Glassy polymers show “strain hardening”: at constant extensional load, their flow first accelerates, then arrests. Recent experiments under such loading have found this to be accompanied by a striking dip in the segmental relaxation time. This can be explained by a minimal nonfactorable model combining flow-induced melting of a glass with the buildup of stress carried by strained polymers. Within this model, liquefaction of segmental motion permits strong flow that creates polymer-borne stress, slowing the deformation enough for the segmental (or solvent) modes then to re-vitrify. Here, we present new results for the corresponding behavior under step-stress shear loading, to which very similar physics applies. To explain the unloading behavior in the extensional case requires introduction of a “crinkle factor” describing a rapid loss of segmental ordering. We discuss in more detail here the physics of this, which we argue involves non-entropic contributions to the polymer stress, and which might lead to some important differences between shear and elongation. We also discuss some fundamental and possibly testable issues concerning the physical meaning of entropic elasticity in vitrified polymers. Finally, we present new results for the startup of steady shear flow, addressing the possible role of transient shear banding.

1.
T. C. B.
McLeish
,
Adv. Phys.
51
,
1379
1527
(
2002
).
2.
R. G.
Larson
,
Constitutive Equations for Polymer Melts and Solutions
(
Butterworth-Heinemann
,
Boston
,
1988
).
3.
J. M.
Brader
,
M. E.
Cates
, and
M.
Fuchs
,
Phys. Rev. Lett.
101
,
138301
(
2008
).
4.
J. M.
Brader
,
M. E.
Cates
, and
M.
Fuchs
,
Phys. Rev. E
86
,
021403
(
2012
).
5.
P. G.
Debenedetti
and
F. H.
Stillinger
,
Nature (London)
410
,
259
267
(
2001
).
6.
D. C.
Hofmann
 et al,
Nature (London)
451
,
1085
1089
(
2008
).
7.
P.
Schall
,
D. A.
Weitz
, and
F.
Spaepen
,
Science
318
,
1895
1899
(
2007
).
8.
H.-N.
Lee
,
K.
Paeng
,
S. F.
Swallen
, and
M. D.
Ediger
,
Science
323
,
231
234
(
2009
).
9.
L. C. E.
Struik
,
Physical Aging in Amorphous Polymers and Other Materials
(
Elsevier
,
New York
,
1978
).
10.
J. M.
Brader
,
T.
Voigtmann
,
M.
Fuchs
,
R.
Larson
, and
M. E.
Cates
,
Proc. Natl. Acad. Sci. U.S.A.
106
,
15186
15191
(
2009
).
11.
E. T. J.
Klompen
,
T. A. P.
Engels
,
L. E.
Govaert
, and
H. E. H.
Meijer
,
Macromolecules
38
,
6997
7008
(
2005
).
12.
L. C. A.
van Breemen
,
E. T. J.
Klompen
,
L. E.
Govaert
, and
H. E. H.
Meijer
,
J. Mech. Phys. Solids
59
,
2191
1107
(
2011
).
13.
R. N.
Haward
and
G.
Thackray
,
Proc. R. Soc. London, Ser. A
302
,
453
372
(
1968
).
14.
E. M.
Arruda
and
M. C.
Boyce
,
Int. J. Plast.
9
,
697
720
(
1993
).
15.
O. A.
Hasan
and
M. C.
Boyce
,
Polymer
34
,
5085
5092
(
1993
).
16.
O. A.
Hasan
and
M. C.
Boyce
,
Polym. Eng. Sci.
35
,
331
344
(
1995
).
17.
S. M.
Fielding
,
P.
Sollich
, and
M. E.
Cates
,
J. Rheol.
44
,
323
369
(
2000
).
18.
K.
Chen
and
K. S.
Schweizer
,
EPL
79
,
26006
(
2007
).
19.
M. L.
Falk
and
J. S.
Langer
,
Annu. Rev. Condens. Matter Phys.
2
,
353
373
(
2010
).
20.
C.
Derec
,
A.
