly with depth, as the size
effects are quite remarkable. It is also known that at depths
below 200 nm, the glass transition temperature actually
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2 C.-S. Han etal.: Origin of indentation size effects in polymers
drops [25] and corresponding softening has been observed
in polymer surfaces [26] and thin films, which would rep-
resent the opposite behavior than increased stiffness with
decreasing indentation depth seen in [1, 2].
The goal of this study is to examine whether the elastic
modulus is in fact depth dependent, or the change in H
and E with respect to indentation depth can be explained
by higher order displacement gradients. As a model mate-
rial, PDMS is chosen as strong increases in the elastic
modulus have been observed in the literature [1, 2] and
as it exhibits strong depth dependent hardness at a very
wide range of h [19, 20] and far above 200 nm. As indenta-
tion size effects in PDMS – in contrast to epoxy [27] – are
significant far above several microns, its surface rough-
ness will not affect the depth dependent deformation to
any appreciable extent.
2 Materials and methods
2.1 Methodology
To examine the relevance of higher order gradients (here
gradients in the rotations [21, 22]), indentation tests with
two different indenter tips – (i) a pyramidal three-sided
Berkovich tip which is geometrically self-similar and
(ii) a spherical tip with a curvature radius of 250 μm –
are performed with which the elastic modulus E is then
determined by various approaches of the literature. Here,
one needs to note that the hardness H is not suitable for
comparison because, in contrast to the Berkovich tip,
the spherical tip is not self-affine at different depths (see
Figure 1). Consequently, the spherical tip will yield differ-
ent hardness values at different h even for homogeneous
materials as the strain and stress fields – according to local
continuum mechanics – will not be similar/affine and will
not scale with h. Since the applied spherical tip has a rela-
tively large curvature radius, the rotation gradients and
other second order gradients in the displacements should
be small and more importantly, should be essentially
independent of the indentation depth [6, 7]. Thus, if the
dependence of E on h is related to higher order displace-
ment gradients, then E determined with the spherical
indenter tip will yield considerably different results than
with the Berkovich tip, which is examined here.
2.2 Sample preparation
The PDMS sample was fabricated with the Sylgard 184
components of Dow Corning (Midland, MI, USA) with
a 10% curing agent by volume fraction, where the base
agent has a weight density of 1.03 g/ml and the curing
agent has a weight density of 0.97 g/ml. The material was
prepared by mixing the base agent and the curing agent
thoroughly with a Speed Mixer DAC 150 FVE-K and then
placing the compound into a vacuum oven for 15min to
remove any air bubbles formed during mixing. The com-
pound was then poured into glass Petri dishes, cured at
150°C for 20min and then allowed to cure at room temper-
ature for 1 week. The final material is smooth, transparent
and free of visible defects.
2.3 Nanoindentation testing
The indentations were performed by an Agilent Nano XP
indenter system (Knoxville, TN, USA). Experiments were
execued at room temperature of about 22°C with a linear
relationship between time and indentation force (during
both loading and unloading) without any holding time.
Figure 1: Affine (left) and non-affine (right) indenter tips with respect to indentation depth.
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C.-S. Han etal.: Origin of indentation size effects in polymers 3
Force controlled loading was applied and corresponding
displacements of the indenter tip, h, after initial contact
with the sample, were determined by the system. Loading
times of 5, 20 and 80s were used with the same unload-
ing times in force controlled tests. In addition to the force
controlled experiments, nanoindentation tests were also
performed with a constant loading rate of 1 mN/s. Two
indenter tips were employed: (a) a three-sided Berkovich
tip and (b) a spherical tip with a radius of 250 μm.
