We study theoretically the role of aging in the rheology of soft materials. We define several generalized rheological response functions suited to aging samples (in which time translation invariance is lost). These are then used to study aging effects within a simple scalar model (the “soft glassy rheology” or SGR model) whose constitutive equations relate shear stress to shear strain among a set of elastic elements, with distributed yield thresholds, undergoing activated dynamics governed by a “noise temperature,” x. (Between yields, each element follows affinely the applied shear.) For 1 < x < 2 there is a power-law fluid regime in which transients occur, but no aging. For x < 1, the model has a macroscopic yield stress. So long as this yield stress is not exceeded, aging occurs, with a sample’s apparent relaxation time being of order its own age. The (age-dependent) linear viscoelastic loss modulus G(ω,t) rises as frequency is lowered, but falls with age t, so as to always remain less than G(ω,t) (which is nearly constant). Significant aging is also predicted for the stress overshoot in nonlinear shear startup and for the creep compliance. Though obviously oversimplified, the SGR model may provide a valuable paradigm for the experimental and theoretical study of rheological aging phenomena in soft solids.

1.
Barnes, H. A., J. F. Hutton, and K. Walters, An Introduction to Rheology (Elsevier, Amsterdam, 1989).
2.
Bernstein
,
B.
,
E. A.
Kearsley
, and
L. J.
Zapas
, “
A study of stress relaxation with finite strain
,”
Trans. Soc. Rheol.
7
,
391
410
(
1963
).
3.
Bouchaud
,
J. P.
, “
Weak ergodicity breaking and aging in disordered-systems
,”
J. Phys. I
2
,
1705
1713
(
1992
).
4.
Bouchaud, J. P., L. F. Cugliandolo, J. Kurchan, and M. Mézard, “Out of equilibrium dynamics in spin-glasses and other glassy systems,” in Spin Glasses and Random Fields, edited by A. P. Young (World Scientific, Singapore, 1998).
5.
Bouchaud
,
J. P.
and
D. S.
Dean
, “
Aging on Parisi’s tree
,”
J. Phys. I
5
,
265
286
(
1995
).
6.
Cates
,
M. E.
and
S. J.
Candau
, “
Statics and dynamics of wormlike surfactant micelles
,”
J. Phys.: Condens. Matter
2
,
6869
6892
(
1990
).
7.
Copson, E. T., An Introduction to the Theory of Functions of a Complex Variable (Oxford University Press, New York, 1962).
8.
Cugliandolo
,
L. F.
and
J.
Kurchan
, “
Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model
,”
Phys. Rev. Lett.
71
,
173
176
(
1993
).
9.
Cugliandolo
,
L. F.
and
J.
Kurchan
, “
On the out-of-equilibrium relaxation of the Sherrington-Kirkpatrick model
,”
J. Phys. A
27
,
5749
5772
(
1994
).
10.
Cugliandolo
,
L. F.
and
J.
Kurchan
, “
Weak ergodicity breaking in mean-field spin-glass models
,”
Philos. Mag. B
71
,
501
514
(
1995
).
11.
Cugliandolo
,
L. F.
,
J.
Kurchan
,
P.
LeDoussal
, and
L.
Peliti
, “
Glassy behaviour in disordered systems with nonrelaxational dynamics
,”
Phys. Rev. Lett.
78
,
350
353
(
1997a
).
12.
Cugliandolo
,
L. F.
,
J.
Kurchan
, and
L.
Peliti
, “
Energy flow, partial equilibration, and effective temperatures in systems with slow dynamics
,”
Phys. Rev. E
55
,
3898
3914
(
1997b
).
13.
Dickinson, E., An Introduction to Food Colloids (Oxford University Press, Oxford, 1992).
14.
Doi, M. and S. F. Edwards, The Theory of Polymer Dynamics (Clarendon, Oxford, 1986).
15.
Evans
,
R. M. L.
,
M. E.
Cates
, and
P.
Sollich
, “
Diffusion and rheology in a model of glassy materials
,”
Eur. Phys. J. B
10
,
705
718
(
1999
).
16.
Gardon
,
R.
and
O. S.
Narayanaswamy
, “
Stress and volume relaxation in annealing flat glass
,”
J. Am. Ceram. Soc.
53
,
380
385
(
1970
).
