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Basic characteristics of uniaxial extension rheology:
Comparing monodisperse and bidisperse polymer melts
Yangyang Wang, Shiwang Cheng, and Shi-Qing Wanga)
Department of Polymer Science and Maurice Morton Institute of
Polymer Science, The
University of Akron, Akron, Ohio 44325-3909
Synopsis
We have carried out continuous and step uniaxial extension
experiments on monodisperse
and bidisperse styrene-butadiene random copolymers (SBR) to
demonstrate that their
nonlinear rheological behavior can be understood in terms of
yielding through
disintegration of the chain entanglement network and rubber-like
rupture via
non-Gaussian chain stretching leading to scission. In continuous
extension, the sample
with bidisperse molecular weight distribution showed greater
resistance, due to
double-networking, against the yielding-initiated failure. An
introduction of 20 % high
molecular weight (106 g/mol) SBR to a SBR matrix (2.4×10
5 g/mol) could postpone the
onset of non-uniform extension by as much as two Hencky strain
units. In step
extension, the bidisperse blends were also found to be more
resistant to elastic breakup
than the monodisperse matrix SBR. Rupture in both monodisperse
and bidisperse SBR
samples occurs when the finite chain extensibility is reached at
sufficiently high rates. It
is important to point out here that these results along with the
concept of yielding allow
us to clarify the concept of strain hardening in extensional
rheology of entangled
polymers for the first time.
a) Electronic mail: [email protected]
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I. INTRODUCTION
Extensional rheological behavior of entangled polymer melts has
been studied for
several decades. A clear understanding of failure behavior and
material cohesive
strength in extensional deformation is important for material
designs. In the 1970s and
80s, Vinogradov and coworkers carried out extensive studies on
the failure behavior of
monodisperse entangled polymer melts in uniaxial extension
[Vinogradov (1975);
Vinogradov et al. (1975a, 1975b); Vinogradov (1977); Vinogradov
and Malkin (1980);
Malkin and Vinogradov (1985)]. During the same period, the
failure behavior of
various commercial polydisperse polymers were also studied by
several teams [Takaki
and Bogue (1975); Ide and White (1977, 1978); Pearson and
Connelly (1982)]. It has
been realized that polydispersity in the molecular weight
distribution significantly affects
the failure behavior in uniaxial extension [Takaki and Bogue
(1975)]. Since most of the
commercial synthetic polymers are polydisperse, a general
understanding of the influence
of polydispersity on failure behavior is of great industrial
value.
A first step towards a better understanding of the molecular
weight distribution
effect is to study bimodal blends where the behavior and
dynamics of each individual
component can be readily established. Most of the previous
investigations of such
systems [Minegishi et al. (2001); Ye et al. (2003); Nielsen et
al. (2006)] focused on the
"strain-hardening" characteristic during uniform extension,
leaving the failure phenomena
largely unexplored.
Our recent studies on a series of monodisperse linear SBR melts
[Wang et al.
(2007b); Wang and Wang (2008)] have suggested that certain
failure behavior of highly
entangled polymers in rapid uniaxial extension is analogous to
the shear inhomogeneity
revealed by particle-tracking velocimetry [Tapadia and Wang
(2006); Wang (2007)], and
can be understood in terms of the disintegration of the chain
entanglement network.
Specifically, in both startup shear and extension, entangled
polymers exhibit the same
scaling characteristics associated with the yield point in the
elastic deformation regime
[Wang et al. (2007b)], which is the point where the shear stress
and engineering stress
peak. Beyond the yield point, structural inhomogeneity develops
in the form of
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3
non-uniform spatial distribution of chain entanglement. The
purpose of this study is to
demonstrate that the notion of yielding can be readily applied
to explain some basic
rheological characteristics of entangled bimodal blends in
uniaxial extension, including
their failure behaviors. Moreover, this concept of yielding
allows us to clarify when
strain hardening really happens in uniaxial extension of
entangled polymers.
II. MATERIALS AND METHODS
The bimodal blends in the current investigation were made from
three of
near-monodisperse linear styrene-butadiene random copolymers
(SBR) provided by Dr.
Xiaorong Wang at the Bridgestone Americas Center for Research
and Technology. The
molecular characteristics of three SBR melts can be derived from
small amplitude
oscillatory shear measurements using a Physica MCR 301
rotational rheometer equipped
with 25mm parallel plates. Two samples involve 20 wt. % of SBR1M
in two different
"matrices" of SBR240K and SBR70K respectively, and are labeled
as 240K/1M (80:20)
and 70K/1M (80:20) respectively. A third mixture has the
composition of SBR240K
and SBR1M given by 240K/1M (90:10). The small amplitude
oscillatory measurements
of the bimodal blends are shown in Fig. 1, from which some basic
information is
obtained as shown in Table I, where the equilibrium melt shear
modulus Geq is
determined as the value of G' at the frequency, at which G"
shows a minimum. The
Rouse relaxation time τR of the sample was estimated as τ/3Z,
according to the tube
model [Doi and Edwards (1986)]. Due to their slight
microstructure and polydispersity
differences, the terminal relaxation time τ of these samples
does not scale with the
molecular weight M as: τ ~ M3.4
.
TABLE I. The Molecular Characteristics of SBR Melts
Sample Mn (kg/mol) Mw/Mn Geq (MPa) Z τ (s) τR (s)
SBR70K 70 1.05 0.74 25 0.67 0.0089
SBR240K 241 1.10 0.82 98 34 0.12
SBR1M 1068 1.23 0.85 510 11000 7.2
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103
104
105
106
0.01 0.1 1 10 100
240K/1M (80:20) G'
240K/1M (80:20) G"
70K/1M (80:20) G'
70K/1M (80:20) G"
Sto
rage/
loss
modulu
s (
Pa)
Angular frequency (rad/s)
0.002
Figure 1 Small amplitude oscillatory shear measurements of the
bimodal blends at room
temperature. The storage modulus and the loss modulus are
represented by the open and filled
symbols respectively.
The uniaxial extension experiments were carried out using a
first generation SER
fixture [Sentmanat (2004); Sentmanat et al. (2005)] mounted on
the Physica MCR 301
rotational rheometer. The failure behaviors of pure melts and
blends were investigated
in two different types of testing. One was the failure during
startup continuous uniaxial
extension at a constant Hencky strain rate . The other was the
failure during step
extension where the sample was suddenly subjected to a certain
amount of strain. The
specimen failure is video-recorded to allow post-experiment
analyses.
