- Email: [email protected]
Microstructure characterization of polyethylene using thermo-rheological methods
Accepted Manuscript Microstructure characterization of polyethylene using thermo-rheological methods Maziar Derakhshandeh, Mahmoud Ansari, Antonios K. Doufas, Savvas G. Hatzikiriakos PII:
S0142-9418(17)30086-7
DOI:
10.1016/j.polymertesting.2017.03.010
Reference:
POTE 4957
To appear in:
Polymer Testing
Received Date: 25 January 2017 Revised Date:
2 March 2017
Accepted Date: 9 March 2017
Please cite this article as: M. Derakhshandeh, M. Ansari, A.K. Doufas, S.G. Hatzikiriakos, Microstructure characterization of polyethylene using thermo-rheological methods, Polymer Testing (2017), doi: 10.1016/j.polymertesting.2017.03.010. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT 1
Microstructure Characterization of Polyethylene using Thermo-rheological Methods Maziar Derakhshandeh1, Mahmoud Ansari1, Antonios K. Doufas2 and Savvas G. Hatzikiriakos1,∗ Department of Chemical and Biological Engineering, The University British Columbia, Vancouver, BC, Canada 2
RI PT
1
ExxonMobil Chemical Company, Baytown, TX, USA
Abstract
SC
In the present work, we use the viscoelastic moduli of a large number of industrially available polyethylenes in order to evaluate/test some of the previously proposed correlations between levels of long chain branching and
M AN U
polydispersity with the rheological properties. These correlations together with some new ones can be used to correct for the effects of polydispersity or long chain branching in order to assess the effect of these two molecular features on the rheological properties independently. The effects of short and long chain branching are studied providing a methodology to detect rheologically levels of short and long chain branching.
TE D
Introduction
Polymers are extensively used in industry to fabricate plastic products using various processing techniques such as film blowing, thermoforming, and injection molding. Molecular and morphological characteristics such as molecular weight (Mw), molecular weight distribution (MWD), and branching along the backbone of the
EP
polymer chains govern the melt rheology of entangled polymers. To put this into a perspective, in the film blowing process a certain degree/distribution of long chain branching (LCB) within the matrix leads to a stable bubble. To overcome the issue of bubble instability, linear low density polyethylene (LLDPE) resins are often
AC C
blended with small amount of low density polyethylene (LDPE) to increase the extensional viscosity inducing a more stable bubble. Since LCB characterization is inherently a challenging topic of vital importance in the plastic industry, it has drawn significant attention of various researchers previously. The difficulty in microstructure characterization using rheology is due to two main factors. First, the production of polyethylenes of high Mw with well-defined levels, types, and distribution of long chain branching along the backbone is challenging. To this end, several studies were initiated on detecting subtle changes in polymer microstructure using rheology i.e. to differentiate between star, comb, pom-pom, comb-star, and dendritic long chain branched polymers [1–14]. Hierarchical relaxation was proposed as the mechanism by
∗ Corresponding author:[email protected]
ACCEPTED MANUSCRIPT 2
which the chains complete their relaxation [10,15]. The entanglements belonging to topologically different parts of the macromolecules relax in accordance to seniority rules [10,15]. The outer parts of molecules relax first including the branches on the backbone of chains. During this step the backbone is nearly frozen without any relaxation. As ends are freely dangling, backbone relaxation begins. The relaxed portion of the chain act as an
RI PT
effective solvent for the frozen backbone inducing enhanced relaxation by increasing the effective tube diameter as proposed in the tube model [16,17]. It was observed that the molecular weight of arms, the molecular weight between branch points, number of branches per backbone, functionality of branched point, and type of branching affects the rheological response extensively. Due to the complexity of industrial polymerization
SC
processes, often different type, degree, and distribution of branches along the backbone is obtained. This along with the polydispersity complicate the interplay of different relaxation modes and rheological responses [7,18– 21]. Therefore, more research on this end is still needed to characterize long chain branching and other
M AN U
microstructure features of industrially important polymers, in our case polyethylenes. The second factor which makes the microstructure assessment difficult lays in the fact that the melt rheology of entangled polymers is influenced more or less not only by LCB but also by Mw, its distribution and other microstructural features. In particular, the effects of polydispersity and long chain branching are pointing to the same direction making the analysis difficult [21]. It is highly desirable to develop a route for separating
TE D
these effects and if possible to assess whether or not possible effects in rheology or processing are due to long chain branching or polydispersity or other microstructural features. There is lately a considerable interest in detecting small level of long chain branching in metallocene polyethylenes (PE) [22–24]. This family of resins is a major technological advancement in the polyolefins
EP
industry i.e., the development of homogeneous metallocene catalysts for polymerization reactions. Metallocene catalysts produce polymers with narrow molecular weight and comonomer distributions, which, combined with
AC C
controlled amounts of long chain branching, are claimed to lead to both excellent processability and superior mechanical properties [25–27].
As mentioned, the presence of long branches in the structure of polymers affects the rheological properties in different ways. The zero-shear viscosity of polymer melts with long branches has been found to scale exponentially with Mw [28]. This makes their zero-shear viscosity much higher than those of their linear counterparts. In addition, branched polymers exhibit higher degrees of shear thinning [29], the onset of shear thinning occurs at smaller shear rates, they exhibit larger extrudate swell, and extensional viscosities [30–35]. Similar effects to these of long chain branching are also exhibited by polydispersity, although to a milder degree [36–39].
ACCEPTED MANUSCRIPT 3
Various authors have developed/proposed correlations in the literature between long chain branching and rheological properties. These include the relationships between the long chain branching and (i) flow energy of activation [38–40], (ii) a normalized elevation of the flow energy of activation [24], (iii) the area in the van Gurp plot, tan −1 (G ''/ G ') vs G* , included between the curve corresponding to the branched polymer and
RI PT
its linear counterpart, and finally (iv) the shift in the Mw-location that corresponds to the maximum in the predicted molecular weight distribution from rheological data with respect to the Mw-location of the maximum from GPC data. It is worthwhile to mention that different studies have found the shift factors to fall on the same curve with that of linear polymers when they are shifted with reference to glass transition temperature [5,7].
SC
In the present work, we use the linear viscoelastic moduli of a large number of industrially available polyethylenes in order to evaluate/test previously proposed correlations between levels of long chain branching
M AN U
and polydispersity with the rheological properties. These correlations together with some new ones in a graphical form can be used to correct for the effects of polydispersity or long chain branching in order to assess the effect of these two molecular features on the rheological properties independently. The effects of short and long chain branching are studied providing a methodology to detect levels of short and long chain branching.
TE D
Materials and Experimental Methods
Eleven PE resins have been studied with different molecular weight characteristics as listed in Table 1. The codes indicate the type (LL for linear low, H for high-density and L for low-density polyethylene), the Mw (kg/mol) and the polydispersity index (PDI) i.e. L-149-8p2 indicates a low-density polyethylene of Mw close to
EP
149 kg/mol (the exact value can be obtained from table 1) and PDI=8.2. The Melt Flow Index (MFI) was measured according to ASTM D1238 (190°C, 2.16 kg weight).
AC C
The linear viscoelastic properties of all resins have been determined using an Anton Paar MCR-502 (strain and stress-controlled rheometer) equipped with a cone-and-plate geometries. Four types of tests have been performed; small amplitude oscillatory shear (SAOS) or frequency sweep, step shear strain relaxation, creep/recovery and uniaxial extension tests. All shear experiments have been performed with cone-and-plate of 25mm in diameter and 4° in angle. To avoid degradation, all experiments have been run under nitrogen. Time sweep experiments confirmed the stability of samples during each measurement. The frequency sweep tests were carried out at temperatures ranging from 140°C to 220°C over 20°C increments. The various curves were shifted by means of applying the time-temperature superposition (tTS) in order to generate the master curves at the reference temperature of 180°C. The activation energies for the vertical and horizontal shift factors were determined, an indication of the level of LCB and polydispersity.
ACCEPTED MANUSCRIPT 4 Table 1. Polymers used in this study and their molecular characteristics.
#
Resin
Type
1
L-149-8p2
LDPE
149.5
8.2
2.8
0.22
2
LL-97-3p5
LLDPE
97.2
3.5
2.1
0.91
3
LL-113-2p4
LLDPE
113.2
2.4
4
LL-117-3p3
LLDPE
117.3
3.3
5
LL-124-4p5
LLDPE
124.1
4.5
6
LL-124-5p3
LLDPE
124.3
5.3
7
LL-143-6p0
LLDPE
143.2
8
H-121-32p0
HDPE
9
H-151-12p1
HDPE
10 H-158-12p3
HDPE
RI PT 0.90
2.9
0.52
3.1
0.58
3.5
0.38
6.0
3.2
0.33
120.8
32.0
5.2
0.09
151.1
12.1
5.3
0.26
157.6
12.3
5.2
0.24
287.0
14.2
4.5
0.04
M AN U
SC
1.8
TE D
11 bH-287-14p2 HDPE-bimodal
Mw (kg/mol) Mw/Mn Mz/Mw MFI (gr/10 min)
Although the experimental data extended to lower frequencies by applying the tTS technique, the terminal zone at low frequencies (zero-shear viscosity) for most of the cases here could not be achieved. One way to extend the data to even lower frequencies is using a step strain experiment and converting the data to the dynamic
EP
moduli (G’, G”). This can be done by calculating the parsimonious relaxation spectrum from the experimental relaxation modulus. The level of strain was chosen to be as low as 5% to ensure that the data are collected
AC C
within the linear viscoelastic envelope. Furthermore, creep experiments followed by recovery measurements have been performed with stress level of 5 Pa as another means of determining the zero-shear viscosity. The zero-shear viscosity values were found to be in agreement with those generated by step relaxation showing consistency of the obtained experimental results. The extensional behavior in simple uniaxial extension is also studied using the Sentmanat Extensional Rheometer (SER2) housed on the Anton Paar MCR-502 rotational rheometer [41]. The resins were molded using a hot press at 180°C in order to make a plaque with a thickness of around 0.7mm. Then rectangular strips were cut using a dual blade cutter. The dimensions of these strips were about 8 mm in width and 15 mm in length. All the uniaxial tests were conducted at 180°C over a wide range of Hencky strain rates, namely 0.01, 0.05, 0.5, 5.0 and 10.0 s-1.
ACCEPTED MANUSCRIPT 5
Results and Discussion Linear Viscoelasticity
RI PT
Figure 1 depicts the LVE master curve of LL-124-4p5 at the reference of 180oC. Experiments were run over the temperature range from 140oC to 220oC in order to generate the master curve extending the data over the frequency range that covers lower frequencies. However, this is not enough to reach the terminal zone so as to determine the zero-shear viscosity important in many applications, i.e., relate to molecular weight and its
SC
distribution. This terminal (low frequency) zone is limited by polymer degradation at high temperatures, while
EP
TE D
M AN U
the data at higher frequency cannot be obtained trivially due to the polymer crystallization at lower temperature.
Figure 1. The master curve for resin LL-124-4p5 at 180°C.
AC C
In order to extend the data to even lower frequencies, step strain relaxation measurements have been performed. Figure 2 shows such results for resin LL-124-4p5 for the strain level of 5%. The continuous line is the prediction of Maxwell relaxation model (Eq. 1) with 6 modes. The fitting has been performed by an inhouse Matlab code applying the Levenberg-Marquardt nonlinear optimization method. The relaxation spectra values were used into Eq. 2a & b in order to convert the relaxation modulus data to the dynamic ones [42]. N
G ( t ) = ∑Gi e − t / λi i =1
Eq. 1
ACCEPTED MANUSCRIPT 6
Gi (ωλi )
N
i =1
1 + (ωλi )
i =1
Eq. 2a
2
Gi (ωλi )
N
G '' (ω ) = ∑
2
1 + (ωλi )
Eq. 2b
2
RI PT
G ' (ω ) = ∑
N
η0 = ∑Gi λi
Eq. 2c
TE D
M AN U
SC
i =1
Figure 2. Relaxation modulus after imposing strain of 5% for resin LL-124-4p5 at 180°C.
EP
Figure 3 depicts the creep and recoverable compliance data for resin LL-124-4p5. For the creep experiment, a stress level of 5 Pa was imposed on the material and after a sufficiently long time (when a constant Newtonian slope has been achieved), the stress removed and the recoverable strain was measured. From linear
AC C
viscoelasticity, it is known that Equations 3a & b are valid for creep and recoverable compliances [42]: N
(
)
J ( t ) = J 0 + ∑J i 1 − e −t /τ i + t / η0 i =1
N
(
J r ( t ) = J 0 + ∑J i 1 − e−t /τ i i =1
)
Eq. 3a Eq. 3b
N
J e0 = ∑J i
Eq. 3c
i =1
where
is the zero-shear viscosity,
the instantaneous elastic compliance,
the retardation times, and
the
retardation strengths. Table 2 summarizes the obtained values of zero-shear viscosity, steady state creep compliances, and relaxation time of the different resins studied. Using these equations, it is expected that for a
ACCEPTED MANUSCRIPT 7
sufficiently long time, the creep compliance data fall into a straight line with a constant slope, which is the inverse of
(Equation 3a). The steady state creep compliance,
, can also be determined using both the
intercept of that line and the ultimate value of the recoverable creep compliance. It is also possible to calculate it by using the retardation spectrum from Eq. 3c. From Eq. 3a & b, it is possible to calculate the
from ( ). Figure 4 shows the comparison between the recoverable creep compliance
TE D
M AN U
SC
data obtaining from this calculation and the experimental data.
