To meet the challenge of efficient modeling of film blowing with realistic constitutive equations for commercial thermoplastic melts, we present a multistage optimization modeling framework that integrates polymerization reaction modeling, rheology modeling, and bubble-shape prediction. A direct link is thereby created between the polymer architecture and the bubble shape of low-density polyethylene (LDPE) through a three-stage modeling protocol. Stage 1 aims to get complete polymer structure information from a limited set of linear and nonlinear rheological data and the measured averaged molecular weight. An optimization loop uses the Tobita algorithm for polymer reaction and the BoB model for rheology to minimize the deviation between experimental data and model predictions. Stage 2 is designed to obtain a representative reduced ensemble of LDPE in the Rolie-double-poly (RDP) model to reduce the computational cost of rheology calculations during processing. The parameters of the reduced molecular components are obtained by fitting the RDP model to a wide range of rheology data predicted by the BoB model applied to the full ensemble of polymer architectures obtained in stage 1. In stage 3, the reduced-ensemble RDP model is coupled to measured temperature profiles using time–temperature superposition, and the bubble shape and strain rate history of a fluid particle in the bubble are obtained by minimizing error in the momentum balance equations. We show that each stage of the process yields successful fitting, and at the end, we obtain an a priori prediction of height-dependent bubble radius and velocity in agreement with experiment. With this multistage optimization strategy, we link the polymer compositions to the bubble properties during the film blowing of LDPE.

1.
Laverde
,
G.
, “
Agricultural films: Types and applications
,”
J. Plastic Film Sheeting
18
,
269
277
(
2002
).
2.
Zhang
,
Q.
,
W.
Chen
,
H.
Zhao
,
Y.
Ji
,
L.
Meng
,
D.
Wang
, and
L.
Li
, “
In-situ tracking polymer crystallization during film blowing by synchrotron radiation X-ray scattering: The critical role of network
,”
Polymer
198
,
122492
(
2020
).
3.
Münstedt
,
H.
,
T.
Steffl
, and
A.
Malmberg
, “
Correlation between rheological behaviour in uniaxial elongation and film blowing properties of various polyethylenes
,”
Rheol. Acta
45
,
14
22
(
2005
).
4.
Zhang
,
X.
,
A.
Ajji
, and
V.
Jean-Marie
, “
Processing–structure–properties relationship of multilayer films. 1. Structure characterization
,”
Polymer
42
,
8179
8195
(
2001
).
5.
Kundu
,
P. P.
,
J.
Biswas
,
H.
Kim
, and
S.
Choe
, “
Influence of film preparation procedures on the crystallinity, morphology and mechanical properties of LLDPE films
,”
Eur. Polym. J.
39
,
1585
1593
(
2003
).
6.
La Mantia
,
F. P.
,
R.
Scaffaro
,
G.
Carianni
, and
P.
Mariani
, “
Rheological properties of different film blowing polyethylene samples under shear and elongational flow
,”
Macromol. Mater. Eng.
290
,
159
164
(
2005
).
7.
Sidiropoulos
,
V.
, and
J.
Vlachopoulos
, “
Temperature gradients in blown film bubbles
,”
Adv. Polym. Technol.
24
,
83
90
(
2005
).
8.
Sidiropoulos
,
V.
, and
J.
Vlachopoulos
, “
Numerical study of internal bubble cooling (IBC) in film blowing
,”
Int. Polym. Process.
16
,
48
53
(
2001
).
9.
Sidiropoulos
,
V.
, and
J.
Vlachopoulos
, “
The effects of dual-orifice air-ring design on blown film cooling
,”
Polym. Eng. Sci.
40
,
1611
1618
(
2000
).
10.
Huang
,
T. A.
, and
G. A.
Campbell
, “
Deformational history of LLDPE/LDPE blends on blown film equipment
,”
Adv. Polym. Technol.
5
,
181
192
(
1985
).
11.
van Drongelen
,
M.
,
D.
Cavallo
,
L.
Balzano
,
G.
Portale
,
I.
Vittorias
,
W.
Bras
,
G. C.
Alfonso
, and
G. W. M.
Peters
, “
Structure development of low-density polyethylenes during film blowing: A real-time wide-angle X-ray diffraction study
,”
Macromol. Mater. Eng.
