Rheologica Acta

, Volume 44, Issue 2, pp 174–187

Non Linear Rheology for Long Chain Branching characterization, comparison of two methodologies : Fourier Transform Rheology and Relaxation.

Authors

  • Guillaume Fleury
    • LIPHT (Laboratoire d’Ingénierie des Polymères pour les Hautes Technologies)ECPM (Ecole Européenne de Chimie Polymères et Matériaux de Strasbourg)
  • René Muller
Original paper

DOI: 10.1007/s00397-004-0394-3

Cite this article as:
Fleury, G., Schlatter, G. & Muller, R. Rheol Acta (2004) 44: 174. doi:10.1007/s00397-004-0394-3

Abstract

In this study we compare three rheological ways for Long Chain Branching (LCB) characterization of a broad variety of linear and branched polyethylene compounds. One method is based on dynamical spectrometry in the linear domain and uses the van Gurp Palmen plot. The two other methods are both based on non linear rheology (Fourier Transform Rheology (FTR) and chain orientation/relaxation experiments). FTR consists in the Fourier analysis of the shear stress signal due to large oscillatory shear strains. In the present work we focus on the third and the fifth harmonics of the shear stress response. Chain orientation/relaxation experiment consists in the analysis of the polymer relaxation after a large step strain obtained by squeeze flow. In this method, relaxation is measured by dynamical spectrometry and is characterized by two relaxation times related to LCB. All methods distinguish clearly the group of linear polyethylene from the group of branched polyethylene. However, FTR and Chain orientation/relaxation experiments show a better sensitivity than the van Gurp Palmen plot. Non linear experiments seem suitable to distinguish long branched polyethylene between themselves.

Keywords

Long Chain BranchingLCBFourier Transform RheologyFTRChain orientation

Introduction

It is well known that the molecular structure of Long Chain Branching (LCB) induces drastic effects on the properties of molten polymers. Thus, LCB increases the viscosity, involves the onset of shear thinning at much lower shear rate and is the cause of strain hardening in elongation. Whereas rheological properties of linear and branched polymers are significantly different, it is however difficult to differentiate low level LCB polymers between themselves. This is actually a crucial point of research. Our work, presented here, tries to contribute to the development of new rheological methodologies able to characterize the LCB effects.

Numerous methods have been developed for determining directly the amount of LCB. One method consists in combining size exclusion chromatography (SEC) with intrinsic viscosity measurement (Pang and Rudin (1991), Vega et al. (1999)). Thanks to the Zimm-Stockmayer equation it is possible to estimate the reduction of the hydrodynamic volume caused by the branches. Unfortunately, for small LCB amounts (<<1 LCB / 103 Carbon atoms), the intrinsic viscosity of the branched polymer is very close to the intrinsic viscosity of the linear polymer of the same molecular weight. In that way, SEC seems to be suitable for high amounts of short chain branching characterization but is unusable for low levels of LCB. More recently, Hadjichristidis et al. (2000) have used the Zimm-Stockmayer model to estimate the degree of branching for well defined long chain branched polyethylene like stars, H-shaped, pom-pom and comb polymers. The authors have shown that this technique seems to be well adapted for simple architectures like stars (symmetric or not) but fails for complex architectures like combs. Another method consists in the estimation of LCB by high-resolution nuclear magnetic resonance 13C-NMR (Yan et al. (1999)). Although 13C-NMR is known to detect side chain branches up to 8 carbons, the authors estimate LCB densities (LCBD) less then 1 branch / 104 carbon atoms of several metallocene polyethylene. The LCBD was defined as the ratio of the integral area of α-CH2 resonance to 3 times the total intensity of carbon atoms. Nevertheless, this technique required a long acquisition time (more than 20 h) to obtain an exploitable spectrum and is of course limited by the chemical composition and side chain length of the polymer.

