We develop a rheological model to approximate the nonlinear rheology of wormlike micelles using two constitutive models to represent a structural transition at high shear rates. The model is intended to describe the behavior of semidilute wormlike micellar solutions over a wide range of shear rates whose parameters can be determined mainly from small-amplitude equilibrium measurements. Length evolution equations are incorporated into reactive Rolie-Poly entangled-polymer rheology and dilute reactive-rod rheology, with a kinetic exchange between the two models. Although the micelle length is remarkably reduced during flow, surprisingly, we propose that they are not shortened by stress-enhanced breakage, which remains thermally driven. Instead, we hypothesize that stretching energy introduces a linear potential that decreases the rate of recombination and reduces the mean micelle length. This stress-hindered recombination approach accurately describes transient stress-growth upon start-up shear flow, and it predicts a transition of shear viscosity and alignment response observed at high shear rates. The proposed mechanism applies only when self-recombination occurs frequently. The effect of varying the relative rate of self-recombination on the rheology of wormlike micelles at high shear rates is yet to be explored.

1.
Jafari Nodoushan
,
E.
,
T.
Yi
,
Y. J.
Lee
, and
N.
Kim
, “
Wormlike micellar solutions, beyond the chemical enhanced oil recovery restrictions
,”
Fluids
4
,
173
(
2019
).
2.
Rothstein
,
J. P.
, and
H.
Mohammadigoushki
, “
Complex flows of viscoelastic wormlike micelle solutions
,”
J. Non-Newtonian Fluid Mech.
285
,
104382
(
2020
).
3.
Gaudino
,
D.
,
S.
Costanzo
,
G.
Ianniruberto
,
N.
Grizzuti
, and
R.
Pasquino
, “
Linear wormlike micelles behave similarly to entangled linear polymers in fast shear flows
,”
J. Rheol.
64
,
879
888
(
2020
).
4.
Adams
,
A. A.
,
M. J.
Solomon
, and
R. G.
Larson
, “
A nonlinear kinetic-rheology model for reversible scission and deformation of unentangled wormlike micelles
,”
J. Rheol.
62
,
1419
1427
(
2018
).
5.
Tan
,
G.
,
W.
Zou
,
M.
Weaver
, and
R. G.
Larson
, “
Determining threadlike micelle lengths from rheometry
,”
J. Rheol.
65
,
59
71
(
2021
).
6.
Tan
,
G.
, and
R. G.
Larson
, “
Quantitative modeling of threadlike micellar solution rheology
,”
Rheol. Acta
61
,
443
457
(
2022
).
7.
Peterson
,
J. D.
, and
L. G.
Leal
, “
Predictions for flow-induced scission in well-entangled living polymers: The “living Rolie-Poly” model
,”
J. Rheol.
65
,
959
982
(
2021
).
8.
Sato
,
T.
, and
R. G.
Larson
, “
Nonlinear rheology of entangled wormlike micellar solutions predicted by a micelle-slip-spring model
,”
J. Rheol.
66
,
639
656
(
2022
).
9.
Salmon
,
J. B.
,
A.
Colin
,
S.
Manneville
, and
F.
Molino
, “
Velocity profiles in shear-banding wormlike micelles
,”
Phys. Rev. Lett.
90
,
228303
(
2003
).
10.
Rehage
,
H.
, and
H.
Hoffmann
, “
Rheological properties of viscoelastic surfactant systems
,”
J. Phys. Chem.
92
,
4712
4719
(
1988
).
11.
Turner
,
M. S.
, and
M. E.
Cates
, “
Linear viscoelasticity of wormlike micelles—A comparison of micellar reaction-kinetics
,”
J. Phys. II France
2
,
503
519
(
1992
).
12.
Berret
,
J.-F.
,
J.
Appell
, and
G.
Porte
, “
Linear rheology of entangled wormlike micelles
,”
Langmuir
9
,
2851
2854
(
1993
).
13.
Yang
,
J.
, “
Viscoelastic wormlike micelles and their applications
,”
Current Opinion Coll. Inter. Sci.
7
(
5–6
),
276
281
(
2002
).
14.
Kotenko
,
M.
,
H.
Oskarsson
,
C.
