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Numerical Studies of the Effect of Constitutive Model Parameters
as Reflecting Polymer Molecular Structure on Extrudate Swell
Nattapong Nithi-Uthai, Ica Manas-Zloczower*
Case Western Reserve University, Department of Macromolecular Science
2100 Adelbert Road, Cleveland, OH 44106, USA
*Email: ixm@po.cwru.edu
Fax: x1.216.368.4202
Received: 16.4.2002, Final version: 8.8.2002
Abstract:
PolyFlow, a software package based on the finite element method was employed to simulate the extrudate swell
for polybutadiene of various molecular weight (Mw) and molecular weight distribution (MWD). We calculated
the relaxation spectra for the different samples and then inserted the spectra into a standard K-BKZ constitutive model used in the numerical simulations. Accurate predictions of MWD confirm the completeness of frequency range in the oscillatory shear experimental data. In turn, the wholeness of relaxation spectra as substantiated by MWD predictions, sustain the level of confidence when using constitutive models based on these
spectra. We demonstrate the importance of using the full range of relaxation spectrum rather than a short range
around typical shear rates for the accuracy of the numerical predictions. We found extrudate swell ratio (ESR)
to be strongly dependent on MWD and stress conditions at the die exit.
Zusammenfassung:
Das auf der Finite-Element-Methode basierende Softwarepaket PolyFlow wird hier eingesetzt, um das Schwellen
von Polybutadien bei verschiedenen Molekulargewichten (Mw) und Molekulargewichtverteilungen (MWD) im
Austritt aus einer Röhre zu simulieren. Wir errechnen die Relaxationsspektren für unterschiedliche Proben und
setzten dann die Spektren in ein Standard K-BKZ-konstitutives Modell ein, das in den numerischen Simulationen benutzt wurde. Genaue Vorhersagen der MWD bestätigen die Vollständigkeit des Frequenzbereichs in den
experimentellen Daten. Umgekehrt unterstützt die Gesamtheit der Relaxationsspektren, wie durch MWD
Vorhersagen bestätigt, das Vertrauen in die konstitutiven Modelle. Wir demonstrieren auch, wie wichtig es ist,
den vollen Frequenzbereich des Spektrums zu nutzen, um genaue numerische Voraussagen zu erhalten. Wir
finden eine starke Abhängigkeit der Austrittsschwellung (ESR) von der MWD und dem Spannungszustand am
Röhrenausgang.
Résumé:
Polyflow, un logiciel basé sur la méthode d’élément fini, a été utilisé pour simuler le gonflement d’extrudats
dans le cas de polybutadiènes de masses moléculaires variées (Mw) et de distributions en masses moléculaires
variées (MWD). Nous avons calculé les spectres de relaxation pour les différents échantillons, puis nous avons
insérer ces spectres dans un modèle constitutif standard de type K-BKZ utilisé dans les simulations numériques.
Les prédictions précises des MWD sont en accord avec la gamme entière de fréquences des données expérimentales de cisaillement dynamique oscillatoire. De manière réciproque, les spectres de relaxation, qui ont été
validés par les prédictions de MWD, supportent le niveau de précision dans toute la gamme de fréquence lorsque
des modèles constitutifs basés sur ces spectres sont utilisés. Nous démontrons l’importance d’utiliser la totalité du spectre de relaxation plutôt que une courte plage de fréquences autour des vitesses de cisaillement typiques, afin que les prédictions numériques soient précises. Nous trouvons que les rapports de gonflement d’extrudats (ESR) dépendent fortement des MWD et des conditions de contrainte en sortie d’extrusion.
Key Words: K-BKZ constitutive model; Extrudate swell; Numerical simulation; Finite element method; Relaxation time spectrum; Molecular weight distribution
© Appl. Rheol. 12 (2002) 252-259
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1
Sample
Mw(*10-3) Mw/Mn
50K
200K
400K
64
244
347
Table 1: Molecular characteristics of polybutadiene
samples.