Another way to characterize branched polymers is rheology. The difficulty is to distinguish the effect of LCB on rheological properties from the effect of molecular weight distribution. Nevertheless, rheology presents the big advantage that it is not limited by the chemical composition of the characterized polymer. Fujimoto et al. (1970) then Roovers (1984) have experimentally studied the rheology of well defined comb polymers. The authors have shown long-time relaxation processes that are not present for linear polymers. The G’, G’’ versus ω plots show relaxation processes dominated by two main characteristic times. The shortest one has been allotted to the dynamic of the arms and the longest one to the relaxation of the whole macromolecule characterized by the zero shear viscosity. McLeish et al. (1999, 1988), Read and McLeish (2001), Blackwell et al. (2001) and Groves et al. (2000) have studied both theoretically and experimentally the rheology of well controlled H and branched polymer melts. They have measured and simulated G’ and G’’ for well defined H-polymers in a wide range of frequencies. The authors have clearly explained the typical two peak shape of the G’’(ω) plot. They have shown that the peak at high frequencies is allotted to the relaxation of the arms of H-polymers. On the other hand, they have shown that increasing the length of the backbone (i.e. increasing the number of entanglement units) shifts the second peak to lower frequencies. The linear dynamics of H-polymers can be characterized by retraction of dangling arms at short times then by reptation/fluctuation of the backbone at long times. Furthermore, they also have shown that polydispersity in the arm molecular weight drastically shifts the retraction phenomenon to long relaxation times. Vega et al. (1999) and Yan et al. (1999) have studied the rheological properties of different branched polymers which were beforehand characterized by SEC and 13C-NMR respectively. The authors have shown significant effects of LCBD on the shear thinning effect. For the same molecular weight, higher LCBD give higher viscosity at low shear rates and lower viscosity at high shear rates. This particular behaviour, compared to linear polymers, can be explained by the two following opposite factors : firstly, LCB increases chain entanglements and secondly, at constant molecular weight, LCB decreases the polymer hydrodynamic volume. Vega et al. (1999) have tested 13 metallocene catalyzed polyethylene. By plotting the η’- η’’ Cole-Cole diagram, the authors have determined η0, the limiting extrapolated viscosity and λc, the characteristic relaxation time corresponding to the frequency when the function η’’ is maximum. After plotting λc versus Mw and η0 versus Mw, three groups of polyethylene samples clearly appear, one group corresponding to linear samples and two groups corresponding to the branched polyethylene. The two latter groups have been allotted to long chain branched polymers which differ in the amount of branches. Vega et al. (1999) reported also the results of the activation energy versus ν (degree of hexyl branching per 1000 carbon atoms). One more time, the group of linear polymers differ from the other two groups of branched polymers which have higher flow activation energy. These results confirm those presented by Malmberg et al. (1998, 2000) and Baird et al. (2000). Unfortunately, this method is not able to differentiate between the two groups of branched polymers. An original way to analyze the LCB effect by linear dynamical measurements has been proposed by Trinkle et al. (2001, 2002). It consists in analyzing the rheological experiments by plotting the loss angle δ ( = tan-1[G’’/G’] ) versus a reduced modulus Gred defined as the magnitude of the complex modulus divided by the plateau modulus\(G^{0}_{N} \)(obtained when δ→0). This technique, the so-called van Gurp-Palmen analysis (Van Gurp and Palmen (1998)) allows to highlight structural effects, such as molecular weight distribution and LCB, on rheological properties while minimizing the molecular weight effect. Furthermore, the authors have built a map plotted from characteristic points Pc(Gred,c, δc) on which typical branched polymer structures cover distinct areas. Garcia-Franco et al. (2001) then Lohse et al. (2002) also have used this methodology to analyze well defined long chain branched polyethylene. Particularly, by using the experimental data of Wood-Adams et al. (2000), Garcia-Franco et al. (2001) have explained that branching exhibits physical gel-like behaviour near the critical gelation point. Rheology in the non linear domain (high strains) has also been investigated in order to characterize LCB. McLeish et al. (1999) and Heinrich et al. (2002) have studied the dynamic of H-polymers during step-strain experiments and have confirmed their results by means of small angle neutron scattering SANS. The authors have shown the effect of the rearrangement of the branches on SANS which corresponds to a characteristic relaxation time allotted to the stretch of the “cross-bar” of the H-polymers. Kimura et al. (1981) and Chai et al. (1999) have used birefringence induced during steady start-up shear flow to measure indirectly the shear stress and the first normal stress difference. The authors have compared the behaviour of a linear polyethylene, a branched one and their blends with narrow molecular weight distribution in order to highlight LCB effects. The results clearly show that overshoots in the normal forces occur at lower shear rates for branched polymers. An original work has been presented by Schedenig and Schausberger (2000). The authors propose to uncouple the effect of molar mass distribution to the effect of LCB. First, a large non linear deformation is applied on the sample. Then, after a certain measured time, a classical stress relaxation experiment is carried out. From these experiments, the authors determine a relaxation time mainly correlated to LCB. Another way to characterize LCB should be non linear dynamical oscillatory rheology (Fourier Transform Rheology FTR). MacSporran and Spiers (1982, 1984) have studied the non linear response for polymer in solution. Chavanne (1996) used the methodology for EPDM. Wilhelm et al. (2002, 1998, 1999) have studied the non linear dynamical response for different well defined linear polymers. They have particularly correlated the relative intensity of the third harmonic as a function of the shear amplitude with the molar mass of the tested polymer. In conclusion, FTR seems to be a suitable method to characterize LCB.