Bojesen
, and
M. P.
Nielsen
, “
An experimental study of the drag reducing surfactant for district heating and cooling
,”
Energy
178
,
72
78
(
2019
).
15.
Elter
,
J. K.
,
S.
Quader
,
J.
Eichhorn
,
M.
Gottschaldt
,
K.
Kataoka
, and
F. H.
Schacher
, “
Core-cross-linked fluorescent worm-like micelles for glucose-mediated drug delivery
,”
Biomacromolecules
22
,
1458
1471
(
2021
).
16.
Cates
,
M. E.
, and
S. J.
Candau
, “
Statics and dynamics of worm-like surfactant micelles
,”
J. Phys.: Condens. Matter
2
,
6869
6892
(
1990
).
17.
Hommel
,
R. J.
, and
M. D.
Graham
, “
Constitutive modeling of dilute wormlike micelle solutions: Shear-induced structure and transient dynamics
,”
J. Non-Newtonian Fluid Mech.
295
,
104606
(
2021
).
18.
Dutta
,
S.
, and
M. D.
Graham
, “
Mechanistic constitutive model for wormlike micelle solutions with flow-induced structure formation
,”
J. Non-Newtonian Fluid Mech.
251
,
97
106
(
2018
).
19.
Germann
,
N.
,
L. P.
Cook
, and
A. N.
Beris
, “
Nonequilibrium thermodynamic modeling of the structure and rheology of concentrated wormlike micellar solutions
,”
J. Non-Newtonian Fluid Mech.
196
,
51
57
(
2013
).
20.
Vasquez
,
P. A.
,
G. H.
McKinley
, and
L. P.
Cook
, “
A network scission model for wormlike micellar solutions I: Model formulation and homogeneous flow predictions
,”
J. Non-Newtonian Fluid Mech.
144
,
122
139
(
2007
).
21.
Bautista
,
F.
,
J. F. A.
Soltero
,
J. H.
Perez-Lopez
,
J. E.
Puig
, and
O.
Manero
, “
On the shear banding flow of elongated micellar solutions
,”
J. Non-Newtonian Fluid Mech.
94
,
57
66
(
2000
).
22.
Bautista
,
F.
,
J. F. A.
Soltero
,
E. R.
Macias
,
J. E.
Puig
, and
O.
Manero
, “
Irreversible thermodynamics approach and modeling of shear-banding flow of wormlike micelles
,”
J. Phys. Chem. B
106
,
13018
13026
(
2002
).
23.
Cromer
,
M.
,
L. P.
Cook
, and
G. H.
McKinley
, “
Extensional flow of wormlike micellar solutions
,”
Chem. Eng. Sci.
64
,
4588
4596
(
2009
).
24.
Cromer
,
M.
,
L. P.
Cook
, and
G. H.
McKinley
, “
Pressure-driven flow of wormlike micellar solutions in rectilinear microchannels
,”
J. Non-Newtonian Fluid Mech.
166
,
180
193
(
2011
).
25.
Cromer
,
M.
,
L. P.
Cook
, and
G. H.
McKinley
, “
Interfacial instability of pressure-driven channel flow for a two-species model of entangled wormlike micellar solutions
,”
J. Non-Newtonian Fluid Mech.
166
,
566
577
(
2011
).
26.
Cromer
,
M.
, and
L.
Pamela Cook
, “
A study of pressure-driven flow of wormlike micellar solutions through a converging/diverging channel
,”
J. Rheol.
60
,
953
972
(
2016
).
27.
Zhou
,
L.
,
P. A.
Vasquez
,
L. P.
Cook
, and
G. H.
McKinley
, “
Modeling the inhomogeneous response and formation of shear bands in steady and transient flows of entangled liquids
,”
J. Rheol.
52
,
591
623
(
2008
).
28.
Zhou
,
L.
,
L. P.
Cook
, and
G. H.
McKinley
, “
Probing shear-banding transition of a model of entangled wormlike micellar solutions using large amplitude oscillatory shearing (LAOS) deformation
,”
J. Non-Newtonian Fluid Mech.
165
,
1462
1472
(
2010
).
29.
Germann
,
N.
,
A. K.
Gurnon
,
L.
