Full text of DOI:10.1557/JMR.2000.0169

Measurement of the loss tangent of low-density polyethylene
with a nanoindentation technique
J.L. Loubet
Ecole Central de Lyon, Laboratoire de Tribologie et Dynamique des Systemes, Centre National de la
Recherche Scientifique, BP 163, F-69131 Ecully Cedex, France
W.C. Oliver and B.N. Lucas
MTS Systems Corporation, Nano Instruments Innovation Center, 1001 Larson Drive,
Oak Ridge, Tennessee 37830
(Received 23 September 1999; accepted 28 February 2000)
This paper describes experimental measurements of the linear viscoelastic behavior of
the surface of low-density (LD) polyethylene in contact with a pyramidal Berkovich
diamond indenter. The experiments were carried out at two different temperatures, 15.9
and 27.2 °C, between frequencies of 0.1 and 800 Hz. Using the shift of the loss tangent
between the two temperatures at frequencies lower than 20 Hz and an Arrhenius
equation, an activation energy of 105 ± 2 kJ/mol was obtained. This value is in good
agreement with the bulk value of the ␣ relaxation of LD polyethylene reported in the
literature.
I. INTRODUCTION
energy for the indentation process can be deduced this
way. The value obtained for the small surface volume
will then be compared to the one obtained with conven-
tional bulk mechanical techniques.
The viscoelastic behavior of surfaces is fundamental in
fields dealing with interfaces, such as adhesion, polymer
blending, composite materials, and tribology. There has
been considerable recent effort devoted to investigating
the viscoelastic properties of thin films using surface
1
–5
II. EXPERIMENTAL
apparati (SFA)
and atomic force microscopy
6,7
(AFM). One of the most advanced techniques avail-
Two different samples were analyzed, fused silica and
LD polyethylene. Fused silica is a bulk specimen of a
few millimeters in thickness. It is an optically flat sub-
strate material. It is known to have low internal mechani-
able for the measurement of spatially resolved mechani-
cal properties of solids is the depth-sensing indentation
technique. Both static and dynamic mechanical proper-
ties can be measured using this technique. In addition to
hardness, techniques for measuring the elastic modulus
15
cal losses and is the sample usually taken as reference
9
in nanoindentation experiments. The LD polyethylene
8
–10
are well documented.
Recently, there has been
sample was a thin sheet of 100-␮m thickness from the
Goodfellow Corp. low-density polyethylene (LDPE) is
typically made by a high-pressure process using a free
radical initiator/catalyst and is a polymer with a high
degree of chain branching. The technical data sheet for
much progress in the development of techniques for
measuring time-dependent properties. The strain rate
sensitivities and apparent activation energies for inden-
11–13
tation creep
for a number of materials were accu-
3
rately measured. The results compared well with more
conventional measurements. Furthermore, recent results
indicate that both the storage and loss modulus can also
the material reported a density of 0.92 g/cm and crys-
tallinity of 50–60%. The tensile modulus for the material
was reported to be in the range of 0.1 to 0.3 GPa. No
polishing procedure was used on this material. The
roughness of samples, as taken, was below 0.05 ␮m peak
to peak. Before the experiments were conducted, both
samples were carefully cleaned with propanol.
A nanoindentation instrument, Nano Indenter௡ II,
(MTS Systems Corp., Oak Ridge, TN) at Oak Ridge
National Laboratories with a diamond Berkovich in-
denter was used for all experiments. The indenter was
situated in a temperature-controlled room, with the tem-
perature controllable to 0.5 °C, to prevent any response
of the depth-sensing instrumentation to thermal expan-
14
be measured. Specifically, it was shown that it is pos-
sible to measure the linear viscoelastic properties of an
unvulcanized, natural rubber using small, oscillatory mo-
tions of the indenter, yielding a shear field in the material
from which the complex elastic modulus (G*) of this
tacky polymer could be deduced.
The goal of this work is to extend the dynamic inden-
tation technique to the measurements of the loss tangent
of a nonadherent solid polymer, low-density (LD) poly-
ethylene. This study will be conducted as a function of
frequency and temperature to determine if an activation
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J.L. Loubet et al.: Measurement of the loss tangent of low-density polyethylene with a nanoindentation technique
sion of the machine or sample during the test. After a
change in room temperature, both the indenter and
samples were allowed to equilibrate to the new tempera-
ture for a period of 48 h. The actual temperature of the
indenter and the samples was measured with a platinum
resistance temperature detector which was placed in the
instrument box. The experiments were carried out at tem-
peratures of 15.9 and 27.2 °C. The temperature fluctua-
tion in 24 h was less than 0.05 °C. The thermal drift rate
during the experiments was less than 0.02 nm/s.
