Relative quantification of long chain branching in essentially linear polyethylenes

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European Polymer Journal 44 (2008) 376–391

EUROPEAN POLYMER JOURNAL www.elsevier.com/locate/europolj

Relative quantification of long chain branching in essentially linear polyethylenes Ce´sar A. Garcı´a-Franco a,*, David J. Lohse b, Christopher G. Robertson c, Olivier Georjon d a

ExxonMobil Chemical Co., Baytown Technology and Engineering Complex/West. 5200 Bayway Dr., Baytown, TX 77522, USA b ExxonMobil Research and Engineering Co., Corporate Strategic Research, 1545 Route 22 East, Annandale, NJ 08801, USA c Bridgestone Americas, Center for Research and Technology, 1200 Firestone Parkway, Akron, OH 44317-001, USA d ExxonMobil Chemical Europe Inc., European Technology Center, Hermeslaan 2, B-1831, Belgium Received 3 August 2007; received in revised form 24 October 2007; accepted 31 October 2007 Available online 17 November 2007

Abstract The aim of this work is to describe a method whereby low levels of long chain branching, LCB, can be quantified on a relative basis for whole, unfractionated, and essentially linear ethylene/a-olefin copolymers. The method is based on a well established, relatively fast and robust experiment, namely the measurement of the linear viscoelastic properties by a single, isothermal, small amplitude oscillatory shear experiment. The analysis of the data is predicated on the use of the so-called van Gurp–Palmen plots (the phase angle, d (=tan1(G00 /G0 )), plotted against the absolute value of the dynamic complex modulus, |G*| = (G0 2+G00 2)1/2). From this plot, the value of d at |G*| = 10 kPa is recorded, and it is demonstrated that the amount of LCB inversely correlates with such value of the phase angle, d. Depending on the desired frequency range, the experiment duration varies between 15 and 60 min rendering this technique well suited for high throughput parallel testing. Its applicability is critically examined with a wide variety of commercial ethylene/a-olefin copolymers. Moreover, we have improved on the long chain branching index (LCBI) proposed by Shroff and Mavridis [Shroff RN, Mavridis H. Long-chain-branching index for essentially linear polyethylenes. Macromolecules 1999;32:8454–64] by basing it on data of truly linear polyethylenes (hydrogenated anionically synthesized polybutadienes) instead of apparently linear commercial polyethylenes. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Rheology; Long chain branching; Polyethylene; Dynamic complex modulus; Phase angle

1. Introduction The identification and quantification of long chain branching (LCB) in essentially linear polyeth*

Corresponding author. Tel.: +1 281 834 2447; fax: +1 281 834 1793. E-mail address: [email protected] (C.A. Garcı´a-Franco).

ylenes (i.e., where there are only a very few long branches, if any) has received considerable attention because of its commercial and scientific implications. Fundamentally linear polyethylenes are those in which the LCB amount is less than ca. 0.3/1000 carbon atoms. At these levels of LCB, the conventional method for determining LCB based on the combined results of gel permeation chromatography and intrinsic viscosity measurements using the

0014-3057/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.eurpolymj.2007.10.030

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Zimm–Stockmayer method is of little practical use [2]. The Zimm–Stockmayer method for estimating LCB frequency is predicated on the reduction of the hydrodynamic volume caused by the presence of LCB. This hydrodynamic volume reduction is measured from the contraction factor g0 which is defined as the ratio of the intrinsic viscosity of the branched polymer, [g]B, to that of the linear polymer, [g]L, of equivalent molecular weight. For small LCB contents, such as less than 0.3/1000 carbon atoms, the contraction factor g0 is quite close to 1, within experimental error, and the determination of LCB becomes infeasible [1]. Small variations in the ratio g0 , can produce large variations in the LCB determination. Furthermore it should be kept in mind that the derivation of the Zimm–Stockmayer equation is based on the assumption that the SEC solution is in the theta state, whereas SEC is usually performed in good solvents. High-resolution nuclear magnetic resonance (NMR) is also used in the determination of LCB [3]. However, small amounts of LCB also pose serious difficulties for this technique, such as the required correction for short chain branching (two to six carbons long) present in commercial ethylene-a-olefin copolymers [3]. Furthermore, the NMR method cannot differentiate branch length for branches equal to or larger than six carbon atoms. This means it cannot be used to characterize LCB in ethylene/octene-1 copolymers, whose short chain branches are six carbons long. On the other hand, the rheological properties of polymer melts are quite sensitive to the presence of even small amounts of LCB [4]. However, it is necessary to separate the effects caused by the presence of LCB from those due to molecular weight (M) and molecular weight distribution (MWD). Our objective is to describe a method whereby low levels of long chain branching can be detected and quantified on a relative basis for whole, unfractionated, and essentially linear ethylene/a-olefin copolymers. The method is based on the measurement of the linear viscoelastic properties by a single isothermal small amplitude oscillatory shear experiment. A van Gurp–Palmen plot (whereby the phase angle, d (=tan1(G00 /G0 )) is plotted against the absolute value of the complex modulus, |G*| = (G0 2+G00 2)1/2 [5–10]) is prepared and the value of the phase angle, d, at |G*| = 10 kPa is recorded. This work will demonstrate that the amount of LCB inversely correlates with this value of d. Its applicability will be critically examined with a number of