RI PT
subtracting values of ⁄
( ) by
Figure 3. Creep and recoverable compliances for resin LL-124-4p5 at 180°C.
It is clear from Figure 4 that both creep and recoverable compliances for resin LL-124-4p5 reach the ultimate values. It is again possible to convert these data to dynamic data in order to extend them to very low
EP
frequencies. This can be done by considering Eq. 4a & b [42]: N
AC C
J ' (ω ) = ∑
J ′′ (ω ) =
where:
i =1
1
ωη0
G' (ω ) = G '' (ω ) =
Ji
1 + (ωτ i ) N
+∑ i =1
Eq. 4a
2
Gi (ωτ i )
1 + (ωτ i )
J ' (ω )
J ' (ω ) + J '' (ω ) 2
2
J '' (ω )
J ' (ω ) + J '' (ω ) 2
2
2
Eq. 4b
Eq. 5a
Eq. 5b
ACCEPTED MANUSCRIPT
M AN U
SC
RI PT
8
Figure 4. Comparison between experimental data of recoverable creep compliance and calculated ones from creep data for resin LL-124-4p5 at 180°C.
Figure 5 illustrates the extended LVE data by converting the stress relaxation, creep and recovery data along with the master curve for resin LL-124-4p5. All sets of data show excellent consistency at low frequencies.
TE D
Moreover, the Newtonian region (zero-shear viscosity) has been reached for this polymer. The solid line is fitting of Eq. 2a & 2b with 10 modes. In the master curve obtained, by increasing frequency, the loss modulus always remains above storage modulus until very higher frequencies are reached. After the terminal regime, both modulus increase in parallel for about two decades (with the slope of around 0.8), this behavior is known
EP
as “Zimm-like” behavior and was observed for dendritically branched polystyrene [43]. As soon as the branches are relaxed they act as effective solvent and thus the backbone tube diameter increases. The increase in tube
AC C
diameter reduces the number of entanglements efficiently contributing to the observed Zimm-like behavior, which is similar to the behavior of untangled polymers. The longest and segmental relaxation times are determined as 2287 and 0.01s, respectively. The viscoelastic behavior between segmental and terminal relaxation times was used previously to obtain information about the relaxation spectrum of macromolecules and long chain branching characteristics [44].
ACCEPTED MANUSCRIPT
SC
RI PT
9
M AN U
Figure 5. A generalized master curve by converting data from different types of measurements for resin LL-124-4p5 at 180°C. The intersection of lines passing through the terminal region defines the longest relaxation time of the sample.
Table 2: Linear viscoelastic characteristics of the PE resins obtained from various methods. zero-shear
Steady state compliance
Relaxation time
Zimm-like
viscosity (kPa.s)
(1/kPa)
(s)
regime
Creep (Eq. 3a)
Relax.
Ret. Spec.
Ret. Spec.
(Creep,
(Recovery
Eq. 3c)
, Eq. 3c)
N/D
23.30
0.74
0.75
9.9
N/D
226.8
Spec. (Eq. 2c)
TE D
Resin
Creep
Recovery
(Eq. 3a)
(Eq. 3a)
13.01
Longest
Segmental
slope
breadth
N/D
N/D
2.95
0.54
0.77
13.47
0.0089
N/D
N/D
N/D
0.1216
0.0083
42.75
22.50
47.82
23.26
8953
0.0074
0.67
3.17
327.9
339.9
LL-97-3p5
26.3
27.0
LL-113-2p4
N/D
LL-117-3p3
236.4
LL-124-4p5
53.5
56.2
22.31
22.95
59.54
23.08
2287
0.0104
0.88
2.87
LL-124-5p3
896.8
880.6
16.90
4.54
15.19
4.65
1509
6.54
0.41
2.67
LL-143-6p0
265.2
329.2
3.50
13.89
3.56
14.07
598
0.36
N/A
H-121-32p0
1016.9
1001.8
4.84
6.60
4.48
8.57
3824
1.63
N/A
H-151-12p1
433.0
448.4
12.83
12.08
23.59
12.46
9616
0.098
0.75
2.15
H-158-12p3
301.4
318.0
36.03
20.29
42.05
31.86
10108
0.156
0.72
1.76
bH-287-14p2
1729.5
2337.9
12.75
13.77
12.80
14.03
19519
1.48
0.69
2.96
AC C
EP
L-149-8p2
N/A 0.68
2.00 N/A
ACCEPTED MANUSCRIPT 10
Thermo-rheological behavior (Energy of activation) The time-Temperature superposition (tTS) principle was used previously to obtain information on LCB characteristics. The vertical shift factors were obtained by using the available density functions. It was found that, all horizontal shift factors were fallen on the same curve if the superposition is done with respect to glass
RI PT
temperature transition of each resins indicating similar WLF model constants [5,7]. However, obtaining reliable rheological data for semicrystalline polymers below their crystallization temperatures is challenging. Therefore, in the current study both horizontal and vertical shift factors are obtained by shifting the data to a reference temperature higher than the melting peak temperature determined by calorimetry techniques. It is worth noting
SC
that a protocol for measuring the complete relaxation spectrum of semicrystalline polymers is established [7]. The presence of LCB, which is defined as side chains consist of 7 or higher carbon atoms [45], is
M AN U
determined by analyzing the vertical and horizontal shift factors in terms of the activation energies [8,11,16,46– 57]. High horizontal activation energies (usually accompanied by vertical shift factors, bT) is a sign of long chain branching. The benefit of this approach is that unlike other methods, the presence of high molecular weight tails in the MWD does not have an effect. However, the shortcomings include its insensitivity to the diminishing low level of LCB and also effects of side branch length. The typical values of horizontal activation energy (Eact,H) for HDPE and LLDPE are 27-28 and 33-35 kJ/mol, respectively. Moreover, such linear HDPE
TE D
and LLDPE polymers are thermo-rheologically simple vertical activation (Eact,V) close to zero. However, for the case of commercial LDPEs which contains high levels of long chain branches, either thermo-rheological complexity or simplicity with high Eact,H (>40) and Eact,V (>6) have been observed [58]. Based on the discussion
EP
provided, a first assessment for presence of LCB can be seen in Table 3. Table 3. Branching assessment based on activation energy values.
Eact,H (kJ/mol)
AC C
Resin
Eact,V (kJ/mol)
Branching
L-149-8p2
72.7
9.1
LCB
LL-97-3p5
52.9
10.2
LCB
LL-113-2p4
33.1
2.0
NB/SCB
LL-117-3p3
82.6
20.3
LCB
LL-124-4p5
30.0
2.6
NB/SCB
LL-124-5p3
40.7
6.2
LCB or SCB
LL-143-6p0
70.3
15.1
LCB
H-121-32p0
27.2
0.7 ≈ 0
NB
H-151-12p1
27.6
0.3 ≈ 0
NB
ACCEPTED MANUSCRIPT 11
H-158-12p3
31.9
2.5
SCB
bH-287-14p2
29.2
1.5
NB/SCB
NB: no branching; SCB: short chain branching; LCB: long chain branching
RI PT
The Mw-η0 relationship Viscosity at small shear rates (zero-shear viscosity) is typically correlated with molecular weight. A linear relation with Mw is observed for un-entangled polymers (very low Mw). For linear entangled polymers with narrow MWD, there is a power-law relationship between zero-shear viscosity and molecular weight. Increasing the MWD or having LCB would cause deviation from this power-law relationship. Although initially it was
SC
claimed that polydispersity has no effect on the Mw-??0 relationship [59–61], Karjala et al. [62] has confirmed that only for Mw/Mn<10, the polydispersity has no effect on this relationship [62]. Wasserman & Graessly [34]
M AN U
have shown that the MWD effect can be adequately excluded by modifying the zero-shear viscosity over a factor of Mz/Mw. This was confirmed in one of our previous studies of some very broad MWD metallocene HDPEs [63]. Therefore, it is possible to detect LCB for these resins by checking the modified ??0-Mw relationship with available data on PE in the literature (Figure 6). The solid line represents the power-law relationship obtained by Karjala et al. [62] for some linear PE at 190°C. As it can be seen from this figure, for
⁄
>1, where
LL-113-2p4,
the
other
resins
is zero-shear viscosity of the polymer and
TE D
except
lay
above
this
line
with
is the zero-shear viscosity obtained using
the proposed Mw-η0 power-law relationship in the literature. It is an indication that these resins possess LCB. The arrow on top of the viscosity value of resin L-149-8p2 indicates that the actual zero-shear viscosity is higher than the reported value. This is due to the fact that the terminal regime for this particular resin (the only
EP
LDPE in this study) was not accessible even using long time creep/recovery experiments. The relaxation of linear polymer chains is governed by entanglements, however, for hyperbranched polymers,
AC C
the role of entanglements becomes negligible due to spatial barrier and limited interpenetration of the neighboring chains [44]. Therefore, for hyperbranched/dendritic type of LCB, the deviation from the reference line should be negative [43,44] which is not the case for any of the resins studied. Hence it can be concluded that there is no polymer possessing such a long chain dendritic structure. Table 4 lists the assessment for these resins based on the above analysis.
ACCEPTED MANUSCRIPT 12 Table 4. Branching assessment based on Mw-η0 relationship.
Mz/Mw
⁄
Branching
L-149-8p2
8.2
2.8
12.0
LCB
LL-97-3p5
3.5
2.1
5.1
LCB
LL-113-2p4
2.4
1.8
1.3
NB
LL-117-3p3
3.3
2.9
17.3
LCB
LL-124-4p5
4.5
3.1
5.1
LCB
LL-124-5p3
5.3
3.5
74.6
LCB
LL-143-6p0
6.0
3.2
10.8
LCB
H-121-32p0
32.0
5.2
H-151-12p1
12.1
5.3
H-158-12p3
12.3
5.2
bH-287-14p2
14.2
SC
LCB
3.9*
Low LCB
2.3*
Low LCB
1.7*
Low LCB
M AN U
20.5*
4.5
zero-shear viscosities are corrected by a factor of Mz/Mw according to Wasserman & Graessly [34].
AC C
EP
TE D
∗
RI PT
Mw/Mn
Resin
Figure 6. Modified η0 for the effect of polydispersity vs. Mw, for available resins. The modification is applied for the resins with Mw/Mn>10 (closed symbols). The arrow indicates that the actual η0 can be higher. The solid line represents the power-law relationship for linear PEs obtained by Karjala et al. [62] and the dashed line represents 3.65 ±0.02 to include uncertainty effect in Mw measurements (190°C).
ACCEPTED MANUSCRIPT 13
van Gurp-Palmen Graphs van Gurp-Palmen graph plots the phase angle as a function of complex modulus. It was proposed as an alternative method to check for the tTS failure [64]. At the low values of complex modulus, phase angle should approach 90o indicating complete viscous behavior. As the complex modulus increases, the phase angle
RI PT
decreases and it approaches 0o when the complex modulus approaches the plateau modulus indicating ideal elastic response. This limiting behavior is clearly observed in Figures 7a&b indicating the accuracy of data. The potential of van Gurp-Palmen graph in determining the branching characteristics was not known initially. This method has been used later by other authors to obtain information about the relaxation behavior of the desired
SC
polymers with the aim to reveal branching characteristics [65–67]. Different minimum within the plot shows relaxation of different origin (branch and backbone). It was also shown that the shape of van Gurp-Palmen
M AN U
graph is different if relaxation characteristics of branches are different (star shape, arm retraction vs comb, chain reptation) [65,66]. It is well-understood that both MWD and LCB shift the values of the phase angle to smaller angles at given levels of the complex modulus. Therefore, for resins with PDI greater than 10 it cannot be concluded clearly if the reduction is a result of LCB or MWD. It is worth noting that for these commercial
AC C
EP
TE D
polymers the distinct minimums which are often observed in control branched polymers are not revealed.
Figure 7. van Gurp-Palmen plots for resins with a) Mw/Mn<10 and b) Mw/Mn>10 at 180°C.
Table 5. Branching assessment based on van Gurp-Palmen graph.