This paper presents two methodologies, based on non linear rheology, for assessing the differences between several branched polydispersed polymers. The first method consists of FTR. The second method, is based on the idea of Schedenig and Schausberger (2000) and consists in studying the relaxation of polymers after strong non linear deformation. The two methods will then be compared and discussed in the light of linear dynamical rheology (by means of van Gurp-Palmen analysis) and GPC results.

Experimental

Equipment

Dynamical rheometry

Our experimental set-up is based on a Rheometrics-ARES strain controlled rheometer. This ARES-rheometer is equipped with two torque trancducers (2 K FRTN1 and 2 K FRTN1E) that can detect torques within the range 0.002–200 mN.m and normal forces within the range 0.02–20 N. An air oven with N2 cooling allows a temperature range of between 150 °C and 600 °C to be covered. The motor is a STD motor with a strain amplitude of 0.05 to 500 mrad and a rotation frequency varying between 10-5 to 500 rad.s-1.

The geometry chosen for small deformations mechanical spectrometry (van Gurp-Palmen plot and relaxation after non linear deformation) were parallel plates of 25 mm diameter with a gap of 0.5 mm. For Fourier Transform Rheology (FTR), the configuration of the test was cone-plate geometry with a diameter of 25 mm and an angle of 5°.

Non linear signal storage

During Fourier Transform Rheology, we used an ADC-212 virtual oscilloscope from Virtual Instrument to record simultaneously the shear amplitude and the shear torque versus time. The data from the transducers are digitized with a 12-bit analog to a digital converter which is able to operate up to a sampling rate of 3 million samples per second. The connections between the ARES-rheometer and the virtual oscilloscope are realized with BNC-type cables. The Picolog software from Virtual Instrument enablesto collect the data directly on the computer in the form of tables.

Materials

Macromolecular structure of the samples

A wide variety of polyethylene has been used to perform these experiments. They were synthesized and characterized by SEC at the Atofina laboratory of Feluy. Table 1presents the supposed chain structure and SEC results (molecular weights and polydispersity) of some studied polymers. Intrinsic viscosity and light scattering detections were used to carry out the measurements.
Table 1

Chain structure and polydispersity of the studied polyethylene

PE

LM1

LM2

BM1

BM2

BM3

BM4

BM5

BM6

BR1

Chain Structure

Linear

Long branches

Many short branches

Mn (kg/mol)

30.4

18.5

15.1

16.9

18.1

15.4

Mw (kg/mol)

74

92

219

145

241

116

Mz (kg/mol)

143

532

2330

1222

2357

902

Mw/Mn

2.4

5

14.5

8.6

13.3

7.5

We have used nine different products with a large range of polydispersity but with molecular weight Mw of the same order. LM1 and BM1 to BM6 are polyethylene synthesized by chrome metallocene catalysis. LM2 is by polyethylene synthesized Ziegler-Natta technique whereas BR1 has been performed by radical synthesis. LM1 is a linear polyethylene with a low degree of polydispersity whereas LM2 is linear but has a broader polydispersity. BM1 to BM6 are branched polyethylenes including long chain branching. BR1 is a polyethylene with many short and long branches. All samples have been stabilized by antioxidants during the commercial pelletization operations.

Mechanical spectrometry in the linear domain, van Gurp Palmen plot

In order to process rheological tests on ARES rheometer, samples were first carried out at 150 °C under a pressure of 20 MPa into 15 mm diameter disks with a thickness of 3 mm. We have checked, within an error margin of 10%, the non-degradation of the samples in order to avoid chain extension or cross-linking during experiments. We have performed dynamical measurements of dynamical modulus at 150 °C and 180 °C for all the samples on a large time scale. Then, we noticed that the thermal degradation of the polymers at 150 °C could be excluded within an error margin simply because G’ and G” did not change over several hours at 150 °°C with a frequency of 0.5 rad.s-1 and a strain amplitude of 10%.