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commercial ethylene/a-olefin copolymers. Moreover, we have extended the long chain branching index (LCBI) proposed by Shroff and Mavridis [1] by basing it on data of truly linear polyethylenes (hydrogenated anionically synthesized polybutadienes) instead of apparently linear commercial polyethylenes. 2. Previous relevant studies A number of groups have searched for ways to characterize low levels of LCB in polyethylene. Low-pressure polyethylene fractions of molecular weight above 1000 kg/mol were studied by Tung [11] using light scattering, and the presence of long chain branching in fractions of molecular weight above 300 kg/mol was observed by intrinsic viscosity, [g], versus molecular weight data. Hogan et al. [12] inferred the presence of LCB in low pressure, unfractionated Phillips high-density polyethylene (HDPE) and noticed that the LCB caused a rise of the low shear melt viscosity with little effect on [g]. These workers also observed that as result of LCB the parison sag and draping in a blow molding operation and sheet sag in thermoforming were greatly improved. Servotte and DeBruille [13] detected LCB in HDPE samples by direct measurement of the intrinsic viscosity on gel permeation chromatography effluents using the hydrodynamic volume as a universal calibration parameter. Locati and Gargani [14] proposed a linear logarithmic expression involving [g], the zero shear viscosity, g0, and a characteristic relaxation time k0 to calculate an index sensitive to the presence of LCB for high pressure LDPE and irradiated HDPE samples. Agarwal et al. [15], characterized fractions of industrial HDPE by light scattering, gel permeation chromatography (GPC), and viscometry and concluded that the high molecular weight fractions contained branched structures. Bersted et al. [16] showed that branched high-density polyethylene exhibited dramatic viscosity enhancement relative to linear polyethylenes and described this enhancement in terms of a logarithmic rule of mixtures for blends of branched and linear materials. Hughes [17] studied high-density polyethylene with long chain branching due to peroxide treatment. They proposed a relation for the LCB content in terms of the flow activation energy and also reported that small amounts of LCB have large effects on the rheological properties of HDPE melts at low, but not high, frequencies. Vega et al. [18] studied the dynamic

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viscoelasticity of a number of noncommercial metallocene-catalyzed polyethylenes and poly(ethylene/ 1-hexene) copolymers at several temperatures, and defined a ‘‘LCB index associated to activation energy of flow” as a possibility of appraising the presence of LCB in polyethylenes. Vega et al. [19] quantified LCB using the scaling of the zero shear viscosity with the weight average molecular weight. In LCB polymers, higher exponents than the g0 = KMw3.6 well known for linear polymers are obtained. Wood-Adams et al. [20] and Yan et al. [3] evaluated the effect of the amount of LCB on g0 enhancement. They found enhancement of g0 at LCB content of 0.02/1000C and g0 increase is more pronounced as the amount of LCB grows. 3. Experimental 3.1. Materials Two series of commercial ethylene/a-olefin copolymers were studied in this work. The first series is composed by 12 commercial metallocene copolymers (abbreviated here as mLLDPE) made in a gas-phase reactor. Among these were three ethylene/butene copolymers, two ethylene/hexene copolymers, and seven ethylene/octene copolymers. The second series consisted of nine commercial ethylene/octene copolymers manufactured by a solution process using constrained geometry catalysts. Tables 1 and 2 describe the characterization of these copolymers. The data from these commercial polymers was compared with a number of model poly-

ethylene samples for which the degree and type of LCB was well known from the manner of their synthesis [10,21]. The characterization data for these model polymers is shown in Tables 3–5. 3.2. Linear viscoelastic properties Dynamic shear melt rheological data was measured with an Advanced Rheometrics Expansion System (ARES) using parallel plates (diameter = 25 mm) at several temperatures (150, 170, 190 and 210 °C) using a pristine compression molded sample at each temperature. The measurements were made over the angular frequency range 0.01–100 rad/s. Depending on the molecular weight and temperature, strains of 10% and 15% were used and linearity of the response was verified. A N2 stream was circulated through the sample oven to minimize chain extension or cross-linking during the experiments. All the samples were received as pellets, compression molded at 190 °C and no further stabilizers were added. The analysis presented in this work is based on the viscoelastic data taken at 190 °C. 3.3. GPC triple detector The GPC measurements were performed on a Polymer Laboratories Model 220 high temperature SEC with on-line differential refractive index (DRI), light scattering, and viscometer detectors. It used three Polymer Laboratories PLgel 10 m Mixed-B columns for separation, a flow rate of

Table 1 Ethylene copolymers: gas-phase polymerization Polymer

EBG11 EBG25 EHG17 EBG07 EOG18 EOG27 EOG16 EOG30 EOG36 EHG06 EOG23 EOG32 a

Comonomer Type

(wt%)

(mol%)

Butene Butene Hexene Butene Octene Octene Octene Octene Octene Hexene Octene Octene

11.1 24.5 16.5 7.3 17.5 26.6 15.7 30 36.1 6 22.6 32

5.8 13.9 6.2 3.8 5 8.3 4.4 9.7 12.4 2.1 6.8 10.6

Mn (kg/mol)

Mw (kg/mol)

Mz (kg/mol)

Mw/Mn

g0

[g] (dL/g)