Resin
Branching
ACCEPTED MANUSCRIPT 14
LCB
LL-97-3p5
LCB
LL-113-2p4
NB/SCB
LL-117-3p3
LCB
LL-124-4p5
LCB
LL-124-5p3
LCB
LL-143-6p0
LCB
H-121-32p0
LCB or MWD
H-151-12p1
LCB or MWD
H-158-12p3
LCB or MWD
bH-287-14p2
LCB or MWD
M AN U
SC
RI PT
L-149-8p2
Normalized Complex Viscosity
The normalized complex viscosities are plotted versus frequency for all resins in Figure 8. In this way, it is possible to compare the shear thinning behavior of these resins. One analysis which can be done here is to check the Dow Rheology Index (DRI) [68–71]. The idea behind this analysis is that if the Cross model (Eq. 7a) is used
TE D
to fit the viscosity data, the relaxation time (τ0) in this model should obey Eq. 7b for a linear chain polymer. Therefore, the DRI index can be defined based on deviation from this relationship (Eq. 7c):
| ∗ ( )| =
. ) = 3.65 × 10!
EP
(
AC C
"#$ =
⁄ 1+(
)
3.65 × 10! ( ⁄ ) − 1 10
Eq. 7a Eq. 7b Eq. 7c
ACCEPTED MANUSCRIPT 15 Figure 8. Normalized complex viscosity versus frequency for resins with a) Mw/Mn<10 and b) Mw/Mn>10 at 180°C.
Solid lines in Figure 8 are Cross model fittings. Table 6 collects the DRI and LCB branching based on
reflected on it. Therefore, it just can be used for narrow MWD resins. Table 6. Branching assessment & DRI based on normalized complex rheology.
RI PT
normalized complex viscosity data. The problem with DRI index is that the broadness of MWD can also be
Mw/Mn
η0 (Pa.s)
τ0 (s)
n
L-149-8p2
8.2
339755*
182
0.610
LL-97-3p5
3.5
26966
2.114
0.527
2.8
LCB
LL-113-2p4
2.4
9922
0.02529
0.768
0.0
NB
LL-117-3p3
3.3
226701
872.7
0.433
140.4
LCB
LL-124-4p5
4.5
56178
21.28
0.463
13.7
LCB
LL-124-5p3
5.3
879731
1088
0.559
45.0
LCB
LL-143-6p0
6.0
243339
149.4
0.551
22.3
LCB
H-121-32p0
32.0
1001725
60.56
0.679
2.1
LCB or MWD
H-151-12p1
12.1
448307
787.1
0.503
64.0
LCB or MWD
H-158-12p3
12.3
317510
319.9
0.528
36.7
LCB or MWD
bH-287-14p2
14.2
2334785
1084
0.620
16.8
LCB or MWD
DRI
Branching
19.5
LCB
SC
M AN U
TE D
Resin
* zero-shear viscosity couldn’t be obtained experimentally and it is calculated from the Maxwell model (Eq. 2c).
EP
Quantitative approach
Janzen and Colby proposed the following equations for the effect of LCB on the Mw-η0 relationship[72]:
AC C
= &'( 1 + ('( ⁄') )*.+ (', ⁄'( )-
3 9 '( . = / 0 11, + 678 9 => 2 8 90':;< ?=
' 1 1 9 − = 2 '( ',
Eq. 6a Eq. 6b Eq. 6c
where K is a coefficient having units of Pa.s/(g/mol), Mb is an average molecular weight between branch points, Mc is the critical molecular weight for entanglements and Mw is the molecular weight. “α“ is also the fraction of long-chain branch points. Having K=5.22×10-6 (Pa.s)/(g/mol), B=6, M0=14.027 g/mol, Mc=2100 g/mol and MKuhn=145.9 g/mol, it is possible to solve the above equations for Mb and α [72].
ACCEPTED MANUSCRIPT 16
As discussed earlier, polydispersity has an effect on η0-Mw relationship, particularly for resins with Mw/Mn>10. This correction has been performed according to Wasserman & Graessly who have suggested to divide the zeroshear viscosity by Mz/Mw [34]. Table 7 collects the values of Cb=Mb/M0 (average number of monomers between branch points) and α from both approaches. Figure 9 shows Cb versus Mw/M0 for all the resins. The distance of
RI PT
the values from the continuous unit line depicts the degree of LCB present. The LCB assessment presented in
TE D
M AN U
SC
Table 7 is based on such analogy.
AC C
EP
Figure 9. Cb versus Mw/M0 for all the resins
Table 7. LCB analysis using Eq. 6a-c based on Janzen-Colby [72] relationship.
Cb
α
LCB Assessment
L-149-8p2
8642
1.09×10-5
LCB
LL-97-3p5
5910
1.24×10-5
LCB
7968
-7
SCB/NB
-4
LCB
-6
SCB/NB
-5
LCB
-5
LCB
Resin
LL-113-2p4 LL-117-3p3 LL-124-4p5 LL-124-5p3 LL-143-6p0
1657 7789 5867 8370
7.91×10
2.42×10 7.68×10
2.88×10 1.08×10
ACCEPTED MANUSCRIPT 17
H-121-32p0 H-151-12p1 H-158-12p3
6668
1.69×10-5
9799
-6
SCB/NB
-6
SCB/NB
-6
SCB/NB
10594 19437
4.61×10
2.69×10 1.29×10
RI PT
bH-287-14p2
LCB
Uniaxial Extension
Extensional experiments are solid in predicting the degree and type of different branches on the backbone. It
SC
was shown previously that only small amount of combs (5%) on the backbone is sufficient to induce extensional strain hardening [65]. Extensional rheology of well-defined comb polymers have reported that increasing the average number of entanglements on the branches or between branch points on the backbone
M AN U
induces stronger hardening at lower strain rates [18]. Of note, the inverse of the onset rate for strain hardening is referred to as the effective stretch time which is related to the Rouse time of chains [18,73]. As shown in Figure 10, the onset of strain hardening at lowest studied rate for LL-124-4p5 and LL-143-6p0 occurred at elapsed time of ~40 and ~65s, respectively. Since from Janzen and Colby analysis [72], “α“ (fraction of long-chain branch points) is larger for LL-143-6p0 and both of resins have similar Mw, LL-124-4p5
TE D
should possess longer branches (more entangled branches) in order to induce the onset of hardening at lower elapsed time. This finding is in agreement with the study of Lentzakis et al. [18] who find the key parameter to be the number of entanglements per branch. For H-121-32p0 very small upturn in torque signal was observed. This can be either an indication of the presence of small degree of LCB on the backbone or due to its very broad
EP
molecular weight distribution. For bH-287-14p2, the hardening happens for all the investigated rates even at the smallest deformation rates employed. These upturns are not abrupt when compared to those of LL-143-6p0 and
AC C
LL-124-4p5. Similar conclusion as for resin H-121-32p0 can be drawn here, but the effect of polydispersity is more dominant due to its bimodal MWD [74].
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
18
Figure 10a-c. Uniaxial stress growth coefficient versus time for all the resins at 180°C.
ACCEPTED MANUSCRIPT 19 Table 8. Branching assessment based on extensional rheology.
Resin
Branching LCB
LL-97-3p5
NB/low LCB
LL-113-2p4
No LCB
LL-117-3p3
LCB
LL-124-4p5
LCB
LL-124-5p3
NB/low LCB
LL-143-6p0
LCB
H-121-32p0
NB/low LCB
H-151-12p1
NB/low LCB
H-158-12p3
NB/low LCB
SC low LCB
TE D
Conclusions
M AN U
bH-287-14p2
RI PT
L-149-8p2
In this paper, eleven PE resins with different molecular characteristics are examined using rheological methods in an attempt to obtain information on their branching characteristics. Table 9 summarizes the conclusions made using various methods mentioned in the results and discussion. Although thermo-rheological method benefits
EP
from insensitivity to the presence of high molecular weight tails in the MWD, it should be used along with other methods to obtain the correct LCB characteristics. On the other hand, other rheological methods lack the
AC C
insensitivity to MWD. Therefore, as shown, different methods should be compared simultaneously in order to infer the correct long chain branching characteristics.
ACCEPTED MANUSCRIPT 20
Table 9. Comparison of branching assessments obtained by using different rheological methods. η0 Mw-η
van Gurp plot
relationship
Resin
Normalized |η η*|
Extensional Rheology
Janzen-Colby Conclusion
RI PT
Thermorheology
Branching
Branching
Branching
Branching
Branching
L-149-8p2
LCB
LCB
LCB
LCB
LCB
LCB
LCB
LL-97-3p5
LCB
LCB
LCB
LCB
No/low LCB
LCB
Low LCB
LL-113-2p4
SCB
NB
No LCB
SCB/NB
SCB
LL-117-3p3
LCB
LCB
LCB
LCB
LCB
LCB
LL-124-4p5
NB/SCB
LCB
LCB
LCB
LCB
SCB/NB
LCB
LL-124-5p3
LCB or SCB
LCB
LCB
LCB
NB/low LCB
LCB
Low LCB
LL-143-6p0
LCB
LCB
LCB
LCB
LCB
LCB
LCB
H-121-32p0
NB
LCB
LCB or MWD
NB/low LCB
LCB
NB
H-151-12p1
NB
Low LCB
LCB or MWD
NB/low LCB
SCB/NB
NB
H-158-12p3
NB/SCB
Low LCB
LCB or MWD
NB/low LCB
SCB/NB
NB/low LCB
bH-287-14p2
NB/SCB
Low LCB
LCB or MWD
low LCB
SCB/NB
NB/low LCB
NB
LCB
M AN U
ND (LCB or MWD)
ND (LCB or
TE D
AC C
Acknowledgements
Branching/SCB
MWD)
ND (LCB or MWD)
ND (LCB or MWD)
EP
ND: not determined.
No
SC
Branching
The authors would like to acknowledge ExxonMobil Chemical for providing the materials, their GPC characterization and funding for this project.
ACCEPTED MANUSCRIPT 21
REFERENCES [1]
E. Van Ruymbeke, M. Kapnistos, D. Vlassopoulos, T. Huang, D.M. Knauss, Linear melt rheology of pom-pom polystyrenes with unentangled branches, Macromolecules. 40 (2007) 1713–1719. T.C.B. McLeish, S.T. Milner, Entangled dynamics and melt flow of branched polymers, in: Branched
RI PT
[2]
Polym. II, Springer, 1999: pp. 195–256. [3]
J.H. Lee, L.J. Fetters, L.A. Archer, Stress relaxation of branched polymers, Macromolecules. 38 (2005)
[4]
SC
10763–10771.
D.R. Daniels, T.C.B. McLeish, B.J. Crosby, R.N. Young, C.M. Fernyhough, Molecular rheology of
[5]
M AN U
comb polymer melts. 1. Linear viscoelastic response, Macromolecules. 34 (2001) 7025–7033. M. Kapnistos, D. Vlassopoulos, J. Roovers, L.G. Leal, Linear rheology of architecturally complex macromolecules: Comb polymers with linear backbones, Macromolecules. 38 (2005) 7852–7862. [6]
N.J. Inkson, R.S. Graham, T.C.B. McLeish, D.J. Groves, C.M. Fernyhough, Viscoelasticity of monodisperse comb polymer melts, Macromolecules. 39 (2006) 4217–4227. M. Kapnistos, G. Koutalas, N. Hadjichristidis, J. Roovers, D.J. Lohse, D. Vlassopoulos, Linear rheology
TE D
[7]
of comb polymers with star-like backbones: melts and solutions, Rheol. Acta. 46 (2006) 273–286. [8]
A.L. Frischknecht, S.T. Milner, A. Pryke, R.N. Young, R. Hawkins, T.C.B. McLeish, Rheology of three-
[9]
EP
arm asymmetric star polymer melts, Macromolecules. 35 (2002) 4801–4820. R.J. Blackwell, O.G. Harlen, T.C.B. McLeish, Theoretical linear and nonlinear rheology of symmetric
AC C
treelike polymer melts, Macromolecules. 34 (2001) 2579–2596. [10]
T.C.B. McLeish, Tube theory of entangled polymer dynamics, Adv. Phys. 51 (2002) 1379–1527.
[11]
T.C.B. McLeish, R.G. Larson, Molecular constitutive equations for a class of branched polymers: The pom-pom polymer, J. Rheol. 42 (1998) 81–110.
[12]
R.S. Graham, T.C.B. McLeish, O.G. Harlen, Using the pom-pom equations to analyze polymer melts in exponential shear, J. Rheol. 45 (2001) 275–290.
[13]
L.A. Archer, Juliani, Linear and nonlinear viscoelasticity of entangled multiarm (Pom-Pom) polymer liquids, Macromolecules. 37 (2004) 1076–1088.
ACCEPTED MANUSCRIPT 22
[14]
C.D. Chodankar, J.D. Schieber, D.C. Venerus, Pom--Pom theory evaluation in double-step strain flows, J. Rheol. 47 (2003) 413–427.
[15]
T.C.B. McLeish, Hierarchical relaxation in tube models of branched polymers, EPL (Europhysics Lett. 6
[16]
RI PT
(1988) 511. G. Marrucci, Relaxation by reptation and tube enlargement: a model for polydisperse polymers, J. Polym. Sci. Polym. Phys. Ed. 23 (1985) 159–177. [17]
S.T. Milner, T.C.B. McLeish, Arm-length dependence of stress relaxation in star polymer melts,
[18]
SC
Macromolecules. 31 (1998) 7479–7482.