3 Determination of elastic modulus
For metals, the effect of different indenter tip geometries
on indentation size effect has been studied in [6–8] where
size effects are usually associated with plastic deforma-
tion, plastic strain gradients, and corresponding geo-
metrically necessary dislocations. These concepts are not
applicable to the highly elastic PDMS. Due to differences
in the tip geometry, different theories should be applied
for the determination of the elastic moduli. The Hertzian
contact theory [28] and Johnson-Kendall-Roberts’ (JKR)
theory [29] are employed for obtaining the elastic moduli
with the spherical tip, whereas theories by Oliver and
Pharr [30] and Sneddon [31] are applied to determine the
elastic moduli of the PDMS sample using the Berkovich
tip. These theories will be briefly reviewed in the following
for the convenience of the reader.
3.1 Probing with spherical tip
3.1.1 Hertzian theory
The elastic contact of a sphere with a half space is consid-
ered in the Hertzian theory. Assuming linear elasticity and
a small ratio of the contact area to the curvature radius of
the spherical tip, the Hertzian contact theory states [28]:
3/2
2
r
4,
3
h
FRE
R

=

(1)
where F is the applied force, R the radius of the sphere,
h the probing or indentation depth which is more spe-
cifically defined as the displacement of the tip after
first contact with the sample, and Er the reduced elastic
modulus which is related to the elastic modulus of the
polymer, Ep, via:
22
pi
rp i
1- 1-
1,
vv
EE E
=+
(2)
where vp is the Poisson’s ratio of the polymer, and Ei and vi
are the elastic modulus and Poisson’s ratio of the indenter
tip, respectively. As Ei of the diamond tip is much higher
than Ep, the elastic modulus of polymer determined by the
Hertzian contact theory, Hertz
p,
E
can be obtained as:
()
Hert
z2
pp
r
1- ,EvE= (3)
without hardly any loss of accuracy.
3.1.2 Adhesion
The Hertzian contact theory does not account for adhe-
sion. For shallow indentation depths, however, adhesion
forces between tip and sample can be significant. Particu-
larly, for shallow indentation depths of compliant materi-
als considered here, adhesion can become very significant
[32–40], so that adhesion should be considered in the
determination of mechanical properties of soft polymers
such as PDMS at small indentation depths. The arguably
most common theories to account for adhesion between
bodies in contact are: (i) Derjaguin-Muller-Toporov [32]
(DMT), (ii) Maugis-Dugdale [33] (MD), and (iii) the already
mentioned JKR theory. To determine the appropriate
theory, a dimensionless parameter μ can be evaluated as
a criterion [35, 41]:
1/3
2
A
23
r0
,
RW
EZ
µ

=


(4)
where Z0 is the equilibrium separation distance of the
surfaces with respect to Lennard-Jones potential (usually
taken to be < 1 nm) and WA the work of adhesion. Accord-
ing to Cao etal. [36], the JKR theory is applicable for μ > 5,
DMT for μ < 0.1 and MD for the range in between. In our
case, μ is around 800 and therefore the JKR theory would
be the most appropriate. The elastic modulus applying
JKR, is obtained as [40]:
3
JKRHertz
pp
po po po po
21
31 /221/
F
E E
FF FF
FF
F

=+

++++

(5)
where Hertz
p
E
is determined as before [see Eqs. (1)–(3)] and
the second term in Eq. (5) is related to adhesion with the
pull-off force, Fpo, at the interface of the sample and tip. Fpo
can be related to the work of adhesion WA via:
Ap
o
2
,
3
WF
Rπ
=
(6)
which is consequently assumed to be independent of h.
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4 C.-S. Han etal.: Origin of indentation size effects in polymers
3.2 Indentation with Berkovich tip
3.2.1 Inelastic materials
For the determination of the elastic modulus of polymers,
the approach by Oliver and Pharr [30] is often applied
with the Berkovich tip where the reduced modulus Er is
determined as:
r
c
,
2
S
E
A
π
β
=
(7)
in which Er is related to Ep by
()
2
pp
r
1-E
vE
= [analogous
to Eq. (3)], S is the stiffness at initial unloading, Ac the
projected contact area, and β a constant which depends
on the tip geometry. For the stiffness determination, the
unloading is described as [30]:
()
=f
-,
m
FAhh (8)
where the parameters A, m, and hf are obtained by least
square fitting. Therefore, S is obtained by taking the
derivative of F with respect to h in Eq. (8) at the maximum
applied load.