III. RESULTS
A. Continuous Extension
Two types of failure mode were observed during startup
continuous extension of the
pure SBR melts. During low-rate extension of SBR240K and SBR1M
melts the sample
breakage is found to initiate from non-uniform extension. At
sufficiently high rates
rubber-like rupture was found for SBR1M when the sample broke
sharply and the
original cross-sectional dimensions returned after rupture,
indicating that no part of the
specimen suffered much irrecoverable deformation (i.e., yielding
or flow)
Fig. 2(a) and Fig. 3 present the stress-strain curves of SBR240K
and SBR1M melts
at various rates. The experiments ending in rupture are
represented by open symbols.
In filled symbols we see that the engineering stress engr always
exhibits a maximum such
that (engr)max is linearly proportional to max. This
characteristic has been reported
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5
before and suggested to be a signature of yielding [Wang et al.
(2007b), Wang and Wang
(2008)], beyond which chains mutually slide past one another
[Wang and Wang (2009)].
The engineering stress is also presented as a function of time
in Fig. 2(b) for
SBR240K. It is easy to see that the previously reported scaling
behavior in the elastic
deformation regime σengr ~ t1/2
is valid for SBR240K as well [Wang et al. (2007b)].
The end of each stress-strain curve corresponds to the onset of
non-uniform extension by
visual inspection, i.e., by examining the recorded images of the
stretched specimen. On
the other hand, at high rates rupture of SBR1M truncates the
monotonic rise in the
engineering stress engr. In other words, engr never had a chance
to decline before
rupture. Note that the data at 1 s-1
in Fig. 3 approaches the borderline between yielding
and rupture.
0
0.5
1
1.5
2
2.5
3
3.5
4
0 1 2 3 4
en
gr (
MP
a)
Hencky strain,
15 s-110 s
-16.0 s-1
3.0 s-1
1.0 s-1
0.3 s-1
0.1 s-1
SBR240K
max
(a)
0.01
0.1
1
10
0.01 0.1 1 10Time (s)
(b)
15 s-1
10 s-1
6.0 s-1
3.0 s-1
1.0 s-1
SBR240K
0.3 s-1 0.1 s
-1
-0.5
en
gr (
MP
a)
Figure 2 Engineering stress as a function of (a) Hencky strain
and (b) time for SBR240K at
various strain rates. The straight dashed line in (a) provides
an indication of how the
engineering stress increases more weakly than linearly with the
Hencky strain.
0
2.5
5
7.5
10
0 1 2 3 4
6.0 s-1
3.0 s-1
2.0 s-1
1.0 s-1
0.6 s-1
0.3 s-1
Hencky strain
en
gr
(M
Pa)
SBR1M
neo-Hookean
Figure 3 Engineering stress as a function of Hencky strain for
SBR1M at various strain rates.
The stretching, ending in yielding-initiated failure and
rupture, is represented by filled and open
symbols respectively. The dotted curve is the neo-Hookean
formula of Eq. (1) with Geq = 0.85
MPa from Table 1, showing exponential growth with the Hencky
strain .
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The engineering stress-strain curves for startup continuous
stretching of bimodal
SBR blends are shown in Fig. 4, Fig. 5 and Fig. 6 respectively.
The shape of the
engineering stress-strain curve for bimodal SBR blends is
qualitatively different from that
of the pure SBR melts. At various rates, all three blends, i.e.,
240K/1M (90:10),
240K/1M (80:20) and 70K/1M (80:20), exhibited an engineering
stress-strain curve
containing two maxima and only showed non-uniform extension
beyond the second
maximum. At sufficiently high rates, the uniaxial extension of
240K/1M (80:20) and
70K/1M (80:20) was terminated abruptly without displaying a
second maximum in the
engineering stress when the specimens ruptured. The rate
dependence of strains at
yielding-initiated failure and rupture for SBR240K, 240K/1M
(90:10), and 240K/1M
(80:20) is shown in Fig. 7. The onset of both failures in the
bimodal blends is
significantly postponed at the highest four rates relative to
those of the pure components.
0
0.5
1
1.5
2
0 1 2 3 4 5
15 s-1
10 s-1
6.0 s-1
3.0 s-1
1.0 s-1
0.3 s-1
0.1 s-1
240K/1M (90:10)
Hencky strain
en
gr (
MP
a)
Figure 4 Engineering stress as a function of Hencky strain at
various strain rates for the
240K/1M (90:10) blend.
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5
240K/1M (80:20) 15 s-1
10 s-1
6.0 s-1
3.0 s-1
2.0 s-1
1.0 s-1
0.6 s-1
0.3 s-1
Hencky strain
en
gr (
MP
a)
Figure 5 Engineering stress as a function of Hencky strain at
various strain rates for the
240K/1M (80:20) blend. The stretching, yielding-initiated
failure and rupture, is represented by
filled and open symbols respectively.
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0
0.5
1
1.5
2
0 1 2 3 4 5
70K/1M (80:20)
15 s-1
10 s-1
6.0 s-1
3.0 s-1
2.0 s-1
1.0 s-1
0.6 s-1
Hencky strain
en
gr (
MP
a)
Figure 6 Engineering stress as a function of Hencky strain at
various strain rates for the
70K/1M (80:20) blend. The stretching, yielding-initiated failure
and rupture, is represented by
filled and open symbols respectively.
1
10
0.1 1 10 100
SBR240K
240K/1M (90:10)
240K/1M (80:20)
Elastic rupture (80:20)
Elastic rupture (0:100)
Fail
ure
Hencky
str
ain
Strain rate (1/s)
Figure 7 Failure Hencky strain as a function of applied strain
rate for SBR(240K) and the
two bimodal blends. The solid symbols represent ductile failure
through yielding. The
dashed lines show the borderline between viscoelastic and
elastic regimes for the two blends.
The open symbols denote strains at rupture for the pure SBR1M
(triangles) and the 240K/1M
(80:20) blend (diamond).
B. Step Extension
The engineering stress as a function of time during and after
step extension of the
pure SBR1M melt is first presented in Fig. 8 at three different
amplitudes all high enough
to produce elastic yielding, i.e., failure during relaxation.
The induction times, before
which the specimens appear intact, are all longer than the Rouse
relaxation time R of 7.2
s. After this period, one portion of the specimen started to
shrink in its dimensions,
leading to the sample breakage. Fig. 9(a) shows the elastic
breakup behavior of not only
the pure SBR240K in solid symbols but also the bimodal SBR
blends of 240K/1M
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(90:10). Here the critical Hencky strain for the breakup is just
beyond 0.6 for the pure
SBR240K, which is slightly lower than the vinyl-rich SBR melts
previously reported
[Wang et al. (2007b)]. The critical strain for SBR 240K/1M
(90:10) is apparently the
same as the pure SBR240K melt, although the induction time to
break is markedly longer.
0.01 0.1 1 10 100
1.20.90.75
en
gr
(M
Pa)
Applied rate: 15 s-1
strain
SBR1M10
0.1
1
0.01
Time (s)
Figure 8 Engineering stress as a function of time in step
relaxation experiments for the
SBR240K melt and the 240K/1M (90:10) blend.