Figure 1shows the van Gurp Palmen curves (phase angle δ versus absolute value of the complex modulus |G*|) for each polyethylene. As we can see, there are three distinct groups: the group of linear polymers (LM1 and LM2), the group of branched polymers (BM1 to BM6) and the group of the branched polyethylene BR1. The curves of LM1 and LM2 have a classical shape: From low to high |G*|, the phase angle δ starts from a plateau at 90° and then decreases. Because of the crystallization, the plateau modulus\(G^{0}_{N} \;{\left( {\; = G'{\left( {\omega \left| {{}_{{\delta \;at\;\min }}} \right.} \right)}} \right)}\)is not reached for our samples. As Trinkle and Friedrich (2001) have shown, the higher degree of polydispersity (DPLM2>DPLM1), the more the curves are stretched at low |G*|. The second group, corresponding to the branched polyethylene BM1 to BM6, have lower δ then the first group of linear PE which is in agreement with the results of Trinkle et al. (2002). The curve of BM4 is the most stretched. This result means that BM4 should have the longest arms or should have a large number of long arms. Unfortunately, the van Gurp Palmen plot is not able to differentiate between the other branched PE between themselves. The last PE, BR1, obtained by radical synthesis, has a different behaviour: δ starts from a plateau near 90° and then decreases very rapidly. This result should be explained by the fact that the molecular structure of BR1 is distinguished from BM1 to BM6 by a high number of short branches.
Fig. 1

van Gurp Palmen plots of linear and branched polyethylene obtained at 150 °C

Fourier Transform Rheology

Principle and methodology

Fourier Transform Rheology, as called by Wilhelm (2002, 1998, 1999), consists in analyzing the frequency spectrum of the torque signal obtained during mechanical spectrometry at high strains γ(t) = γ0 sin(ω1t). Figure 2shows the effect of the strain amplitude γ0 on the normalized torque (τ(t)=T(t) / Tmax). Although the torque keeps its periodicity one can see that the response can not be represented by a single harmonic function. Thus, two non linear effects are shown: firstly, the signal of the torque is squeezed near the maximum and the minimum and secondly, the signal shows a dissymmetry compared to the vertical line passing by the maximum value of the torque (see Fig. 2). The first effect is due to non Newtonian behaviour occurring for shear rate above a critical value γ̇c. Since the maximum shear rate during spectrometry is equal to\(\ifmmode\expandafter\dot\else\expandafter\.\fi{\gamma }_{{\max }} = \gamma _{0} \omega _{1} \), non linear effects occur when γ0ω1 >γ̇c. The dissymmetry of the torque might be allotted to the viscoelastic behaviour of the polymers combined with the successive load-unload cycles of the strain. Furthermore, due to polymer elasticity, it must be noticed that periodicity is not established at the beginning of the experiment. Thus, in order to ensure the quasi steady state flow, each recording is carried out after a time equal to ten times the fundamental period T1 = 2π/ω1. As the torque Τ(t) is represented by a periodic function, it can be represented thanks to Fourier series.
Fig. 2

Normalized torque T(t)/Tmax as a function of time of BM6 obtained at γ0 = 200% and 450%, 2πf1 = 0.193 rad.s-1 and T = 150 °C

The frequency spectrum of the torque has been obtained by discrete Fourier transformation (DFT). For each test, we have chosen to collect a sampling number NS=4096 during a sampling period TS=357 s (i.e. a sampling interval ΔtS=87.2 ms). In order to increase the accuracy of DFT, the fundamental frequency of the torque f1 (i.e. the frequency of the imposed strain) has been chosen as a whole number k of the frequency step ΔfS:
$$f_{1} = k\;\;\Delta f_{S} = \frac{k} {{N_{S} \;\;\Delta t_{S} }}$$
Consequently, the pulsation ω1 has been fixed at 0.193 rad.s-1 for each FTR experiment. Thus, the choice of the different experimental parameters allows us to obtain a DFT which verifies the Nyquist criterion until at least the 11th harmonic. Moreover, a Hanning windowing has been used to filter the signal of the torque.
Figure 3shows the frequency spectrum of the torque obtained by DFT. The intensity of the spectrum I(f) is represented relatively to the intensity I1 (=I(f1)=I(ω1)) of the fundamental frequency ω1. Even for the highest strain amplitude (γ0 = 500%), it appears that the level of I2/I1 (=I(2ω1)/I1) was every time negligible compared to I5/I1. More generally, due the relationship between the viscosity and the absolute value of the shear rate\(\eta {\left( {{\left| {\ifmmode\expandafter\dot\else\expandafter\.\fi{\gamma }} \right|}} \right)} = \eta {\left( {\ifmmode\expandafter\dot\else\expandafter\.\fi{\gamma }} \right)} = \eta {\left( { - \ifmmode\expandafter\dot\else\expandafter\.\fi{\gamma }} \right)}\), the spectrum shows only odd harmonics (Wilhelm (1998, 2002). Consequently, the torque can be written according to the following expression:
Fig. 3

