understanding the effect of short-chain branches by analyzing viscosity functions of linear and...

Korea-Australia Rheology Journal December 2011 Vol. 23, No. 4 185

Korea-Australia Rheology JournalVol. 23, No. 4, December 2011 pp. 185-193DOI: 10.1007/s13367-011-0023-5

Understanding the effect of short-chain branches by analyzing viscosity functions of

linear and short-chain branched polyethylenes

Florian J. Stadler1,2

and Tahmineh Mahmoudi1

1Chonbuk National University, School of Semiconductor and Chemical Engineering, Baekjero 567Deokjin-gu, Jeonju, Jeonbuk, 561-756, Republic of Korea

2Institute of Polymer Materials, Friedrich-Alexander University Erlangen-Nürnberg, Martensstr. 7, D-91058 Erlangen, Germany

(Received April 7, 2011; final revision received May 20, 2011; accepted June 3, 2011)

Abstract

In this paper, we assess the frequency dependent rheological data of several different linear and short-chainbranched metallocene catalyzed polyethylenes with respect to the description of the viscosity functions. Theaim is to derive material specific relaxation time λ from the fit. This characteristic relaxation time followsa similar dependence to the molar mass Mw as the zero shear-rate viscosity, but slightly depends on bothmolar mass distribution and comonomer content. The transition between the shear-thinning region and theterminal regime is broader for mHDPE than for mLLDPE and widens with increasing molar mass Mw. Thisfinding is explained by the differences in the plateau modulus GN

0 and the increase of the normalized ter-minal relaxation time in comparison to the characteristic relaxation time, respectively.

Keywords : Carreau-Yasuda fit, characteristic relaxation time, polyethylene, metallocene catalyst, normal-

ization

1. Introduction

In the last 50 years, polyethylene has made a remarkable

career, despite its relatively high undefinedness in terms of

molecular architecture. This has changed with the advent

of metallocene catalysts, which can be controlled very well

and, thus, allow for molecular tailoring.

Metallocene catalysts allow the molar mass to be

adjusted in a broad range while retaining a narrow molar

mass distribution (MMD) (Böhm, 2003; Jordens et al.,

2000; Mülhaupt, 2003; Piel et al., 2006a). Metallocene cat-

alysts are highly stereo- and regiospecific, hence, highly

tactic and tailored copolymers may be produced (Coates,

2000; Heuer and Kaminsky, 2005; Resconi et al., 2000).

Furthermore, metallocene catalysts also have a high affin-

ity to incorporate α-olefins into growing chains, and are

even able to produce homopolymers of these higher α-ole-

fins (Arikan et al., 2007; Hoff and Kaminsky, 2004; Koi-

vumäki et al., 1994; Sperber and Kaminsky, 2003).

Metallocene catalysts were first invented in the late

1950s (Breslow and Newberg, 1957), however, their prac-

tical usability was restricted until suitable cocatalysts were

discovered (Sinn and Kaminsky, 1980).

In the early 1990s, metallocene catalysts making sparsely

branched polyethylenes were discovered (Brant et al.,

1994; Lai et al., 1993). The rheological characterization

PEs started some years later with the main focus being on

long-chain branched (LCB-) mPE (Gabriel and Münstedt,

1999; Malmberg et al., 1998; Malmberg et al., 1999; Sta-

dler et al., 2006a; Vega et al., 1996; Wood-Adams et al.,

2000), as long-chain branches are of much more com-

mercial interest.

In order to properly characterize LCB-mPE, however,

the knowledge of the behavior in linear viscoelasticity is

required. There are three possibilities for obtaining such a

linear reference.

On the one hand, the characterization of a linear sample

with identical molecular structure is possible, which, how-

ever, is quite difficult to do. The high sensitivity of the

rheological properties to the molar mass distribution are

responsible for this.

Alternatively, a molar mass normalized reference could

be used, whose establishment is one of the aims of this arti-

cle. The background of this is the connection between

molar mass distribution and rheological data of linear poly-

mers (Guzman et al., 2005; Tuminello, 1986; van Ruym-

beke et al., 2002). Currently, the molecular models are in

the process of reaching a high level of maturity in describ-

ing the rheological data of linear samples, while for non-

model long-chain branched architectures the general

understanding is still incomplete.