0.01
0.1
1
10
0.01 0.1 1 10 100
0.9
0.75
0.9
0.75
240K
240K/1M
(90:10)
Applied rate: 15 s-1
Time (s)
strain
en
gr
(M
Pa)
(a)
0.1
1
0.01 0.1 1 10 100 1000
1.2
0.9
0.6
240K/1M (80:20)
Strain
Time (s)
Applied rate: 15 s-1
No breakup
Elastic breakup
2
en
gr (
MP
a)
(b)
Figure 9 Engineering stress as a function of time in step
relaxation experiments for (a) the
SBR240K melt and the 240K/1M (90:10) blend, and (b) 240K/1M
(80:20) blend.
In contrast, the critical Hencky strain c for SBR 240K/1M
(80:20) shifted to c = 0.9,
significantly higher than c ~ 0.6 for the pure SBR240K. Thus,
the bimodal blends are
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also more resistant to breakup after a step extension than the
corresponding pure
monodisperse matrix.
IV. DISCUSSION
A. Continuous Extension
A.1 Yielding and rupture of Monodisperse Melts
Yielding and non-uniform extension
During startup deformation at rates involving Weissenberg number
Wi = , after
the initial elastic deformation, the entanglement network is
forced to disintegrate when an
imbalance emerges between the elastic retraction force and
intermolecular locking force
[Wang et al. (2007a), Wang and Wang (2009)]. In other words, the
tensile force
represented by engr is expected to display a maximum. The
present study confirms that
pure monodisperse melts exhibit yielding in constant-rate
uniaxial extension. The
yielding is signified by the emergence of a maximum in the
engineering stress as shown
in Fig. 2(a). Consistent with the reported scaling behavior of
yielding in simple shear
[Boukany et al. (2009a)], the solid symbols of Fig. 2(a) and
Fig. 3 also show a shift of the
yielding strain (corresponding to the engineering stress
maximum) εy to higher values
upon increasing the applied rate. The broad maxima at lower
rates in Fig. 3 are due to
the large polydispersity in the molecular weight distribution of
SBR1M.
There is strong evidence that the observed sample necking is not
due to an elastic
Considère-type instability, advocated recently by Hassager and
coworkers [Lyhne et al.
(2009); Hassager et al. (2010)]: When set to a stress-free state
after the engineering
stress maximum, the specimen cannot return to its original
dimensions [Wang and Wang
(2008)]. The plastic deformation and the corresponding decline
in the engineering
stress are a result of the entanglement network disintegration
in presence of the
continuing extension. Apparently, the eventual outcome is, not
surprisingly, that one
portion of the stretched specimen reaches a point of network
disintegration and
irreversible deformation before the rest of the specimen does,
resulting in necking or lack
of uniform extension. It is interesting to note from Fig. 2(a)
that the specimen can
extend uniformly well after the yield point for the group of
data involving the higher rates
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10
of 6.0 s-1
and above. This dividing line coincides with the boundary
between the
viscoelastic and elastic deformation regimes, given by the
condition of the Rouse
Weissenberg number WiRouse = τR ~ 1 [Wang and Wang (2008)]. In
the viscoelastic
regime, the entanglement network could undergo irreversible
deformation even before
reaching the engineering stress maximum, so that the specimen is
more ready to fall apart
beyond the stress maximum. Conversely, in the elastic
deformation regime, the network
would be largely intact and fully recoverable up to the yield
point (engineering stress
maximum) [Wang and Wang (2008)]. As a consequence, the onset of
structural collapse
is delayed markedly beyond the yield point.
We close this subsection by emphasizing that nearly monodisperse
linear melts such
as SBR240K only show significant strain softening in the rate
range bounded by WiRouse <
10. Fig. 2(a) shows, consistent with previous analyses [Wang et
al. (2007b), Wang and
Wang (2008)], that engr initially grows with the Hencky strain =
t linearly. This may
be expected because even the neo-Hookean model prescribes such a
linear response at
small extensions:
engr = G(1/2),
which has a limiting form of engr ~ 3Gfor
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III and IV. However, our recent study [Wang and Wang (2010a)]
has revealed that for
highly entangled polymers, such a yielding-to-rupture transition
is located in the middle
of the rubbery plateau region, and does not involve any
transformation to the glassy state.
In other words, it would be misleading to associate the rupture
with the term “glass-like
zone”, as classified in the Malkin-Petrie master curve.
The ductile failure arises from yielding of the entanglement
network, during which
linear chains mutually slide past one another [Wang et al.
(2007a); Wang and Wang
(2009)], whereas the origin of the rupture could have something
to do with chain scission.
At the very high rates, the mode of yielding, i.e., chain mutual
sliding, is no longer
available when sufficient tension in the chains can build up to
cause chain scission. In
other words, chain scission would occur before chains have
reached the point of force
imbalance [Wang and Wang (2009) to allow mutual sliding. This
appears to occur in
SBR1M at an extensional rate = 2.0 s-1
and above as shown in Fig. 3 where the tensile
force (i.e., the engineering stress) rises monotonically. This
rapid rise in the engineering
stress at higher rates is plausibly due to non-Gaussian
stretching [Wang and Wang
(2010a)]. In the explored rate range, SBR240K cannot be
stretched fast enough at room
temperature to produce the non-Gaussian stretching that leads to
the upward rise in
tensile force and ends in rupture via chain scission. In
passing, we note that some
alternative theoretical explanation for specimen failure during
uniaxial extension has
been made in the literature [Joshi and Denn (2003, 2004)]. We do
not elaborate on this
theoretical study because it appears to have oversimplified the
process leading to the
yielding-initiated failure.
A.2 Failure Behaviors of the Bimodal Blends
Effect of the high molecular weight component on yielding
We have seen that the incorporation of a small amount of SBR1M
to SBR240K
significantly delays the sample failure. For example, at = 6.0
s-1
, the SBR240K melt
would yield at ≈ 1.4 and eventually undergo ductile failure at ≈
2.6. The presence of
10% or 20% 1M SBR postpones the onset of the ductile failure, as
shown in Fig. 4 and
Fig. 5, to much higher strains. A closer examination shows that
there exists a critical
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rate, beyond which the strength of the blends is greatly
improved. As evident from Fig.
7, the failure behavior of the 240K/1M (90:10) blend is not so
different until reaching c
~ 6.0 s-1
. Similarly, c ~ 2.0 s-1
can be found for the 240K/1M (80:20) blend.