1.08
1.19
1.59
INTRODUCTION
Extrusion is one of the major processing methods in the polymer industry. Polymeric materials
when extruded are commonly found to have
larger dimensions than the die. This phenomenon is known as die swell or extrudate swell.
Extrudate swell in polymers is a complex combination of hydrodynamic boundary conditions at
the die and bulk viscoelastic properties of the
polymers. The bulk viscoelastic properties of
polymers are ultimately determined by their
structure, which in turn can be characterized
through weight average molecular weight (Mw),
molecular weight distribution (MWD), and the
amount of long chain branching (LCB). On the
other hand, viscoelastic material properties control flow behavior and dictate extrudate swell
demeanor.
Extrudate swell has been extensively
investigated in experiments [1 - 9] as well as
through numerical simulations [10 - 27]. Since a
systematic change in rheological behavior is
rather difficult to obtain even in well-controlled
experiments [4 - 6, 28, 29], numerical simulations
appear to be the method of choice under certain
circumstances. However a detailed study of the
relationships between molecular parameters,
rheological behavior and expected extrudate
swell is still lacking.
Koopmans [4 - ] carried out a series of
extensive investigations into the characteristics
of extrudate swell using high-density polyethylene. His results indicate that molecular structure,
and in particular long chain branching, plays a
vital role in the observed die swell behavior. He
concluded also that extrudate swell is a function
of time, and that the maximum swell is highly
dependent on the molecular weight distribution
of the polymer. On the other hand, Yang et al. [8]
did comprehensive studies on extrudate swell
behavior of high-density and linear low-density
polyethylene. They reported that the elusive
apparent molecular weight dependence of the
transient extrudate swell ratio arises from the
different molecular relaxation rates. Giving polyethylene of similar Mw, the extrudate swell
varies strongly with the detailed features of the
MWD. The effect of molecular structure on the
rheology of high-density polyethylene, was also
studied by Ariawan et al. [28]. It has been documented that tails in the high- or low-end regions
of the MWD can result in significantly different
rheological and processing behaviors.
The current study aims to examine simulated extrudate swell behavior as influenced by
the constitutive equation parameters, reflecting
intrinsically, its dependence on molecular weight
and molecular weight distribution. One of the
main objectives of this work is to be able to provide not only numerical results of extrudate swell
but also elucidate the effect of molecular structure on extrudate swell. We will also illustrate
the benefits of using a molecular weight distribution (MWD) calculation routine in determining
a reliable constitutive model for simulations. Our
studies allow us to optimize relaxation spectra,
obtained from oscillatory shear experiments by
comparing calculated MWD with that from sizeexcluded chromatography.
2
MATERIALS
Three relatively monodisperse polybutadiene
samples were used. Their molecular characteristics are described in Table 1. To demonstrate the
model ability to differentiate bi-modal distributions of MWD, we also studied two binary blends.
One is 75% by weight of 400K with 25% of 50K
(75% 400K). The other is 30% by weight of 400K
and 70% of 50K (30% 400K). Rheological data for
all the samples were obtained using a Rheometrics RMS-800 system. Oscillatory shear master
curves were measured based on a time-temperature superposition with a reference temperature of 50∞C. These data were processed to obtain
the relaxation spectra using the Tikhonov regularization method [30]. Details on determining
the relaxation spectra can be found elsewhere
[31]. Calculated relaxation spectra are shown in
Table 2.
50K
Table 2: Relaxation
spectra of samples used in
this work.