The approach by Oliver and Pharr [30] was developed
for hard materials exhibiting plastic deformation and may
not be appropriate for the highly elastic material consid-
ered here [1] (see also below). As a comparison, a different
approach for the elastic modulus determination will be
briefly described in the following.
3.2.2 Elastic materials
For elastic materials and a conical tip, Sneddon’s theory
[31] for frictionless indentation can be applied where the
load-displacement relation is described with:
()
p2
2
p
2tan
,
1-
E
F h
v
α
π
=
(9)
where α = 70.3° is the half angle defining a cone equivalent
to a Berkovich tip. As PDMS is highly elastic – as shown
in Figure 2 where the inelastic indentation work (i.e., area
enclosed by the load displacement curve) is quite small
compared to the total indentation work – Eq.(9) should be
valid for PDMS if conventional local small strain elasticity
would apply.
3.3 Surface detection
The accurate determination of initial contact or detection
of the surface poses some challenges for soft materials
Figure 2: Typical load-displacement curve of polydimethylsiloxane
(PDMS) with the Berkovich tip.
and can result in significant overestimation of the elastic
modulus [42]. In the performed indentation experiments,
the surface is detected when the stiffness of the load-dis-
placement curve exceeds a prescribed value, KL, which is
called stiffness limit hereafter. Only the spherical tip is
considered here in detail. For the Berkovich indenter tip,
Alisafaei etal. [20] performed indentations using differ-
ent stiffness limit values on PDMS samples that have been
prepared in the same way as described in Section 2.2. A
stiffness limit of 50 N/m resulted in good accuracy using
a Berkovich tip and lower values of stiffness limit only
improved the accuracy insignificantly [20].
To assess the contact detection error, applying this
stiffness criterion with a spherical tip the Hertzian contact
theory of Subsection 3.1 is considered here for simplicity
where the contact stiffness of the sample in contact with
the sphere yields in view of Eq. (1):
r
2
dF
K
ER
h
dh
==
(10)
Consequently, the contact stiffness K is directly
proportional to h starting at K = 0 at h = 0 and increas-
ing with h as illustrated in Figure 3, where the average
elastic modulus of 20 indentations with 20s loading time
was used, resulting in a standard error of 3%. In view of
Figure3, the lower KL is chosen, the lower the error would
be, but the error of contact detection would never be zero
unless a non-feasible value of zero would be assigned
for KL (a low value for KL increases the likelihood of false
surface detection because of detecting noise or airflow in
lieu of the actual surface). Although an error in detecting
the surface cannot be avoided, this error should be small
compared to the considered h for a reliable determination
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C.-S. Han etal.: Origin of indentation size effects in polymers 5
of material properties. For instance, applying the common
stiffness limit KL = 200 N/m (usually used for metals),
Figure 3 would indicate that the indenter tip will travel
more than 4.5 μm into the sample before the contact stiff-
ness criterion detects the surface, which results in signifi-
cant overestimation of the elastic modulus for h < 100 μm.
4 Results
The elastic moduli of PDMS determined by Hertzian and
JKR theories are plotted with respect to hmax (the inden-
tation depth h at maximum load Fmax) in Figure 4, where
vp = 0.5 is assumed (elastomers like PDMS are known to be
incompressible). As can be seen in Figure 4, the elastic
moduli determined by both theories remain approxi-
mately constant from hmax = 25 down to about 5 μm. Below
about hmax = 5 μm, the elastic moduli determined with Hert-
zian and JKR theories start to divert as adhesion becomes
more significant. In Figure 4, each data point represents
one test and the minimum distance between two individ-
ual indents was 150 μm. For the evaluation of the elastic
modulus according to JKR’s theory, the pull-off force Fpo
was determined in such a way that the resulting elastic
modulus JK
R
p
E
[calculated from Eq. (5)] roughly stays
constant in hmax. Following this procedure, Fpo was deter-
mined as 60.3 μN, yielding the adhesion work WA for the
tip radius of R = 250 μm to be 51.2mJ/m2 [see Eq. (6)].