In this paper, we want to show the effect of molar mass

on the dynamic-mechanical properties of short-chain*Corresponding author: [email protected], [email protected]© 2011 The Korean Society of Rheology and Springer

Florian J. Stadler and Tahmineh Mahmoudi

186 Korea-Australia Rheology Journal

branched polyethylenes. The aim is to establish a linear

reference for a narrow MMD (Mw/Mn=2) used for further

analysis with long-chain branched samples. Furthermore, it

is aimed to understand better which influence short-chain

branches have on the rheological properties.

2. Experimental

The rheological characterization was performed on a

Bohlin/Malvern Gemini, a Rheometric Scientific ARES, or a

TA Instruments AR-G2. Frequency sweeps were performed

at 150oC. If the thermal stability was found to be sufficient,

creep recovery tests were used to extend the frequency

regime towards the terminal regime using the method of

Kaschta and Schwarzl (1994a, b). A deeper description of the

experimental methods used and also of the SEC-MALLS-

analysis is given elsewhere (Stadler et al., 2006b; Stadler et

al., 2006c). The information about the samples is collected in

Table 1 (molecular data) and 2 (rheological data).

This paper contains rheological previously published by

us before under the same designations (Stadler et al., 2007;

Stadler and Münstedt, 2008c; Stadler et al., 2006b). The

synthesis conditions of the materials are given by Kamin-

sky et al. (2005), Piel et al. (2006a), Piel et al. (2006b).

The complete characterization by SEC-MALLS and NMR

is given elsewhere in detail (Stadler and Münstedt, 2008c;

Stadler et al., 2006b; Stadler et al., 2006c).

3. Results and Discussion

3.1. Carreau-Yasuda fitsThe analysis of viscosity functions |η*(ω)| has been one

Table 1. Molecular data of the linear ethene-/α-olefin copolymers

NamePolymer

type

Mw

[kg/mol]

Mw/Mn

[-]

Mw

[kg/mol]

lb

[CnH2n]

ncc

[mol%]

wc d

[wt.%]

sce

[wt.%]method

C1 HDPE 42 3.0 42 - 0 0 0 -

E5 HDPE 52 2.0 52 - 0 0 0 -

C2 HDPE 114 16.0 114 - 0 0 0 -

C3 HDPE 120 2.0 120 - 0 0 0 -

A7 HDPE 178 4.0 178 - 0 0 0 -

C4 HDPE 224 3.0 224 - 0 0 0 -

A5 HDPE 403 2.6 403 - 0 0 0 -

A1 HDPE 665 3.5 665 - 0 0 0 -

A4 HDPE 564 4.3 564 - 0 0 0 -

A2 HDPE 923 3.8 923 - 0 0 0 -

L8 LLDPE 86 2.1 83.5 8 about 1a

about 3.9a

about 2.9a

DSC

L4b LLDPE 114 2.0 106.2 4 6.8 12.7 6.4 melt-state NMR

L6b LLDPE 114.6 2.2 101.8 6 5.9 15.8 10.6 FT-IR

F18C LLDPE 161 1.9 136.4 18 2.2 16.8 15.0 melt-state NMR

F26C LLDPE 174 2.1 135.7 26 2.3 23.4 21.6 melt-state NMR

F12Fc LLDPE 210 2.0 178.6 12 3.4 17.4 14.5 solution NMR

F18Fd LLDPE 216 1.9 176.8 18 2.7 20.0 17.8 solution NMR

F26Fe LLDPE 233.5 2.1 170.3 26 3 28.7 26.5 solution NMR

a Approximation of comonomer content from melting point and crystallinity (Stadler et al. (2005)), b comonomer length, c number

comonomer content, d weight comonomer content, e side chain content

Understanding the effect of short-chain branches by analyzing viscosity functions of linear and short-chain branched polyethylenes


of the important objectives in rheology, as they have a very

high practical relevance for processing operations. It was

previously shown that the “Carreau-Yasuda” model

(1)

leads to very good results in the description and analysis of

viscosity functions. This model was chosen, as it allows

the numerical description of various types of viscosity

functions flexibly (Carreau, 1972; Yasuda et al., 1981). λ

is the characteristic relaxation time, “a” is the parameter

describing the width of the transition between the terminal

and the shear thinning regime, whose double logarithmic

slope is described by the parameter n. λ is approximately

equivalent to the reciprocal crossover frequency 1/ωc

(Kazatchkov et al., 1999; Stadler et al., 2006b).