H. Lentzakis, D. Vlassopoulos, D.J. Read, H. Lee, T. Chang, P. Driva, N. Hadjichristidis, Uniaxial
[19]
M AN U
extensional rheology of well-characterized comb polymers, J. Rheol. 57 (2013) 605–625. M. Ahmadi, C. Bailly, R. Keunings, M. Nekoomanesh, H. Arabi, E. Van Ruymbeke, Time marching algorithm for predicting the linear rheology of monodisperse comb polymer melts, Macromolecules. 44 (2011) 647–659. [20]
K.M. Kirkwood, L.G. Leal, D. Vlassopoulos, P. Driva, N. Hadjichristidis, Stress relaxation of comb
[21]
TE D
polymers with short branches, Macromolecules. 42 (2009) 9592–9608. S.G. Hatzikiriakos, Long chain branching and polydispersity effects on the rheological properties of polyethylenes, Polym. Eng. Sci. 40 (2000) 2279–2287. A. Malmberg, C. Gabriel, T. Steffl, H. Münstedt, B. Löfgren, Long-chain branching in metallocene-
EP
[22]
catalyzed polyethylenes investigated by low oscillatory shear and uniaxial extensional rheometry,
[23]
AC C
Macromolecules. 35 (2002) 1038–1048. C. Piel, F.J. Stadler, J. Kaschta, S. Rulhoff, H. Münstedt, W. Kaminsky, Structure-Property Relationships of Linear and Long-Chain Branched Metallocene High-Density Polyethylenes Characterized by Shear Rheology and SEC-MALLS, Macromol. Chem. Phys. 207 (2006) 26–38. [24]
P.M. Wood-Adams, J.M. Dealy, Using rheological data to determine the branching level in metallocene polyethylenes, Macromolecules. 33 (2000) 7481–7488.
[25]
J. Scheirs, W. Kaminsky, Metallocene-based Polyolefins: Preparation, properties, and technology, Wiley, 2000.
[26]
K.W. Swogger, G.M. Lancaster, S.Y. Lai, T.I. Butler, Improving Polymer Processability Utilizing
ACCEPTED MANUSCRIPT 23
Constrained Geometry Single Sue Catalyst Technology, J. Plast. Film Sheeting. 11 (1995) 102–112. [27]
S.-Y. Lai, J.R. Wilson, G.W. Knight, J.C. Stevens, P.-W.S. Chum, Elastic substantially linear olefin polymers, (1993). S.-Y. Lai, J.R. Wilson, G.W. Knight, J.C. Stevens, Elastic substantially linear olefin polymers, (1995).
[29]
J.M. Delay, K.F. Wissbrun, Melt rheology and its role in plastics processing: Theory and applications,
RI PT
[28]
(1990).
M.S. Jaćović, D. Pollock, R.S. Porter, A rheological study of long branching in polyethylene by blending,
SC
[30]
J. Appl. Polym. Sci. 23 (1979) 517–527.
B.H. Bersted, On the effects of very low levels of long chain branching on rheological behavior in
M AN U
[31]
polyethylene, J. Appl. Polym. Sci. 30 (1985) 3751–3765. [32]
V.R. Raju, G.G. Smith, G. Marin, J.R. Knox, W.W. Graessley, Properties of amorphous and crystallizable hydrocarbon polymers. I. Melt rheology of fractions of linear polyethylene, J. Polym. Sci. Polym. Phys. Ed. 17 (1979) 1183–1195.
R.J. Koopmans, Extrudate swell of high density polyethylene. Part I: Aspects of molecular structure and
TE D
[33]
rheological characterization methods, Polym. Eng. Sci. 32 (1992) 1741–1749. [34]
S.H. Wasserman, W.W. Graessley, Prediction of linear viscoelastic response for entangled polyolefin
[35]
EP
melts from molecular weight distribution, Polym. Eng. Sci. 36 (1996) 852–861. M.G. Lachtermacher, A. Rudin, Reactive processing of LLDPEs in corotating intermeshing twin-screw extruder. I. Effect of peroxide treatment on polymer molecular structure, J. Appl. Polym. Sci. 58 (1995)
[36]
AC C
2077–2094.
C. Tzoganakis, A rheological evaluation of linear and branched controlled-rheology polypropylenes, Can. J. Chem. Eng. 72 (1994) 749–754.
[37]
H. Münstedt, S. Kurzbeck, L. Egersdörfer, Influence of molecular structure on rheological properties of polyethylenes, Rheol. Acta. 37 (1998) 21–29.
[38]
C. Gabriel, J. Kaschta, H. Münstedt, Influence of molecular structure on rheological properties of polyethylenes, Rheol. Acta. 37 (1998) 7–20.
[39]
S.G. Hatzikiriakos, I.B. Kazatchkov, D. Vlassopoulos, Interfacial phenomena in the capillary extrusion of
ACCEPTED MANUSCRIPT 24
metallocene polyethylenes, J. Rheol. 41 (1997) 1299–1316. [40]
J.M. Carella, J.T. Gotro, W.W. Graessley, Thermorheological effects of long-chain branching in entangled polymer melts, Macromolecules. 19 (1986) 659–667. M.L. Sentmanat, Dual windup extensional rheometer, (2003).
[42]
Y. Doi, F.M. Gray, F.L. Buchholz, A.T. Graham, A. Striegel, W.W. Yau, J.J. Kirkland, D.D. Bly, M.J.
RI PT
[41]
Gordon Jr, R.D. Archer, Rheology: principles, measurements, and applications, (1994).
J.R. Dorgan, D.M. Knauss, H.A. Al-Muallem, T. Huang, D. Vlassopoulos, Melt rheology of dendritically
SC
[43]
branched polystyrenes, Macromolecules. 36 (2003) 380–388.
P.F.W. Simon, A.H.E. Müller, T. Pakula, Characterization of highly branched poly (methyl methacrylate)
M AN U
[44]
by solution viscosity and viscoelastic spectroscopy, Macromolecules. 34 (2001) 1677–1684. [45]
M. Dennis, Introduction To Industrial Polyethylene: Properties, Catalysts and Processes, 2010. doi:10.1002/9781118463215.
[46]
L.J. Fetters, D.J. Lohse, D. Richter, T.A. Witten, A. Zirkel, Connection between polymer molecular
TE D
weight, density, chain dimensions, and melt viscoelastic properties, Macromolecules. 27 (1994) 4639– 4647.
J.D. Ferry, Viscoelastic properties of polymers, John Wiley & Sons, 1980.
[48]
D. Walsh, P. Zoller, Standard pressure volume temperature data for polymers, CRC Press, 1995.
[49]
H.H. Winter, M. Mours, Rheology of polymers near liquid-solid transitions, in: Neutron Spin Echo
EP
[47]
[50]
AC C
Spectrosc. Viscoelasticity Rheol., Springer, 1997: pp. 165–234. C.A. Garc’\ia-Franco, S. Srinivas, D.J. Lohse, P. Brant, Similarities between gelation and long chain branching viscoelastic behavior, Macromolecules. 34 (2001) 3115–3117. [51]
C.G. Robertson, C.A. Garc’\ia-Franco, S. Srinivas, Extent of branching from linear viscoelasticity of long-chain-branched polymers, J. Polym. Sci. Part B Polym. Phys. 42 (2004) 1671–1684.
[52]
R.C. Ball, T.C.B. McLeish, Dynamic dilution and the viscosity of star-polymer melts, Macromolecules. 22 (1989) 1911–1913.
[53]
R.H. Colby, M. Rubinstein, Two-parameter scaling for polymers in $θ$ solvents, Macromolecules. 23 (1990) 2753–2757.
ACCEPTED MANUSCRIPT 25
[54]
D.R. Daniels, T.C.B. McLeish, R. Kant, B.J. Crosby, R.N. Young, A. Pryke, J. Allgaier, D.J. Groves, R.J. Hawkins, Linear rheology of diluted linear, star and model long chain branched polymer melts, Rheol. Acta. 40 (2001) 403–415.
[55]
V.R. Raju, E. V Menezes, G. Marin, W.W. Graessley, L.J. Fetters, Concentration and molecular weight
RI PT
dependence of viscoelastic properties in linear and star polymers, Macromolecules. 14 (1981) 1668– 1676. [56]
H. Tao, C. Huang, T.P. Lodge, Correlation length and entanglement spacing in concentrated
[57]
SC
hydrogenated polybutadiene solutions, Macromolecules. 32 (1999) 1212–1217.
A. Miros, D. Vlassopoulos, A.E. Likhtman, J. Roovers, Linear rheology of multiarm star polymers
[58]
M AN U
diluted with short linear chains, J. Rheol. 47 (2003) 163–176.
U. Keßner, J. Kaschta, H. Münstedt, Determination of method-invariant activation energies of long-chain branched low-density polyethylenes, J. Rheol. (N. Y. N. Y). 53 (2009) 1001. doi:10.1122/1.3124682.
[59]
J.F. Vega, A. Santamaria, A. Munoz-Escalona, P. Lafuente, Small-amplitude oscillatory shear flow measurements as a tool to detect very low amounts of long chain branching in polyethylenes,
[60]
TE D
Macromolecules. 31 (1998) 3639–3647.
J.L. White, H. Yamane, A collaborative study of the stability of extrusion, melt spinning and tubular film extrusion of some high-, low-and linear-low density polyethylene samples, Pure Appl. Chem. 59 (1987)
[61]
EP
193–216.
H. Mavridis, R. Shroff, Appraisal of a molecular weight distribution-to-rheology conversion scheme for
[62]
AC C
linear polyethylenes, J. Appl. Polym. Sci. 49 (1993) 299–318. T.P. Karjala, R.L. Sammler, M.A. Mangnus, L.G. Hazlitt, M.S. Johnson, J. Wang, C.M. Hagen, J.W.L. Huang, K.N. Reichek, Detection of low levels of long-chain branching in polydisperse polyethylene materials, J. Appl. Polym. Sci. 119 (2011) 636–646. [63]
M. Ansari, S.G. Hatzikiriakos, A.M. Sukhadia, D.C. Rohlfing, Rheology of Ziegler--Natta and metallocene high-density polyethylenes: broad molecular weight distribution effects, Rheol. Acta. 50 (2011) 17–27.
[64]
M. Van Gurp, J. Palmen, Time-temperature superposition for polymeric blends, Rheol. Bull. 67 (1998).
[65]
D.J. Lohse, S.T. Milner, L.J. Fetters, M. Xenidou, N. Hadjichristidis, R.A. Mendelson, C.A. Garcia-
ACCEPTED MANUSCRIPT 26
Franco, M.K. Lyon, Well-defined, model long chain branched polyethylene. 2. Melt rheological behavior, Macromolecules. 35 (2002) 3066–3075. [66]
L.J. Fetters, A.D. Kiss, D.S. Pearson, G.F. Quack, F.J. Vitus, Rheological behavior of star-shaped
[67]
S. Trinkle, P. Walter, C. Friedrich, Van Gurp-Palmen plot II--classification of long chain branched polymers by their topology, Rheol. Acta. 41 (2002) 103–113.
B.A. Story, G.W. Knight, The new family of polyolefins from insite technology, Proc. Metcon ‘93. (1993) 112.
[69]
S. Lai, others, CGCT: New Rules for Ethylene alpha-Olefin Interpolymers-Controlled Melt Rheology
M AN U
Polyolefins, in: ANTEC, 1993: pp. 1188–1192. [70]
SC
[68]
RI PT
polymers, Macromolecules. 26 (1993) 647–654.
S. Lai, T.A. Plumley, T.I. Butler, G.W. Knight, Dow rheology index (DRI) for insite technology polyolefins (ITP): unique structure-processing relationships, in: Tech. Pap. Annu. Tech. Conf. Plast. Eng. Inc., 1994: p. 1814.
[71]
Y.S. Kim, C.I. Chung, S.Y. Lai, K.S. Hyun, Melt rheological and thermodynamic properties of
TE D
polyethylene homopolymers and poly (ethylene/$α$-olefin) copolymers with respect to molecular composition and structure, J. Appl. Polym. Sci. 59 (1996) 125–137. [72]
J. Janzen, R.H. Colby, Diagnosing long-chain branching in polyethylenes, J. Mol. Struct. 485 (1999)
[73]
EP
569–584.
D. Auhl, P. Chambon, T.C.B. McLeish, D.J. Read, Elongational flow of blends of long and short
[74]
AC C
polymers: Effective stretch relaxation time, Phys. Rev. Lett. 103 (2009) 136001. M. Ansari, Y.W. Inn, A.M. Sukhadia, P.J. Deslauriers, S.G. Hatzikiriakos, Melt fracture of HDPEs: Metallocene versus Ziegler-Natta and broad MWD effects, Polym. (United Kingdom). 53 (2012) 4195– 4201. doi:10.1016/j.polymer.2012.07.030.