With respect to the determined WA, it should be noted
that the PDMS samples have been probed with the spheri-
cal tip at the same or similar locations multiple times. It
has been found that the fresh, unprobed PDMS samples
may exhibit different adhesion behavior than samples that
0 2 4 6 8 10
0
50
100
150
200
250
300
h (µm)
K (N/m)
Figure 3: Contact stiffness of polydimethylsiloxane (PDMS) based
on Hertzian contact theory as a function of indentation depth for a
spherical tip with 250 μm radius.
0 5 10 15 20 25 30
1.5
2
2.5
3
hmax (µm)
E (MPa)
Hertz
JKR
Figure 4: Comparison of elastic modulus obtained from Hertzian
and JKR theories for 5s loading time with the spherical tip.
0 5 10 15 20 25 30
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
hmax (µm)
E (MPa)
5 s
20 s
80 s
Figure 5: Elastic modulus of polydimethylsiloxane (PDMS) with
respect to hmax for three different loading times using the spherical
tip and Hertzian theory.
have been probed multiple times as in our case [43]. For
PDMS, however, a rather wide range of WA can be found
in the literature (ranging from 36.4 and 42mJ/m2 to 197.6
and 227 mJ/m2) [36, 40, 44, 45]. As can be seen in Figure 4,
adhesion becomes significant at shallow probing depths.
Subsequently, the material properties will be determined
in the following at hmax > 5 μm where adhesion is negligible
and the Hertzian theory can be applied.
The Hertz
p
E
vs. hmax data for three different loading times
(5 s, 20 s, 80 s) is shown in Figure 5 (data below 5 μm has
been omitted) where the elastic modulus is approximately
constant with hmax but slightly dependent on the loading
time, indicating the presence of little viscous deforma-
tion. To investigate the tip dependence of the determined
elastic moduli, the elastic moduli have been plotted with
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6 C.-S. Han etal.: Origin of indentation size effects in polymers
respect to hmax in Figure 6, applying both Berkovich and
spherical tips (KL = 50 N/m was applied for the indenta-
tions with the Berkovich indenter tip – lower values in
KL did not significantly improve contact detection for
h above 5 μm [20]). As can be seen, the elastic modulus
determined with the Berkovich tip is highly dependent on
hmax and increases sharply as hmax decreases, whereas the
elastic modulus obtained from the spherical tip, Hertz
p,
E
remains approximately constant in hmax. It should be
noted that the data from nanoindentation tests presented
in this study (including results depicted in Figure 6) have
been obtained without any holding time. While results
obtained with the theories of Hertz, JKR and Sneddon [31]
are not affected by holding time (as these methods are
applied to the loading curve), the elastic modulus deter-
mined according to Oliver and Pharr [30] with a Berkovich
tip (shown in Figure 6) can be affected by holding time
for some materials. However, this influence of the holding
time should not be significant for the highly elastic PDMS
considered here, as also a previous study [20] illustrates
that time/rate dependent behavior in PDMS is rather neg-
ligible for hmax > 1 μm, which is also in agreement with the
results in [1].
As the elastic modulus of PDMS is slightly time-
dependent in Figure 5, the effect of loading time on the
average elastic modulus of 10 indentations (with standard
error of < 1%) has been plotted in Figure 7. It is seen that
the average elastic modulus decreases slightly (6%) from
2.27MPa to 2.12MPa by increasing loading time from 5s
to 1000 s.