Zhou
,
L. P.
Cook
,
A. N.
Beris
,
N. J.
Wagner
,
A. K.
Gurnon
, and
L. P.
Cook
, “
Validation of constitutive modeling of shear banding, threadlike wormlike micellar fluids
,”
J. Rheol.
60
,
983
999
(
2016
).
30.
Zhou
,
L.
,
G. H.
McKinley
, and
L. P.
Cook
, “
Wormlike micellar solutions: III. VCM model predictions in steady and transient shearing flows
,”
J. Non-Newtonian Fluid Mech.
211
,
70
83
(
2014
).
31.
Varchanis
,
S.
,
S. J.
Haward
,
C. C.
Hopkins
,
J.
Tsamopoulos
, and
A. Q.
Shen
, “
Evaluation of constitutive models for shear-banding wormlike micellar solutions in simple and complex flows
,”
J. Non-Newtonian Fluid Mech.
307
,
104855
(
2022
).
32.
Giesekus
,
H.
, “
A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility
,”
J. Non-Newtonian Fluid Mech.
11
,
69
109
(
1982
).
33.
Wiest
,
J. M.
, and
R. B.
Bird
, “
Molecular extension from the Giesekus model
,”
J. Non-Newtonian Fluid Mech.
22
,
115
119
(
1986
).
34.
Cates
,
M. E.
, “
Reptation of living polymers: Dynamics of entangled polymers in the presence of reversible chain-scission reactions
,”
Macromolecules
20
,
2289
2296
(
1987
).
35.
Likhtman
,
A. E.
, “
Single-chain slip-link model of entangled polymers: Simultaneous description of neutron spin-echo, rheology, and diffusion
,”
Macromolecules
38
,
6128
6139
(
2005
).
36.
Liu
,
C.
, and
D. J.
Pine
, “
Shear-induced gelation and fracture in micellar solutions
,”
Phys. Rev. Lett.
77
,
2121
–2124 (
1996
).
37.
Hu
,
Y. T.
,
P.
Boltenhagen
, and
D. J.
Pine
, “
Shear thickening in low-concentration solutions of wormlike micelles. I. Direct visualization of transient behavior and phase transitions
,”
J. Rheol.
42
,
1185
1208
(
1998
).
38.
Wang
,
Y.
,
T. F.
Zhang
, and
C.
Liu
, “
A two species micro–macro model of wormlike micellar solutions and its maximum entropy closure approximations: An energetic variational approach
,”
J. Non-Newtonian Fluid Mech.
293
,
104559
(
2021
).
39.
Wang
,
S. Q.
, “
Growth of dynamic polymers (micelles) in shear flow
,”
Macromolecules
24
,
3004
3009
(
1991
).
40.
Rothstein
,
J. P.
, “
Transient extensional rheology of wormlike micelle solutions
,”
J. Rheol.
47
,
1227
1247
(
2003
).
41.
Mandal
,
T.
, and
R. G.
Larson
, “
Stretch and breakage of wormlike micelles under uniaxial strain: A simulation study and comparison with experimental results
,”
Langmuir
34
,
12600
12608
(
2018
).
42.
Salipante
,
P. F.
,
V. L.
Dharmaraj
, and
S. D.
Hudson
, “
Entrance effects and high shear rate rheology of shear-banding wormlike micelle fluids in a microcapillary flow
,”
J. Rheol.
64
,
481
492
(
2020
).
43.
Padding
,
J. T.
,
E. S.
Boek
, and
W. J.
Briels
, “
Dynamics and rheology of wormlike micelles emerging from particulate computer simulations
,”
J. Chem. Phys.
129
,
74903
(
2008
).
44.
Weston
,
J.
,
D.
Seeman
,
D.
Blair
,
P.
Salipante
,
S.
Hudson
, and
K.
Weigandt
, “
Simultaneous slit rheometry and in situ neutron scattering
,”
Rheol. Acta
57
,
241
250
(
2018
).
45.
Murphy
,
R. P.
,
Z. W.
Riedel
,
M. A.
Nakatani
,
P. F.
Salipante
,
J. S.
Weston
,
S. D.
Hudson
, and
K. M.
Weigandt
, “
Capillary RheoSANS: Measuring the rheology and nanostructure of complex fluids at high shear rates
,”
Soft Matter
16
,
6285
6293
(
2020
).