LCB1000C

g0a (kPa s)

na (ms)

aa

44.5 41.7 40.3 62.9 41.8 34.5 41.2 108 68.9 64.3 122 93.0

82.3 86.9 69.3 116 96.6 68.4 80.9 203 118 122 212 172

127 139 104 173 189 116 145 312 181 195 326 276

1.85 2.08 1.72 1.84 2.31 1.98 1.97 1.88 1.72 1.89 1.75 1.85

0.981 0.977 0.974 0.976 0.929 0.867 0.907 0.818 0.843 0.978 0.822 0.899

1.261 1.11 1.077 1.663 1.262 0.867 1.12 1.703 1.139 1.75 1.9 1.63

0.05 0.07 0.09 0.04 0.30 0.89 0.40 0.45 0.63 0.04 0.34 0.26

1.991 1.942 0.978 7.019 17.090 1.069 4.102 41.473 4.366 8.055 148.500 52.900

8.9 10.2 4.08 23.4 2140 25.0 136 3142 108 22.6 515,000 113,000

0.651 0.729 0.935 0.604 0.490 0.614 0.518 0.502 0.550 0.740 0.436 0.466

g0 Cross equation parameters [32]: g ðxÞ ¼ 1þðkxÞ a.

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Table 2 Ethylene octene copolymers: solution polymerization Polymer

EOS31 EOS36a EOS12 EOS39 EOS32 EOS17 EOS19 EOS36b EOS18 a

Comonomer Type

(wt%)

(mol%)

Octene Octene Octene Octene Octene Octene Octene Octene Octene

30.81 35.7 12.4 39.4 31.6 16.6 18.8 35.6 18.1

10 12.2 3.4 14 10.4 4.7 5.5 12.1 5.2

Mn (kg/mol)

Mw (kg/mol)

Mz (kg/mol)

Mw/Mn

g0

[g] (dL/g)

LCB1000C

g0a (kPa s)

na (ms)

aa

97.1 96.9 45.5 105 71.4 52.4 33.9 50.5 35.9

171 172 87.7 182 141 91.7 73.0 91.5 112

260 261 143 266 223 147 127 138 159

1.76 1.77 1.93 1.74 1.98 1.75 2.15 1.81 3.12

0.889 0.936 0.925 0.924 0.892 0.925 0.932 0.913 0.737

1.635 1.645 1.253 1.632 1.422 1.254 1.046 1.036 0.945

0.27 0.16 0.26 0.18 0.36 0.25 0.36 0.42 1.84

26.671 20.852 22.115 17.083 11.283 11.832 3.028 1.923 1.755

627 255 2260 168 192 823 79.0 18.0 64.0

0.554 0.608 0.485 0.641 0.566 0.473 0.520 0.631 0.507

g0 Cross equation parameters [32]: g ðxÞ ¼ 1þðkxÞ a.

Table 3 Model polyethylene copolymers – hydrogenated linear polybutadienes Polymer

Equivalent butene contenta (wt%)

Mwb (kg/mol)

Mw/Mnb

[g]c (dL/g)

g0d (kPa s)

PEL19 PEL90 PEL123 PEL125 PEL147 PEL193 PEL243 PEL280

8.6 6.4 7.8 7.9 7.5 7.5 8.1 7.3

19.3 90.2 124 127 148 195 255 290

1.02 1.01 1.01 1.02 1.01 1.02 1.05 1.04

0.52 1.42 1.84 1.86 2.10 2.59 3.08 3.30

0.0103 2.240 6.890 6.580 10.100 28.500 49.300 92.200

a b c d

By By By By

1

H NMR. SEC-MALLS. SEC-VIS. small amplitude oscillatory shear.

Table 4 Model polyethylene copolymers – hydrogenated polybutadiene stars Polymer

Equivalent butene contenta (wt%)

Mwb (kg/mol)

Mw/Mnb

[g]c (dL/g)

g0d (kPa s)

PES(43)3e PES(50)3e PES(49)2(5)f PES(50)2(5)f PES(50)2(15)f PES(50)2(25)f PES(15)2(85)f PES(40)2(60)f

9.0 8.5 11 8.9 7.9 7.5 9.9 9.0

133 194 105 130 138 131 129 132

1.13 1.02 1.04 1.03 1.04 1.07 1.18 1.02

1.61 2.06 1.49 1.77 1.76 1.61 1.74 1.55

5700 350,000 299 925 8560 2280 46,000 6420

a b c d e f

By 1H NMR. By SEC-MALLS. By SEC-VIS. By small amplitude oscillatory shear. Symmetric three-arm star PES(X)3 with arms of Mw = X kg/mol. Asymmetric three-arm star PES(X)2(Y) with two arms of Mw = X kg/mol and one arm of Mw = Y kg/mol.

0.54 cm3/min, and a nominal injection volume of 300 lL. The detectors and columns are contained in an oven maintained at 135 °C. The light scattering detector is a high temperature miniDAWN (Wyatt Technology, Inc.). The

primary components are an optical flow cell, a 30 mW, 690 nm laser diode light source, and an array of three photodiodes placed at collection angles of 45°, 90°, and 135°. The stream emerging from the SEC columns is directed into the

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Table 5 Model polyethylene copolymers – hydrogenated polybutadiene combs Polymer

Equivalent butene contenta (wt%)

Mwb (kg/mol)

Mw/Mnb

[g]c (dL/g)

g0d (kPa s)

PEC(101)-g-(7)30e PEC(100)-g-(5)2e PEC(100)-g-(5)12e

9.0 9.0 10.0

426 105 171

1.07 1.01 1.03

1.75 1.62 1.62

1070 7.48 4740

a b c d e

By 1H NMR. By SEC-MALLS. By SEC-VIS. By small amplitude oscillatory shear. Comb PEC(X)-g-(Y)Z with backbone of Mw = X kg/mol and an average of Z arms of Mw = Y kg/mol.