The relaxation times of the two individual components in the
240K/1M blends are
widely separated. As indicated in a preceding subsection, the
Rouse relaxation times of
240K and 1M are different by a factor of 60, whereas the
difference in their terminal
relaxation times is even larger, approaching 330. The wide
separation of relaxation
times causes the high and the low molecular weight components to
stay in different
deformation regimes under a given applied strain rate. In other
words, the incorporated
SBR1M chains actually formed a second entanglement network,
which is highly
plausible given the huge separation of its relaxation time scale
from that of the "matrix"
SBR240K. Specifically, the number of entanglements per chain in
such a second
network can be estimated according to the empirical scaling law
for the entanglement
molecular weight: Me() ~ Me,0-1.3
[Yang et al. (1999)]. At the volume fractions of 0.1
and 0.2, the second network involves respectively 32 and 73
entanglement points per
SBR1M chain.
Our previous study on the scaling characteristics of yielding
[Wang and Wang
(2009)] has revealed, as also demonstrated by Fig. 2(a), that
the onset of yielding moves
to higher strains with increasing Rouse Weissenberg Number
WiRouse. Thus, at a given
rate, the second entanglement network formed by the SBR1M would
yield at a much
higher strain. The blend could retain its integrity well beyond
the first engineering
stress maximum where the (first) matrix network has collapsed.
This evidently
indicates that the second entanglement network made of SBR1M did
not give in until
much higher strains. Since the second network involves much
greater entanglement
spacing, the non-Gaussian stretching sets in at significantly
higher strains around a
Hencky strain of 3.0 as read from Fig. 5 [Wang and Wang (2010b)]
than a Hencky strain
of 2.0 read from Fig. 3 for the neat SBR1M. More discussion on
non-Gaussian
stretching is deferred to IV.D.4.
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13
Rupture
With only 10% SBR1M, the first blend did not undergo rupture in
the explored rate
range as shown in Fig. 4. It appears that at these rates the
second network associated
with 10% SBR1M never got extended sufficiently to suffer chain
scission. As a
consequence, it yields in the form of mutual chain sliding.
It is truly remarkable that the two blends of 240K/1M (80:20)
and 70K/1M (80:20)
containing only 20 % of SBR1M would undergo rupture in the range
of Hencky rates
where the pure "matrices" only show yielding-initiated
non-uniform extension at the
same rates. Apparently at these rates there is not sufficient
intermolecular gripping
force to produce full chain extension and enough chain tension
to cause chain scission in
either pure SBR240K or SBR70K. Thus, the 80% matrix chains only
yield in the form
of mutual chain sliding. On the other hand, the second network
due to SMR1M can be
fully stretched to reach the point of chain scission.
Apparently, the 80% matrix chains
at the point of rupture have reached such a fully disengaged
state that the creation of the
new surface during rupture costs no more energy than that
estimated from the surface
tension. In other words, the incorporation of 20% SBR1M as a
second network
provided so much structural stability that the matrix, i.e., the
first network made of
SBR240K or SBR70K, was able to disintegrate fully.
The rupture occurred at much higher strains in the blends than
in the neat SBR1M
because at the volume fraction of 0.2, a strand (made of SBR1M)
between two
neighboring entanglement points of the second network is of a
much longer chain length.
Our recent study has shown that a higher stretching ratio is
required to produce full chain
extension, which appears to be a necessary condition for chain
scission [Wang and Wang
(2010b)]. The difference between the open diamond and triangles
in Fig. 7 indicates
rupture at rather different stretching ratios due to the
difference in entanglement density.
Finally, it is more than interesting to note that the blend of
240K/1M (80:20) barely
shows rupture at = 15 s-1
, whereas the blend with SBR70K as the matrix would
undergo rupture at a rate as low as = 6 s-1
. We know that at the same rate SBR70K
yields at a lower strain than SBR240K because the yield strain
decreases with lowering
WiRouse, which is lower for SBR70K. Consequently, SBR70K can
reach a fully
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14
disengaged state at a lower strain. As suggested above, the
disappearance of any
entanglement networking in the matrix appears to be a
prerequisite for rupture.
Apparently at = 6 s-1
, when SBR1M suffers chain scission, the matrix is also already
in
a state of full disengagement, allowing rupture to take place in
70K/1M (80:20).
Conversely, in 240K/1M (80:20) that is being extended at = 6
s-1
, even when a
significant fraction of SBR1M chains have become fully extended,
evidenced by the
upturn in the engr vs. curve of Fig. 5, and reached the
condition for chain scission, and
even if these chains do suffer scission, the matrix apparently
has not fully collapsed at
these strains. The engineering stress engr could further build
up until most SBR1M
chains have undergo chain scission. Beyond this point, the
matrix chains further
mutually slide past one another until the system lose uniform
structural support from
chain entanglement. This appears to be what is happening in the
blend involving the
stronger first network made of SBR240K.
B. Step Extension
Recent studies [Wang et al. (2007b); Wang and Wang (2008)] on
the relaxation
behavior of monodisperse entangled linear polymers after step
uniaxial extension have
revealed that the entanglement network suffers cohesive
breakdown at large strains.
Here the cohesion refers to that due to chain entanglement [Wang
et al. (2007a)]. Such
elastic breakup is analogous to those disclosed by the
particle-tracking velocimetric
observations of entangled polymer solutions [Wang et al. (2006);
Ravindranath and Wang
(2007)] and melts [Boukany et al. (2009b)] in startup simple
shear. The dynamics of
such elastic yielding appear to depend on the molecular weight
of the polymer melt and
the amplitude of the step strain. For both uniaxial extension
[Wang et al. (2007b); Wang
and Wang (2008)] and simple shear [Boukany et al. (2009b)], the
elastic yielding takes
place faster with increasing step strain amplitude. For the same
amplitude of step strain,
a sample with higher molecular weight takes a longer time to
lose the cohesive integrity
[Boukany et al. (2009b)]. In fact, the elastic yielding appears
to be related to the Rouse
chain dynamics in the sense that the induction time for the
breakdown scales with the
longest Rouse relaxation time [Wang et al. (2007a), Boukany et
al. (2009b)]. Because
-
15
the Rouse relaxation time of SBR1M is 60 times of SBR240K (note
that this ratio is far
greater than Z2 ~ 25, due to the polydispersity of SBR1M), the
incorporation of
incorporation of SBR1M may delay the specimen breakup after step
extension as shown
in Fig. 9(a) for the 240K/1M (90:10) blend. Upon a step
extension of Hencky strain of
0.75 and 0.9, the pure SBR240K undergoes elastic yielding that
leads to the sample
failure at 3.5 and 10 s respectively as shown by the filled
symbols in Fig. 9(a). The
blend (open symbols) fails at appreciably later times of 6.6 and
13 s respectively. It
appears that the specimen would not undergo failure until the
second network associated
with the high molecular weight SBR1M also collapse on some
longer time scale.
