The ability to accurately predict the rheological behavior of the blends of two incompatible polymers is critical to the polymer industry. The constitutive modeling of incompatible polymer blends requires understanding the structure and dynamics of the blends across different length scales. The polydispersity of chain length at the molecular level and nonuniformity of flow field due to dispersed domains at the mesoscopic level present significant challenges to this industrially relevant problem. This work proposes a modeling framework for linear and nonlinear rheology of industrial incompatible polymer blends with sea-island morphology. For the individual components, we adopt the Rolie-Double-Poly model and generate the relaxation spectrum from an optimized molecular weight distribution. We derive a new mixing rule without empirical parameters from the flow field analysis inside and outside the droplets. The phase interface, modeled by an ellipsoidal model, contributes to the apparent rheology only at low shear rates. Our modeling approach is verified by the shear and extensional rheology of eight polymer blends with a broad range of viscosity ratios (0.01–100). We also show that the model has the ability to predict the nonlinear rheological behaviors of incompatible polymer blends with known molecular weight distributions and phase morphology.

1.
Doi
,
M.
, and
S. F.
Edwards
,
The Theory of Polymer Dynamics
(
Oxford University
,
Oxford
,
1986
).
2.
Narimissa
,
E.
, and
M. H.
Wagner
, “
Review on tube model based constitutive equations for polydisperse linear and long-chain branched polymer melts
,”
J. Rheol.
63
,
361
375
(
2019
).
3.
Mykhaylyk
,
O. O.
,
C. M.
Fernyhough
,
M.
Okura
,
J. P. A.
Fairclough
,
A. J.
Ryan
, and
R.
Graham
, “
Monodisperse macromolecules – A stepping stone to understanding industrial polymers
,”
Eur. Polym. J.
47
,
447
464
(
2011
).
4.
McLeish
,
T. C. B.
,
J.
Allgaier
,
D. K.
Bick
,
G.
Bishko
,
P.
Biswas
,
R.
Blackwell
,
B.
Blottière
,
N.
Clarke
,
B.
Gibbs
,
D. J.
Groves
et al, “
Dynamics of entangled H-polymers:  Theory, rheology, and neutron-scattering
,”
Macromolecules
32
,
6734
6758
(
1999
).
5.
Daniels
,
D. R.
,
T. C. B.
McLeish
,
B. J.
Crosby
,
R. N.
Young
, and
C. M.
Fernyhough
, “
Molecular rheology of comb polymer melts. 1. Linear viscoelastic response
,”
Macromolecules
34
,
7025
7033
(
2001
).
6.
Inkson
,
N. J.
,
R. S.
Graham
,
T. C. B.
McLeish
,
D. J.
Groves
, and
C. M.
Fernyhough
, “
Viscoelasticity of monodisperse comb polymer melts
,”
Macromolecules
39
,
4217
4227
(
2006
).
7.
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
).
8.
Han
,
C. D.
,
Rheology and Processing of Polymeric Materials: Volume 2: Polymer Processing
(
Oxford University
,
New York
,
2006
).
9.
Cloizeaux
,
J. d.
, “
Double reptation vs. simple reptation in polymer melts
,”
Europhys. Lett.
5
,
437
442
(
1988
).
10.
van Ruymbeke
,
E.
,
R.
Keunings
,
V.
Stéphenne
,
A.
Hagenaars
, and
C.
Bailly
, “
Evaluation of reptation models for predicting the linear viscoelastic properties of entangled linear polymers
,”
Macromolecules
35
,
2689
2699
(
2002
).
11.
Boudara
,
V. A. H.
,
J. D.
Peterson
,
L. G.
Leal
, and
D. J.
Read
, “
Nonlinear rheology of polydisperse blends of entangled linear polymers: Rolie-Double-Poly models
,”
J. Rheol.
63
,
71
91
(
2019
).
12.
Zhang
,
J.
,
A.
Jurzyk
,
M. E.
Helgeson
, and
L. G.
Leal
, “
Modeling orthogonal superposition rheometry to probe nonequilibrium dynamics of entangled polymers
,”
J. Rheol.
65
,
983
998
(
2021
).
13.
Yu
,
W.
,
M.
Bousmina
,
M.
Grmela
,
J.-F.