miniDAWN optical flow cell and then into the DRI detector. The DRI detector is an integral part of the Polymer Laboratories SEC. The viscometer is a high temperature viscometer purchased from Viscotek Corporation. It consists of four capillaries arranged in a Wheatstone bridge configuration with two pressure transducers. One transducer measures the total pressure drop across the detector, and the other, positioned between the two sides of the bridge, measures a differential pressure. The specific viscosity gsp(c) = (g(c)  gs)/gs for the solution then flowing through the viscometer is calculated from their outputs. The viscometer is inside the SEC oven, positioned after the DRI detector. 3.3.1. Calibration The DRI detector was calibrated by integrating the signal obtained by injecting a known quantity of NBS1475, a polyethylene standard. The DRI response, is directly proportional to the product of polymer concentration and refractive index increment c(dn/dc). The miniDAWN detector was calibrated by measuring the 90° scattering intensity of the elution solvent TCB at 135 °C. We normalized the photodetectors by injecting a low molecular weight polymer which scatters light nearly isotropically. The polymer used for this purpose was NBS1482, a polyethylene standard fraction with certificate Mw = 13.6 kg/mol. Based on extrapolations of our own and literature data for polyethylenes of higher molecular weight, we estimate Rg = 5 nm for NBS1482. That size corresponds to an expected 1% asymmetry in scattering intensity between the lowest and highest scattering angles. The viscometer was calibrated by injecting three standard polyethylenes (NBS1482, NBS1483 and NBS1484), for which [g] in TCB at 130 °C is supplied with an expected limit of systematic error of 1%. The viscometer cal-

ibration constant was then adjusted to match, on average, the stated intrinsic viscosity values. Tables 1 and 2 give a summary of the GPC triple detector data for the commercial polymers made with singlesite catalysts. 4. Van Gurp–Palmen plots Van Gurp and Palmen [5] presented a novel approach to verify the validity of the time temperature superposition principle (tTSP) whereby the phase angle d of the measured dynamic rheological data at various temperatures is plotted against the corresponding absolute value of the complex modulus |G*|. Such a plot reveals that isothermal frequency curves merge into a common line if the tTSP holds. It is argued [6–8] that this verification of the tTSP for a given polymeric melt is predicated on the exclusion of the temperature dependent characteristic time k0 as well as those properties based on such characteristic time according to the Doi– Edwards [22] scaling laws. Recently, Trinkle and Friedrich [6–8] have discussed and extended the applicability of these plots and established them as a useful and reliable tool for characterizing polymer melts rheological properties. Next, a brief discussion of the effects of molecular weight, molecular weight distribution and branching on the van Gurp–Palmen plots will be presented. 4.1. Molecular weight effects Linear polymers can be clearly identified by their characteristic curves in the van Gurp–Palmen plot. Fig. 1 shows the van Gurp–Palmen plot of three nearly monodisperse linear model polyethylenes of different molecular weights obtained by the hydrogenation of anionically polymerized polybutadiene [10]. The characteristics of these model linear polyethylenes are shown in Table 3.

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381

90 80

δ (degrees)

70 60

PEL 193 PEL 90

50 PEL 123 40 30 20 10 1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

G* (Pa)

Fig. 1. Van Gurp–Palmen plots of linear model polyethylenes (hydrogenated anionically polymerized polybutadiene). Curves are molecular weight invariant and a modest extrapolation of the curves to low values of the phase angle d, clearly shows the plateau modulus of polyethylene Gn0 = 2.3 MPa.

The curves extend from the terminal region (low values of the |G*| and d values at or close to 90°) to the plateau region (large values of |G*| and d values decreasing towards zero degrees). The curves monotonically decrease as the absolute value of the complex modulus, |G*| increases. Several features of these curves are worthwhile to mention. First, the plots of these three polyethylenes are practically on top of each other, i.e., the van Gurp–Palmen plots are independent of the molecular weight. Secondly, a modest extrapolation of the data to low

values of the phase angle marks the plateau modulus, Gn0 = 2.3 MPa of polyethylene. Third, for semi-crystalline polymers such as polyethylene only data to the left of Gn0 can be monitored by melt rheology, since crystallization occurs at the low temperatures that are necessary to reach the minimum region of the van Gurp–Palmen plot (unlike amorphous polymers such as polybutadiene [5–7]). Fourth, as the molecular weight increases the data extend to lower values of the phase angle, i.e., a closer approach to the plateau modulus is achieved by

90

δ (degrees)

80 70

SHORT CHAINS

60

LONG CHAINS

50 40 30 20 10 0 0.00001

0.0001

0.001

0.01

0.1

1

10

Reduced Modulus Fig. 2. Van Gurp–Palmen plots of two-monodisperse polymer melts. Calculations based on double reptation mixing rules. Reduced modulus = |G*|/GN0. Short chain kS = 104 s, long chain kL = 101 s. ML = 6.81 MS. Calculations also show molecular weight invariance of the Van Gurp–Palmen plots.