Normalized frequency spectrum of the torque of BM6 obtained at γ0 = 450%, 2πf1 = 0.193 rad.s-1 and T = 150 °C

$$T(t) = I_{1} e^{{i\omega _{1} t}} + I_{3} e^{{3i\omega _{1} t}} + I_{5} e^{{5i\omega _{1} t}} + I_{7} e^{{7i\omega _{1} t}} + ...$$
(1)
Furthermore, Figure 3 shows that the relative intensities of odd harmonics decrease with respect of frequency. Thus, the relative intensity I3/I1 gives an idea of non linear effects at the first order. Our Fourier Transform Rheology (FTR) experiments consist in characterizing non linear effects by the study of the relative intensities I3/I1, I5/I1 with respect to strain amplitude γ0.

Results

Strain amplitude dependence.

Figure 4 shows the evolution of the torque amplitude as a function of the strain amplitude γ0. For each polymer, the amplitude of the torque increases faster for low γ0 before reaching a plateau. Figure 5 compares the relative first harmonic I3/I1 versus γ0 of each polyethylene. Firstly, one can see that the relative position of the curves is different from the relative position observed for the torque in Fig. 4. Secondly, Fig. 5shows two distinct groups : the group of linear polymers LM1 and LM2 and the group of branched polymers. Thus, for LM1 and LM2, non linear effects are below 1% at γ0 = 300%, whereas they are more than three times higher for branched polymers. One can explain these two results by the fact that the torque is equivalent, at the first order, to the fundamental intensity I1 which simultaneously depends on molecular weight, molecular distribution and polymer structure. As LM1 and LM2 are linear polymers, their relative first harmonic I3/I1 is only due to the effects of molecular weight and its distribution. As molecular weights of each polymer are of the same order, the drastic increase of non linear effects for polymers BM1-BM6 and BR1 can only be explained by the effect of their LCB. Consequently, the relative first harmonic I3/I1 seems to be a good indicator of branching. Figure 6shows the second relative harmonic I5/I1 versus γ0 for each polyethylene. As mentioned before, one can see that I5/I1 is clearly lower than I3/I1. However, the relative order of each curve is the same as observed for I3/I1 but the differentiation between each polymer is more pronounced. Indeed, different groups of polymers appear. The group LM1, LM2 and BR1: I5/I1 stays at a low level up to strain amplitudes of 500%. This result seems to indicate that BR1 would have a chain structure closer to linear polymer. A second group consists of BM1, BM3 and BM5. For these polymers, I5/I1 starts to increase drastically from a strain amplitude of 300%. This effect is more pronounced for BM5. The latter group is made of BM6, BM2 and BM4. For these three polymers, the non linear effects characterized by I5/I1 start to increase drastically from a strain amplitude of around 150%. Thus, BM4 seems to have the LCB which generates the highest non linear effects.
Fig. 4

Amplitude of the torque Tmax and corresponding shear stress amplitude as a function of shear strain amplitude γ0 of linear and branched polyethylene obtained at 2πf1 = 0.193 rad.s-1 and T = 150 °C

Fig. 5

Relative intensity of the third harmonic I3/I1 as a function of strain amplitude γ0 of linear and branched polyethylene obtained at 2πf1 = 0.193 rad.s-1 and T = 150 °C

Fig. 6

Relative intensity of the fifth harmonic I5/I1 as a function of strain amplitude γ0 of linear and branched polyethylene obtained at 2πf1 = 0.193 rad.s-1 and T = 150 °C