299
,
1494
1512
(
2014
).
12.
Troisi
,
E. M.
,
M.
van Drongelen
,
H. J. M.
Caelers
,
G.
Portale
, and
G. W. M.
Peters
, “
Structure evolution during film blowing: An experimental study using in-situ small angle X-ray scattering
,”
Eur. Polym. J.
74
,
190
208
(
2016
).
13.
Zhao
,
H.
,
Q.
Zhang
,
L.
Li
,
W.
Chen
,
D.
Wang
,
L.
Meng
, and
L.
Li
, “
Synergistic and competitive effects of temperature and flow on crystallization of polyethylene during film blowing
,”
ACS Appl. Polym. Mater.
1
,
1590
1603
(
2019
).
14.
Tadmor
,
Z.
, and
C. G.
Gogos
,
Principles of Polymer Processing
(
John Wiley & Sons
, Hoboken,
2006
).
15.
Pearson
,
J. R. A.
, and
C. J. S.
Petrie
, “
The flow of a tubular film part 2. Interpretation of the model and discussion of solutions
,”
J. Fluid Mech.
42
,
609
625
(
1970
).
16.
Pearson
,
J. R. A.
, and
C. J. S.
Petrie
, “
The flow of a tubular film. Part 1. Formal mathematical representation
,”
J. Fluid Mech.
40
,
1
–19 (
1970
).
17.
Beaulne
,
M.
, and
E.
Mitsoulis
, “
Effect of viscoelasticity in the film-blowing process
,”
J. Appl. Polym. Sci.
105
,
2098
2112
(
2007
).
18.
Tas
,
P. P.
,
Film Blowing: From Polymer to Product
(
Technische Universiteit Eindhoven
, Eindhoven,
1994
).
19.
Sarafrazi
,
S.
, and
F.
Sharif
, “
Non-isothermal simulation of the film blowing process using multi-mode extended Pom-Pom model
,”
Int. Polym. Process.
23
,
30
37
(
2008
).
20.
Doufas
,
A. K.
, “
A microstructural flow-induced crystallization model for film blowing: Validation with experimental data
,”
Rheol. Acta
53
,
269
293
(
2014
).
21.
Doufas
,
A. K.
, and
A. J.
McHugh
, “
Simulation of film blowing including flow-induced crystallization
,”
J. Rheol.
45
,
1085
1104
(
2001
).
22.
Dietz
,
W.
, “
Phase transition during film blowing of LDPE. Part I: From viscoelastic melt to Neo-hookean solid
,”
J. Rheol.
62
,
1515
1532
(
2018
).
23.
Campbell
,
G. A.
, and
B.
Cao
, “
The interaction of crystallinity, elastoplasticity, and a two-phase model on blown film bubble shape
,”
J. Plastic Film Sheeting
3
,
158
170
(
1987
).
24.
Read
,
D. J.
,
D.
Auhl
,
C.
Das
,
J.
Den Doelder
,
M.
Kapnistos
,
I.
Vittorias
, and
T. C. B.
McLeish
, “
Linking models of polymerization and dynamics to predict branched polymer structure and flow
,”
Science
333
,
1871
1874
(
2011
).
25.
Das
,
C.
,
D. J.
Read
,
D.
Auhl
,
M.
Kapnistos
,
J.
Den Doelder
,
I.
Vittorias
, and
T. C. B.
McLeish
, “
Numerical prediction of nonlinear rheology of branched polymer melts
,”
J. Rheol.
58
,
737
757
(
2014
).
26.
Tobita
,
H.
, “
Simultaneous long-chain branching and random scission: I. Monte carlo simulation
,”
J. Polym. Sci. Part B: Polym. Phys.
39
,
391
403
(
2001
).
27.
McLeish
,
T. C. B.
, and
R. G.
Larson
, “
Molecular constitutive equations for a class of branched polymers: The Pom-Pom polymer
,”
J. Rheol.
42
,
81
110
(
1998
).
28.
Gong
,
Y.
,
V.
Ginzburg
,
S.
Vervoort
,
J.
Den Doelder
, and
R. G.
Larson
, “
Strategy for reducing molecular ensemble size for efficient rheological modeling of commercial polymers
,”
J. Rheol.
65
,
43
57
(
2021
).