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382

90 80

100% SHORT CHAINS 70

δ (degrees)

60 50

100% LONG CHAINS 20% LONG CHAINS 40% LONG CHAINS 60% LONG CHAINS

40

80% LONG CHAINS

30 20 10 0 1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Reduced Modulus

Fig. 3. Simulated van Gurp–Palmen plots of binary linear blends based on double reptation mixing rules. The characteristics of the parent chains are the same as in Fig. 2. Reduced modulus = |G*|/GN0.

high molecular weight polymers. Finally, the phase angle of linear monodisperse polymers, exhibits a relatively long flat span (at d ffi 90°) before it rapidly falls at higher values of |G*|. Similar conclusions can be drawn from theoretical considerations. Van Gurp–Palmen plots can be prepared from linear viscoelastic data obtained via the double reptation mixing rules [23–26]. Fig. 2 shows the van Gurp–Palmen plot of two perfectly monodisperse polymer melts one with a relaxation time of 104 s and the other 101 s, which corre-

sponds to Mlong chains = 6.81 Mshort chains (assuming k / M 3.6), and as expected the plots are molecular weight invariant. 4.2. Molecular weight distribution effects Fig. 3 shows the results of simulated rheological behavior of a series of bimodal blends based on the double reptation model [23–26]. The behavior of d as a function of |G*| of mixtures of monodisperse linear long chains (klong chain = 101 s) and

90

80

δ (degrees)

70

60

EHG06 20% -LDPX1

50 40% -LDPX1 60%-LDPX1 40 80%-LDPX1 30

20 1.E+02

LDPX1

1.E+03

1.E+04

1.E+05

1.E+06

G* (Pa)

Fig. 4. Van Gurp–Palmen plots of linear mLLDPE EHG06, high pressure LDPE LDPX1 and their blends.

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monodisperse linear short chains (kshort chain = 104 s) with different proportions is depicted in the above-mentioned figure. The bimodality introduces two relaxation regions corresponding to the long and short chains. The dome between the two minima corresponds to the relaxation of the short chains. As the short chain population decreases, one minimum moves to higher values of the reduced modulus and lower values of the phase angle d, which means that long chains enhance the polymer elasticity. The second minimum stays in the region of the plateau modulus independently of long–short chains proportions because Gn0 is independent of the molecular weight for well entangled polymers. The effect of blending linear and highly branched chains on the evolution of d in terms of |G*| is shown in Fig. 4 whereby the van Gurp–Palmen plots of mLLDPE EHG06 (Mw = 122. kg/mol, Mw/Mn = 1.89), high pressure low density polyethylene LDPX1, and their blends are presented. As the amount of long chain branched polymer increases, the blend curves remain quite close to the LDPX1 plot at low values of |G*|, but at high values of |G*| the effects of the linear chains become much more evident. To avoid overcrowding, the blend curves are shown in 20% increments. A similar set of curves was also obtained for LDPX1, a Ziegler–Natta catalyzed EB copolymer, ZNEB (Mw = 121 kg/mol, and Mw/Mn = 3.28), and their blends. It is evident from Fig. 4 that the areas limited by the linear polymer curve and the blends curves scale with the proportions of the blends. These areas can be determined quantitatively by

383

numerical integration. However, since the limits of integration are not the same for all curves, it is advisable to fit each curve (a 4th order polynomial does an excellent job), select the limits, and perform the formal integration. In Fig. 5 the relationship between the phase angle at |G*| = 30 kPa and the composition of LDPX1 is shown for both sets of blends. This figure clearly shows a linear relationship between the LDPE composition in the blends and the phase angle at this value of the complex modulus, |G*|. However, the linear relationship corresponding to the mLLDPE, EHG06, is steeper than that corresponding to the Ziegler product, ZNEB, which implies that MWD effects are also present in these relationships. 4.3. Model polyethylenes Within the context of this work it is of great relevance to present the van Gurp–Palmen plots of model, nearly monodisperse linear, star, and comb polyethylenes, which have been made by hydrogenating anionically synthesized polybutadienes. The characterization data for these polymers is shown in Tables 3–5, and further details on these polymers can be found in Refs. [10] and [27]. Because we know the LCB architecture of these polymers by the way they were made, they provide an excellent way to establish new characterization methods. Fig. 6 shows the van Gurp–Palmen plot for model linear, star, and comb PEs. Clearly, as the long chain branching structure becomes more extensive the departure of the curve from the behavior of

90 EHG06

70

ZNEB

4

(degrees) @ G*=3X10 (Pa)

80

60 50 40 30 20 Lines are a guide to the eye.

10 0 0

10

20

30

40

50

60

70

80

90

100

% weight LDPE

Fig. 5. Relationship between the phase angle at |G*| = 30 kPa and the composition for the blends of LDPX1 with EHG06 (see Fig. 4) and ZNEB. The lines are aids to the eye.

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384

90 80 70

δ (degrees)

60 50 40 30 20 10 0 1E+3

PEL147 PES(43)3 PEC(101)-g-(7)30 1E+4

1E+5

1E+6

1E+7

G* (Pa)

Fig. 6. Van Gurp–Palmen plots for a model linear, star, and a comb PE. These examples show the effects of LCB type and amount on rheology. Note the inability to superpose the data of the LCB polymers obtained at different temperatures. For a description of these model polymers see Refs. [10,21].