17.
Hodge
,
I. M.
, “
Physical aging in polymer glasses
,”
Science
267
,
1945
1947
(
1995
).
18.
Hoffmann
,
H.
and
A.
Rauscher
, “
Aggregating systems with a yield stress value
,”
Colloid Polym. Sci.
271
,
390
395
(
1993
).
19.
Holdsworth
,
S. D.
, “
Rheological models used for the prediction of the flow properties of food products
,”
Trans. Inst. Chem. Eng.
71
,
139
179
(
1993
).
20.
Hopkins
,
L. L.
, “
Stress relaxation or creep of linear viscoelastic substances under varying temperature
,”
J. Polym. Sci.
28
,
631
633
(
1958
).
21.
Ketz
,
R. J.
,
R. K.
Prudhomme
, and
W. W.
Graessley
, “
Rheology of concentrated microgel solutions
,”
Rheol. Acta
27
,
531
539
(
1988
).
22.
Khan
,
S. A.
,
C. A.
Schnepper
, and
R. C.
Armstrong
, “
Foam rheology. 3: Measurement of shear-flow properties
,”
J. Rheol.
32
,
69
92
(
1988
).
23.
Kob
,
W.
and
J. L.
Barrat
, “
Aging effects in a Lennard-Jones glass
,”
Phys. Rev. Lett.
78
,
4581
4584
(
1997
).
24.
Kossuth
,
M. B.
,
D. C.
Morse
, and
F. S.
Bates
, “
Viscoelastic behavior of cubic phases in block copolymer melts
,”
J. Rheol.
43
,
167
196
(
1999
).
25.
Kurchan
,
J.
, “
Rheology, and how to stop aging
,” Preprint cond-mat/9812347 (
1998
);
Proceedings of Jamming and Rheology: Constrained Dynamics on Microscopic and Macroscopic Scales, workshop at ITP, Santa Barbara, 1997 (to be published).
26.
Larson, R. G., The Structure and Rheology of Complex Fluids (Oxford University Press, Oxford, 1999).
27.
Mackley
,
M. R.
,
R. T. J.
Marshall
,
J. B. A.
F Smeulders
, and
F. D.
Zhao
, “
The rheological characterization of polymeric and colloidal fluids
,”
Chem. Eng. Sci.
49
,
2551
2565
(
1994
).
28.
Mason
,
T. G.
and
D. A.
Weitz
, “
Linear viscoelasticity of colloidal hard-sphere suspensions near the glass-transition
,”
Phys. Rev. Lett.
75
,
2770
2773
(
1995
).
29.
Mason
,
T. G.
,
J.
Bibette
, and
D. A.
Weitz
, “
Elasticity of compressed emulsions
,”
Phys. Rev. Lett.
75
,
2051
2054
(
1995
).
30.
Monthus
,
C.
and
J. P.
Bouchaud
, “
Models of traps and glass phenomenology
,”
J. Phys. A
29
,
3847
3869
(
1996
).
31.
Narayanaswamy
,
O. S.
, “
Model of structural relaxation in a glass
,”
J. Am. Ceram. Soc.
54
,
491
498
(
1971
).
32.
Odagaki
,
T.
, “
Glass-transition singularities
,”
Phys. Rev. Lett.
75
,
3701
3704
(
1995
).
33.
Panizza
,
P.
,
D.
Roux
,
V.
Vuillaume
,
C. Y. D.
Lu
, and
M. E.
Cates
, “
Viscoelasticity of the onion phase
,”
Langmuir
12
,
248
252
(
1996
).
34.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, Cambridge, 1992).
35.
Scherer, G. W., Relaxation in Glass and Composites (Wiley, New York, 1986).
36.
Sollich
,
P.
, “
Rheological constitutive equation for a model of soft glassy materials
,”
Phys. Rev. E
58
,
738
759
(
1998
).
37.
Sollich
,
P.
,
F.
Lequeux
,
P.
Hébraud
, and
M. E.
Cates
, “
Rheology of soft glassy materials
,”
Phys. Rev. Lett.
78
,
2020
2023
(
1997
).
38.
Struik, L. C. E., Physical Aging in Amorphous Polymers and Other Materials (Elsevier, Houston, 1978).
This content is only available via PDF.
You do not currently have access to this content.