200K
400K
li
[s]
Gi
[Pa]
li
[s]
Gi
[Pa]
li
[s]
Gi
[Pa]
1.98*10-5
1.16*10-4
6.78*10-4
3.97*10-3
2.32*10-2
1.36*10-1
7.94*10-1
2.52*105
2.28*105
3.19*105
5.29*105
3.51*104
2.39*103
2.38*102
5.47*10-6
1.30*10-4
3.11*10-3
7.40*10-2
1.76*100
4.20*101
1.00*103
1.59*105
1.21*105
2.08*105
6.79*105
1.31*105
9.75*102
2.27*101
8.49*10-6
2.76*10-4
8.97*10-3
2.91*10-1
9.47*100
3.08*102
1.00E*104
1.15*105
7.59*104
1.47*105
4.21*105
4.42*105
9.58*103
3.57*102
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75% 400K & 25% 50K
li
Gi
[Pa]
[s]
[Pa]
1.00*10-7
2.15*10-6
4.64*10-5
1.00*10-3
2.15*10-2
4.64*10-1
1.00*101
2.16E*106
2.07*105
2.13*105
4.96*105
2.92*105
5.48*104
2.39*104
2.00*10-7
1.21*10-5
7.37*10-4
4.47*10-2
2.71*100
1.65*102
1.00*104
1.79*106
1.58*105
3.41*105
2.91*105
3.40*105
7.65*103
1.48*102
Table 2 (left above):
Relaxation spectra of
samples used in this work.
Figure 1 (right above):
MWD of 400K polybutadiene (solid line:
GPC, symbols: calculations).
Figure 2 (right below):
MWD of binary mixture
(75% 400K + 25% 50K,
solid line: GPC, symbols:
calculations).
0.6
0.5
Mi/hw
[s]
0.7
0.4
0.3
0.2
0.1
0
103
104
105
106
107
108
Mi
2.1 DETERMINATION OF MOLECULAR WEIGHT
DISTRIBUTION FROM POLYMER RHEOLOGICAL
DATA
We utilized Baumgaertel and Winter method to
derive a discrete relaxation spectrum [32]. The
range of relaxation times was selected to ensure
complete coverage of the molecular weight distribution. This procedure is based on the double
reptation theory and a monodisperse relaxation
function F(M,t) characterized by a single time
constant t(M):
0.8
0.7
0.6
Mi/hw
30% 400K & 70% 50K
Gi
li
0.5
0.4
0.3
0.2
0.1
0
103
104
105
106
107
Mi
F ( M, t ) = e {− t τ ( M )}
τ ( M) = KMα
(1)
The time constant t is related to the molecular
weight through the above relation. The exponent
a is typically around 3.4 and the constant K depends
on the precise chemical structure of the polymer
and temperature. An approximate analytical relationship is used to determine the polymer molecular weight distribution described by the relaxation
spectrum [33, 34]. The relationship is:
w (m) =
1 α
Gent (m)
2 GN0
1
Gent (m′)
∫m m′ dm′
∞
(2)
Parameter a is the same as in Eq. 1 and Gent is
essentially a relaxation spectrum with regard to
the molecular weight, m. In our study the range
of MW is directly related to the range of frequencies in the dynamic moduli spectrum. For
example, the high molecular weight limit to the
solution of Eq. 2 is estimated in terms of convenient parameters by:
 G

Mmax = 
λ max 
K
1
.
7
n


0
N
1α
(3)
The molecular weight corresponding to the high
frequency end of the data is similarly
254
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September/October 2002
1α
Mmin
 G0

=  N λ min  ,
 1.7K n

M ≥ Mc
(4)
where Mc is the critical molecular weight in the
common h0(M) relation.
Calculated molecular weight distributions derived from relaxation spectra obtained
using dynamic moduli data are compared with
data from size excluded chromatography and are
shown in Figs. 1 and 2. We encountered convergence difficulties in the numerical simulations
when using a higher than 7 number of relaxation
modes (in PolyFlow, a maximum of eight relaxation modes is allowed). Therefore we selected
only seven relaxation modes in the numerical
simulations covering the whole range of polymer
molecular weight. We can see that for the 400K
and binary mixtures samples spreads of relaxation modes are larger than one decade apart.
Models fitting of these samples are slightly poorer than those of 50K and 200K.