Palierne
, and
C.
Zhou
, “
Quantitative relationship between rheology and morphology in emulsions
,”
J. Rheol.
46
,
1381
1399
(
2002
).
14.
Palierne
,
J. F.
, “
Linear rheology of viscoelastic emulsions with interfacial tension
,”
Rheol. Acta
29
,
204
214
(
1990
).
15.
Yu
,
W.
,
M.
Bousmina
,
C.
Zhou
, and
C. L.
Tucker
III
, “
Theory for drop deformation in viscoelastic systems
,”
J. Rheol.
48
,
417
438
(
2004
).
16.
Yu
,
W.
,
C.
Zhou
, and
M.
Bousmina
, “
Theory of morphology evolution in mixtures of viscoelastic immiscible components
,”
J. Rheol.
49
,
215
236
(
2005
).
17.
Cox
,
R. G.
, “
The deformation of a drop in a general time-dependent fluid flow
,”
J. Fluid Mech.
37
,
601
623
(
1969
).
18.
Guo
,
Y.
,
W.
Yu
,
Y.
Xu
, and
C.
Zhou
, “
Correlations between local flow mechanism and macroscopic rheology in concentrated suspensions under oscillatory shear
,”
Soft Matter
7
,
2433
2443
(
2011
).
19.
Frankel
,
N. A.
, and
A.
Acrivos
, “
The constitutive equation for a dilute emulsion
,”
J. Fluid Mech.
44
,
65
78
(
1970
).
20.
Yu
,
W.
, and
M.
Bousmina
, “
Ellipsoidal model for droplet deformation in emulsions
,”
J. Rheol.
47
,
1011
1039
(
2003
).
21.
Lamb
,
H.
,
Hydrodynamics, 6th ed.
(
Cambridge University
,
Cambridge
,
1932
).
22.
Choi
,
S. J.
, and
W. R.
Schowalter
, “
Rheological properties of nondilute suspensions of deformable particles
,”
Phys. Fluids
18
, 420–427 (
1975
).
23.
Mwasame
,
P. M.
,
N. J.
Wagner
, and
A. N.
Beris
, “
On the macroscopic modelling of dilute emulsions under flow
,”
J. Fluid Mech.
831
,
433
473
(
2017
).
24.
Deng
,
L.
,
S.
Fan
,
Y.
Zhang
,
Z.
Huang
,
H.
Zhou
,
S.
Jiang
, and
J.
Li
, “
Multiscale modeling and simulation of polymer blends in injection molding: A review
,”
Polymers
13, 3783 (2021).
25.
Doi
,
M.
, and
T.
Ohta
, “
Dynamics and rheology of complex interfaces. I
,”
J. Chem. Phys.
95
,
1242
1248
(
1991
).
26.
Yu
,
W.
,
M.
Bousmina
,
M.
Grmela
, and
C.
Zhou
, “
Modeling of oscillatory shear flow of emulsions under small and large deformation fields
,”
J. Rheol.
46
,
1401
1418
(
2002
).
27.
Almusallam
,
A. S.
, and
T. B.
Bini
, “
Modeling vorticity stretching of viscoelastic droplets during shearing flow
,”
J. Rheol.
65
,
1327
1345
(
2021
).
28.
Minale
,
M.
, “
Deformation of a non-Newtonian ellipsoidal drop in a non-Newtonian matrix: Extension of Maffettone–Minale model
,”
J. Non-Newtonian Fluid Mech.
123
,
151
160
(
2004
).
29.
Dressler
,
M.
, and
B. J.
Edwards
, “
Rheology of polymer blends with matrix-phase viscoelasticity and a narrow droplet size distribution
,”
J. Non-Newtonian Fluid Mech.
120
,
189
205
(
2004
).
30.
Yu
,
W.
, and
C.
Zhou
, “
A simple constitutive equation for immiscible blends
,”
J. Rheol.
51
,
179
194
(
2007
).
31.
Huneault
,
M. A.
,
Z. H.
Shi
, and
L. A.
Utracki
, “
Development of polymer blend morphology during compounding in a twin-screw extruder. Part IV: A new computational model with coalescence
,”
Polym. Eng. Sci.
35
,
115
127
(
1995
).
32.