46.
Couillet
,
I.
,
T.
Hughes
,
G.
Maitland
,
F.
Candau
, and
S.
Jean Candau
, “
Growth and scission energy of wormlike micelles formed by a cationic surfactant with long unsaturated tails
,”
Langmuir
20
,
9541
9550
(
2004
).
47.
Kröger
,
M.
, and
R.
Makhloufi
, “
Wormlike micelles under shear flow: A microscopic model studied by nonequilibrium-molecular-dynamics computer simulations
,”
Phys. Rev. E
53
,
2531
–2536 (
1996
).
48.
Arenas-Gómez
,
B.
,
C.
Garza
,
Y.
Liu
, and
R.
Castillo
, “
Alignment of worm-like micelles at intermediate and high shear rates
,”
J. Colloid Interface Sci.
560
,
618
625
(
2020
).
49.
Koide
,
Y.
, and
S.
Goto
, “
Effect of scission on alignment of nonionic surfactant micelles under shear flow
,”
Soft Matter
19
,
4323
4332
(
2023
).
50.
Vasquez
,
P. A.
,
G. H.
McKinley
, and
L.
Pamela Cook
, “
A network scission model for wormlike micellar solutions. I. Model formulation and viscometric flow predictions
,”
J. Non-Newtonian Fluid Mech.
144
,
122
139
(
2007
).
51.
Steller
,
R.
, “
Determination of the first normal stress difference from viscometric data for shear flows of polymer liquids
,”
Rheol. Acta
55
,
649
656
(
2016
).
52.
Hommel
,
R. J.
, and
M. D.
Graham
, “
Constitutive modeling of dilute wormlike micelle solutions: Shear-induced structure and transient dynamics
,”
J. Non-Newtonian Fluid Mech.
295
,
104606
(
2021
).
53.
Berret
,
J. F.
, “Rheology of wormlike micelles: Equilibrium properties and shear banding transitions,” in Molecular Gels: Materials with Self-Assembled Fibrillar Networks (Springer, Dordrecht, 2006), pp. 667–720.
54.
O’Shaughnessy
,
B.
, and
J.
Yu
, “
Rheology of wormlike micelles: Two universality classes
,”
Phys. Rev. Lett.
74
,
4329
4332
(
1995
).
55.
Nyrkova
,
I. A.
, and
A. N.
Semenov
, “
Correlation effects in dynamics of living polymers
,”
Europhys. Lett.
79
,
66007
(
2007
).
56.
Huang
,
C. C.
,
J. P.
Ryckaert
, and
H.
Xu
, “
Structure and dynamics of cylindrical micelles at equilibrium and under shear flow
,”
Phys. Rev. E
79
,
041501
(
2009
).
57.
Padding
,
J. T.
, and
E. S.
Boek
, “
Evidence for diffusion-controlled recombination kinetics in model wormlike micelles
,”
Europhys. Lett.
66
,
756
–762 (
2004
).
58.
Granek
,
R.
, and
M. E.
Cates
, “
Stress relaxation in living polymers: Results from a poisson renewal model
,”
J. Chem. Phys.
96
,
4758
4767
(
1992
).
59.
Hassan
,
P. A.
,
S. J.
Candau
,
F.
Kern
, and
C.
Manohar
, “
Rheology of wormlike micelles with varying hydrophobicity of the counterion
,”
Langmuir
14
,
6025
6029
(
1998
).
60.
Schuss
,
Z.
,
A.
Singer
, and
D.
Holcman
, “
The narrow escape problem for diffusion in cellular microdomains
,”
Proc. Natl. Acad. Sci. U. S. A.
104
,
16098
16103
(
2007
).
61.
Berret
,
J. F.
, “
Transient rheology of wormlike micelles
,”
Langmuir
13
,
2227
2234
(
1997
).
62.
Moghadam
,
S.
,
I.
Saha Dalal
, and
R. G.
Larson
, “
Unraveling dynamics of entangled polymers in strong extensional flows
,”
Macromolecules
52
,
1296
1307
(
2019
).
63.