90 80

δ (degree)

70 60 50 40 30 20 10 0 1E+3

PEL147 5/95 PES(43)3/PEL147 10/90 PES(43)3/PEL147 50/50 PES(43)3/PEL147 PES(43)3 1E+4

1E+5

1E+6

1E+7

|G*| (Pa)

Fig. 7. Van Gurp–Palmen plots for the model linear and star PE shown in Fig. 10, plus three blends of the two. These examples show the effect of the amount of LCB chains on rheology. Note how difficult superposition of the data obtained at different temperatures becomes for the 50/50 blend. For a description of these model polymers see Refs. [10,21].

a linear polymer is greater. This can also be seen in Fig. 7, which shows the van Gurp–Palmen plots for blends of a linear and star polymer. As the level of LCB added increases, d at a given value of |G*|, say 10 kPa drops. We thus can see that we stand on solid ground to believe that this parameter provides a measure of LCB. 4.4. Polyolefins made with single-site catalysts Fig. 8 shows a graph of the phase angle, d as a function of the absolute value of the complex mod-

ulus, |G*|, of a series of commercial ethylene/octene copolymers synthesized by the solution process using constrained geometry catalysts. The characteristics of these polymers are given in Table 2. Fig. 8 also shows the van Gurp–Palmen plot of a monodisperse linear polyethylene (PEL193, one of the model hydrogenated anionically polymerized polybutadienes in Table 3) included as a reference. These data clearly show that the polymer EOS12 contains the largest amount of long chain branching, LCB. A plot of the data for commercial metallocene polyethylene copolymers produced in a gas

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385

90 80 70

δ (degrees)

60 50 40 30

EOS32 EOS31 EOS36Aa EOS39 EOS36b EOS19 EOS12 EOS17 EOS18 PEL 193

20 10 0 1e+1

T=190 C 1e+2

1e+3

1e+4

1e+5

1e+6

1e+7

G* (Pa) Fig. 8. Van Gurp–Palmen plot commercial ethylene/octene copolymers synthesized by solution polymerization using metallocene catalyst. The linear polyethylene PEL-193 (hydrogenated polybutadiene) is also included as reference.

phase reactor (Table 1) showed similar behavior. For both sets of commercial copolymers, the drop in d correlates with the level of LCB, indicating that this is a good measure of this feature.

5. Establishing a long chain branching index (LCBI) Long chain branching has two opposite effects on viscosity. The non-linear nature of the molecules means that the coil dimensions are reduced compared to a linear chain of the same molecular weight. This drop in molecular size reduces viscosity. On the other hand, the entanglement of the long branches in the melt means that the dynamics of these molecules are not due solely (or even chiefly) to reptation, but must incorporate mechanisms such as arm retraction. This will greatly increase relaxation times and so viscosity. For dilute solutions, only the first effect is operative since the polymer chains do not entangle with each other, so [g] always drops as LCB is introduced. However, in the melt both effects are in play. For very light branching, g0 may be lower than that of a linear chain, but in general it is much larger. Thus, comparing the changes to g0 and [g] can reveal much about the LCB of a polymer. Recently, Shroff and Mavridis [1] (s&m) derived a long chain branching index (LCBI) from zero shear viscosity, g0, and intrinsic viscosity, [g], data. This approach is justified on fundamental grounds as follows. It is well known that the dependency

of g0 on molecular weight for linear polymers can be expressed [4,31] by g0 ðMÞ ¼ k 0 M a0

ð1Þ

where k0 and a0 are constants. Similarly, the intrinsic viscosity of a linear polymer is related to its molecular weight through ½g ¼ k i M ai

ð2Þ

where ki and ai are the Mark–Houwink constants. Eqs. (1) and (2) can be used to derive the relationship between the zero-shear viscosity and the intrinsic viscosity of linear polymers, [g], is given by ga0m ¼ k m ½g

ð3Þ ðk a0i =a0 =k i Þ.

where av = ai/a0 and k m ¼ Thus, for linear polymers the quantity ðga0m =k m ½gÞ should equal 1; if it is larger, this can be taken as an effect of long chain branching. One can then define a long chain branching index as LCBI ¼

ga0m 1 ½gk m

ð4Þ

so that LCBI = 0 for linear polymers with no long chain branching. The larger LCBI is, the greater the influence the LCB has on its rheology. Shroff and Mavridis [1] defined the constants in Eq. (4) by selecting linear polyethylenes and preparing a log-log plot of the intrinsic viscosity against the zero shear viscosity, which according to Eq. (3) should yield a straight line from which the

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386

parameters kv and av can be easily obtained. Accordingly these authors found the following relation: LCBIðs&mÞ ¼

g0:179 0 1 3:179½g

ð5Þ

It should be pointed out that Shroff and Mavridis [1] reported this equation in terms of poises as the units of the zero shear viscosity, whereas Eq. (5) is written in terms of the zero shear viscosity expressed in Pa-s. As usual, the [g] is expressed in dL/g. 6. LCBI based on model linear polyethylenes The Shroff and Mavridis development of an LCBI makes good sense for a single polymer type. However, we know that the rheological parameters of polymers in general and polyolefins in particular are functions of the chemical nature of their repeating units as well as of molecular weight [27–31]. This means that the parameters that define the molecular weight dependence of [g] and g0 (k0, a0, ki, and ai) should be regarded as functions of such features as copolymer composition. While we know well how some parameters, such as the plateau modulus [27–31] depend on comonomer type and content, we do not yet have a general model of [g] and g0 as a function of composition. However, we do have data on model polymers that more closely resemble linear low density PE than do the polyethylenes Shroff

and Mavridis used to derive their expression (Eq. (5)). We now use these data to derive an alternative expression for LCB index. The selection of strictly linear polyethylenes from commercial grades is not trivial because a priori one cannot rule out the presence of a small level of long chain branching which might be present due to catalyst/process characteristics, or generated in the pelletization stage. Therefore, having the parameters of Eqs. (1)–(3) calculated from data of known strictly linear monodisperse polyethylenes represents a great advantage over the data calculated by Shroff and Mavridis [1] based upon commercial polyethylenes. In the course of our work concerning the effects of long chain branching on the rheology of polyethylenes, model LCB and strictly linear monodisperse polyethylenes have been made by hydrogenating anionically synthesized polybutadiene, and their rheology measured [10,21]. This allows us to derive the following expression for the relation between [g] and g0: g0 ¼ 298:52½g

4:818

ð6Þ

This is shown in Fig. 9. Thus the definition of LCBI given by Eq. (4) becomes LCBIðHPBÞ ¼

g0:208 0 1 3:272½g

ð7Þ

As above, g0 is expressed in Pa s, and [g] is expressed in dL/g.