3
SIMULATIONS
We used the commercial package PolyFlow in our
simulations. For an incompressible fluid under
isothermal, laminar flow conditions, the field
equations read:
−∇p + ∇ ⋅ τ = 0
(5)
Applied Rheology Vol.12/5.qxd 31.10.2002 12:01 Uhr Seite 255
∇ ⋅v = 0
(6)
where v is the velocity vector, t is the stress tensor,
and p is the scalar pressure. Polymer rheological
behavior was modeled using a K-BKZ model with
an irreversible Wagner damping function. Several
studies found that multi-modes integral type constitutive equations are better suited for describing
viscoelastic behavior of polymeric liquids [35].
Moreover, the integral constitutive equation of the
K-BKZ type was found to adequately predict experimental data of extrudate swell [18, 19, 25, 26] as
well as other viscoelastic flow phenomena [23, 24,
35]. The stress tensor t is defined as
(7)
The kernel function m(t’) takes the usual form
m(t ′) = ∑
i
Gi ( − t ′ λ i )
e
λi
(8)
where the values of Gi and of li are given by the
relaxation spectrum obtained from oscillatory
shear data. Ct(t-t’) stands for the right CauchyGreen relative strain tensor. The damping function h is a function of the invariants I1 and I2 of
Ct(t-t’) defined as follows,
[
]
I1 = tr Ct−1 (t − t ′)
I2 = tr[Ct (t − t ′)]
(9)
We use a generalized strain invariant [18],
(10)
In simple shear flow, I1, I2 and thus I are identical.
The Wagner damping function has an exponential form
h(I ) = e − n
ing the material flow behavior. In simple shear,
the n parameter, which contains information on
the polymer non-linear viscoelastic properties,
can be obtained using
N
.
η(γ ) = ∑
i =1
Gi λ i
.
(1 + nλiγ )2
(12)
We used the Cox-Merz rule to arrive at h(g· ) from
the oscillatory shear data. However, information is
missing for the selection of the parameter b. It was
shown that a small value of b enhances extensional viscosity effects while a higher value accelerates
the decrease of the damping function in extensional flows. Theoretically, parameters n and b
should remain unchanged while changing only Mw.
In contrast, by increasing polydispersity, both parameters will change. Adding high molecular weight
molecules and hence increasing polydispersity, Ariawan found that extensional viscosity greatly
increases [28]. These findings were intuitively in
agreement with the experimental results by Yang
et al. [8], and simulation results by Goublomme et
al. [18]. In short, increasing polydispersity by
increasing the high molecular weight tail will
increase extensional viscosity, which in turn will
lower n and b, thus increasing extrudate swell.
However the change in b with polydispersity is very
small [18] and therefore we have selected a constant value for b (0.3) in all our simulations.
Details on the iterative procedure and
convergence strategy can be found elsewhere
[18]. Simulations are done under isothermal conditions for an incompressible fluid and neglecting inertia and body forces. No-slip boundary
conditions were applied. We considered the
axisymmetric problem for a 2 millimeters diameter die of 30 millimeters length (L/D = 15). With
a die of this length, upstream sections of the
channel do not affect the final shape of the
extrudate. The central portion of the mesh is
shown in Fig. 3.
( I −3 )
(11)
where n is a material coefficient. Thus, using the
relaxation spectrum, we need to specify only two
material parameters (n, b) for fully characteriz-
Figure 3:
Central portion of mesh.
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ESR
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Figure 5 (right above):
Typical ESR plot against
downstream distance.
Figure 6 (right below):
Effect of relaxation spectrum range on extrudate
swell results.
256
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September/October 2002
0.1
1
10
Axial distance [mm]
exiting the die and up to reaching the equilibrium state. Comparison between the numerical
results and the experimental data are valid only
for the “true” equilibrium values. Otherwise stated, only the final extrudate cross-section in an
experiment (equilibrium value) should be compared with the extrudate swell ratio in the
plateau region of the numerical simulations.