Something more dramatic occurs in the second blend containing
20% SBR1M.
Here the presence of the SBR1M can prevent the sample from
undergoing the cohesive
failure. For a step extension of Hencky strain = 0.9, the blend
does not suffer failure.
Perhaps before the second network made of SBR1M undergoes any
disentanglement the
first network has already recovered its entanglement structure
after experiencing elastic
yielding. At the higher strain of 1.2, the disintegration of the
second network perhaps
occurred so quickly that the first network had no chance to
return to its entangled state to
provide any adequate structural support. Consequently, the
sample failure was observed
at this higher strain. Fig. 8 indeed shows that the breakup of
the pure SBR1M
accelerates from an induction of 28 s for the step extension of
=0.9 to 11s for = 1.2.
C. Elastic Instability or Yielding
Over the past a few decades, there have been extensive studies
of entangled polymer
melts and solutions in extensional deformation, because of their
theoretical and practical
importance. Among all these studies, significant efforts have
been made toward
understanding the nature of specimen failure. Vincent (1960) was
the first to use the
Considère criterion [Considère (1885)] to analyze the necking
instability in elongation
and cold flow of solid plastics (PE and unplasticized PVC). Such
an analysis was
subsequently extended to polymer melts in the rubbery state
[Cogswell and Moore (1974);
Pearson and Connelly (1982)] and polymer solutions [Hassager et
al. (1998); Yao et al.
(1998)]. Doi and Edwards (1979) also suggested an elastic
necking instability in their
-
16
original tube theory based on the Considère reasoning. McKinley
and Hassager (1999)
applied the Doi-Edwards theory and pom-pom model [McLeish and
Larson (1998)] to
predict the critical Hencky strain for failure in linear and
branched polymer melts during
fast stretching.
0
0.5
1
1.5
2
0 1 2 3 4 5
15 s-1
10 s-1
6.0 s-1
3.0 s-1
15 s-1
10 s-1
6.0 s-1
3.0 s-1
Hencky strain
240K
240K/1M
(90:10)
en
gr (
MP
a)
Figure 10 Comparison of the engineering stress-strain curves for
the SBR240K melt and the
240K/1M (90:10) blend.
In our view, the emergence of a maximum in the engineering
stress implies
weakening of the underlying structure. Beyond the maximum, the
material is no longer
the original elastic body. In rigid solids such as metals
studied originally by Considère
(1885), when the measured force passes a maximum, non-uniform
extension take place
immediately due to the very limiting amount of extensibility.
The debate in the
literature about how to apply the Considère criterion has been
around the issue of whether
necking could occur immediately after the force maximum [Barroso
et al. (2010); Petrie
(2009); Joshi and Denn (2004b)]. For rubbery materials including
entangled melts, the
force maximum occurs at a significant stretching ratio, and
non-uniform extension
usually does not occur right after the force maximum. Fig. 2(a)
shows that the
maximum occurs for = 15 s-1
at a Hencky strain of = 1.8, and the failure strain is at
3.4. This behavior renders the application of the Considère
criterion irrelevant. The
force peak is the yield point when the sample is weakening in
its resistance against
further external deformation. The entanglement network takes
some further extension
before disintegration [Wang and Wang (2009)]. In other words,
uniform extension
could persist for a while until the network structure of the
sample becomes
inhomogeneous. The eventual "necking" and failure are anything
but a mechanical
-
17
(elastic) instability in our judgment. So far no theory can
describe quantitatively how
such a localization of yielding takes place. Any continuum
mechanical depiction based
on a non-monotonic relation between the tensile force and degree
of extension would
have to first explain the microscopic physics responsible for
the tensile force decline with
increasing stretching.
More specifically, the phenomenon of the specimen failure after
yielding is a result
of localized cohesive failure, having to do with how the initial
elastic deformation turns
into plastic deformation in an inhomogeneous manner.
Introduction of 10 or 20 %
SBR1M long chains into the SBR240K produces similar engineering
stress versus
Hencky strain characteristics as the pure SBR240K up to some
significant strains well
beyond the maxima, as shown in Fig. 10. For example, at = 6.0
s-1
, both the pure
SBR240K and the blend show nearly identical stress-strain curve
till = 2.7. A
continuum mechanical analysis based on such curves would have
predicted the onset of
necking instability, i.e., non-uniform extension, at the same
strain for both the pure
SBR240K and its blend with SBR1M. This is far from truth: the
presence of only 10 %
SBR1M long chains significantly extended the range of uniform
stretching till = 4.0,
well beyond the strain where the necking and failure took place
in the pure SBR240K.
We assert that this suppression of non-uniform extension by the
second entanglement
network is something well beyond any existing theoretical
description based on
continuum mechanical calculations.
More recently, Hassager and coworkers also tried to explain the
delayed failure after
rapid uniaxial extension (i.e., the elastic yielding according
to Wang et al. (2007b)) on the
basis of the Doi-Edwards model and a Considère-type analysis
[Lyhne et al. (2009);
Hassager et al. (2010)]. Since the observed
elastic-yielding-initiated failures after step
extension only involved a level of extension well below the
tensile force maxima that
occur around = 2.0, it is rather difficult to understand that
this failure discussed in Fig.
9(a)-(b) has something to do with the emergence of the
non-monotonic engineering stress
vs. strain curves shown in Fig. 2(a) and 5.
-
18
D. Where Is Strain Hardening
Since strain hardening is a frequently invoked expression of
extensional rheological
behavior of polymer melts, we will discuss thoroughly this idea
in the present context.
In particular, we will review and comment on four ways in which
the concept of "strain
hardening" may get used to describe uniaxial extension of
polymers.
10-1
100
101
102
103
104
0 1 2 3 4 5
106.03.02.0
1.00.60.3
Str
ess
(M
Pa)
Hencky strain
Strain rate (s-1)
240K/1M
(80:20)
neo-Hookean
Figure 11 Cauchy stress as a function of Hencky strain for the
240K/1M (80:20) blend. The
dotted line represents the stress-strain curve from the
neo-Hookean model of Eq. (1) with Geq =
0.82 MPa from Table 1.
D.1 Strain hardening with stress-strain curve above neo-Hookean
line
One of the earlier references to strain hardening came from high
extension of natural
rubbers when the stress-strain curve appears above the
prediction of the classical rubber
elastic theory, or the neo-Hookean model for a network of
Gaussian chains [Treloar
(1944)]. This strain hardening at high stretching ratios is
partially due to the finite chain
extensibility, i.e., the system has reached the point of
non-Gaussian stretching.
Moreover, the strain-induced crystallization in natural rubber
can also enhance strain
hardening [Smith et al. (1964)]. However, entangled melts that
do not undergo
strain-induced crystallization, such as the present SBR melts
and blends, only appear to
show stress-strain curves beneath the neo-Hookean model
prediction as shown in Fig. 3
for the SBR1M and Fig. 11 for the 240K/1M (80:20) blend.