29.
Zhang
,
W.
, and
R. G.
Larson
, “
Effect of flow-induced nematic order on polyethylene crystal nucleation
,”
Macromolecules
53
,
7650
7657
(
2020
).
30.
Zhang
,
W.
,
E. D.
Gomez
, and
S. T.
Milner
, “
Predicting nematic phases of semiflexible polymers
,”
Macromolecules
48
,
1454
1462
(
2015
).
31.
Bourque
,
A.
,
C. R.
Locker
, and
G. C.
Rutledge
, “
Molecular dynamics simulation of surface nucleation during growth of an alkane crystal
,”
Macromolecules
49
,
3619
3629
(
2016
).
32.
Read
,
D. J.
,
C.
McIlroy
,
C.
Das
,
O. G.
Harlen
, and
R. S.
Graham
, “
PolySTRAND model of flow-induced nucleation in polymers
,”
Phys. Rev. Lett.
124
,
147802
(
2020
).
33.
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
).
34.
Warner
, Jr.,
H. R.
, “
Kinetic theory and rheology of dilute suspensions of finitely extendible dumbbells
,”
Ind. Eng. Chem. Fundam.
11
,
379
387
(
1972
).
35.
Likhtman
,
A. E.
, and
T. C. B.
McLeish
, “
Quantitative theory for linear dynamics of linear entangled polymers
,”
Macromolecules
35
,
6332
6343
(
2002
).
36.
Blackwell
,
R. J.
,
T. C.
McLeish
, and
O. G.
Harlen
, “
Molecular drag–strain coupling in branched polymer melts
,”
J. Rheol.
44
,
121
136
(
2000
).
37.
Verbeeten
,
W. M. H.
,
G. W. M.
Peters
, and
F. P. T.
Baaijens
, “
Differential constitutive equations for polymer melts: The extended Pom–Pom model
,”
J. Rheol.
45
,
823
843
(
2001
).
38.
Das
,
C.
,
N. J.
Inkson
,
D. J.
Read
,
M. A.
Kelmanson
, and
T. C. B.
McLeish
, “
Computational linear rheology of general branch-on-branch polymers
,”
J. Rheol.
50
,
207
234
(
2006
).
39.
Bennett
,
J. C.
,
Mathematical Analysis of Film Blowing
(
RMIT University
, Melbourne,
2008
).
40.
Leonova
,
T. I.
, Film Blowing Modeling to Enhance Film Property Prediction (Technische Universiteit Eindhoven, Eindhoven, 2015).
41.
Muke
,
S.
,
H.
Connell
,
I.
Sbarski
, and
S. N.
Bhattacharya
, “
Numerical modelling and experimental verification of blown film processing
,”
J. Non-Newtonian Fluid Mech.
116
,
113
138
(
2003
).
42.
Dealy
,
J. M.
,
D. J.
Read
, and
R. G.
Larson
,
Structure and Rheology of Molten Polymers: From Structure to Flow Behavior and Back Again
(Hanser Publishers, Munich, 2018).
43.
Kanai
,
T.
, and
G. A.
Campbell
, Film Processing Advances (Hanser Publications, Cincinnati, 2014).
44.
Fuller
,
G. G.
,
C. A.
Cathey
,
B.
Hubbard
, and
B. E.
Zebrowski
, “
Extensional viscosity measurements for low-viscosity fluids
,”
J. Rheol.
31
,
235
249
(
1987
).
45.
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.
Boudara
,
V. A. H.
,
D. J.
Read
, and
J.
Ramírez
, “
Reptate rheology software: Toolkit for the analysis of theories and experiments
,”
J. Rheol.
64
,
709
722
(
2020
).
48.
Luo
,
X.-L.
, and
R. I.
Tanner
, “
A computer study of film blowing
,”
Polym. Eng. Sci.
25
,
620
629
(
1985
).
49.
Kurtz
,
S. J.
, “
Relationship of stresses in blown-film processes
,”
Int. Polym. Process.
10
,
148
154
(
1995
).
50.
See supplementary material online three different methods for evaluating λ max , i in the Rolie-double-poly (RDP) model with the “representative reduced ensemble” (RRE). It also compares rheological predictions with different molecular ensemble sizes in the “Branch-on-Branch” (BoB) model.

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