Another common property determined from nanoin-
dentation is the universal hardness HU defined as:
0 5 10 15 20 25 30 35
0
1
2
3
4
5
6
E (MPa)
Hertz
Sneddon
Oliver & Pharr
hmax (µm)
Figure 6: Elastic modulus of polydimethylsiloxane (PDMS) with
respect to hmax for 80s loading times using the spherical (Hertzian)
and Berkovich (Sneddon and Oliver and Pharr) indenter tips.
0 200 400 600 800 1000 1200
2
2.05
2.1
2.15
2.2
2.25
2.3
t (s)
E (MPa)
Figure 7: Elastic modulus of polydimethylsiloxane (PDMS) vs.
loading time determined with the spherical tip and Hertzian theory
(the upper and lower values are the maximum and minimum values
of 10 tests at indentation depths between about 15 μm and 25 μm,
the point in between is the average of these tests).
0 10 20 30 40
0
0.2
0.4
0.6
0.8
1
HU (MPa)
20 s
80 s
hmax (µm)
Figure 8: Universal hardness of polydimethylsiloxane (PDMS)
determined by the Berkovich tip for two different loading times.
U2,
F
Hch
= (11)
with a constant c = 26.43 for the Berkovich tip [46] evalu-
ated at h = hmax (more details on the indentation tests with
the Berkovich tip on PDMS are discussed by Alisafaei etal.
[20]). The universal hardness HU has been plotted against
hmax in Figure 8, where HU rises as hmax decreases. Interest-
ingly, the loading time dependence is not significant for
HU at the studied length scale range which is consistent
with results of Alisafaei at al. [20], where hardly any time
dependence of HU was observed for hmax > 1 μm. It should
be mentioned that nanoindentation testing on a soft mate-
rial (like PDMS) with a sharp tip is always affected by the
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C.-S. Han etal.: Origin of indentation size effects in polymers 7
surface detection error particularly at small h (even if a
very small stiffness limit KL is selected) [20].
For determining mechanical properties over a
range of h, a different but common application of the
load is to keep the loading rate constant rather than the
loading time. Hertz
p
E
with respect to hmax for a loading
rate of 1 mN/s is shown in Figure 9, where similar to
the previous experiments, E remains roughly constant
for a wide range of hmax (there is a slight increase in the
elastic modulus from E = 2.18MPa at hmax = 140.7 μm to
E = 2.27MPa at hmax = 20.71 μm).
5 Analysis and discussion
With the different trends of E with respect to h for the dif-
ferent probing tip geometries in Figure 6, the notion of
changing material properties through the depth cannot
be applied without causing contradictions. For metals,
Swadener etal. [6] also observed different trends in h for
the hardness applying Berkovich and spherical tips. In
their results, the hardness determined with the Berkovich
tip also increased with decreasing h, whereas the use of
spherical tips showed no increase in the hardness which
is similar to the present findings in PDMS. This phenom-
enon observed in metals was related to gradients in the
plastic strain and geometrically necessary dislocations [6]
which cannot be applied for polymers. Nikolov etal. [21]
and Han and Nikolov [22] considered the depth dependent
deformation of polymers as a result of an increase in rota-
tion gradient Xij (with decreasing h) which may be moti-
vated by the finite bending stiffness of polymer chains and
0 50 100 150
0
0.5
1
1.5
2
2.5
3
E (MPa)
hmax (µm)
Figure 9: Elastic modulus of polydimethylsiloxane (PDMS) deter-
mined by the spherical tip and Hertzian theory with the loading rate
of 1 mN/s.