Larson
,
R. G.
, and
P. S.
Desai
, “
Modeling the rheology of polymer melts and solutions
,”
Ann. Rev. Fluid Mech.
47
,
47
65
(
2015
).
64.
Likhtman
,
A. E.
, and
R. S.
Graham
, “
Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie-Poly equation
,”
J. Non-Newtonian Fluid Mech.
114
,
1
12
(
2003
).
65.
Graham
,
R. S.
,
A. E.
Likhtman
,
T. C. B.
McLeish
, and
S. T.
Milner
, “
Microscopic theory of linear, entangled polymer chains under rapid deformation including chain stretch and convective constraint release
,”
J. Rheol.
47
,
1171
1200
(
2003
).
66.
Bird
,
R. B.
,
R. C.
Armstrong
, and
O.
Hassager
,
Dynamics of Polymeric Liquids: Vol. 1, Fluid Mechanics
, 2nd ed. (
Wiley
,
New York
,
1987
).
67.
Larson
,
R. G.
,
T.
Sridhar
,
L. G.
Leal
,
G. H.
McKinley
,
A. E.
Likhtman
, and
T. C. B.
McLeish
, “
Definitions of entanglement spacing and time constants in the tube model
,”
J. Rheol.
47
,
809
818
(
2003
).
68.
Doi
,
M.
, and
S. F.
Edwards
,
The Theory of Polymer Dynamics
(
Oxford University
, Oxford
1986
).
69.
Yesilata
,
B.
,
C.
Clasen
, and
G. H.
McKinley
, “
Nonlinear shear and extensional flow dynamics of wormlike surfactant solutions
,”
J. Non-Newtonian Fluid Mech.
133
,
73
90
(
2006
).
70.
Peterson
,
J. D.
,
M.
Cromer
,
G. H.
Fredrickson
, and
L.
Gary Leal
, “
Shear banding predictions for the two-fluid Rolie-Poly model
,”
J. Rheol.
60
,
927
951
(
2016
).
71.
Stephanou
,
P. S.
,
I. C.
Tsimouri
, and
V. G.
Mavrantzas
, “
Flow-induced orientation and stretching of entangled polymers in the framework of nonequilibrium thermodynamics
,”
Macromolecules
49
,
3161
3173
(
2016
).
72.
Peterson
,
J. D.
,
M. E.
Cates
, and
M. E.
Cates
, “
A full-chain tube-based constitutive model for living linear polymers
,”
J. Rheol.
64
,
1465
1496
(
2020
).
73.
Frischknecht
,
A. L.
, and
S. T.
Milner
, “
Diffusion with contour length fluctuations in linear polymer melts
,”
Macromolecules
33
,
5273
5277
(
2000
).
74.
Shikata
,
T.
,
S. J.
Dahman
, and
D. S.
Pearson
, “
Rheo-optical behavior of wormlike micelles
,”
Langmuir
10
,
3470
3476
(
1994
).
75.
Pincus
,
I.
,
A.
Rodger
, and
J. R.
Prakash
, “
Viscometric functions and rheo-optical properties of dilute polymer solutions: Comparison of FENE-Fraenkel dumbbells with rodlike models
,”
J. Non-Newtonian Fluid Mech.
285
,
104395
(
2020
).
76.
Hinch
,
E. J.
, and
L. G.
Leal
, “
The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles
,”
J. Fluid Mech.
52
,
683
712
(
1972
).
77.
Brenner
,
H.
, “
Rheology of a dilute suspension of axisymmetric Brownian particles
,”
Int. J. Multiphase Flow
1
,
195
341
(
1974
).
78.
Batchelor
,
G. K.
, “
Slender-body theory for particles of arbitrary cross-section in Stokes flow
,”
J. Fluid Mech.
44
,
419
440
(
1970
).
79.
Larson
,
R. G.
,
The Structure and Rheology of Complex Fluids
(
Oxford University
,
New York
,
1999
).
80.
Petrie
,
C. J.
, “
The rheology of fibre suspensions
,”
J. Non-Newtonian Fluid Mech.
87
,
369
402
(
1999
).
81.
Jack
,
D. A.
, and
D. E.
Smith
, “
The effect of fibre orientation closure approximations on mechanical property predictions
,”
Compos. A: Appl. Sci. Manuf.