S0142-9418(17)30086-7
DOI:
10.1016/j.polymertesting.2017.03.010
Reference:
POTE 4957
To appear in:
Polymer Testing
Received Date: 25 January 2017 Revised Date:
2 March 2017
Accepted Date: 9 March 2017
Please cite this article as: M. Derakhshandeh, M. Ansari, A.K. Doufas, S.G. Hatzikiriakos, Microstructure characterization of polyethylene using thermo-rheological methods, Polymer Testing (2017), doi: 10.1016/j.polymertesting.2017.03.010. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT 1
Microstructure Characterization of Polyethylene using Thermo-rheological Methods Maziar Derakhshandeh1, Mahmoud Ansari1, Antonios K. Doufas2 and Savvas G. Hatzikiriakos1,∗ Department of Chemical and Biological Engineering, The University British Columbia, Vancouver, BC, Canada 2
RI PT
1
ExxonMobil Chemical Company, Baytown, TX, USA
Abstract
SC
In the present work, we use the viscoelastic moduli of a large number of industrially available polyethylenes in order to evaluate/test some of the previously proposed correlations between levels of long chain branching and
M AN U
polydispersity with the rheological properties. These correlations together with some new ones can be used to correct for the effects of polydispersity or long chain branching in order to assess the effect of these two molecular features on the rheological properties independently. The effects of short and long chain branching are studied providing a methodology to detect rheologically levels of short and long chain branching.
TE D
Introduction
Polymers are extensively used in industry to fabricate plastic products using various processing techniques such as film blowing, thermoforming, and injection molding. Molecular and morphological characteristics such as molecular weight (Mw), molecular weight distribution (MWD), and branching along the backbone of the
EP
polymer chains govern the melt rheology of entangled polymers. To put this into a perspective, in the film blowing process a certain degree/distribution of long chain branching (LCB) within the matrix leads to a stable bubble. To overcome the issue of bubble instability, linear low density polyethylene (LLDPE) resins are often
AC C
blended with small amount of low density polyethylene (LDPE) to increase the extensional viscosity inducing a more stable bubble. Since LCB characterization is inherently a challenging topic of vital importance in the plastic industry, it has drawn significant attention of various researchers previously. The difficulty in microstructure characterization using rheology is due to two main factors. First, the production of polyethylenes of high Mw with well-defined levels, types, and distribution of long chain branching along the backbone is challenging. To this end, several studies were initiated on detecting subtle changes in polymer microstructure using rheology i.e. to differentiate between star, comb, pom-pom, comb-star, and dendritic long chain branched polymers [1–14]. Hierarchical relaxation was proposed as the mechanism by
∗ Corresponding author:[email protected]
ACCEPTED MANUSCRIPT 2
which the chains complete their relaxation [10,15]. The entanglements belonging to topologically different parts of the macromolecules relax in accordance to seniority rules [10,15]. The outer parts of molecules relax first including the branches on the backbone of chains. During this step the backbone is nearly frozen without any relaxation. As ends are freely dangling, backbone relaxation begins. The relaxed portion of the chain act as an
RI PT
effective solvent for the frozen backbone inducing enhanced relaxation by increasing the effective tube diameter as proposed in the tube model [16,17]. It was observed that the molecular weight of arms, the molecular weight between branch points, number of branches per backbone, functionality of branched point, and type of branching affects the rheological response extensively. Due to the complexity of industrial polymerization
SC
processes, often different type, degree, and distribution of branches along the backbone is obtained. This along with the polydispersity complicate the interplay of different relaxation modes and rheological responses [7,18– 21]. Therefore, more research on this end is still needed to characterize long chain branching and other
M AN U
microstructure features of industrially important polymers, in our case polyethylenes. The second factor which makes the microstructure assessment difficult lays in the fact that the melt rheology of entangled polymers is influenced more or less not only by LCB but also by Mw, its distribution and other microstructural features. In particular, the effects of polydispersity and long chain branching are pointing to the same direction making the analysis difficult [21]. It is highly desirable to develop a route for separating
TE D
these effects and if possible to assess whether or not possible effects in rheology or processing are due to long chain branching or polydispersity or other microstructural features. There is lately a considerable interest in detecting small level of long chain branching in metallocene polyethylenes (PE) [22–24]. This family of resins is a major technological advancement in the polyolefins
EP
industry i.e., the development of homogeneous metallocene catalysts for polymerization reactions. Metallocene catalysts produce polymers with narrow molecular weight and comonomer distributions, which, combined with
AC C
controlled amounts of long chain branching, are claimed to lead to both excellent processability and superior mechanical properties [25–27].
As mentioned, the presence of long branches in the structure of polymers affects the rheological properties in different ways. The zero-shear viscosity of polymer melts with long branches has been found to scale exponentially with Mw [28]. This makes their zero-shear viscosity much higher than those of their linear counterparts. In addition, branched polymers exhibit higher degrees of shear thinning [29], the onset of shear thinning occurs at smaller shear rates, they exhibit larger extrudate swell, and extensional viscosities [30–35]. Similar effects to these of long chain branching are also exhibited by polydispersity, although to a milder degree [36–39].
ACCEPTED MANUSCRIPT 3
Various authors have developed/proposed correlations in the literature between long chain branching and rheological properties. These include the relationships between the long chain branching and (i) flow energy of activation [38–40], (ii) a normalized elevation of the flow energy of activation [24], (iii) the area in the van Gurp plot, tan −1 (G ''/ G ') vs G* , included between the curve corresponding to the branched polymer and
RI PT
its linear counterpart, and finally (iv) the shift in the Mw-location that corresponds to the maximum in the predicted molecular weight distribution from rheological data with respect to the Mw-location of the maximum from GPC data. It is worthwhile to mention that different studies have found the shift factors to fall on the same curve with that of linear polymers when they are shifted with reference to glass transition temperature [5,7].
SC
In the present work, we use the linear viscoelastic moduli of a large number of industrially available polyethylenes in order to evaluate/test previously proposed correlations between levels of long chain branching
M AN U
and polydispersity with the rheological properties. These correlations together with some new ones in a graphical form can be used to correct for the effects of polydispersity or long chain branching in order to assess the effect of these two molecular features on the rheological properties independently. The effects of short and long chain branching are studied providing a methodology to detect levels of short and long chain branching.
TE D
Materials and Experimental Methods
Eleven PE resins have been studied with different molecular weight characteristics as listed in Table 1. The codes indicate the type (LL for linear low, H for high-density and L for low-density polyethylene), the Mw (kg/mol) and the polydispersity index (PDI) i.e. L-149-8p2 indicates a low-density polyethylene of Mw close to
EP
149 kg/mol (the exact value can be obtained from table 1) and PDI=8.2. The Melt Flow Index (MFI) was measured according to ASTM D1238 (190°C, 2.16 kg weight).
AC C
The linear viscoelastic properties of all resins have been determined using an Anton Paar MCR-502 (strain and stress-controlled rheometer) equipped with a cone-and-plate geometries. Four types of tests have been performed; small amplitude oscillatory shear (SAOS) or frequency sweep, step shear strain relaxation, creep/recovery and uniaxial extension tests. All shear experiments have been performed with cone-and-plate of 25mm in diameter and 4° in angle. To avoid degradation, all experiments have been run under nitrogen. Time sweep experiments confirmed the stability of samples during each measurement. The frequency sweep tests were carried out at temperatures ranging from 140°C to 220°C over 20°C increments. The various curves were shifted by means of applying the time-temperature superposition (tTS) in order to generate the master curves at the reference temperature of 180°C. The activation energies for the vertical and horizontal shift factors were determined, an indication of the level of LCB and polydispersity.
ACCEPTED MANUSCRIPT 4 Table 1. Polymers used in this study and their molecular characteristics.
#
Resin
Type
1
L-149-8p2
LDPE
149.5
8.2
2.8
0.22
2
LL-97-3p5
LLDPE
97.2
3.5
2.1
0.91
3
LL-113-2p4
LLDPE
113.2
2.4
4
LL-117-3p3
LLDPE
117.3
3.3
5
LL-124-4p5
LLDPE
124.1
4.5
6
LL-124-5p3
LLDPE
124.3
5.3
7
LL-143-6p0
LLDPE
143.2
8
H-121-32p0
HDPE
9
H-151-12p1
HDPE
10 H-158-12p3
HDPE
RI PT 0.90
2.9
0.52
3.1
0.58
3.5
0.38
6.0
3.2
0.33
120.8
32.0
5.2
0.09
151.1
12.1
5.3
0.26
157.6
12.3
5.2
0.24
287.0
14.2
4.5
0.04
M AN U
SC
1.8
TE D
11 bH-287-14p2 HDPE-bimodal
Mw (kg/mol) Mw/Mn Mz/Mw MFI (gr/10 min)
Although the experimental data extended to lower frequencies by applying the tTS technique, the terminal zone at low frequencies (zero-shear viscosity) for most of the cases here could not be achieved. One way to extend the data to even lower frequencies is using a step strain experiment and converting the data to the dynamic
EP
moduli (G’, G”). This can be done by calculating the parsimonious relaxation spectrum from the experimental relaxation modulus. The level of strain was chosen to be as low as 5% to ensure that the data are collected
AC C
within the linear viscoelastic envelope. Furthermore, creep experiments followed by recovery measurements have been performed with stress level of 5 Pa as another means of determining the zero-shear viscosity. The zero-shear viscosity values were found to be in agreement with those generated by step relaxation showing consistency of the obtained experimental results. The extensional behavior in simple uniaxial extension is also studied using the Sentmanat Extensional Rheometer (SER2) housed on the Anton Paar MCR-502 rotational rheometer [41]. The resins were molded using a hot press at 180°C in order to make a plaque with a thickness of around 0.7mm. Then rectangular strips were cut using a dual blade cutter. The dimensions of these strips were about 8 mm in width and 15 mm in length. All the uniaxial tests were conducted at 180°C over a wide range of Hencky strain rates, namely 0.01, 0.05, 0.5, 5.0 and 10.0 s-1.
ACCEPTED MANUSCRIPT 5
Results and Discussion Linear Viscoelasticity
RI PT
Figure 1 depicts the LVE master curve of LL-124-4p5 at the reference of 180oC. Experiments were run over the temperature range from 140oC to 220oC in order to generate the master curve extending the data over the frequency range that covers lower frequencies. However, this is not enough to reach the terminal zone so as to determine the zero-shear viscosity important in many applications, i.e., relate to molecular weight and its
SC
distribution. This terminal (low frequency) zone is limited by polymer degradation at high temperatures, while
EP
TE D
M AN U
the data at higher frequency cannot be obtained trivially due to the polymer crystallization at lower temperature.
Figure 1. The master curve for resin LL-124-4p5 at 180°C.
AC C
In order to extend the data to even lower frequencies, step strain relaxation measurements have been performed. Figure 2 shows such results for resin LL-124-4p5 for the strain level of 5%. The continuous line is the prediction of Maxwell relaxation model (Eq. 1) with 6 modes. The fitting has been performed by an inhouse Matlab code applying the Levenberg-Marquardt nonlinear optimization method. The relaxation spectra values were used into Eq. 2a & b in order to convert the relaxation modulus data to the dynamic ones [42]. N
G ( t ) = ∑Gi e − t / λi i =1
Eq. 1
ACCEPTED MANUSCRIPT 6
Gi (ωλi )
N
i =1
1 + (ωλi )
i =1
Eq. 2a
2
Gi (ωλi )
N
G '' (ω ) = ∑
2
1 + (ωλi )
Eq. 2b
2
RI PT
G ' (ω ) = ∑
N
η0 = ∑Gi λi
Eq. 2c
TE D
M AN U
SC
i =1
Figure 2. Relaxation modulus after imposing strain of 5% for resin LL-124-4p5 at 180°C.
EP
Figure 3 depicts the creep and recoverable compliance data for resin LL-124-4p5. For the creep experiment, a stress level of 5 Pa was imposed on the material and after a sufficiently long time (when a constant Newtonian slope has been achieved), the stress removed and the recoverable strain was measured. From linear
AC C
viscoelasticity, it is known that Equations 3a & b are valid for creep and recoverable compliances [42]: N
(
)
J ( t ) = J 0 + ∑J i 1 − e −t /τ i + t / η0 i =1
N
(
J r ( t ) = J 0 + ∑J i 1 − e−t /τ i i =1
)
Eq. 3a Eq. 3b
N
J e0 = ∑J i
Eq. 3c
i =1
where
is the zero-shear viscosity,
the instantaneous elastic compliance,
the retardation times, and
the
retardation strengths. Table 2 summarizes the obtained values of zero-shear viscosity, steady state creep compliances, and relaxation time of the different resins studied. Using these equations, it is expected that for a
ACCEPTED MANUSCRIPT 7
sufficiently long time, the creep compliance data fall into a straight line with a constant slope, which is the inverse of
(Equation 3a). The steady state creep compliance,
, can also be determined using both the
intercept of that line and the ultimate value of the recoverable creep compliance. It is also possible to calculate it by using the retardation spectrum from Eq. 3c. From Eq. 3a & b, it is possible to calculate the
from ( ). Figure 4 shows the comparison between the recoverable creep compliance
TE D
M AN U
SC
data obtaining from this calculation and the experimental data.