It was shown before that the double logarithmic slope

dlog |η*(ω)|/dlog ω can be directly correlated to the phase

angle d for the further analysis using

, (2)

thus, δ can be described as

(3)

with this relation, it is also possible to fit G’(ω) and G”(ω)

using

(4)

and

η∗ ω( ) η0 1 λ ω⋅( )a+[ ]n 1–

a----------

= δ ω( ) 90o d η∗log

d ωlog------------------- 1+⎝ ⎠⎛ ⎞⋅=

δ ω( ) 90o 1 λ ω⋅( )a λ ω⋅( )a–+

1 λ ω⋅( )a+----------------------------------------------⎝ ⎠⎛ ⎞⋅

90o

1 λ ω⋅( )a+------------------------= =

G' ω( ) ω η0 1 λ ω⋅( )a+[ ]n 1–

a----------

90o

1 λ ω⋅( )a+------------------------⎝ ⎠⎛ ⎞cos⋅=

Table 2. Rheological data of the linear ethene-/α-olefin copolymers

Name Polymer typeη0 at 150oC

[Pa s]

ωc

[s−1]

λ[s]

“a”

[-]

Je0 at 150oC

[10-4 Pa−1]

C1 HDPE 520h i 0.00144 h 0.717 h n.d.

E5 HDPE 730 h i 0.00104 h 0.634 h n.d.

C2 HDPE 16,130h 35 0.03185 h 0.468 h n.d.

C3 HDPE 17,800h 60 0.01602 h 0.480 h 5.0

A7 HDPE 70,500h 14.95 0.59426 h 0.331 h n.d.

C4 HDPE 113,200h 12.4 4.1E-4 h 0.301h n.d.

A5 HDPE 1,570,000h 1.66 1.31948 h 0.294 h n.d.

A1 HDPE 4,233,000h 0.15 7.23915 h 0.371 h n.d.

A4 HDPE 6,732,310h 0.25 4.715 h 0.362 h n.d.

A2 HDPE 26,100,000h 0.038 21.95 h 0.339 h n.d.

L8 LLDPE 4840g 211 0.0068 0.67844 0.6g

L4b LLDPE 14800f 63 0.014 0.58549 0.7f

L6b LLDPE 14200f 56 0.022 0.72 0.3f

F18C LLDPE 56620g 18.8 0.052 0.59719 1.4g

F26C LLDPE 86520g 14.8 0.075 0.647 1.6g

F12Fc LLDPE 124200g 9.58 0.11 0.6073 2.0g

F18Fd LLDPE 145800g 6.1 0.143 0.57626 3.4g

F26Fe LLDPE 174400g 5.6 0.182 0.56056 4.0g

f Gabriel and Münstedt (2002),

g Stadler and Münstedt (2008c),

h Stadler et al. (2006b),

i outside measurement range, n.d. – not deter-

mined



. (5)

This extension is based on the same arguments as the

extension of the modified CY-model published elsewhere

(Stadler and Münstedt, 2008a; Stadler and Münstedt,

2008b).

An example for the quality of the fit is given in Fig. 1

for the hexacosene LLDPE F26C. The fit is basically

undistinguishable from the measured data except for

G"(ω>100 s-1). Two factors contribute to this imperfec-

tion. The CY-model, on one hand, is unable to fit the rub-

bery plateau correctly; it simply assumes that δ → 0°,

which of course is not true in the strict sense, as the

Rouse-motions start to play a role in this regime and,

therefore, an upturn in G”(ω) is expected, which the CY-

model cannot describe. On the other hand, it is well

known that at high frequencies, inertia starts playing a

role, thus, changing the phase angle. Therefore, it is con-

cluded that the deviations of G”(ω) at ω>100 s-1 in Fig. 1

are caused by the first traces of the Rouse motions.

Considering the simplicity of the function and the lim-

itations of the model, the fit is excellent.

3.2. Influence of short-chain branches on the char-

acteristic relaxation timeThe dependence of the parameter λ and the reciprocal

crossover frequency 1/wc of the mLLDPEs is found to fol-

low the same dependence as found for the mHDPEs (Sta-

dler et al., 2006b). However, the absolute values of these

quantities lie somewhat higher than the correlation

(6)

found for mHDPE. Depending on the comonomer length a

prefactor between 0.84 and 1.3×10−20 is found. Later, the

parameter λlin will be discussed, which is the characteristic

relaxation λ expected from molar mass Mw, being calcu-

lated from Eq. (6).

G'' ω( ) ω η0 1 λ ω⋅( )a+[ ]n 1–

a----------

90

o

1 λ ω⋅( )a+------------------------sin⋅=

λ 8.4 1021–

Mw

3.6⋅ ⋅=

Fig. 1. Viscosity |η*(ω)| and moduli functions G’(ω) and G”(ω)

of F26C along with the fits using the Carreau-Yasuda

equations.