Frequency and temperature dependences

Figure 7shows the effects of frequency on the two first relative harmonics I3/I1 and I5/I1 for BM2. These experiments have been carried out at 170 °C with a strain amplitude of 200% in order to stay below the maximum admissible level of the torque sensor. The frequencies starts from 0.053 rads-1 up to 2.812 rad.s-1. At higher frequencies a strong decrease of the torque as a function of time was observed during experiments. This effect could be due to viscoelastic flow instabilities. Each experiment has been carried out four times. The bars of error show a good repeatability of the experiments. Figure 7shows that I3/I1 and I5/I1 increase quickly at low frequencies which characterises the transition from linear to non linear viscoelasticity. From 0.77 rad.s-1 the relative harmonics I3/I1 and I5/I1 reach a plateau. These results were also observed for other polymers. Unfortunately, due to Weissenberg effects, it was not possible to carry out experiments for higher frequencies. On the other hand, Fig. 8shows the two first relative harmonics I3/I1 and I5/I1 versus γ0 for BM6 at different temperatures (140°°C, 150°°C, 160°°C and 170 °C). It clearly appears that temperature does not affect the curves of I3/I1 and I5/I1. One should notice that no time-temperature-superposition was carried out since the same frequency was used for all temperatures. If reduced frequencies were used almost no difference would be observed due to the relative low activation energies and to the small influence of the frequencies on the relative harmonics I3/I1 and I5/I1 as shown in Fig. 7. These results legitimate comparisons between all FTR experiments which were carried out at 0.193 rad.s-1 for 150 °C.
Fig. 7

Third and fifth relative harmonic (I3/I1 and I5/I1) of BM2 as a function of frequency obtained at γ0 = 200% and T = 170 °C

Fig. 8

Third and fifth relative harmonic (I3/I1 and I5/I1) of BM6 as a function of strain amplitude obtained at 2πf1 = 0.193 rad.s-1 and T = 140, 150, 160 and 170 °C

Relaxation after chain orientation

Principle and methodology

This method of characterization of the LCB has been inspired by the work of Schedenig and Schausberger (2000). It consists in studying the relaxation of the polymer melt after a large step strain. The test, based on the monitoring of the relaxation by mechanical spectrometry of the polymer melts after chain orientation, was developed in order to obtain information about the LCB. The purpose of this method is to determine characteristic relaxation times in order to obtain information about their branching.

The methodology can be split in two main steps (see Fig. 9):
Fig. 9

Experimental procedure carried out in order to obtain normalized relaxation times λ1c and λ2c characteristic to the branching

i)

Step 1. After the chain orientation carried out by large deformations, the relaxation of the polymer chains is characterized by the measurement of the elastic modulus G’(t) during time. Then, the fitting of a model with the experiment allows us to compute two characteristic relaxation times λ1 and λ2.

ii)

Step 2. The second step is a linear characterization of the relaxed chains by measurements of dynamic moduli G’(ω) and G”(ω). This step allows us to compute the Cole-Cole relaxation time λc which is characteristic to the whole chain polymer.

After these two steps, an analysis allows finally to obtain information about the branching rate thanks to the study of normalized relaxation times λ1c and λ2c.

Results

The orientation step
The method is based on the measurement of G’(t) by mechanical spectrometry after chain orientation. Two methods have been considered in order to obtain a chain orientation. A first method consists to impose a constant pre-shear between the parallel plates of the ARES-rheometer. Thus, this type of solicitation involves a tangential average orientation of the chains (see Fig. 10). The parameters of pre-shear have been chosen to avoid the Weissenberg effect (duration of pre-shear : 60°s with a shear rate fixed to 1°s-1). A second method consists of imposing a squeeze flow namely a compression of the molten sample between the ARES-plates during the gap procedure. This compression is carried out with a constant force of 10 N from a 3 mm gap down to 0.5 mm. As a consequence, the normal stress obtained by this compression was confined between 57 kPa for the initial diameter (15 mm) of the sample and 20 kPa for the ARES plate diameter (25 mm). Thus, we obtain a radial average chain orientation (see Fig. 10).
Fig. 10

Transverse and radial orientation of the chains obtained respectively by pre-shear and squeeze flow

In order to choose the most efficient orientation procedure we have tested the sensitivity of the dynamical measurements of G’(t) after chain orientation. After each procedure of chain orientation, mechanical spectrometry has been carried out at 150 °C with a shear frequency of 0.5 rad.s-1 and a strain amplitude of 10% between parallel plates with a gap of 0.5 mm. The relaxation monitoring starts just after the end of the chain orientation (by pre-shear or by squeeze flow). The results of the experiments are shown in Fig. 11. Time evolution of G’ during relaxation is shown for polyethylene BR1 in Fig. 11. It appears that the return to equilibrium\({G}\ifmmode{'}\else$'$\fi_{\infty } \)takes more time after squeeze flow than after pre-shear. Consequently, the measurement of G’(t) during relaxation after squeeze flow is more precise than after pre-shear. Thus, during this study, squeeze flow has been chosen for the orientation procedure.
Fig. 11

Relaxation experiments after transverse and radial orientation of the chains characterized by G’(t) obtained at ω = 0.5 rad.s-1, γ0 = 10% and T = 150 °C