RI PT
subtracting values of ⁄
( ) by
Figure 3. Creep and recoverable compliances for resin LL-124-4p5 at 180°C.
It is clear from Figure 4 that both creep and recoverable compliances for resin LL-124-4p5 reach the ultimate values. It is again possible to convert these data to dynamic data in order to extend them to very low
EP
frequencies. This can be done by considering Eq. 4a & b [42]: N
AC C
J ' (ω ) = ∑
J ′′ (ω ) =
where:
i =1
1
ωη0
G' (ω ) = G '' (ω ) =
Ji
1 + (ωτ i ) N
+∑ i =1
Eq. 4a
2
Gi (ωτ i )
1 + (ωτ i )
J ' (ω )
J ' (ω ) + J '' (ω ) 2
2
J '' (ω )
J ' (ω ) + J '' (ω ) 2
2
2
Eq. 4b
Eq. 5a
Eq. 5b
ACCEPTED MANUSCRIPT
M AN U
SC
RI PT
8
Figure 4. Comparison between experimental data of recoverable creep compliance and calculated ones from creep data for resin LL-124-4p5 at 180°C.
Figure 5 illustrates the extended LVE data by converting the stress relaxation, creep and recovery data along with the master curve for resin LL-124-4p5. All sets of data show excellent consistency at low frequencies.
TE D
Moreover, the Newtonian region (zero-shear viscosity) has been reached for this polymer. The solid line is fitting of Eq. 2a & 2b with 10 modes. In the master curve obtained, by increasing frequency, the loss modulus always remains above storage modulus until very higher frequencies are reached. After the terminal regime, both modulus increase in parallel for about two decades (with the slope of around 0.8), this behavior is known
EP
as “Zimm-like” behavior and was observed for dendritically branched polystyrene [43]. As soon as the branches are relaxed they act as effective solvent and thus the backbone tube diameter increases. The increase in tube
AC C
diameter reduces the number of entanglements efficiently contributing to the observed Zimm-like behavior, which is similar to the behavior of untangled polymers. The longest and segmental relaxation times are determined as 2287 and 0.01s, respectively. The viscoelastic behavior between segmental and terminal relaxation times was used previously to obtain information about the relaxation spectrum of macromolecules and long chain branching characteristics [44].
ACCEPTED MANUSCRIPT
SC
RI PT
9
M AN U
Figure 5. A generalized master curve by converting data from different types of measurements for resin LL-124-4p5 at 180°C. The intersection of lines passing through the terminal region defines the longest relaxation time of the sample.
Table 2: Linear viscoelastic characteristics of the PE resins obtained from various methods. zero-shear
Steady state compliance
Relaxation time
Zimm-like
viscosity (kPa.s)
(1/kPa)
(s)
regime
Creep (Eq. 3a)
Relax.
Ret. Spec.
Ret. Spec.
(Creep,
(Recovery
Eq. 3c)
, Eq. 3c)
N/D
23.30
0.74
0.75
9.9
N/D
226.8
Spec. (Eq. 2c)
TE D
Resin
Creep
Recovery
(Eq. 3a)
(Eq. 3a)
13.01
Longest
Segmental
slope
breadth
N/D
N/D
2.95
0.54
0.77
13.47
0.0089
N/D
N/D
N/D
0.1216
0.0083
42.75
22.50
47.82
23.26
8953
0.0074
0.67
3.17
327.9
339.9
LL-97-3p5
26.3
27.0
LL-113-2p4
N/D
LL-117-3p3
236.4
LL-124-4p5
53.5
56.2
22.31
22.95
59.54
23.08
2287
0.0104
0.88
2.87
LL-124-5p3
896.8
880.6
16.90
4.54
15.19
4.65
1509
6.54
0.41
2.67
LL-143-6p0
265.2
329.2
3.50
13.89
3.56
14.07
598
0.36
N/A
H-121-32p0
1016.9
1001.8
4.84
6.60
4.48
8.57
3824
1.63
N/A
H-151-12p1
433.0
448.4
12.83
12.08
23.59
12.46
9616
0.098
0.75
2.15
H-158-12p3
301.4
318.0
36.03
20.29
42.05
31.86
10108
0.156
0.72
1.76
bH-287-14p2
1729.5
2337.9
12.75
13.77
12.80
14.03
19519
1.48
0.69
2.96
AC C
EP
L-149-8p2
N/A 0.68
2.00 N/A
ACCEPTED MANUSCRIPT 10
Thermo-rheological behavior (Energy of activation) The time-Temperature superposition (tTS) principle was used previously to obtain information on LCB characteristics. The vertical shift factors were obtained by using the available density functions. It was found that, all horizontal shift factors were fallen on the same curve if the superposition is done with respect to glass
RI PT
temperature transition of each resins indicating similar WLF model constants [5,7]. However, obtaining reliable rheological data for semicrystalline polymers below their crystallization temperatures is challenging. Therefore, in the current study both horizontal and vertical shift factors are obtained by shifting the data to a reference temperature higher than the melting peak temperature determined by calorimetry techniques. It is worth noting
SC
that a protocol for measuring the complete relaxation spectrum of semicrystalline polymers is established [7]. The presence of LCB, which is defined as side chains consist of 7 or higher carbon atoms [45], is
M AN U
determined by analyzing the vertical and horizontal shift factors in terms of the activation energies [8,11,16,46– 57]. High horizontal activation energies (usually accompanied by vertical shift factors, bT) is a sign of long chain branching. The benefit of this approach is that unlike other methods, the presence of high molecular weight tails in the MWD does not have an effect. However, the shortcomings include its insensitivity to the diminishing low level of LCB and also effects of side branch length. The typical values of horizontal activation energy (Eact,H) for HDPE and LLDPE are 27-28 and 33-35 kJ/mol, respectively. Moreover, such linear HDPE
TE D
and LLDPE polymers are thermo-rheologically simple vertical activation (Eact,V) close to zero. However, for the case of commercial LDPEs which contains high levels of long chain branches, either thermo-rheological complexity or simplicity with high Eact,H (>40) and Eact,V (>6) have been observed [58]. Based on the discussion
EP
provided, a first assessment for presence of LCB can be seen in Table 3. Table 3. Branching assessment based on activation energy values.
Eact,H (kJ/mol)
AC C
Resin
Eact,V (kJ/mol)
Branching
L-149-8p2
72.7
9.1
LCB
LL-97-3p5
52.9
10.2
LCB
LL-113-2p4
33.1
2.0
NB/SCB
LL-117-3p3
82.6
20.3
LCB
LL-124-4p5
30.0
2.6
NB/SCB
LL-124-5p3
40.7
6.2
LCB or SCB
LL-143-6p0
70.3
15.1
LCB
H-121-32p0
27.2
0.7 ≈ 0
NB
H-151-12p1
27.6
0.3 ≈ 0
NB
ACCEPTED MANUSCRIPT 11
H-158-12p3
31.9
2.5
SCB
bH-287-14p2
29.2
1.5
NB/SCB
NB: no branching; SCB: short chain branching; LCB: long chain branching
RI PT
The Mw-η0 relationship Viscosity at small shear rates (zero-shear viscosity) is typically correlated with molecular weight. A linear relation with Mw is observed for un-entangled polymers (very low Mw). For linear entangled polymers with narrow MWD, there is a power-law relationship between zero-shear viscosity and molecular weight. Increasing the MWD or having LCB would cause deviation from this power-law relationship. Although initially it was
SC
claimed that polydispersity has no effect on the Mw-??0 relationship [59–61], Karjala et al. [62] has confirmed that only for Mw/Mn<10, the polydispersity has no effect on this relationship [62]. Wasserman & Graessly [34]
M AN U
have shown that the MWD effect can be adequately excluded by modifying the zero-shear viscosity over a factor of Mz/Mw. This was confirmed in one of our previous studies of some very broad MWD metallocene HDPEs [63]. Therefore, it is possible to detect LCB for these resins by checking the modified ??0-Mw relationship with available data on PE in the literature (Figure 6). The solid line represents the power-law relationship obtained by Karjala et al. [62] for some linear PE at 190°C. As it can be seen from this figure, for
⁄
>1, where
LL-113-2p4,
the
other
resins
is zero-shear viscosity of the polymer and
TE D
except
lay
above
this
line
with
is the zero-shear viscosity obtained using
the proposed Mw-η0 power-law relationship in the literature. It is an indication that these resins possess LCB. The arrow on top of the viscosity value of resin L-149-8p2 indicates that the actual zero-shear viscosity is higher than the reported value. This is due to the fact that the terminal regime for this particular resin (the only
EP
LDPE in this study) was not accessible even using long time creep/recovery experiments. The relaxation of linear polymer chains is governed by entanglements, however, for hyperbranched polymers,
AC C
the role of entanglements becomes negligible due to spatial barrier and limited interpenetration of the neighboring chains [44]. Therefore, for hyperbranched/dendritic type of LCB, the deviation from the reference line should be negative [43,44] which is not the case for any of the resins studied. Hence it can be concluded that there is no polymer possessing such a long chain dendritic structure. Table 4 lists the assessment for these resins based on the above analysis.
ACCEPTED MANUSCRIPT 12 Table 4. Branching assessment based on Mw-η0 relationship.
Mz/Mw
⁄
Branching
L-149-8p2
8.2
2.8
12.0
LCB
LL-97-3p5
3.5
2.1
5.1
LCB
LL-113-2p4
2.4
1.8
1.3
NB
LL-117-3p3
3.3
2.9
17.3
LCB
LL-124-4p5
4.5
3.1
5.1
LCB
LL-124-5p3
5.3
3.5
74.6
LCB
LL-143-6p0
6.0
3.2
10.8
LCB
H-121-32p0
32.0
5.2
H-151-12p1
12.1
5.3
H-158-12p3
12.3
5.2
bH-287-14p2
14.2
SC
LCB
3.9*
Low LCB
2.3*
Low LCB
1.7*
Low LCB
M AN U
20.5*
4.5
zero-shear viscosities are corrected by a factor of Mz/Mw according to Wasserman & Graessly [34].
AC C
EP
TE D
∗
RI PT
Mw/Mn
Resin
Figure 6. Modified η0 for the effect of polydispersity vs. Mw, for available resins. The modification is applied for the resins with Mw/Mn>10 (closed symbols). The arrow indicates that the actual η0 can be higher. The solid line represents the power-law relationship for linear PEs obtained by Karjala et al. [62] and the dashed line represents 3.65 ±0.02 to include uncertainty effect in Mw measurements (190°C).
ACCEPTED MANUSCRIPT 13
van Gurp-Palmen Graphs van Gurp-Palmen graph plots the phase angle as a function of complex modulus. It was proposed as an alternative method to check for the tTS failure [64]. At the low values of complex modulus, phase angle should approach 90o indicating complete viscous behavior. As the complex modulus increases, the phase angle
RI PT
decreases and it approaches 0o when the complex modulus approaches the plateau modulus indicating ideal elastic response. This limiting behavior is clearly observed in Figures 7a&b indicating the accuracy of data. The potential of van Gurp-Palmen graph in determining the branching characteristics was not known initially. This method has been used later by other authors to obtain information about the relaxation behavior of the desired
SC
polymers with the aim to reveal branching characteristics [65–67]. Different minimum within the plot shows relaxation of different origin (branch and backbone). It was also shown that the shape of van Gurp-Palmen
M AN U
graph is different if relaxation characteristics of branches are different (star shape, arm retraction vs comb, chain reptation) [65,66]. It is well-understood that both MWD and LCB shift the values of the phase angle to smaller angles at given levels of the complex modulus. Therefore, for resins with PDI greater than 10 it cannot be concluded clearly if the reduction is a result of LCB or MWD. It is worth noting that for these commercial
AC C
EP
TE D
polymers the distinct minimums which are often observed in control branched polymers are not revealed.
Figure 7. van Gurp-Palmen plots for resins with a) Mw/Mn<10 and b) Mw/Mn>10 at 180°C.
Table 5. Branching assessment based on van Gurp-Palmen graph.