Fig. 2. Characteristic relaxation time λ and reciprocal crossover

frequency 1/ωc as a function of Mw.

Fig. 3. Average of the ratio of the characteristic relaxation time λ

and the reciprocal crossover frequency 1/ωc to the char-

acteristic relaxation time expected from the molar mass

λlin (Eq. (6)) as a function of side chain content sc. An

error of ±20% results from the high exponent of 3.6 and

the experimental uncertainty of ±5% in the determination

of Mw.

1As can be seen from Fig. 2, there are some deviations between

the values of ωc, which are due to statistical errors. Hence, the

two quantities – being theoretically identical – were averaged to

reduce the scatter. It can be seen from the error bars in Fig. 3 that

the correlation would be significantly worse, if only one of the

two quantities would be plotted.



When averaging the deviation of the characteristic

relaxations times λ and the reciprocal crossover fre-

quency λ/ωc from the correlation between λ-Mw estab-

lished by Stadler et al. (2006b),1 a clear trend towards a

higher λ for longer comonomers is found, which is plot-

ted in Fig. 3 as (1/ωc+ λ)/2λlin

as a function of side chain

content sc. The introduction of a small amount

(<3.5 mol%) of hexacosene into the polymer increases

the λ by about 45% (Fig. 3).

Fig. 3 shows that the samples F12F, F18F and F26F

deviate to smaller (1/ωc+λ)/2λlin, which comes from the

fact that these samples were taken into the evaluation later

than the others and were, thus, measured by SEC about 12

months later on a new set of GPC-columns. Due to the

high complexity of an SEC-MALLS, it is assumed that the

new calibration of the SEC-MALLS-setup delivers molar

masses, which are about 5% higher than on the previous

calibration. For this reason, F12F, F18F, and F26F are

included in Fig. 3 also taking a molar mass 5% lower than

measured into account (○), which increases (1/ωc+λ)/2λlin

by 20%, due to λlin~Mw

3.6. After this adjustment, the data

fit the trend (gray shaded area) of the other samples with

high precision.

Although this effect is rather small and only marginally

larger than the experimental error of the SEC-MALLS, it is

quite clear that it is not an artifact as a clear correlation

with an average deviation of about 10% was found.

Chen et al. (2010) stated that this effect is due to an

almost perfect canceling out of the effect of increase of

entanglement molar mass Me and characteristic entangle-

ment relaxation time τe (which is equivalent to the mono-

meric density reequilibration process (Stadler and van

Ruymbeke, 2010)).

3.3. Transition parameter “a”The molar mass dependence of Je

0 (Stadler and Mün-

stedt, 2008c) raises the question if this finding can be con-

firmed by looking at a different rheological quantity. For

this purpose the viscosity functions were evaluated by fits

of the Carreau-Yasuda-equation.

Another important fitting parameter is “a”, the quantity

describing the width of the transition between the New-

tonian and the shear thinning regime. Stadler et al. (2006b)

found that “a” is a function of the molar mass Mw for the

mHDPEs (reprinted in Fig. 4). The same correlation was

also found for the mLLDPEs, however, the absolute values

found for these samples are higher by about 0.12 at the

same molar mass Mw.

Due to these similarities in the trend of a(Mw) and

Je

0(Mw) a comparison between these quantities is compel-

ling. The plot of the bending parameter “a” as a function of

the equilibrium compliance Je

0 shows a clear correlation

between these quantities (Fig. 5). The data points can be

described sufficiently well by the correlation:

(7)

Thus it is, in theory, possible to estimate the elastic com-

pliance from the viscosity function. However, this corre-

lation is only determined for mLLDPEs with a Schulz-

Flory-type MMD and, thus, cannot be generalized to all

MMDs and is only valid for samples without LCBs. Espe-

cially the limitations for the proper application of a Car-

reau-Yasuda fit, the absence of long-chain branches and a

molar mass distribution without distinct high or low molec-

≈

a 0.693 455 Je

0⋅–=

Fig. 4. Transition parameter “a” as a function of Mw (data of the

mHDPEs are taken from Stadler et al. (2006b)), lines

added to guide the eye.