38
,
975
982
(
2007
).
82.
Dhont
,
J. K. G.
, and
W. J.
Briels
, “
Viscoelasticity of suspensions of long, rigid rods
,”
Colloids Surf. A
213
,
131
156
(
2003
).
83.
Kröger
,
M.
,
A.
Ammar
, and
F.
Chinesta
, “
Consistent closure schemes for statistical models of anisotropic fluids
,”
J. Non-Newtonian Fluid Mech.
149
,
40
55
(
2008
).
84.
Kim
,
Y. C.
,
A.
Bénard
, and
C. A.
Petty
, “
Microstructure and rheology of rigid rod suspensions
,”
Ind. Eng. Chem. Res.
54
,
4497
4504
(
2015
).
85.
Corona
,
P. T.
,
K.
Dai
,
M. E.
Helgeson
, and
L. G.
Leal
, “
Testing orientational closure approximations in dilute and non-dilute suspensions with Rheo-SANS
,”
J. Non-Newtonian Fluid Mech.
315
,
105014
(
2023
).
86.
Adams
,
J. M.
,
S. M.
Fielding
, and
P. D.
Olmsted
, “
Transient shear banding in entangled polymers: A study using the Rolie-Poly model
,”
J. Rheol.
55
,
1007
1032
(
2011
).
87.
Adams
,
J. M.
, and
P. D.
Olmsted
, “
Nonmonotonic models are not necessary to obtain shear banding phenomena in entangled polymer solutions
,”
Phys. Rev. Lett.
102
,
067801
(
2009
).
88.
Lonetti
,
B.
,
J.
Kohlbrecher
,
L.
Willner
,
J. K.
Dhont
, and
M. P.
Lettinga
, “
Dynamic response of block copolymer wormlike micelles to shear flow
,”
J. Phys.: Condens. Matter
20
,
404207
(
2008
).
89.
Castelletto
,
V.
,
P.
Parras
,
I. W.
Hamley
,
P.
Bäverbäck
,
J. S.
Pedersen
, and
P.
Panine
, “
Wormlike micelle formation and flow alignment of a pluronic block copolymer in aqueous solution
,”
Langmuir
23
,
6896
6902
(
2007
).
90.
Moorcroft
,
R. L.
, and
S. M.
Fielding
, “
Shear banding in time-dependent flows of polymers and wormlike micelles
,”
J. Rheol.
58
,
103
147
(
2014
).
91.
Inoue
,
T.
,
Y.
Inoue
, and
H.
Watanabe
, “
Nonlinear rheology of CTAB/NaSal aqueous solutions: Finite extensibility of a network of wormlike micelles
,”
Langmuir
21
,
1201
1208
(
2005
).
92.
Pasquino
,
R.
,
P. R.
Avallone
,
V.
Ianniello
,
G.
Ianniruberto
,
G.
Marrucci
,
N.
Grizzuti
,
P. R.
Avallone
,
S.
Costanzo
,
I.
Inbal
, and
D.
Danino
, “
On the startup behavior of wormlike micellar networks: The effect of different salts bound to the same surfactant molecule
,”
J. Rheol.
67
,
353
364
(
2023
).
93.
Ianniruberto
,
G.
, and
G.
Marrucci
, “
Convective constraint release (CCR) revisited
,”
J. Rheol.
58
,
89
102
(
2014
).
94.
Thien
,
N. P.
, and
R. I.
Tanner
, “
A new constitutive equation derived from network theory
,”
J. Non-Newtonian Fluid Mech.
2
,
353
365
(
1977
).
95.
Bishko
,
G.
,
T. C. B.
McLeish
,
O. G.
Harlen
, and
R. G.
Larson
, “
Theoretical molecular rheology of branched polymers in simple and complex flows: The pom-pom model
,”
Phys. Rev. Lett.
79
,
2352
–2355 (
1997
).
96.
Salipante
,
P. F.
,
M.
Cromer
, and
S. D.
Hudson
, “
Two-species model for nonlinear flow of wormlike micelle solutions. II: Experiment
,”
J. Rheol.
68
,
895
–911 (
2024
).
You do not currently have access to this content.