Ajdari
, and
F.
Lequeux
,
Eur. Phys. J. E
4
,
355
361
(
2001
).
21.
P.
Coussot
,
Q. D.
Nguyen
,
H. T.
Huynh
, and
D.
Bonn
,
Phys. Rev. Lett.
88
,
175501
(
2002
).
22.
B.
Rinn
,
P.
Maass
, and
J.-P.
Bouchaud
,
Phys. Rev. Lett.
84
,
5403
5406
(
2000
).
23.
S. M.
Fielding
,
R. G.
Larson
, and
M. E.
Cates
,
Phys. Rev. Lett.
108
,
048301
(
2012
).
24.
H.-N.
Lee
and
M. D.
Ediger
,
Macromolecules
43
,
5863
5873
(
2010
).
25.
R. A.
Riggleman
,
H. N.
Lee
,
M. D.
Ediger
, and
J. J.
de Pablo
,
Phys. Rev. Lett.
99
,
215501
(
2007
).
26.
H. N.
Lee
,
R. A.
Riggleman
,
J. J.
de Pablo
, and
M. D.
Ediger
,
Macromolecules
42
,
4328
4336
(
2009
).
27.
M.
Warren
and
J.
Rottler
,
Phys. Rev. E
76
,
031802
(
2007
).
28.
M.
Warren
and
J.
Rottler
,
Phys. Rev. Lett.
104
,
205501
(
2010
).
29.
R. A.
Riggleman
,
H. N.
Lee
,
M. D.
Ediger
, and
J. J.
de Pablo
,
Soft Matter
6
,
287
291
(
2010
).
30.
K.
Chen
and
K. S.
Schweizer
,
Phys. Rev. Lett.
102
,
038301
(
2009
).
31.
K.
Chen
and
K. S.
Schweizer
,
Phys. Rev. E
82
,
041804
(
2010
).
32.
K.
Chen
,
E. J.
Saltzman
, and
K. S.
Schweizer
,
J. Phys. Condens. Matter
21
,
503101
(
2009
).
33.
H.
Eyring
,
J. Chem. Phys.
4
,
283
291
(
1936
).
34.
K.
Chen
,
E. J.
Saltzman
, and
K. S.
Schweizer
,
Annu. Rev. Condens. Matter Phys.
1
,
277
300
(
2010
);
K.
Chan
and
K. S.
Schweizer
,
Macromolecules
44
,
3988
4000
(
2011
).
35.
R. S.
Hoy
and
M. O.
Robbins
,
Phys. Rev. Lett.
99
,
117801
(
2007
).
36.
R. S.
Hoy
and
M. O.
Robbins
,
Phys. Rev. E
77
,
031801
(
2008
).
37.
R. S.
Hoy
and
C. S.
O'Hern
,
Phys. Rev. E
82
,
041803
(
2010
).
38.
R. S.
Hoy
and
M. O.
Robbins
,
J. Chem. Phys.
31
,
244901
(
2009
).
39.
M.
Fuchs
and
M. E.
Cates
,
J. Rheol.
53
,
957
1000
(
2009
).
40.
R. L.
Moorcroft
,
M. E.
Cates
, and
S. M.
Fielding
,
Phys. Rev. Lett.
106
,
055502
(
2011
);
[PubMed]
R. L.
Moorcroft
,
M. E.
Cates
, and
S. M.
Fielding
,
Phys. Rev. Lett.
106
,
209903
(
2011
) (Erratum).
41.
J. M.
Brader
,
T.
Voigtmann
,
M. E.
Cates
, and
M.
Fuchs
,
Phys. Rev. Lett.
98
,
058301
(
2007
).
42.
T.
Voigtmann
,
J. M.
Brader
,
M.
Fuchs
, and
M. E.
Cates
,
Soft Matter
8
,
4244
4253
(
2012
).
43.
R. N.
Haward
,
Macromolecules
26
,
5860
5869
(
1993
).
44.
R. G.
Larson
,
Rheol. Acta
29
,
371
384
(
1990
).