4
RESULTS AND DISCUSSIONS
We have demonstrated a close relationship
between relaxation spectra and sample molecular structure and want to investigate its possible
effect on the simulation results. To illustrate this
point, Fig. 6 shows differences brought about by
using relaxation spectra of the same polymer but
covering different parts along the molecular
weight axis of the MWD curve. When using the
full relaxation spectrum in the K-BKZ model we
obtain higher values for the extrudate swell ratio
by comparison with the results obtained when
trimming the relaxation spectrum at either the
low or high end. The number of relaxation times
was kept constant in all the simulations. In general, researchers used only experimentally available dynamic moduli data around the typical
shear rate region (in this case the shear rate at
the die wall) to construct material relaxation
spectra [18, 19, 25, 26, 39].
The rule of thumb frequently used is to
cover a range of frequencies around typical shear
rates encountered in the flow. However when
ESR
Figure 4 (left):
Typical simulation results.
We have done
extensive studies on
mesh design refinement. It is well known
that the continuum
flow at sharp converging and diverging corners generates singularly strong stress
fields. In numerical
simulations [36], a
stress divergence is
encountered as the fluid models are forced to
describe flow behavior on infinitesimal length
scales where the continuum description no
longer applies. Flow birefringence reveals that
the stress in the exit region is larger than in the
die inland [23, 24]. We found that the calculated
stress level is extremely sensitive to mesh density near the die lip. Fortunately, extrudate swell
ratio results are only slightly different even
though the stress levels captured are vastly different. To be able to describe accurately the stress
condition around the die lip, one needs to apply
more rigorous numerical methods to cope with
this singularity [37, 38]. Because of this limitation,
it is very difficult to compare quantitatively simulation results with experimental data even for
equilibrium extrudate cross-sections. Nevertheless, a relative comparison between different
materials can still be carried out. In this research,
we used the highest mesh density around the die
lip that can numerically converge.
We constructed flow curves and selected two flow rates corresponding to wall shear
stresses of 0.155 MPa and 0.205 MPa respectively. These values were selected to avoid any extrudate distortion. Resulting flow rates ranging
from 0.05 to 50 mm3/s were used to simulate
samples of different molecular weight. Typical
simulation results are shown in Figs. 4 and 5.
Extrudate swell ratio (ESR) is the ratio between
extrudate diameter and die diameter.
The numerical results presented in this
work are steady-state results. On the other hand,
extrudate swell is intrinsically a complex time
dependent phenomenon. Experiments performed under isothermal conditions show that
the extrudate exiting the die grows with time
until reaching an equilibrium value [5, 8]. Thus
the results presented here, depict only qualitatively the change in shape of the extrudate upon
1.3
1.25
1.2
1.15
1.1
1.05
1
0.01
1.3
neglect low M
1.25 full relaxationw
1.2 neglect high Mw
1.15
1.1
1.05
1
0.01
0.1
1
200k
10
Axial distance [mm]
100
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Figure 7 (left above): Stress
contour around die exit.
1.4
ESR
1.3
Figure 8 (right above):
ESR of polymer flow at
different stress level.
0.20 MPa
0.16 MPa
1.2
1.1
75% 400K
1
0.01
0.1
1
10
100
Figure 10 (right middle):
ESR comparison between
binary mixtures at constant
wall stress (0.16 MPa).
ESR
ESR
Axial distance [mm]
1. 35
400 k
1. 3
200 k
1. 25
50 k
1. 2
1. 15
1. 1
1. 05
1
0.01
0.1
1
10
Axial distance [mm]
Figure 11 (right below):
ESR comparison of artificial
50K sample with the
original 400K and 50K
samples. The artificial 50K
sample has similar
polydispersity with the
original 400K sample.
400k
75 % 400k
30 % 400k
50 k
0.1
1
10
Axial distance [mm]
ESR
simulating complex problems, such as extrudate
swell, this rule may not apply. Inside the die the
material experiences shear rates of the order of
the shear rate at the wall, whereas upon exiting
the die the shear rate/stress experienced is
orders of magnitude higher (Fig. 7). Thus when
simulating extrudate swell it is important to consider relaxation spectra covering the whole
range of MWD.