Entangled polymer melts,
in absence of chemical cross-linking, apparently always suffer
so much loss of
entanglements during extension that the overall stress-strain
curve could not rise above
the neo-Hookean line even when non-Gaussian stretching takes
place. In other words, a
-
19
significant fraction of entangled points in the network reach
the point of force imbalance
and become ineffective to bear the load during continuous
extension of entangled melts,
which does not occur in the neo-Hookean model. More discussion
on this point is given
in the following IV.D.4
D.2 Monotonic increase of stress in startup extension: false
strain hardening
At high Weissenberg numbers, startup deformation is known to
produce stress
overshoot in simple shear, which has recently been suggested to
indicate the onset of
yielding [Wang et al. (2007a), Wang and Wang (2009)]. Beyond the
yield point, the
shear stress or transient viscosity decreases because the
elastic deformation ceases and
disintegration of the entanglement network leads to reduced
elastic resistance. In
startup uniaxial extension, the so called true stress E, i.e.,
the product of the tensile force
and stretching ratio = exp( t), would usually only monotonically
grow with continuing
extension until the point of non-uniform extension. In other
words, in uniaxial
extension, the extensional (Cauchy) stress E does not exhibit
overshoot, and thus the
transient viscosity E
=/ at a given Hencky strain rate only monotonically rises.
This contrast between simple shear and uniaxial extension has
caused Münstedt and
Kurzbeck (1998) to declare that "The viscosity increase as a
function of time or strain,
respectively, is called strain hardening. It is a special
feature of elongational
deformation of polymer melts."
In our view, this difference between simple shear and uniaxial
extension should be
looked at in a different light. Upon startup deformation the
transient shear viscosity also
initially increases with time, which is just an indication that
the elastic deformation is
dominant at the beginning of startup shear. The elastic
deformation cannot ensue
indefinitely, and yielding must take place, leading to the
observed decrease of the shear
stress as well as viscosity over time. In contrast, during
startup continuous extension at
Hencky rate the dominant reason for the continuous rise in E is
the simple geometric
factor of the exponentially shrinking cross-sectional area
A(t):
= F/A(t) = engr = engr(t) exp( t), (2)
where F is the total tensile force, and
-
20
A(t)=A0 exp( t) (3)
is the time-dependent cross-sectional area with A0 being the
original area. Even if
yielding occurs, i.e., engr (t) starts to decline, and E
would still only increase with
time until the point of non-uniform extension and specimen
failure, provided that the
decline of engr due to yielding is not as fast as the
exponential areal shrinkage, which is
often the case. Thus, the continuous increase of E and E
does not mean that the
uniaxial extension has not experienced the same yielding as seen
in simple shear. In
other words, such increases do not mean that the sample’s
entanglement structure is not
deteriorating and becoming less resistant to the continuous
extension. Actually, strain
softening not hardening has taken place as long as the tensile
force is declining with
continuing extension. As far as we can tell, the point of the
entanglement network
weakening occurs at the engineering stress maximum and cannot be
discerned readily
from such quantities as the Cauchy stress and extensional
viscosity.
D.3 Upward deviation of transient viscosity from zero-rate
limit
There is yet another way in which “strain hardening” is referred
to in uniaxial
extensions of polymer melts. It refers to a specific phenomenon
observed during startup
uniaxial extension when the transient elongational viscosity
shows upward deviation
from the limiting zero-rate-viscosity vs. time curve, in
contrast to the phenomenon in
simple shear where the transient shear viscosity function is
always below the zero-shear
viscosity function. This is perhaps the most widely recognized
signature for “strain
hardening”. The phrase "strain hardening" might have first been
used by Meissner
(1975) in describing the uniaxial extensional behavior of
low-density polyethylenes
(LDPE) [Meissner (1971, 1975); Laun and Münstedt (1978)]. Linear
melts with
bidispersity in molecular weight distribution [Münstedt (1980);
Koyama (1991);
Münstedt and Kurzbeck (1998); Minegishi et al. (2001); Wagner et
al. (2005); Nielsen et
al. (2006)] and even monodisperse melts –stretched at high
enough rates– [Wang and
Wang (2008), and figures below] could also produce this upward
deviation. Since
LDPE shows the most pronounced upward trend, long chain
branching has been thought
-
21
to play some peculiar role in producing this "strain hardening"
[McLeish (2008); van
Ruynbeke et al (2010)].
In our view, this noted difference between simple shear and
uniaxial extension is
perhaps superficial rather than fundamental for entangled
polymer melts. Entangled
polymer melts have been found to exhibit only strain softening
in simple shear as a
consequence of yielding during startup deformation. The maximum
shear stress max
has been found to grow with the applied rate more weakly than
linearly [Boukany et
al. (2009a)] so that the peak transient shear viscosity +
max= max/ can only decrease
with . Moreover, the steady shear viscosity is always lower than
the zero-shear
steady-state viscosity: In the zero-rate limit, the entanglement
network is intact, and it is
the Brownian diffusion that brings the chains past one another.
There is maximum
viscous resistance to terminal flow because the equilibrium
state of chain entanglement is
preserved.
In many cases, uniaxial extension is different in appearance
only because the
cross-sectional area A(t) of the sample keeps shrinking
exponentially with time at a given
Hencky strain rate as shown in Eq. (3). If one insists on
representing the transient
elastic response in terms of the Cauchy extensional stress of
Eq. (2) and the transient
extensional viscosity
E
(t) =/ = engr(t)exp( t)/ , (4)
then the continuous increase of and E
with time largely originates from the
exponential factor associated with the shrinking cross-sectional
area. Let consider the
extension in the beginning in the sense that t < In the
zero-rate limit, i.e., when
Wi, we can estimate Eq. (4) by employing Eq. (1). When engr ~
Eand exp( t) =
1, Eq. (4) turns into
E
(t)|0 = Et, for t 1, E
(t)
is also given by Eq. (5) at short times. But at longer times,
the exponential factor exp(
-
22
t) in Eq. (4) is a rapidly rising function of time. This causes
Eq. (4) to grow above Eq.
(5) by the same exponential factor had engr(t) continued to rise
linearly with the Hencky
strain = t as shown by the dashed line in Fig. 2(a). In reality,
the sample yields
eventually. Thus, before the yield point when engr has not
declined, the exponential
factor in Eq. (4) kicks in to produce a higher E
than the zero-rate curve. In other
words, there is a small window of extension where the transient
viscosity in Eq. (4) ticks
upward as shown in Fig. 12. This upward deviation is largely due
to the geometrical
shrinkage of the transverse dimensions and is not true strain
hardening.