45.
E. J.
Hinch
,
J. Non-Newtonian Fluid Mech.
54
,
209
230
(
1994
).
46.
A. S.
Argon
,
Philos. Mag.
28
,
839
865
(
1973
).
47.
V. M.
Entov
,
J. Non-Newtonian Fluid Mech.
82
,
167
188
(
1999
).
48.
J. P.
Rothstein
and
G. H.
McKinley
,
J. Non-Newtonian Fluid Mech.
108
,
275
290
(
2002
).
49.
I.
Ghosh
,
Y. L.
Joo
,
G. H.
McKinley
,
R. A.
Brown
, and
R. C.
Armstrong
,
J. Rheol.
46
,
1057
1089
(
2002
);
G.
Lielens
,
R.
Keunings
, and
V.
Legat
,
J. Non-Newtonian Fluid Mech.
87
,
179
196
(
1999
).
50.
The resulting model is clearly not quite consistent because the reduced modulus should then also control the early stages of the response after the initial loading before chains become locally stretched. Therefore, at some stage during the transfer of stress from solvent to polymer visible in Fig. 1, the modulus should smoothly rise from θGp to Gp to reflect the onset of local inextensibility. However, we do not expect this drift in modulus to give qualitative changes in the predictions for the loading part of the experiment: what mainly matters there is the modulus within the strain-hardened regime.
51.
S. K.
Ma
,
Statistical Mechanics
(
World Scientific
,
Singapore
,
1985
).
52.
E. D.
Eastman
and
R. T.
Milner
,
J. Chem. Phys.
1
,
444
456
(
1933
).
53.
R. K.
Bowles
and
R. J.
Speedy
,
Mol. Phys.
87
,
1349
1361
(
1996
).
54.
R. L.
Moorcroft
and
S. M.
Fielding
, “
Criteria for shear banding in time-dependent flows of complex fluids
,” preprint arXiv:1201.6259 (
2012
).
55.
Note that this viewpoint would be hard to sustain if one believes that the polymer entropic stress disappears on vitrification of the solvent as mooted in Sec. VI.
56.
K.
Nayak
 et al,
J. Polym. Sci., Part B: Polym. Phys.
49
,
920
938
(
2011
).
57.
M. L.
Manning
,
J. S.
Langer
, and
J. M.
Carlson
,
Phys. Rev. E
76
,
056106
(
2007
).
58.
To solve the system numerically while allowing the flow to be inhomogeneous, small diffusive terms were first added to the governing equations for all conformation tensor components and for τ. The diffusion length ℓd = (Dτ0)1/2 was set at ℓd = 2.5 × 10−2L (Fig. 5) or ℓd = 10−2L (Fig. 6), with L the systems size in the y direction. (See Ref. 54 for the reasoning behind inclusion of such terms.) The system was initialized by adding noise of the form qXcos (πy/L) with y in the shear gradient direction to all conformation tensor components, with q = 0.01 the magnitude of the noise and X a random number −0.5 ⩽ X ⩽ 0.5 chosen separately for each component. To perform the stability analyses, the system is first evolved with flow homogeneity enforced. Small velocity perturbations δvcos (πy/L)exp (ωt) are added to the time-dependent homogeneous state and evolved in time to first order in amplitude, recording
$\delta \dot{\gamma }(y)$
δγ̇(y)
, the perturbation in local shear rate. Our measure of the “degree of banding”
$\dot{\gamma }_{max}-\dot{\gamma }_{min}$
γ̇maxγ̇min
is proportional to δv while the system remains linear; for “young” samples this applies throughout the time evolution. However, for well-aged samples nonlinearities cause the degree of banding to soon become independent of the noise amplitude chosen.
59.
C.
Truesdell
and
W.
Noll
,
The Non-Linear Field Theories of Mechanics
, 3rd ed. (
Springer-Verlag
,
2004
).
60.
R. K.
Bowles
and
R. J.
Speedy
,
Physica A
262
,
76
87
(
1999
).
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