Several different indentation procedures were carried
out at each temperature. These consisted of constant
loading rate segments to the desired loads followed by
hold segments at constant load. Six different loads (P),
divided in two groups were investigated (0.5, 1, 1.5 mN
and 5, 10, 15 mN). The loading rates were 15 ␮N/s for
the first group and 150 ␮N/s for the second group. Dur-
ing the hold segment, after a settling time of 2 min to
allow for rapid, time-dependent processes to occur,
harmonic measurements were made. The experiments
were done 3 to 5 times each in a way to look at the
reproducibility.
where h and F are respectively the amplitude of the har-
monic displacement and the amplitude of the harmonic
force, (K) is equal to (K ) the stiffness of the column
A
support springs (≈23.7 N/m), m is the mass of the moving
part (≈6.53 g), ␻ is the frequency of oscillation (rad/s),
and (C) is equal to (C ) the damping coefficient due to
A
the double gap in the displacement sensing capacitive
gage (≈5.24 N s/m at 15.9 °C and ≈5.46 N s/m at
27.2 °C). Since the damping factor is high, the system is
over-critically damped. This means that no resonant peak
is observed for noncontact experiments.
When a contact is made between the pyramidal Berk-
ovich diamond indenter and a sample, the stiffness and
the damping coefficient involved in Eq. (1) must be re-
written. K now becomes the sum of the stiffness of the
contact, (S), and the stiffness of the column support
springs, (KA):
−1
−1 −1
K ס (S + K A)
,
(2)
and (C) is now the sum of the clamping coefficient of the
contact (C ) and the one of the apparatus (C ):
C
A
C ס C + CA
.
(2Ј)
C
A fast Fourier transform (FFT) analyzer (Z-channel
FFT Network signal analyzer, SR 780, Stanford Re-
search Systems, Sunnyvale, CA) was used to measure the
mechanical transfer function of the contact, that is to say
the ratio of the output spectrum of the contact, the dis-
placement (h*), to the input spectrum, the force (F*).
Both the real and imaginary parts of the transfer function
For a specimen with negligible internal damping in the
range of frequencies investigated (C ≈ 0), as the inden-
A
tation depth increases, the stiffness of the contact in-
creases with no increase in damping. The system then
goes from being overdamped to being underdamped, and
a resonant peak can be observed.
(
h*/F*) as a function of frequency were recorded. The
The maximum amplitude ratio occurs when the fre-
quency (␻, rad/s) is
FFT analyzer was set up in two different ways. The first
one, for low frequencies between 0.1 and 100 Hz, is a
swept sine measurement. The input signal to the contact
2
K
m
C
␻
=
−
.
(3)
ͱ
(F*) is a sine wave whose frequency is swept over the
2
2
m
frequency range of interest. The swept sine technique
only looks at one frequency at a time. Five points per
decade were recorded. These are referred to as swept
experiments. The second one, for high frequencies be-
tween 1 and 801 Hz, is a high-resolution periodic chirp
The maximum amplitude ratio is
h
1
ͩ
ͪ
=
.
(4)
F
2
max
K
C
C
−
ͱ
(800 lines). The input signal to the contact (F*) has a flat
2
m
2
m
spectrum. All the frequencies in the defined window are
analyzed in the same time. These are referred to as FFT
experiments.
Figure 1 shows the ratio of the amplitude of harmonic
displacement over the amplitude of the harmonic force
(h/F) as a function of frequency for noncontact experi-
ments and for contact experiments at different loads (P)
between a pyramidal Berkovich diamond indenter and
the two different samples LD polyethylene and fused
silica. Open symbols correspond to swept-sine records,
while lines correspond to FFT records. One can notice
the good correlation for the two sets of data between 1
and 100 Hz. Figure 1 displays that at low frequencies,
below 50 Hz, (h/F) is constant for fused silica and is
decreasing as a function of frequency for LD polyethyl-
ene. This decrease is the signature of the viscoelastic
behavior of the LD polyethylene sample. One can notice
III. RESULTS AND DISCUSSION
At each temperature, an analysis of the behavior of the
apparatus is made. This is called a noncontact experi-
ment. It is found that the system behaves like a simple
damped harmonic oscillator obeying the following
equation:
h
1
ͩ
ͪ
=
,
(1)
F
2 2
2
͌
͑K − m␻ ͒ + ͑C␻͒
1196
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J.L. Loubet et al.: Measurement of the loss tangent of low-density polyethylene with a nanoindentation technique
that the amplitude of the resonance peak is both depen-
dent upon the applied load and the sample material. The
dashed line at the top of the records, LD polyethylene
frequency, the damped resonance frequency, and at one
indentation depth, the depth at which the system is in
resonance.
(
P ס 5 mN), is a FFT record of a contact made on fused
Knowing the stiffness and the damping coefficient of
the apparatus, it is possible to have access to the stiffness
and to the damping coefficient of the contact and, there-
fore, the loss tangent of the material. The loss tangent is
defined as the ratio of the loss modulus (GЉ) to the stor-
age modulus (GЈ) which is equivalent to
silica. It has the same contact stiffness as the contact
made on LD polyethylene at 5 mN. One can see that
fused silica contact displays a higher resonant peak than
the LD polyethylene contact. This indicates that the con-
tact on LD polyethylene is more damped than the contact
on fused silica [Eq. (4)].