1e+6

1e+5

ηο=298.52 [η]

4.818

ηο (Pa-s)

1e+4

1e+3

1e+2

1e+1

1e+0 0. 1

1

10

[η] (dl/g)

Fig. 9. Zero shear viscosity (T = 190 °C) against the intrinsic viscosity of monodisperse linear model polyethylenes obtained by hydrogenation of anionically polymerized polybutadiene.

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387

0.1

0.0

LCBI

-0.1

-0.2

LCBI-S&M LCBI-HPB

-0.3

-0.4 PEL19 PEL90 PEL123 PEL125 PEL147 PEL193 PEL243 PEL280

SAMPLE Fig. 10. LCBI indexes, of linear monodisperse model polymers based on Shroff and Mavridis (LCBI(s&m)) data, and on linear monodisperse model polymers (LCBI(HPB)). For a description of these model polymers see Table 3 and Refs. [10,21].

7. LCBI results Fig. 10 shows LCBI results of the linear monodisperse model (hydrogenated anionically polymerized polybutadiene) polyethylene samples. The data are given in Table 3 (see Refs. [10,21] for further details). The calculations are based on the expressions proposed by Shroff and Mavridis [1], LCBI(s&m), Eq. (5), and the expression we have calculated from the model linear monodisperse

polyethylenes data, LCBI(HPB), Eq. (7). As the figure shows, all LCBI(s&m) values are negative, whereas the LCBI(HPB) are zero or very small positive (<0.08) and three samples (PEL19, PEL147, and PEL243) show small (<0.06) negative values. In summary, whether positive or negative results, both expressions give very small values indicating that these polymers are indeed linear. The negative values may seem puzzling at first; does such an occurrence mean that the polymers

10 LCBI-S&M LCBI-HPB

8

LCBI

6

4

2

0 PES(43)3

PES(50)3

PES(50)2(5) PES(49)2(5) PES(50)2(15) PES(50)2(25) PES(15)2(85) PES(40)2(60)

SAMPLE Fig. 11. LCBI indexes of model stars polyethylenes based on Shroff and Mavridis (LCBI(s&m)) data, and based on linear monodisperse model polymers (LCBI(HPB)). For a description of these model polymers see Table 4 and Refs. [10,21].

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388

1.00 0.90

LCBI-(s&m) LCBI (HPB)

0.80

LCBI

0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 EOS31 EOS36a EOS12 EOS39 EOS32 EOS17 EOS19 EOS36b EOS18

Polymer Fig. 12. LCBI indexes based on Shroff and Mavridis (LCBI(s&m)) data, and based on linear monodisperse model polymers (LCBI(HPB) of commercial metallocene C2/C8 copolymers manufactured by the solution process. (EOS18 data not available).

have fewer long branches than linear PE (that is, ‘‘more linear than linear”)? This clearly cannot be the case. Rather, it is the relative effects of short chain branches (such as from comonomer) on g0 and [g] that give rise to this. The presence of the short branches has a greater impact on the numerator of Eq. (5) than on its denominator, and so LCBI(s&m) becomes negative. As it is based

on data from model polymers with a fair degree of short branches, Eq. (7) better captures this effect. On the other hand, Fig. 11 shows the results of both long chain branching indexes, LCBI(s&m) and LCBI(HPB), of model monodisperse star polyethylenes (data in Table 4, Refs. [10,21]). As the figure shows, even though the values of these two

1.0 0.9 0.8 LCBI(HPB)=2.775 - 3.093x10-2 δ r2 = 0.921

0.7

LCBI

0.6 0.5 LCBI(s&m)=1.823 - 2.219x10-2 δ r2=0.883

0.4 0.3 0.2

LCBI-HPB LCBI-(s&m) Regression

0.1 0.0

40

50

60

70

80

90

(degrees) Fig. 13. Phase angle, d, measured from Van Gurp–Palmen plot at |G*| = 10 kPa, against the LCBI based on Shroff–Mavridis data (LCBI(s&m)) and the LCBI based on hydrogenated polybutadiene data (LCBI(HPB)) of commercial ethylene/octene copolymers made in solution.

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389

1.0

LCBI(s&m) = 1.4735-1.8088x10-2 δ r2 = 0.9043

0.8

LCBI (s&m)

0.6

0.4

0.2

0.0

-0.2

30

40

50

60

70

80

90

(degrees) Fig. 14. Phase angle, d, measured from Van Gurp–Palmen plot at |G*| = 10 kPa against the LCBI based on the LCBI proposed by Shroff and Mavridis (LCBI(m&s)) of ethylene copolymers made in gas phase.

indexes are different, the same trend is observed by following either one, i.e., the largest LCBI is exhibited by the sample PES(50)3, and the lowest by PES(49)2(5). It should be also pointed out that the numerical values in this figure are much larger than those obtained for the model linear monodisperse polyethylenes shown in Fig. 10. As in the case with stars, the numerical values of LCBI(s&m), and LCBI(HPB) concerning model monodisperse comb polyethylenes (data in Table 5, Refs. [10,21]) are different from each other, but within each set the data follow the same trend.