Resin
Branching
ACCEPTED MANUSCRIPT 14
LCB
LL-97-3p5
LCB
LL-113-2p4
NB/SCB
LL-117-3p3
LCB
LL-124-4p5
LCB
LL-124-5p3
LCB
LL-143-6p0
LCB
H-121-32p0
LCB or MWD
H-151-12p1
LCB or MWD
H-158-12p3
LCB or MWD
bH-287-14p2
LCB or MWD
M AN U
SC
RI PT
L-149-8p2
Normalized Complex Viscosity
The normalized complex viscosities are plotted versus frequency for all resins in Figure 8. In this way, it is possible to compare the shear thinning behavior of these resins. One analysis which can be done here is to check the Dow Rheology Index (DRI) [68–71]. The idea behind this analysis is that if the Cross model (Eq. 7a) is used
TE D
to fit the viscosity data, the relaxation time (τ0) in this model should obey Eq. 7b for a linear chain polymer. Therefore, the DRI index can be defined based on deviation from this relationship (Eq. 7c):
| ∗ ( )| =
. ) = 3.65 × 10!
EP
(
AC C
"#$ =
⁄ 1+(
)
3.65 × 10! ( ⁄ ) − 1 10
Eq. 7a Eq. 7b Eq. 7c
ACCEPTED MANUSCRIPT 15 Figure 8. Normalized complex viscosity versus frequency for resins with a) Mw/Mn<10 and b) Mw/Mn>10 at 180°C.
Solid lines in Figure 8 are Cross model fittings. Table 6 collects the DRI and LCB branching based on
reflected on it. Therefore, it just can be used for narrow MWD resins. Table 6. Branching assessment & DRI based on normalized complex rheology.
RI PT
normalized complex viscosity data. The problem with DRI index is that the broadness of MWD can also be
Mw/Mn
η0 (Pa.s)
τ0 (s)
n
L-149-8p2
8.2
339755*
182
0.610
LL-97-3p5
3.5
26966
2.114
0.527
2.8
LCB
LL-113-2p4
2.4
9922
0.02529
0.768
0.0
NB
LL-117-3p3
3.3
226701
872.7
0.433
140.4
LCB
LL-124-4p5
4.5
56178
21.28
0.463
13.7
LCB
LL-124-5p3
5.3
879731
1088
0.559
45.0
LCB
LL-143-6p0
6.0
243339
149.4
0.551
22.3
LCB
H-121-32p0
32.0
1001725
60.56
0.679
2.1
LCB or MWD
H-151-12p1
12.1
448307
787.1
0.503
64.0
LCB or MWD
H-158-12p3
12.3
317510
319.9
0.528
36.7
LCB or MWD
bH-287-14p2
14.2
2334785
1084
0.620
16.8
LCB or MWD
DRI
Branching
19.5
LCB
SC
M AN U
TE D
Resin
* zero-shear viscosity couldn’t be obtained experimentally and it is calculated from the Maxwell model (Eq. 2c).
EP
Quantitative approach
Janzen and Colby proposed the following equations for the effect of LCB on the Mw-η0 relationship[72]:
AC C
= &'( 1 + ('( ⁄') )*.+ (', ⁄'( )-
3 9 '( . = / 0 11, + 678 9 => 2 8 90':;< ?=
' 1 1 9 − = 2 '( ',
Eq. 6a Eq. 6b Eq. 6c
where K is a coefficient having units of Pa.s/(g/mol), Mb is an average molecular weight between branch points, Mc is the critical molecular weight for entanglements and Mw is the molecular weight. “α“ is also the fraction of long-chain branch points. Having K=5.22×10-6 (Pa.s)/(g/mol), B=6, M0=14.027 g/mol, Mc=2100 g/mol and MKuhn=145.9 g/mol, it is possible to solve the above equations for Mb and α [72].
ACCEPTED MANUSCRIPT 16
As discussed earlier, polydispersity has an effect on η0-Mw relationship, particularly for resins with Mw/Mn>10. This correction has been performed according to Wasserman & Graessly who have suggested to divide the zeroshear viscosity by Mz/Mw [34]. Table 7 collects the values of Cb=Mb/M0 (average number of monomers between branch points) and α from both approaches. Figure 9 shows Cb versus Mw/M0 for all the resins. The distance of
RI PT
the values from the continuous unit line depicts the degree of LCB present. The LCB assessment presented in
TE D
M AN U
SC
Table 7 is based on such analogy.
AC C
EP
Figure 9. Cb versus Mw/M0 for all the resins
Table 7. LCB analysis using Eq. 6a-c based on Janzen-Colby [72] relationship.
Cb
α
LCB Assessment
L-149-8p2
8642
1.09×10-5
LCB
LL-97-3p5
5910
1.24×10-5
LCB
7968
-7
SCB/NB
-4
LCB
-6
SCB/NB
-5
LCB
-5
LCB
Resin
LL-113-2p4 LL-117-3p3 LL-124-4p5 LL-124-5p3 LL-143-6p0
1657 7789 5867 8370
7.91×10
2.42×10 7.68×10
2.88×10 1.08×10
ACCEPTED MANUSCRIPT 17
H-121-32p0 H-151-12p1 H-158-12p3
6668
1.69×10-5
9799
-6
SCB/NB
-6
SCB/NB
-6
SCB/NB
10594 19437
4.61×10
2.69×10 1.29×10
RI PT
bH-287-14p2
LCB
Uniaxial Extension
Extensional experiments are solid in predicting the degree and type of different branches on the backbone. It
SC
was shown previously that only small amount of combs (5%) on the backbone is sufficient to induce extensional strain hardening [65]. Extensional rheology of well-defined comb polymers have reported that increasing the average number of entanglements on the branches or between branch points on the backbone
M AN U
induces stronger hardening at lower strain rates [18]. Of note, the inverse of the onset rate for strain hardening is referred to as the effective stretch time which is related to the Rouse time of chains [18,73]. As shown in Figure 10, the onset of strain hardening at lowest studied rate for LL-124-4p5 and LL-143-6p0 occurred at elapsed time of ~40 and ~65s, respectively. Since from Janzen and Colby analysis [72], “α“ (fraction of long-chain branch points) is larger for LL-143-6p0 and both of resins have similar Mw, LL-124-4p5
TE D
should possess longer branches (more entangled branches) in order to induce the onset of hardening at lower elapsed time. This finding is in agreement with the study of Lentzakis et al. [18] who find the key parameter to be the number of entanglements per branch. For H-121-32p0 very small upturn in torque signal was observed. This can be either an indication of the presence of small degree of LCB on the backbone or due to its very broad
EP
molecular weight distribution. For bH-287-14p2, the hardening happens for all the investigated rates even at the smallest deformation rates employed. These upturns are not abrupt when compared to those of LL-143-6p0 and
AC C
LL-124-4p5. Similar conclusion as for resin H-121-32p0 can be drawn here, but the effect of polydispersity is more dominant due to its bimodal MWD [74].
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
18
Figure 10a-c. Uniaxial stress growth coefficient versus time for all the resins at 180°C.
ACCEPTED MANUSCRIPT 19 Table 8. Branching assessment based on extensional rheology.
Resin
Branching LCB
LL-97-3p5
NB/low LCB
LL-113-2p4
No LCB
LL-117-3p3
LCB
LL-124-4p5
LCB
LL-124-5p3
NB/low LCB
LL-143-6p0
LCB
H-121-32p0
NB/low LCB
H-151-12p1
NB/low LCB
H-158-12p3
NB/low LCB
SC low LCB
TE D
Conclusions
M AN U
bH-287-14p2
RI PT
L-149-8p2
In this paper, eleven PE resins with different molecular characteristics are examined using rheological methods in an attempt to obtain information on their branching characteristics. Table 9 summarizes the conclusions made using various methods mentioned in the results and discussion. Although thermo-rheological method benefits
EP
from insensitivity to the presence of high molecular weight tails in the MWD, it should be used along with other methods to obtain the correct LCB characteristics. On the other hand, other rheological methods lack the
AC C
insensitivity to MWD. Therefore, as shown, different methods should be compared simultaneously in order to infer the correct long chain branching characteristics.
ACCEPTED MANUSCRIPT 20
Table 9. Comparison of branching assessments obtained by using different rheological methods. η0 Mw-η
van Gurp plot
relationship
Resin
Normalized |η η*|
Extensional Rheology
Janzen-Colby Conclusion
RI PT
Thermorheology
Branching
Branching
Branching
Branching
Branching
L-149-8p2
LCB
LCB
LCB
LCB
LCB
LCB
LCB
LL-97-3p5
LCB
LCB
LCB
LCB
No/low LCB
LCB
Low LCB
LL-113-2p4
SCB
NB
No LCB
SCB/NB
SCB
LL-117-3p3
LCB
LCB
LCB
LCB
LCB
LCB
LL-124-4p5
NB/SCB
LCB
LCB
LCB
LCB
SCB/NB
LCB
LL-124-5p3
LCB or SCB
LCB
LCB
LCB
NB/low LCB
LCB
Low LCB
LL-143-6p0
LCB
LCB
LCB
LCB
LCB
LCB
LCB
H-121-32p0
NB
LCB
LCB or MWD
NB/low LCB
LCB
NB
H-151-12p1
NB
Low LCB
LCB or MWD
NB/low LCB
SCB/NB
NB
H-158-12p3
NB/SCB
Low LCB
LCB or MWD
NB/low LCB
SCB/NB
NB/low LCB
bH-287-14p2
NB/SCB
Low LCB
LCB or MWD
low LCB
SCB/NB
NB/low LCB
NB
LCB
M AN U
ND (LCB or MWD)
ND (LCB or
TE D
AC C
Acknowledgements
Branching/SCB
MWD)
ND (LCB or MWD)
ND (LCB or MWD)
EP
ND: not determined.
No
SC
Branching
The authors would like to acknowledge ExxonMobil Chemical for providing the materials, their GPC characterization and funding for this project.
ACCEPTED MANUSCRIPT 21
REFERENCES [1]
E. Van Ruymbeke, M. Kapnistos, D. Vlassopoulos, T. Huang, D.M. Knauss, Linear melt rheology of pom-pom polystyrenes with unentangled branches, Macromolecules. 40 (2007) 1713–1719. T.C.B. McLeish, S.T. Milner, Entangled dynamics and melt flow of branched polymers, in: Branched
RI PT
[2]
Polym. II, Springer, 1999: pp. 195–256. [3]
J.H. Lee, L.J. Fetters, L.A. Archer, Stress relaxation of branched polymers, Macromolecules. 38 (2005)
[4]
SC
10763–10771.
D.R. Daniels, T.C.B. McLeish, B.J. Crosby, R.N. Young, C.M. Fernyhough, Molecular rheology of
[5]
M AN U
comb polymer melts. 1. Linear viscoelastic response, Macromolecules. 34 (2001) 7025–7033. M. Kapnistos, D. Vlassopoulos, J. Roovers, L.G. Leal, Linear rheology of architecturally complex macromolecules: Comb polymers with linear backbones, Macromolecules. 38 (2005) 7852–7862. [6]
N.J. Inkson, R.S. Graham, T.C.B. McLeish, D.J. Groves, C.M. Fernyhough, Viscoelasticity of monodisperse comb polymer melts, Macromolecules. 39 (2006) 4217–4227. M. Kapnistos, G. Koutalas, N. Hadjichristidis, J. Roovers, D.J. Lohse, D. Vlassopoulos, Linear rheology
TE D
[7]
of comb polymers with star-like backbones: melts and solutions, Rheol. Acta. 46 (2006) 273–286. [8]
A.L. Frischknecht, S.T. Milner, A. Pryke, R.N. Young, R. Hawkins, T.C.B. McLeish, Rheology of three-
[9]
EP
arm asymmetric star polymer melts, Macromolecules. 35 (2002) 4801–4820. R.J. Blackwell, O.G. Harlen, T.C.B. McLeish, Theoretical linear and nonlinear rheology of symmetric
AC C
treelike polymer melts, Macromolecules. 34 (2001) 2579–2596. [10]
T.C.B. McLeish, Tube theory of entangled polymer dynamics, Adv. Phys. 51 (2002) 1379–1527.
[11]
T.C.B. McLeish, R.G. Larson, Molecular constitutive equations for a class of branched polymers: The pom-pom polymer, J. Rheol. 42 (1998) 81–110.
[12]
R.S. Graham, T.C.B. McLeish, O.G. Harlen, Using the pom-pom equations to analyze polymer melts in exponential shear, J. Rheol. 45 (2001) 275–290.
[13]
L.A. Archer, Juliani, Linear and nonlinear viscoelasticity of entangled multiarm (Pom-Pom) polymer liquids, Macromolecules. 37 (2004) 1076–1088.
ACCEPTED MANUSCRIPT 22
[14]
C.D. Chodankar, J.D. Schieber, D.C. Venerus, Pom--Pom theory evaluation in double-step strain flows, J. Rheol. 47 (2003) 413–427.
[15]
T.C.B. McLeish, Hierarchical relaxation in tube models of branched polymers, EPL (Europhysics Lett. 6
[16]
RI PT
(1988) 511. G. Marrucci, Relaxation by reptation and tube enlargement: a model for polydisperse polymers, J. Polym. Sci. Polym. Phys. Ed. 23 (1985) 159–177. [17]
S.T. Milner, T.C.B. McLeish, Arm-length dependence of stress relaxation in star polymer melts,
[18]
SC
Macromolecules. 31 (1998) 7479–7482.