their interactions. To account for these effects of rotation
gradients, the local elastic deformation energy density W e
is augmented by a nonlocal rotation gradient term Wc (as
proposed in [21, 22]) yielding the total elastic deformation
energy density W:
e1,
23
ij ijkl kl ij ij
K
WW WC
χ
εχ
χ=+
=+

e
(12)
where εij is the infinitesimal strain tensor, Cijkl the elas-
ticity tensor, and
K
 a material parameter that can be
micromechanically motivated [21]. The rotation gradient
χij in Eq.(12) is defined as
()
,,
1,
2
ij ij ji
χωω
=+
in which ωi is
the rotation vector, and i denotes partial derivatives with
respect to the ith coordinate. With χij in Eq. (12), second
order gradients in the displacements are introduced and
therefore render W to be non-local, with which length
scale-dependent deformation can be described. If rota-
tion gradients are present, additional energy should be
exerted which increases the forces to induce the corre-
sponding deformation. To illustrate the relevance of rota-
tion gradients to the different tips, Figure 10 illustrates
the change in rotation gradient with respect to indenta-
tion depth for spherical and pyramidal tips. Applying a
Berkovich tip, the rotation gradients are inversely pro-
portional to h and therefore with a decrease in h the rota-
tion gradients increase. For a spherical tip, however, the
rotation gradient is inversely proportional to the radius
of the tip and essentially independent of h. It should be
mentioned that similar to Eq. (12), where the total elastic
deformation energy density W is obtained by adding Wx to
the local elastic deformation energy density W e, the total
elastoplastic deformation energy density of a polymer can
be also determined by adding the nonlocal term Wx to the
local elastoplastic deformation energy density W e+W p
[27]. The interested reader is referred to [22], where the
higher order elasticity model Eq. (12) [21] is extended to
elastoplastic materials.
Figure 10: Rotation gradients for conical and spherical tips and
their dependence on indentation depth h.
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8 C.-S. Han etal.: Origin of indentation size effects in polymers
As the existing theories for the determination of the
elastic modulus of Section 3 do not consider rotation gra-
dients (or other second order gradients of displacements),
the determined elastic modulus will be over-predicted with
a tip that induces rotation gradients during testing. In view
of Figure 10, this is clearly the case for the Berkovich tip
with which h dependent E has been determined (see Figure
6). The dependence of E on h can be understood as a result
of the increased forces and corresponding increased stiff-
ness caused by the second term in Eq. (12). For the spheri-
cal tip, the rotation gradients will scale with 1/R and are
therefore not dependent on h, which is in agreement with
the presented experimental data. As h increases, the elastic
moduli determined with the Berkovich tip are approaching
the values determined by the spherical tip in Figure6. Since
the rotation gradients decrease with h, one can expect that
both tips will result in the same elastic modulus at high
values of h. Consequently, common theories [30, 31] for
pyramidal tips may not be accurate for the determination
of E of polymers at shallow indentation depths, as effects
of second order displacement gradients are not reflected in
these theories. Therefore, the effects of second order dis-
placement gradients should be taken into account for the
determination of the elastic modulus of polymers at small
indentation depths when a pyramidal tip is applied. Alter-
natively, the elastic modulus of a polymer can be obtained
by a spherical tip (with large radius) at large depths to
ensure that the obtained results are not affected by higher
order displacement gradient effects. It should also be noted
that the approach by Oliver and Pharr [30] has been devel-
oped for hard materials exhibiting plastic deformation and
may not be very accurate for a highly elastic material such
as PDMS yielding higher elasticity modulus values than
applying Sneddon’s theory which is in agreement with Lim
and Chaudhri [1].
Rearranging Eq. (9), the universal hardness – defined
in Eq. (11) – can be related to the elastic modulus as:
()
Up
2
p
2tan
,
1-
c
H E
v
α
π
=
(13)
where
()
απ 2
p
2tan 1-
cv
is constant and thus HU would
be proportional to Ep. This is also in agreement with
the experimental data, as both HU and Ep increase with
decreasing h (see Figures 6 and 8).
To account for these gradient effects, correspond-
ing to Eq. (12), Han and Nikolov [22] developed a depth-
dependent hardness model:

=+



U0
max
1,
c
H H h
(14)
where H0 is the macroscopic hardness to which H con-
verges at large h and cl is a length scale parameter which
is micromechanically motivated in [22] and can also be
related to the Frank elasticity constant of the polymeric
material. The model Eq. (14) predicts that the hardness is
inversely proportional to hmax and therefore, as can be seen
from Figure 8, the hardness increases as hmax decreases.