The less monodisperse sample of SBR1M can extend more before
non-uniform
extension terminates the experiments. In Fig. 3, although engr
only grows by 40 % at
= 1.0 s-1
from = 3.0 to the point of failure, Fig. 13(a) shows a much
stronger rise in
E
(t) because of the exponential decreasing function A(t) of Eq.
(3) in the dominator of
Eq. (2) for the Cauchy stress E. As a consequence, the data rise
significantly above the
zero-rate envelope. Actually, whenever significant extension
occurs without sample
failure, there would be a great contribution to E
from the cross-sectional areal
shrinkage that has little to do with strain hardening.
Therefore, from now on we shall
call this upward deviation of the transient elongational
viscosity from its zero-rate curve
"pseudo strain hardening".
0.01
0.1
1
10
100
0.01 0.1 1 10 100 1000
15
10
6.0
3.0
1.0
0.3
0.1
Time (s)
3|*(1/)|
Strain rate (s-1)
SBR240K
+ (
MP
a.s)
E
Fig. 12 Transient extensional viscosity as a function of time
for the SBR240K melt, where the
"linear response" data given by ηE+ = 3 |η*(1/ω)| from the small
amplitude oscillatory shear
measurements are also presented as the reference [Gleissle
(1980)].
-
23
Long chain branching (LCB) in entangled melts delays the onset
of non-uniform
extension. For example, LCB in LDPE allows such materials to
display a very shallow
maximum in the engineering stress. In other words, the
engineering stress only
decreases gradually beyond its maximum. This gives the
geometrical exponential factor
a large range of time or strain to boost the transient viscosity
in Eq. (4) above the limiting
zero-rate curve, and caused the phrase "strain hardening" to be
invoked to differentiate
the extensional rheological behavior of LDPE from that of other
linear polymer melts
such as high-density polyethylene [McLeish (2008)]. It is clear
that LCB plays a critical
role to prolong uniform extension relative to entangled melts
made of linear chains.
However, engr does typically decrease with increasing extension
in LDPE. In other
words, there is only evidence of yielding and strain softening
and little sign of strain
hardening. Actually, there is explicit evidence from
birefringence measurements that
LDPE does not suffer non-Gaussian chain stretching during
extension in the typically
explored range of strain rates and temperatures [Koyama and
Ishizuka (1989); Okamoto
et al. (1998)].
Finally, we need to explain why most literature data on linear
melts show little
upward deviation from the limiting zero-rate curve, a fact that
makes LDPE look
somehow special. Most experiments on uniaxial extension of
linear polymer melts have
been conducted in the moderately high (rather than extremely
high) rate regime, and few
have been based on monodisperse samples. For example, up to =
3.0 s-1
,
corresponding to Wi = 100, the data in Fig. 12 hardly tilted
above the limiting curve, and
some unimpressive upward deviation shows up only at the higher
rates. As indicated
above, E
(t) would deviate exponentially fast above the zero-rate
viscosity function if
the sample would maintain linear growth of engr with the Hencky
strain as shown by
the dashed line in Fig. 2(a) In reality, the sample yields. The
deviation of the actual
data in Fig. 2(a) from this linear growth engr ~ E, i.e., the
dashed straight line increases
sharply with time. Beyond the engineering stress maximum, the
deviation actually
increases approximately exponentially, cancelling the
exponential factor of exp( t) in Eq.
(4) associated with the area shrinkage. Thus, at these
intermediate rates, one can hardly
-
24
see any upward deviation in linear melts that readily suffers
yielding. It is the yielding
and the resulting non-uniform extension that makes it impossible
to collect data points at
longer times. As a consequence, it has been impractical to
obtain the extensional
viscosity in the fully developed steady flow state during
startup continuous extension as
noted by Petrie (2006). But Petrie did not know why it is almost
impossible to reach the
flow state [Petrie (2008)], which was due to uniform yielding as
explained by Wang and
Wang (2008).
D.4 A case of “entanglement strain hardening”: non-Gaussian
stretching
The transition from elastic extension to flow, known as yielding
in engineering
terms, occurs during startup uniaxial extension in a wide range
of rates when further
overall elastic deformation of the entanglement network is no
longer possible and the
chains mutually slide past one another. At higher rates, chain
extension could continue
until a fraction of the chains in the sample approaches the
finite chain extensibility limit.
Chains at such a high stretching ratio appear stiffer and
non-Gaussian. This is
non-obvious from a conventional plot like Fig. 13(a). The
mechanical evidence of this
non-Gaussian stretching comes from further analysis of data such
as those in Fig. 3. It
is obvious that some level of yielding, i.e., mutual chain
sliding, can and does occur
before a fraction of the yield-surviving entanglement strands
reaches the finite chain
extensibility limit. Fig. 13(b) shows that when the shear
modulus G in Eq. (1) is
reduced from its equilibrium value of 0.85 MPa to 0.424 MPa, the
neo-Hookean would
emerge onto the data at the applied Hencky strain rate of 6.0
s-1
at a stretching ratio of nG
= 8.5, where the subscript nG stands for non-Gaussian. Beyond
this turning point, the
data deviate upward from the neo-Hookean curve, which can be
taken as a sign of
non-Gaussian stretching. This upward deviation is true strain
hardening at the chain
level, which we shall call "entanglement strain hardening" to
differentiate from the strain
hardening in vulcanized rubbers that produces a stress-strain
curve above the
neo-Hookean limit as discussed in IV.D.1. We see this behavior
as shown in the open
symbols in Fig. 3 for the SBR1M melt and in Fig. 5 and 6 for the
binary mixtures. The
full chain extension leads to chain scission and rupture during
startup continuous
-
25
extension. One key characteristic for this strain hardening is
the emergence of the
upturn seen in Fig. 3 in open symbols in the pure SBR1M, and
more instructively in Fig.
13(a).
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
Time (s)
+ (
MP
a.s)
E
3|*(1/)|
SBR1M
(a) = 1 s-1
0
1
2
3
4
5
6
7
4 8 12 16 20
en
gr
(M
Pa)
1
SBR1M
6.0 s-1
neo-Hookean
(b)
nG
Figure 13 (a) Transient extensional viscosity as a function of
time for the SBR240K melt, where
the "linear response" data given by ηE+ = 3 |η*(1/ω)| from the
small amplitude oscillatory shear
measurements are also presented as a reference [Gleissle
(1980)].
(b) Engineering stress engr versus the stretching ratio at = 6.0
s-1
, relative to a neo-Hookean
curve of Eq. (1) based on G = 0.424 MPa = Geq/2, i.e, half of
the equilibrium shear modulus,
implying that half of the strands in the equilibrium
entanglement network are lost at the
stretching ratio nG ~ 8.5.