Maffettone
,
P. L.
, and
M.
Minale
, “
Equation of change for ellipsoidal drops in viscous flow
,”
J. Non-Newtonian Fluid Mech.
78
,
227
241
(
1998
).
33.
Mark
,
J. E.
,
Physical Properties of Polymers Handbook
(
Springer New York
,
2007
).
34.
Niu
,
H.
,
Y.
Wang
,
X.
Liu
,
Y.
Wang
, and
Y.
Li
, “
Determination of plateau moduli and entanglement molecular weights of ultra-high molecular weight isotactic polypropylene synthesized by Ziegler-Natta catalyst
,”
Polym. Test.
60
,
260
265
(
2017
).
35.
García-Franco
,
C. A.
,
B. A.
Harrington
, and
D. J.
Lohse
, “
Effect of short-chain branching on the rheology of polyolefins
,”
Macromolecules
39
,
2710
2717
(
2006
).
36.
García-Franco
,
C. A.
,
B. A.
Harrington
, and
D. J.
Lohse
, “
On the rheology of ethylene-octene copolymers
,”
Rheol. Acta
44
,
591
599
(
2005
).
37.
van Ruymbeke
,
E.
,
R.
Keunings
, and
C.
Bailly
, “
Determination of the molecular weight distribution of entangled linear polymers from linear viscoelasticity data
,”
J. Non-Newtonian Fluid Mech.
105
,
153
175
(
2002
).
38.
Cox
,
W. P.
, and
E. H.
Merz
, “
Correlation of dynamic and steady flow viscosities
,”
J. Polym. Sci.
28
,
619
622
(
1958
).
39.
Laun
,
H. M.
, “
Prediction of elastic strains of polymer melts in shear and elongation
,”
J. Rheol.
30
,
459
501
(
1986
).
40.
Hansen
,
M. G.
, and
F.
Nazem
, “
Transient normal force transducer response in a modified Weissenberg rheogoniometer
,”
Trans. Soc. Rheol.
19
,
21
36
(
1975
).
41.
Schweizer
,
T.
, and
A.
Bardow
, “
The role of instrument compliance in normal force measurements of polymer melts
,”
Rheol. Acta
45
,
393
402
(
2006
).
42.
Costanzo
,
S.
,
Q.
Huang
,
G.
Ianniruberto
,
G.
Marrucci
,
O.
Hassager
, and
D.
Vlassopoulos
, “
Shear and extensional rheology of polystyrene melts and solutions with the same number of entanglements
,”
Macromolecules
49
,
3925
3935
(
2016
).
43.
Carriere
,
C. J.
, and
H. C.
Silvis
, “
The effects of short-chain branching and comonomer type on the interfacial tension of polypropylene-polyolefin elastomer blends
,”
J. Appl. Polym. Sci.
66
,
1175
1181
(
1997
).
44.
Hinch
,
E. J.
, and
A.
Acrivos
, “
Long slender drops in a simple shear flow
,”
J. Fluid Mech.
98
,
305
328
(
1980
).
45.
Takahashi
,
Y.
, and
I.
Noda
, “
Domain structures and viscoelastic properties of immiscible polymer blends under shear flow
,” in Flow-Induced Structure in Polymers, ACS Symposium Series 597, edited by A. I. Nakatani and M. D. Dadmun (American Chemical Society, Washington, DC, 1995), pp. 140–152.
46.
Li
,
B.
,
W.
Yu
,
X.
Cao
, and
Q.
Chen
, “
Horizontal extensional rheometry (HER) for low viscosity polymer melts
,”
J. Rheol.
64
,
177
190
(
2020
).
47.
See supplementary material online for optimized discrete molecular weight distributions of PP and POE and corresponding viscoelastic master curves; effects of nonlinear parameter β CCR on the startup shear behaviors predicted by the RDP model; the relationship between the Rouse–Weissenberg number and stress overshoot; comparison between the RDP model and the Giesekus model; the distribution of droplet size of the rubber phase; damped fluctuations in transient viscosity due to the tumbling of droplets in blends with a high viscosity ratio; the effect of preshear on transient shear rheology; and blends containing long-chain branched polymer as the dispersed component.

Supplementary Material

You do not currently have access to this content.