It should be noted that some polymeric materials do not
show indentation size effects even using a pyramidal tip
[24]. This can be related to the different molecular proper-
ties of different polymers [24] that affect the length scale
parameter cl in Eq. (14).
Concerning rate/time dependence in the presented
experiment, one should note that for the Berkovich tip,
according to the local continuum mechanics, the stress
and strain distributions should scale with h so that also
the strain rate fields should be affine and scale with h
when the loading time to reach the maximum load is the
same. For the spherical tip, the stress and strain fields are
essentially different at various h so that also the strain
rate fields are different with h as the spherical tip is not
affine with h (see Figure 1). Therefore, a comparison of
the rate effects with spherical and pyramidal tips will be
difficult. In the loading time ranges considered here, the
differences with respect to loading time/rates are argu-
ably smaller than the observed dependence in h when the
Berkovich tip is applied (see Figures 6 and 8). For signifi-
cantly lower or higher loading times, some of the results
presented here may not be valid, like negligible adhesion
with a 250 μm radius spherical tip for hmax < 5 μm.
6 Conclusions
Applying spherical and pyramidal tips, different theo-
ries have been applied to determine the elastic moduli in
PDMS from experiments at indentation depths between
5μm and 25 μm. For a spherical tip with a radius 250μm
and loading times of 5–80 s, comparisons of Hertzian and
JKR contact theories yielded the conclusion that adhesion
effects becomes significant at indentation depths of less
than 5 μm. It was found that with a pyramidal tip, the
elastic modulus increases with decreasing indentation
depth, while tests with the spherical tip yielded essen-
tially constant values for the elastic moduli independent
of indentation depth. The increase in the elastic modulus
with a pyramidal tip should be related to second order
displacement gradients, as these gradients remain con-
stant with a spherical tip, while applying a pyramidal
tip these gradients increase in magnitude with decreas-
ing indentation depth. In addition to PDMS, the effects of
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C.-S. Han etal.: Origin of indentation size effects in polymers 9
second order gradients of displacement is believed to be
of relevance for other polymers that exhibit length scale-
dependent deformation behavior.
Acknowledgments: The material of this article is based on
work supported by the U.S. National Science Foundation
under Grants No. 1102764 and No. 1126860.
References
[1] Lim YY, Chaudhri MM. Mech. Mater. 2006, 38, 1213–1228.
[2] Charitidis C. Indust. Eng. Chem. Res. 2011, 50, 565–570.
[3] Khun NW, Cheng HKF, Li L, Liu E. J. Polym. Eng. 2014, doi:
10.1515/polyeng-2013-0241.
[4] Huang CC, Wei MK, Harmon JP, Lee S. J. Mater. Res. 2012, 27,
2746–2751.
[5] Schilde C, Kwade A. J. Mater. Res. 2012, 27, 672–684.
[6] Swadener JG, George EP, Pharr GM. J. Mech. Phys. Sol. 2002,
50, 681–694.
[7] Tymiak NI, Kramer DE, Bahr DF, Wyrobek TJ, Gerberich WW.
ActaMater. 2001, 49, 1021–1034.
[8] Durst K, Göken M, Pharr GM. J. Phys. D: Appl. Phys. 2008, 41,
074005.
[9] Shen L, Liu T, Lv P. Polym. Test. 2005, 24, 746–749.
[10] Chong ACM, Lam DCC. J. Mater. Res. 1999, 14, 4103–4110.
[11] Zhang TY, Xu WH. J. Mater. Res. 2002, 17, 1715–1720.
[12] Tatiraju RVS, Han CS. J. Mech. Mater. Struct. 2010, 5, 277–288.
[13] Briscoe BJ, Fiori L, Pelillo E. J. Phys. D: Appl. Phys. 1998, 31,
2395–2405.
[14] Fleck NA, Muller GM, Ashby MF, Hutchinson JW. Acta Metal.
Mater. 1994, 42, 475–487.