Relaxation of branched polyethylene
All dynamical measurements of the elastic modulus G’ have been carried out at 150 °C with a shear frequency of 0.5°rad.s-1 and a strain amplitude of 10% in parallel plate configuration. The monitoring of the relaxation starts just after the squeeze flow. As shown in Fig. 11, relaxation curve reaches an equilibrium G’ depending on the polymer. In order to avoid this offset, characteristic relaxation curves are obtained by plotting G’-G’(t) versus time. Figure 12shows these curves for each polymer. We can see two main steps:
Fig. 12

Characteristic relaxation curves G’-G’(t) versus time of linear and branched polyethylene obtained at ω = 0.5 rad.s-1, γ0 = 10% and T = 150 °C

i)

a fast relaxation for times directly after the squeeze flow.

ii)

a slower relaxation until reaching the equilibrium state at the end of the experiment.

The kinetics characterizing the return to equilibrium has been modelled by a sum of exponential functions. Furthermore, these two apparent relaxation steps lead us to consider a model with at least two characteristic relaxation times λ1 and λ2. It gives the following equation:
$$g{\left( t \right)} = G_{1} e^{{ - \frac{t} {{\lambda _{1} }}}} + G_{2} e^{{ - \frac{t} {{\lambda 2}}}} = {G}\ifmmode{'}\else$'$\fi_{\infty } - {G}\ifmmode{'}\else$'$\fi{\left( t \right)}$$
(2)

Where characteristic time λ1 corresponds to a fast relaxation, whereas λ2 corresponds to a lower relaxation mode.

Because G1exp(-t/λ1) is negligible at long times, relaxation time λ2, that describes the second part of function g(t), is obtained from a linear regression of the Ln(G’-G’(t))=Ln(t) curve at long times. Then, time λ1 is computed by fitting function g(t) with experiments at short times.

Table 2 shows the parameters G1, G2, λ1 and λ2 of function g(t) obtained for BR1 for different squeeze flow conditions (Normal force = 4 N, 10 N and 15 N ). For this range of normal force, it appears that the relaxation times λ1 and λ2 are only slightly affected.
Table 2

Relaxation after squeeze flow: results for BR1 for different values of normal forces

Normal force (N)

Range of normal stress during squeeze-flow (kPa)

squeeze time (s)

G1 (Pa)

λ1 (s)

G2 (Pa)

λ2 (s)

λ2 / λ1

4

from 23 down to 8

840

92

119

87

556

4.67

10

from 57 down to 20

360

241

105

229

666

6.34

15

from 85 down to 30

90

103

103

87

558

5.41

Table 3shows that the ratio of the long time over the short time λ21 is nearly constant for all polymers:
Table 3

Relaxation times λ1, λ2, λc and non dimensional relaxation times λ1 / λc, λ2 / λc obtained after a squeeze flow with a normal force of 10 N (i.e. a normal stress during squeeze flow confined between 57 kPa for the initial diameter (15 mm) of the sample and 20 kPa for the ARES plate diameter (25 mm))

LM1

LM2

BM1

BM2

BM3

BM4

BM5

BM6

BR1

G1 (Pa)

170

261

535

479

1069

408

962

461

241

λ1 (s)

36

45

518

353

508

366

549

319

105

G2 (Pa)

195

268

1149

1124

842

2381

1644

439

229

λ2 (s)

279

315

2687

2497

2776

2902

3403

2488

666

λc (s)