Studies [8, 18] have shown that the ESR is larger
when the level of stress is higher (higher flow rate
/ shear rate). Results of extrudate swell ratio are
presented in Fig. 8 for the case of 75% 400K comparing between wall stress levels of 0.16MPa and
0.20 MPa.
The influence of Mw on extrudate swell
is shown in Fig. 9. It is interesting to note that we
obtain similar results for all three materials. The
small differences have their origin in differences
between the polydispersity of these materials.
Indeed in Fig. 10, the effect of polydispersity on
ESR is illustrated, showing strong dependence on
MWD. Similar results were obtained in experiments using polyethylene [4 - 6, 8, 28].
Yang [8] first proposed to express the
extrudate swell ratio for materials with different
molecular weights using normalized times, i.e.
exiting times normalized using the material
relaxation time. We used a similar concept and
employed the 400K sample spectrum shifted
along the relaxation time axis to build a spectrum for a 50K sample of similar polydispersity
1.4
1.35
1.3
1.25
1.2
1.15
1.1
1.05
1
0.01
Figure 9 (left middle):
ESR comparison between
different MW at constant
wall stress (0.16 MPa).
1.4
model 50 k
1.35
400 k
1.3
50 k
1.25
1.2
1.15
1.1
1.05
1
0.01
0.1
1
10
100
1000
Axial distance [mm]
as the 400K sample. We used the “artificial” 50K
spectrum in simulating the extrudate swell ratio.
The results are shown in Fig. 11 in comparison
with the results for the original 50K and 400K
samples. The extrudate swell ratios for the 400K
and “artificial” 50K samples are similar. The differences reflect limitations in our numerical simulations with respect to the mesh refinement at
the die lip and the maximum number of relaxation modes (7) we can use in our model. For the
400K sample, the limited number of relaxation
modes used does not allow for an optimized fitting of the data over 9 decades of relaxation
times. Also, the stress singularity at the die exit
is much better captured in the case of the low
molecular weight material (50K).
To illustrate the importance of stress concentration around the small exit region, Fig. 12
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Figure 12: ESR
comparison between
no-slip and partial slip
boundary condition at
the die exit.
ESR
1.2
Partial slip B.C.
No slip B.C.
1.1
1
0.01
[2]
0.1
1
10
100
Axial distance [mm]
shows ESR results obtained using the same
polybutadiene sample and identical flow rate conditions. The difference is in the boundary condition at die exit. This result illustrates that the stress
singularity at the die exit region highly influences
the extrudate swell. Partial slip conditions at exit
region reduce extrudate swell significantly. Similar findings have been reported before [8, 40].
5
CONCLUSIONS
In this work we analyzed the effect of constitutive equation parameters as linked to sample
molecular weight and molecular weight distribution on the extrudate swell simulation results.
Molecular parameters affect the relaxation spectra for the different samples used in this study,
which in turn alter the extrudate swell results.
Accuracy in the calculated MWD strengthens our level of confidence in the frequency
range completeness for the oscillatory shear
experimental data used in the calculation. On the
other hand, the breadth of relaxation spectra as
substantiated by MWD calculations, augment
our level of confidence in using constitutive models based on such spectra.
We found the extrudate swell ratio to be
strongly affected by the sample polydispersity.
The major limitations in our current numerical
results are primarily related to mesh refinement
in the region of stress singularity at die exit and
the number of relaxation modes in relaxation
spectra especially for high molecular weight
samples. In spite of these limitations, the results
clearly illustrate the strong dependence of extrudate swell ratio on the sample molecular weight
distribution. The results also reiterate the importance of using the full spectrum of relaxation
times for a given sample rather than a limited
spectrum around a characteristic shear rate for
the experiment.
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