[15] Nix WD, Gao H. J. Mech. Phys. Sol. 1998, 46, 411–425.
[16] Han CS, Ma A, Roters F, Raabe D. Int. J. Plast. 2007, 23,
690–710.
[17] Lee MG, Han CS. Comp. Mech. 2012, 49, 171–183.
[18] Lam DCC, Yang F, Chong ACM, Wang J, Tong P. J. Mech. Phys.
Sol. 2003, 51, 1477–1508.
[19] Wrucke A, Han CS, Majumdar P. J. Appl. Polym. Sci. 2013, 128,
258–264.
[20] Alisafaei F, Han CS, Sanei SHR. Polym. Test. 2013, 32, 1220–1228.
[21] Nikolov S, Han CS, Raabe D. Int. J. Sol. Struc. 2007, 44,
1582–1592. Corrigendum: Int. J. Sol. Struc. 2007, 44, 7713.
[22] Han CS, Nikolov S. J. Mater. Res. 2007, 22, 1662–1672.
[23] Garg N, Han CS. Comp. Mech. 2013, 52, 709–720.
[24] Han CS. Mater. Sci. Eng. A 2010, 527, 619–624.
[25] Kim S, Torkelson JM. Macromolecules 2011, 44, 4546–4553.
[26] Priestley RD, Ellison CJ, Broadbelt LJ, Torkelson JM. Science
2005, 309, 456–459.
[27] Alisafaei F, Han CS, Lakhera N. Polym. Test. 2014, 40, 70–78.
[28] Johnson KL. Contact Mechanics, Cambridge University Press,
Cambridge, UK, 1987.
[29] Johnson KL, Kendall K, Roberts AD. Proc. R. Soc. A 1971, 324,
301–313.
[30] Oliver WC, Pharr GM. J. Mater. Res. 1992, 7, 1564–1583.
[31] Sneddon IN. Int. J. Eng. Sci. 1965, 3, 47–57.
[32] Derjaguin BV, Muller VM, Toporov YP. J. Coll. Interf. Sci. 1975,
53, 314–326.
[33] Maugis D. J. Coll. Interf. Sci. 1992, 150, 243–269.
[34] Ebenstein DM. J. Mat. Res. 2011, 26, 1026–1035.
[35] Carrillo F, Gupta S, Balooch M, Marshall SJ, Marshall GW,
PruittL, Puttlitz CM. J. Mat. Res. 2005, 20, 2820–2830.
Erratumin J. Mat. Res. 21, 535.
[36] Cao YF, Yang DH, Soboyejoy W. J. Mater. Res. 2005, 20,
2004–2011.
[37] Chin P, McCullough RL, Wu WL. J. Adhesion 2005, 64, 145–160.
[38] Kröner E, Paretkar DR, McMeeting RM, Arzt E. J. Adhesion 2011,
87, 447–465.
[39] Ebenstein DM, Wahl KJ. J. Coll. Interf. Sci. 2006, 298, 652–662.
[40] Gupta S, Carillo F, Li C, Pruitt L, Puttlitz C. Mater. Lett. 2007, 61,
448–451.
[41] Muller VM, Yushchenko VS, Derjaguin BV. J. Coll. Interf. Sci.
1980, 77, 91–101.
[42] Kaufman JD, Klapperich CM. J. Mech. Beh. Biomed. Mater.
2009, 2, 312–317.
[43] Chin P. The characterization of polymer-solid adhesion. Ph.D.
dissertation, University of Delaware, Newark, Delaware, 1996.
[44] Chin P, McCullough RL, Wu WL. J. Adhesion 1997, 64, 145–160.
[45] Vaenkatesan V, Li Z, Vellinga WP, de Jeu WH. Polymer 2006, 47,
8317–8325.
[46] ISO14577-1 Metallic materials – Instrumented indentation test
for hardness and materials parameters – Part 1: test method.
International Organization for Standardization, 2002, Geneva,
Switzerland.
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