1

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: swang@uakron.edu

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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

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

4

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

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.

7

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

9

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

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.

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

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

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.

29

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