Fig. 5. Transition parameter “a” as a function of Je0. The error

bars for Je0 are equivalent to an error of ±10% but at least

±10-5 Pa−1 based on the average reproducibility of the

creep recovery measurements. The error bars for “a” are

set to 0.03 which is the usual deviation for fits of different

samples. Open symbol: mHDPE C3, closed symbols:

mLLDPEs.



ular tails, have to be considered, as they dramatically influ-

ence Je

0 (Gabriel et al., 2002; Gabriel and Münstedt, 2002)

and might prevent a proper fit of the viscosity function

because of the occurrence of an additional “bending point”

(Stadler et al., 2006b).

A possible explanation of the apparent disagreement

between the correlation between “a” and Mw of mHDPE and

mLLDPE (Fig. 4) and the correlation between “a” and Je

0

(Fig. 5) can be found when taking the plateau modulus into

account. Chen et al. (2010) determined GN

0 for all the sam-

ples. The molar mass distributions are almost identical. Dealy

and Larson (2006) state that the relaxation spectrum index RSI

(8)

only depends on the molar mass distribution for linear

samples. Hence, we should expect that the RSI is constant

for all LLDPE-samples. However, Fig. 6 demonstrates that

the RSI is not constant and scales with “a” in the same way

as Je

0. The reason for that is obviously that Je

0 varies within

a range of roughly factor 15, while GN

0 doesn’t show a

comparable dependence to counter this.

However, as the MMD of all mLLDPE in this article is

comparable, the equation provides an at least qualitative

insight. The direct consequence of this lengthening of the

characteristic relaxation time λ is a decrease of the plateau

modulus GN

0 by up to 1/3 due to the incorporation of up to

26.5 wt.% side chains (side chain content sc). This makes

sense, as these side chains, although quite long, are def-

initely unentangled,2 and, thus, reduces the number of

entanglements per chain, as only the main chain is entan-

gled. A side chain content sc of 25% is equivalent to a

reduction of the amount of entanglements per chain of a

given molar mass by 1/3, which corresponds to a reduction of

GN

0 by the same fraction due to an increase of Me by 1/3.

These considerations are basically a different way to

describe the findings of Garcia-Franco et al. (2005) that the

plateau modulus GN

0 scales with the backbone equivalence

mass.

It was already surprising to find the correlation between

the molar mass Mw and the transition parameter “a” for the

mHDPE (Stadler et al., 2006b), despite distinctive differ-

ences in the MMD. While these could still be attributed to

some kind of MMD-dependence of “a” (which would be

expected), the finding that a comparable correlation is also

found for the mLLDPEs, whose molar mass distribution

does not change, greatly affirms the conclusion that the

correlation between “a” and Mw of mHDPE is not a mere

coincidence. The finding that the values of “a” have a close

correlation to the steady-state elastic recovery compliance

Je

0 (Stadler and Münstedt, 2008c) are another important

evidence that this finding is not an artifact. However, as

already stated in the discussion of the Mw-dependence of

Je

0 before by Stadler and Münstedt (2008c), the origin of

this dependence is unclear.

Another point is, why the correlations between “a” and

Mw are different for mHDPE and mLLDPE. To be able to

understand this, the δ (|G*|)-plot is utilized to establish a

temperature independent data presentation, which is influ-

enced by the MMD, but is also very sensitive with respect

to long-chain branches (Garcia-Franco et al., 2005; Schulze

RSI Je

0GN

0×=

2The generally accepted value for Me of PE is 833 g/mol (Fetters

et al. , 1999) while a C24 side chain has a molar mass of 336 g/

mol. It was established by van Ruymbeke et al. (2007) that such

short branches (in this case of monodisperse pom-poms) slow

down the chain motions but do not lead to a typical behavior of

long-chain branched polymers, i.e. to the appearance of addi-

tional relaxation mechanisms.

Fig. 6. Transition parameter “a” as a function of the relaxation spec-

trum index RSI=GN0×Je

0. The error bars for “a” are set to

0.03 which is the usual deviation for fits of different samples.

Open symbol: mHDPE C3, closed symbols: mLLDPEs.

Fig. 7. Definition of the linear reference of the δ(|G*|)-plot via the

data of the samples C1, C3, C4, and A4.



et al., 2005; Trinkle et al., 2002; van Gurp and Palmen,

1998; Walter et al., 2001).

In order to be able to describe the effects of long-chain

branching in future articles, a linear reference has to be

established being independent of the molar mass Mw (and

in first approximation independent of the MMD). For the δ

(|G*|)-plot the linear reference (thick dotted line in Fig. 7)

was determined from the samples C1, C3, C4, and A4. It

is obvious that the samples agree quite well at the high fre-

quencies → high moduli, low phase angles, while dis-

agreeing at the higher phase angles. This is a consequence

of the Mw dependence of “a”, which broadens the transition

in both |G*| and δ with increasing molar masses.