Fig. 12 shows the values of the LCBI(s&m) and LCBI(HPB) of commercial constrained geometry ethylene/octene copolymers manufactured by a solution process. As in the previous cases, the numerical values of the LCBI(HPB) are larger than those of the LCBI(s&m), but the same trends are given by these indexes. Furthermore, the numerical values obtained with these commercial polymers are substantially lower than those obtained for the model monodisperse stars and comb polyethylenes, but larger than those obtained for the model monodisperse linear polyethylenes.

1.0

LCBI (HPB)

0.8

0.6

0.4

0.2

0.0 30

LCBI(HPB) = 2.5085 - 2.8178x10 -2 δ 2 r = 0.954

40

50

60

70

80

90

(degrees) Fig. 15. Phase angle, d, measured from Van Gurp–Palmen plot at |G*| = 10 kPa against the LCBI based on hydrogenated polybutadiene data (LCBI(HPB)) of ethylene copolymers made in gas phase.

390

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Clearly, while both LCBI(s&m) and LCBI(HPB) correlate with the degree of LCB, the latter is a more preferable measure. LCBI(HPB) also shows a better fit to the data on the model polymers, where we know the LCB. 8. LCBI based on the van Gurp–Palmen plot We have demonstrated that the evolution of the phase angle, d, as a function of |G*|, is strongly dependent upon the molecular architecture of polymer melts, specifically the level of LCB. Consequently, we propose to use the value of the phase angle, d, at a given value of |G*|, say |G*| = 10 kPa as an LCBI. We have chosen this value of the modulus because it is easily accessible for a broad range of commercial resins. This concept is particularly appealing because of its simplicity. The proposed index is predicated on a single isothermal small amplitude oscillatory shear experiment. In Fig. 13 the LCBI based on Shroff and Mavridis data (LCBI(s&m)), as well as the LCBI based on the hydrogenated polybutadiene data (LCBI(HPB)), is plotted against the phase angle d calculated from the Van Gurp–Palmen plots at |G*| = 10 kPa for the commercial solution-polymerized ethylene/octene copolymers of. An acceptable correlation is obtained, with a correlation coefficient r2 = 0.921 and 0.883 based on the LCBI(HPB), and LCBI(s&m), respectively. Figs. 14 and 15 depict plots of LCBI(s&m) and LCBI(HPB) against the value of d calculated from the van Gurp–Palmen plots at |G*| = 10 kPa, respectively, for the commercial grades of the gasphase polymerized ethylene copolymers. An acceptable agreement is also obtained, with r2 = 0.904 and r2 = 0.954 for the data based on LCBI(s&m), and LCBI(HPB), respectively. These results clearly show that the value of d at |G*| = 10 kPa correlates with the presence of very low levels of LCB, ones which could not be detected by conventional means. This is quite powerful, since the van Gurp–Palmen plots can be derived from just a single isothermal measurement of the linear viscoelasticity of the polymers. 9. Conclusions A method is described whereby low levels of long chain branching in polyolefins are detected and quantified on a relative basis. The method is based

on the measurement of the linear viscoelastic properties by a single isothermal small amplitude oscillatory shear experiment. A plot of the phase angle, d, against the absolute value of the complex modulus, |G*|, is prepared and the value of the phase angle at |G*| = 10 kPa is recorded. The amount of LCB inversely correlates with the value of the phase angle. Depending on the frequency range covered by the experiment, the duration of test varies between 12 and 60 min rendering this technique well suited for high throughput parallel testing. Moreover, we have improved the long chain branching index (LCBI) proposed by Shroff and Mavridis by basing it on data of truly linear, model polyethylenes (hydrogenated anionically synthesized polybutadienes) rather than that from commercial polyethylenes that appear to be linear by conventional techniques. Since very low levels of LCB are difficult to detect in such polymers by those means, it is not surprising that a more discriminating index can be obtained by using model polymers where linearity is assured from the synthesis. This improved index of LCB gives us a way to more clearly detect very low levels of such branching, which can be quite important for commercial systems. This new index also correlates well with the van Gurp–Palmen technique described above. Acknowledgments The authors gratefully acknowledge the contributions of M.Y. Amin (Rheology), K. Eulaerts (Rheology), and M.N. Thomas (MWD GPC Triple Detector) to this work. T. Sun kindly provided the description of the GPC-Triple Detector. References [1] Shroff RN, Mavridis H. Long-chain-branching index for essentially linear polyethylenes. Macromolecules 1999;32: 8454–64. [2] Pang S, Rudin A. SEC assessment of long chain branch frequency in polyethylenes. Polym Mater Sci Eng 1991;65:95–6. [3] Yan D, Wang WJ, Zhu S. Effect of long chain branching on rheological properties of metallocene polyethylene. Polymer 1999;40:1737–44. [4] Graessley WW. Effect of long branches on the flow properties of polymers. Acc Chem Res 1977;10:332–9. [5] Van Gurp M, Palmen J. Time–temperature superposition for polymeric blends. Rheol Bull 1998;67:5–8. [6] Trinkle S, Friedrich C. Van Gurp’s plot: a phenomenological approach for the characterization of long chain branching. ACS/PMSE 2000;82:121.

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