From the analysis of these FFT records, “best fit” stiff-
ness (K) and damping (C␻) can be determined using
Eq. (1) and (3). These values of damping (C␻) are plotted
as a function of the corresponding stiffness in Fig. 2. The
continuous line is the prediction of the behavior of
samples with no internal damping. It is drawn by using
Eq. 3 for (␻) and assuming that the damping coefficient
is the same as the one found for the noncontact experi-
ment at the same temperature. One can see that (i) the
fused silica sample is in good agreement with the theo-
retical prediction of samples with no internal damping,
C ␻
C
tan ␦ =
.
(5)
S
(ii) the LD polyethylene sample introduces an additional
damping 10% greater than the theoretical prediction of
samples with no internal damping, and (iii) the good
reproducibility of the method, the scatter is less than 4%.
The values of these best fit parameters are dominated by
the data very near the resonance peak. An important
point is that even if a priori (C␻) and (K) determined this
way mix what occurs as a function of indentation depth
and as a function of frequency, it is still a very accurate
way of measuring the damping of the material at one
FIG. 2. A log–log plot of the damping (C␻) and the stiffness issued of
FFT records analysis [Eq. (1); see text]. The damping displayed is the
sum of the damping of the contact and the damping of the apparatus.
It is shown that (i) fused silica sample is in good agreement with the
theoretical prediction of samples with no damping and (ii) LD poly-
ethylene sample introduces an additional damping 10% greater than
the theoretical prediction of samples with no internal damping.
FIG. 1. A log–log plot for the ratio of the amplitude of harmonic
displacement (h) over the amplitude of the harmonic force (F) as a
function of frequency for noncontact experiment and for contact ex-
periments at different loads (P) between a pyramidal Berkovitch dia-
mond indenter and two different samples, LD polyethylene and fused
silica. Open symbols correspond to swept records. Lines correspond to
FFT records. The dashed line at the top of the LD polyethylene
FIG. 3. Evolution of tan ␦ in function of frequency at two tempera-
tures [␪ ס 15.9 °C (circle symbols) and ␪ ס 27.2 °C (square sym-
bols)] for LD polyethylene. The open symbols correspond to
experiments made at a contact load of 1 mN between a Berkovich
diamond indenter and the sample. They are issued for the swept re-
cords. The filled symbols are issued for the FFT records analysis for
contact loads of 1, 1.5, 5, 10, and 15 mN.
(P ס 5 mN) record is a FFT record of a contact made on fused silica.
These two contacts have the same contact stiffness.
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J.L. Loubet et al.: Measurement of the loss tangent of low-density polyethylene with a nanoindentation technique
Figure 3 shows the evolution of the loss tangent as a
function of frequency at two temperatures [␪ ס 15.9 °C
material. More extensive work including determination
of conservative modulus and loss modulus of the surface
of LD polyethylene with the help of this surface dynamic
viscoelastic technique are currently underway.
(circle symbols) and ␪ ס 27.2 °C (square symbols)] for
LD polyethylene. The open symbols correspond to ex-
periments made at a contact load of 1 mN—that is to say
roughly 10-␮m-deep experiments. They are issued the
results from the swept experiments. In this case the stiff-
ACKNOWLEDGMENTS
ness, (S), and the damping, (C ␻), of the contact are
given by
This research was sponsored in part by the Division of
Materials Sciences, United States Department of Energy,
through the ShaRE Program under Contract No. DE-
AC05-76OR00033 between the United States Depart-
ment of Energy and Oak Ridge Associated Universities
and in part by the Ford Motor Company. We thank Prof.
G.M. Pharr, Prof. C.J. McHargue, and Dr. A.C. Rama-
murthy for helpful discussions and advice.
c
F*
2
S = Reͩ ͪ − ͑K − m␻ ͒
,
(6)
A
h*
and
F*
C ␻ = Imgͩ ͪ − C ␻
,
(6Ј)
c
A
h*
where Re(F*/h*) and Img(F*/h*) are the real and the
imaginary part of the inverse of the transfer function
acquired with this contact, respectively. The filled sym-
bols are assigned to the FFT records analysis for contact
loads of 1, 1.5, 5, 10, and 15 mN. For experiments made
at a contact load of 0.5 mN the peak is not big enough to
perform the global analysis presented previously. The
shape of the loss tangent curve is consistent with data
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Q 1
1
1
n͑a ͒ =
ͩ
−
ͪ
.
(7)
T
R T
T2
1
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16,17
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1
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(
IV. CONCLUSION
1
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indent, using an Arrhenius equation, an activation energy
of 105 kJ/mol can be obtained. This value agrees very
well with the bulk value of the ␣ relaxation of LD poly-
ethylene reported in the literature. This shows that the
few micrometers of the surface of this sample behave,
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1
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1
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1198
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