The addition of comonomer leads to a decrease of the

plateau modulus GN

0 (Chen et al., 2010). Hence, strictly

speaking, the linear reference established in Fig. 7 is not

valid for mLLDPEs.

However, as can be seen from Fig. 8, the differences

between mHDPEs, i.e. the linear reference, and mLLDPEs

is only significant for the low phase angles δ. This is the

consequence of the lower plateau modulus GN

0, which are

caused by the short-chain branching.

The dissimiliarity at δ < 45o between mHDPE-reference

and mLLDPE and the agreement of the mLLDPE to the

mHDPE-reference at δ > 45o indicates that the chain

dynamics of the mLLDPEs are somewhat different from

mHDPEs. The SCBs play a distinctive role only at short

relaxation times, lowering GN

0.

As the introduction of the short-chain branches directly leads

to a reduction of |G*|(δ = 20°) by about a factor of 2, while

|G*|(δ = 60°) is basically uninfluenced by the SCBs, the tran-

sition of the viscosity function |η*(ω)| from the shear-thinning

regime (i.e. low δ) to the terminal regime (δ → 90o) becomes

sharper, thus, increasing the transition parameter “a”.

This influence is different from the influence of the

molar mass Mw, given in both Fig. 7 and Fig. 8, leading to

a systematic lowering of |G*|(δ > 60o) at high molar masses

in comparison to the low Mw-samples.

These reflections explain both the influence of Mw and

the comonomers on the transition parameter “a” (Figs. 4

and 5).

3.4. DiscussionThe rheological data obtained for linear mLLDPE can be

very well described with a Carreau-Yasuda equation,

whose fit parameters can be nicely correlated with molec-

ular properties, although limitations are found at high fre-

quencies, which are due to Rouse modes and probably

inertia problems of the rheometer.

The characteristic relaxation time λ roughly follows the

same correlation with the weight average molar mass Mw

as found for linear mHDPE with an exponent of 3.6. How-

ever, depending on the structure of the short-chain

branches, λ is lengthened by up to 50%. This is in contrast

to the agreement of the correlation between the zero shear-

rate viscosity η0 and Mw published before to the correlation

found for mHDPE and mLLDPE (Stadler and Münstedt,

2008c; Stadler et al., 2006b). This finding affirms the find-

ing of van Ruymbeke et al. (2007) that unentangled

branches slow down the chain mobility. However, unlike

Chen et al. (2010) claimed, while the effect on η0 seems to

be canceled out perfectly (Stadler and Münstedt, 2008c), λ

is comonomer content dependent.

Like mHDPE also a molar mass dependence of the tran-

sition parameter “a” was found, however, at higher values

of “a”. This finding is most probably the consequence of

the lower plateau modulus, which reduces the intensity of

the rubbery relaxation modes and, thus, causes a sharper

transition to the terminal regime. Nevertheless, it is not

possible at the moment to prove this hypothesis.

Although the findings are clear, even up-to-date molec-

ular models (Das et al., 2006; Park et al., 2005; Wang et

al., 2010) are unable to predict the dependence of a(Mw) or

the comonomer content dependence of the δ (|G*|)-plot.

The reasons for these dependencies are, therefore, still

unclear and require a more detailed analysis.

Acknowledgements

The authors would like to thank the German Research

Foundation and the “Human Resource Development

(project name: Advanced track for Si-based solar cell

materials and devices, project number: 201040100660)” of

the Korea Institute of Energy Technology Evaluation and

Planning (KETEP) grant funded by the Korea government

Ministry of Knowledge Economy for the financial support

and Prof. Dr. Helmut Münstedt, Dr. Joachim Kaschta and

Mrs. Inge Herzer (University Erlangen) for the GPC-

MALLS-measurements. The authors would also like to

acknowledge Dr. Christian Piel, Dr. Burçak Arikan-Con-

Fig. 8. δ (|G*|)-plot of the mLLDPEs L8, L4, F26C, and F26F.



ley, and Prof. Dr. Walter Kaminsky (University Hamburg)

for the synthesis of most of the samples used in this article

and Dr. Katja Klimke and Dr. Matthew Parkinson of the

Max-Planck Institute of Polymer Research in Mainz (Prof.

Manfred Wilhelm’s group) for the solid state NMR-mea-

surements.

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understanding the effect of short-chain branches by analyzing viscosity functions of linear and...

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