H. Lentzakis, D. Vlassopoulos, D.J. Read, H. Lee, T. Chang, P. Driva, N. Hadjichristidis, Uniaxial
[19]
M AN U
extensional rheology of well-characterized comb polymers, J. Rheol. 57 (2013) 605–625. M. Ahmadi, C. Bailly, R. Keunings, M. Nekoomanesh, H. Arabi, E. Van Ruymbeke, Time marching algorithm for predicting the linear rheology of monodisperse comb polymer melts, Macromolecules. 44 (2011) 647–659. [20]
K.M. Kirkwood, L.G. Leal, D. Vlassopoulos, P. Driva, N. Hadjichristidis, Stress relaxation of comb
[21]
TE D
polymers with short branches, Macromolecules. 42 (2009) 9592–9608. S.G. Hatzikiriakos, Long chain branching and polydispersity effects on the rheological properties of polyethylenes, Polym. Eng. Sci. 40 (2000) 2279–2287. A. Malmberg, C. Gabriel, T. Steffl, H. Münstedt, B. Löfgren, Long-chain branching in metallocene-
EP
[22]
catalyzed polyethylenes investigated by low oscillatory shear and uniaxial extensional rheometry,
[23]
AC C
Macromolecules. 35 (2002) 1038–1048. C. Piel, F.J. Stadler, J. Kaschta, S. Rulhoff, H. Münstedt, W. Kaminsky, Structure-Property Relationships of Linear and Long-Chain Branched Metallocene High-Density Polyethylenes Characterized by Shear Rheology and SEC-MALLS, Macromol. Chem. Phys. 207 (2006) 26–38. [24]
P.M. Wood-Adams, J.M. Dealy, Using rheological data to determine the branching level in metallocene polyethylenes, Macromolecules. 33 (2000) 7481–7488.
[25]
J. Scheirs, W. Kaminsky, Metallocene-based Polyolefins: Preparation, properties, and technology, Wiley, 2000.
[26]
K.W. Swogger, G.M. Lancaster, S.Y. Lai, T.I. Butler, Improving Polymer Processability Utilizing
ACCEPTED MANUSCRIPT 23
Constrained Geometry Single Sue Catalyst Technology, J. Plast. Film Sheeting. 11 (1995) 102–112. [27]
S.-Y. Lai, J.R. Wilson, G.W. Knight, J.C. Stevens, P.-W.S. Chum, Elastic substantially linear olefin polymers, (1993). S.-Y. Lai, J.R. Wilson, G.W. Knight, J.C. Stevens, Elastic substantially linear olefin polymers, (1995).
[29]
J.M. Delay, K.F. Wissbrun, Melt rheology and its role in plastics processing: Theory and applications,
RI PT
[28]
(1990).
M.S. Jaćović, D. Pollock, R.S. Porter, A rheological study of long branching in polyethylene by blending,
SC
[30]
J. Appl. Polym. Sci. 23 (1979) 517–527.
B.H. Bersted, On the effects of very low levels of long chain branching on rheological behavior in
M AN U
[31]
polyethylene, J. Appl. Polym. Sci. 30 (1985) 3751–3765. [32]
V.R. Raju, G.G. Smith, G. Marin, J.R. Knox, W.W. Graessley, Properties of amorphous and crystallizable hydrocarbon polymers. I. Melt rheology of fractions of linear polyethylene, J. Polym. Sci. Polym. Phys. Ed. 17 (1979) 1183–1195.
R.J. Koopmans, Extrudate swell of high density polyethylene. Part I: Aspects of molecular structure and
TE D
[33]
rheological characterization methods, Polym. Eng. Sci. 32 (1992) 1741–1749. [34]
S.H. Wasserman, W.W. Graessley, Prediction of linear viscoelastic response for entangled polyolefin
[35]
EP
melts from molecular weight distribution, Polym. Eng. Sci. 36 (1996) 852–861. M.G. Lachtermacher, A. Rudin, Reactive processing of LLDPEs in corotating intermeshing twin-screw extruder. I. Effect of peroxide treatment on polymer molecular structure, J. Appl. Polym. Sci. 58 (1995)
[36]
AC C
2077–2094.
C. Tzoganakis, A rheological evaluation of linear and branched controlled-rheology polypropylenes, Can. J. Chem. Eng. 72 (1994) 749–754.
[37]
H. Münstedt, S. Kurzbeck, L. Egersdörfer, Influence of molecular structure on rheological properties of polyethylenes, Rheol. Acta. 37 (1998) 21–29.
[38]
C. Gabriel, J. Kaschta, H. Münstedt, Influence of molecular structure on rheological properties of polyethylenes, Rheol. Acta. 37 (1998) 7–20.
[39]
S.G. Hatzikiriakos, I.B. Kazatchkov, D. Vlassopoulos, Interfacial phenomena in the capillary extrusion of
ACCEPTED MANUSCRIPT 24
metallocene polyethylenes, J. Rheol. 41 (1997) 1299–1316. [40]
J.M. Carella, J.T. Gotro, W.W. Graessley, Thermorheological effects of long-chain branching in entangled polymer melts, Macromolecules. 19 (1986) 659–667. M.L. Sentmanat, Dual windup extensional rheometer, (2003).
[42]
Y. Doi, F.M. Gray, F.L. Buchholz, A.T. Graham, A. Striegel, W.W. Yau, J.J. Kirkland, D.D. Bly, M.J.
RI PT
[41]
Gordon Jr, R.D. Archer, Rheology: principles, measurements, and applications, (1994).
J.R. Dorgan, D.M. Knauss, H.A. Al-Muallem, T. Huang, D. Vlassopoulos, Melt rheology of dendritically
SC
[43]
branched polystyrenes, Macromolecules. 36 (2003) 380–388.
P.F.W. Simon, A.H.E. Müller, T. Pakula, Characterization of highly branched poly (methyl methacrylate)
M AN U
[44]
by solution viscosity and viscoelastic spectroscopy, Macromolecules. 34 (2001) 1677–1684. [45]
M. Dennis, Introduction To Industrial Polyethylene: Properties, Catalysts and Processes, 2010. doi:10.1002/9781118463215.
[46]
L.J. Fetters, D.J. Lohse, D. Richter, T.A. Witten, A. Zirkel, Connection between polymer molecular
TE D
weight, density, chain dimensions, and melt viscoelastic properties, Macromolecules. 27 (1994) 4639– 4647.
J.D. Ferry, Viscoelastic properties of polymers, John Wiley & Sons, 1980.
[48]
D. Walsh, P. Zoller, Standard pressure volume temperature data for polymers, CRC Press, 1995.
[49]
H.H. Winter, M. Mours, Rheology of polymers near liquid-solid transitions, in: Neutron Spin Echo
EP
[47]
[50]
AC C
Spectrosc. Viscoelasticity Rheol., Springer, 1997: pp. 165–234. C.A. Garc’\ia-Franco, S. Srinivas, D.J. Lohse, P. Brant, Similarities between gelation and long chain branching viscoelastic behavior, Macromolecules. 34 (2001) 3115–3117. [51]
C.G. Robertson, C.A. Garc’\ia-Franco, S. Srinivas, Extent of branching from linear viscoelasticity of long-chain-branched polymers, J. Polym. Sci. Part B Polym. Phys. 42 (2004) 1671–1684.
[52]
R.C. Ball, T.C.B. McLeish, Dynamic dilution and the viscosity of star-polymer melts, Macromolecules. 22 (1989) 1911–1913.
[53]
R.H. Colby, M. Rubinstein, Two-parameter scaling for polymers in $θ$ solvents, Macromolecules. 23 (1990) 2753–2757.
ACCEPTED MANUSCRIPT 25
[54]
D.R. Daniels, T.C.B. McLeish, R. Kant, B.J. Crosby, R.N. Young, A. Pryke, J. Allgaier, D.J. Groves, R.J. Hawkins, Linear rheology of diluted linear, star and model long chain branched polymer melts, Rheol. Acta. 40 (2001) 403–415.
[55]
V.R. Raju, E. V Menezes, G. Marin, W.W. Graessley, L.J. Fetters, Concentration and molecular weight
RI PT
dependence of viscoelastic properties in linear and star polymers, Macromolecules. 14 (1981) 1668– 1676. [56]
H. Tao, C. Huang, T.P. Lodge, Correlation length and entanglement spacing in concentrated
[57]
SC
hydrogenated polybutadiene solutions, Macromolecules. 32 (1999) 1212–1217.
A. Miros, D. Vlassopoulos, A.E. Likhtman, J. Roovers, Linear rheology of multiarm star polymers
[58]
M AN U
diluted with short linear chains, J. Rheol. 47 (2003) 163–176.
U. Keßner, J. Kaschta, H. Münstedt, Determination of method-invariant activation energies of long-chain branched low-density polyethylenes, J. Rheol. (N. Y. N. Y). 53 (2009) 1001. doi:10.1122/1.3124682.
[59]
J.F. Vega, A. Santamaria, A. Munoz-Escalona, P. Lafuente, Small-amplitude oscillatory shear flow measurements as a tool to detect very low amounts of long chain branching in polyethylenes,
[60]
TE D
Macromolecules. 31 (1998) 3639–3647.
J.L. White, H. Yamane, A collaborative study of the stability of extrusion, melt spinning and tubular film extrusion of some high-, low-and linear-low density polyethylene samples, Pure Appl. Chem. 59 (1987)
[61]
EP
193–216.
H. Mavridis, R. Shroff, Appraisal of a molecular weight distribution-to-rheology conversion scheme for
[62]
AC C
linear polyethylenes, J. Appl. Polym. Sci. 49 (1993) 299–318. T.P. Karjala, R.L. Sammler, M.A. Mangnus, L.G. Hazlitt, M.S. Johnson, J. Wang, C.M. Hagen, J.W.L. Huang, K.N. Reichek, Detection of low levels of long-chain branching in polydisperse polyethylene materials, J. Appl. Polym. Sci. 119 (2011) 636–646. [63]
M. Ansari, S.G. Hatzikiriakos, A.M. Sukhadia, D.C. Rohlfing, Rheology of Ziegler--Natta and metallocene high-density polyethylenes: broad molecular weight distribution effects, Rheol. Acta. 50 (2011) 17–27.
[64]
M. Van Gurp, J. Palmen, Time-temperature superposition for polymeric blends, Rheol. Bull. 67 (1998).
[65]
D.J. Lohse, S.T. Milner, L.J. Fetters, M. Xenidou, N. Hadjichristidis, R.A. Mendelson, C.A. Garcia-
ACCEPTED MANUSCRIPT 26
Franco, M.K. Lyon, Well-defined, model long chain branched polyethylene. 2. Melt rheological behavior, Macromolecules. 35 (2002) 3066–3075. [66]
L.J. Fetters, A.D. Kiss, D.S. Pearson, G.F. Quack, F.J. Vitus, Rheological behavior of star-shaped
[67]
S. Trinkle, P. Walter, C. Friedrich, Van Gurp-Palmen plot II--classification of long chain branched polymers by their topology, Rheol. Acta. 41 (2002) 103–113.
B.A. Story, G.W. Knight, The new family of polyolefins from insite technology, Proc. Metcon ‘93. (1993) 112.
[69]
S. Lai, others, CGCT: New Rules for Ethylene alpha-Olefin Interpolymers-Controlled Melt Rheology
M AN U
Polyolefins, in: ANTEC, 1993: pp. 1188–1192. [70]
SC
[68]
RI PT
polymers, Macromolecules. 26 (1993) 647–654.
S. Lai, T.A. Plumley, T.I. Butler, G.W. Knight, Dow rheology index (DRI) for insite technology polyolefins (ITP): unique structure-processing relationships, in: Tech. Pap. Annu. Tech. Conf. Plast. Eng. Inc., 1994: p. 1814.
[71]
Y.S. Kim, C.I. Chung, S.Y. Lai, K.S. Hyun, Melt rheological and thermodynamic properties of
TE D
polyethylene homopolymers and poly (ethylene/$α$-olefin) copolymers with respect to molecular composition and structure, J. Appl. Polym. Sci. 59 (1996) 125–137. [72]
J. Janzen, R.H. Colby, Diagnosing long-chain branching in polyethylenes, J. Mol. Struct. 485 (1999)
[73]
EP
569–584.
D. Auhl, P. Chambon, T.C.B. McLeish, D.J. Read, Elongational flow of blends of long and short
[74]
AC C
polymers: Effective stretch relaxation time, Phys. Rev. Lett. 103 (2009) 136001. M. Ansari, Y.W. Inn, A.M. Sukhadia, P.J. Deslauriers, S.G. Hatzikiriakos, Melt fracture of HDPEs: Metallocene versus Ziegler-Natta and broad MWD effects, Polym. (United Kingdom). 53 (2012) 4195– 4201. doi:10.1016/j.polymer.2012.07.030.