10-1
100
101
102
0.01 0.1 1 10 100 103
106.03.02.01.00.60.3
240K/1M
(80:20)
3|*(1/)|
Strain rate (s-1)
Time (s)
+ (
MP
a.s)
E
(a)
0
0.5
1
1.5
2
2.5
10 20 30 40 50
10
6.03.02.0
70K/1M (80:20)
Strain rate (s-1
)
en
gr (
MP
a)
neo-Hookean
(b)
nG
1
Figure 14 (a) Transient extensional viscosity as a function of
time for the 240K/1M (80:20)
blend, where the "linear response" data given by ηE+ = 3
|η*(1/ω)| from the small amplitude
oscillatory shear measurements are also presented as a reference
[Gleissle (1980)].
(b) Engineering stress engr versus the stretching ratio at the
various rates for the 70K/1M
(80:20) blend, relative to a neo-Hookean curve of Eq. (1) based
on G = Geq2.2
= 0.025 MPa
where Geq = 0.85 MPa and = 0.2, where nG ~ 23, far higher than
that of 8.5 for the pure
SBR1M in Fig. 13(b).
This upturn also shows up strongly in Fig. 5 and 6 that depict
the two blends with
the 20% long chains of SBR1M. It occurs whenever the limit of
finite chain
extensibility is approached to cause non-Gaussian stiffening of
the entanglement network.
-
26
When expressed in terms of the transient viscosity E
(t) as shown in Fig. 14(a),
significant upward deviation from the zero-rate curve shows up,
reminiscent of the data
of LDPE. This strong upward deviation arises from both the
exponential factor in Eq.
(4) and the “entanglement strain hardening”, i.e., non-Gaussian
stretching. For LDPE,
there is little evidence of non-Gaussian stretching [Koyama and
Ishizuka (1989);
Okamoto et al. (1998)], yet the similar behavior is observed for
the following reason:
LDPE typically can undergo significant extension without
encountering non-uniform
extension despite the occurrence of yielding, signified by the
emergence of a
non-monotonic relation between the engineering stress and
strain. The prolonged
uniform extension allows the geometric exponential factor in Eq.
(4) to produce the
upward deviation that is well documented in the literature
[Ferry (1980); Laun and
Schuch (1989)].
Actually, it is again more instructive to present the evidence
of non-Gaussian
stretching in the blends by referring to a hypothetical
neo-Hookean behavior of the
second network formed by SBR1M chains. The second network at a
weight fraction of
20 % has a shear modulus given by G = G(=1)2.2
. Fig. 14 (b) shows that the data at
high rates show significant upward deviation from the
neo-Hookean curve. The
deviation occurs at nG = 23, which significantly higher than the
degree of extension
given by nG = 8.5 for the pure SBR1M in Fig. 13(b). The
separation between the blend
and SBR1M is once again exactly a factor of 1 Hencky strain, as
noted above due to the
difference in entanglement spacing. The data at = 3.0 s-1
and below are below the
neo-Hookean curve, indicating that there is significant loss of
entanglement due to
yielding of the second network. The strands at these lower rates
can hardly reach the
fully extended chain limit to cause chain scission. The sample
eventually fails by
mutual chain sliding to reach a state of disengagement.
In summary, the transient viscosity or Cauchy stress involves an
exponentially
decreasing area in the dominator of its definition so that it
tends to grow in time and
becomes greater than its value in the zero-rate limit, in
contrast to the counterpart in
simple shear where the sheared area stays constant. This
geometric difference has
-
27
caused considerable confusion in the literature. We have
examined four situations
where the phase “strain hardening” may have emerged to
characterize the extensional
rheological behavior of entangled polymers with either linear or
branched chain
architecture. In all cases, the definition of the extensional
transient viscosity permits the
exponentially shrinking cross-sectional area to mask the origins
of the physical
phenomena. Systems that resist yielding and structural failure,
such as long-chain
branched LDPE and samples with bimodal molecular weight
distribution, simply show
greater upward deviation from its zero-rate linear response
because at the same moment
during extension the high rate test is in a more stretched state
with a thinner cross-section
than a limiting zero-rate test. This upward deviation may not
imply strain hardening at
all. The real strain hardening involving non-Gaussian stretching
amounts to having an
engineering stress that grows monotonically with increasing
extension and therefore
resist any non-uniform stretching or necking.
V. CONCLUSION
Ductile failure after yielding and rupture after non-Gaussian
stretching have both
been shown to occur in uniaxial extension for monodisperse and
bidisperse entangled
styrene-butadiene rubber (SBR) melts. Within the accessible
range of Hencky strain
rates, the internal chain dynamics of the SBR240K are too fast
to allow full chain
extension and rupture, whereas the SBR1M undergoes cohesive
failure in the form of
yielding and non-uniform extension at low rates, and rupture at
high rates. The
incorporation of a small fraction of SBR1M into a matrix of
SBR240K greatly alters the
characteristic responses to both startup extension and step
extension.
In particular, we have reached the following conclusions. (A)
There is evidence of
double entanglement networking. Following the disintegration of
the faster network
formed by the matrix chains, the second network made of SBR1M
can retain the
structural integrity of the specimen until it also subsequently
yields. The onset of
structural failure is considerably extended beyond that of the
pure SBR240K matrix.
The presence of the SBR1M also altered the kinetics of elastic
yielding and even delayed
the onset of elastic yielding in the blend of 240/1M (80:20).
(B) More surprisingly, in
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28
the same range of extensional rates, under which the pure
SBR240K and SBR70K only
fail through ductile yielding, the blend of 240/1M (80:20) and
70K/1M (80:20) suffer
rupture. (C) The application of the Considère criterion is
irrelevant because the origin
of non-uniform extension appears to be yielding of the
entanglement network, unrelated
to any type of elastic instability. (D) We confirm our previous
assertion [Wang and
Wang (2008)] that entangled linear polymers cannot attain steady
flow during startup
uniaxial extension. In other words, such linear chain systems as
the present pure SBR
melts and their blends fail after yielding over a wide range of
rates in the form of
non-uniform extension without ever reaching a fully developed
flow state. (E) At
various extensional rates beyond the terminal regime, the
monotonic rise of the Cauchy
stress before the onset of non-uniform extension stems from its
definition that involves
the exponentially shrinking area in the denominator. This pseudo
strain hardening has
little to do with the entanglement strain hardening due to the
finite chain extensibility that
produces non-Gaussian stretching of the entanglement
network.
Acknowledgements The authors would like to express their sincere
gratitude to Dr.
Xiaorong Wang from Bridgestone-Americas Center for Research and
Technology for
providing the SBR samples in this study. This work is supported,
in part, by grants
(DMR-0821697 and CMMI-0926522) from the National Science
Foundation.
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29
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