0.02

0.09

2.3

9.5

4.0

13.1

4.1

5.9

4.3

103 λc1

0.61

2.07

4.48

26.76

7.95

35.87

7.42

18.61

40.48

103 λc2

0.08

0.29

0.86

3.79

1.46

4.52

1.20

2.38

6.38

λ2 / λ1

7.83

7.07

5.19

7.07

5.46

7.94

6.20

7.80

6.34

$$\frac{{\lambda ^{{}}_{2} }} {{\lambda ^{{}}_{1} }} = K \approx 6.8$$
(3)
Consequently, λ21 does not depend on the molecular structure. A first idea would consist in allotting short time λ1 to the relaxation of branches and λ2 to the relaxation of the whole chain. But, as the ratio λ21 is the same for branched and linear polymers, this latter idea can surprisingly not be affirmed. Another explanation could be the broad molecular weight distribution. Experiments on narrow molecular weight distribution, such as for instance anionic polystyrene, would be necessary to discriminate between these assumptions. Consequently, the further discussion concerning relaxation time λ1 is transposable to relaxation time λ2. Moreover, comparison of Tables 1and 3 shows that if one classifies the polymers from the lowest molar weight Mw to the highest value, one obtains exactly the same classification for λ1, respectively LM1, LM2, BM6, BM2, BM1, BM5. Thus, although relaxation times λ1 are much lower for linear polymers than for branched one, it appears that, for the group of branched polymers, the effect of molar mass on λ1 seems to dominate the effect of branching on λ1.
Characterization of branching
In order to characterize the branching level of polymers we have to find a rheological parameter more sensitive to the effects of branches than of molecular weight. An idea consists in the evaluation of the Cole-Cole relaxation time λc (see Fig. 13). This characteristic time λc corresponds to the inverse of the frequency ωc for which the Cole-Cole diagram reaches a maximum. When the determination of λc was not directly possible on the Cole-Cole diagram, we used the Schmitt diagram (see Fig. 13). All Cole-Cole diagrams have been obtained from linear spectrometry at 150 °C, in the range [0.01 rad.s-1, 100 rad.s-1] with a strain amplitude of 10% in parallel plate configuration.
Fig. 13

Cole-Cole diagram (η”=f(η’)) and Schmitt diagram (Ln(u/v)=f(Ln(ω))) for BR1 obtained at ω = 0.5 rad.s-1, γ0 = 10% and T = 150 °C

Table 3shows that time λc, which characterises the whole chain structure (molecular weight and its distribution and branching), varies in a broad range. As observed by Vega et al. (1999), λc is much lower for linear than for branched polyethylene. As shown before, relaxation time λ1 is more sensitive to molecular weight, we propose to introduce a dimensionless time λc/ λ1 (respectively λc/ λ2) which would be more sensitive to the effect of branching. Consequently, a classification of λc/ λ1 (respectively λc/ λ2) should be an indication of the branching level. Figure 14 shows dimensionless times λc/ λ2 versus λc/ λ1. Firstly, because of the constant ratio K=λ2/ λ1, Fig. 14shows a straight line with a characteristic slope of 1/K. Secondly, Fig. 14shows a classification of branching level (respectively from linear to highest branching: LM1, LM2, BM1, BM5, BM3, BM6, BM2, BM4, BR1).
Fig. 14

Dimensionless characteristic times λc/ λ2 versus λc/ λ1 of linear and branched polyethylene obtained at ω = 0.5 rad.s-1, γ0 = 10% and T = 150 °C

Discussion and conclusion

We have used three rheological methods to characterize the branching of polymers with molecular weights at the same order (Mwmax/Mwmin ~ 3). The first method is based on dynamical spectrometry in the linear domain and uses the van Gurp Palmen plot. The two other methods, are both based on non linear rheology (Fourier Transform Rheology FTR and chain orientation/relaxation experiments). The relative comparison of the behaviour of each polymer for each rheological method leads to the following conclusions:

Linear polyethylene LM1 and LM2. All methods distinguish clearly the group of linear polyethylene (respectively LM1 and LM2) from the group of branched polyethylene (respectively BM1 to BM6 and BR1). This result has been confirmed by several authors and can be explained by the fact that as polyethylene are characterized far from the plateau zone, the effect of cross linking points are predominant compared to the effects of chain entanglements. Furthermore, each method highlights a difference between LM1 and LM2 which has a higher degree of polydispersity: a) Van Gurp Palmen plots show a curve more stretched for LM2 than for LM1. b) Whereas the measured torque of LM1 and LM2 are identical, FTR shows higher level of non linear effects (characterized by I3/I1 and I5/I1) for LM2. c) Chain relaxation experiments has also located the behaviour of LM2 between those of LM1 and branched polyethylene.

Long chain branched polyethylene BM1 to BM6. This group of polymers has shown typical behaviour. All methods have clearly distinguished BM4 which has the behaviour the more distant from the linear polymers. Whereas van Gurp Palmen plots do not significantly distinguishes branched PE between themselves, the two other methods show surprisingly the same classification (respectively BM1, BM5, BM3, BM6, BM2, BM4). It is interesting to notice the relative classification of BM2 and BM5. Indeed, SEC shows lower molar masses (Mn and Mw) and lower polydispersity for BM2 (see Table 1) than for BM5. These results are confirmed by a lower level of the torque for BM2 measured during FTR experiments (see Fig. 4). Nevertheless, both FTR and chain relaxation experiments seem to indicate that BM2 has a higher level of branching (i.e. the behaviour the more distant from the group of linear polymers).

Short and long branched polyethylene BR1. This polyethylene, performed by radical synthesis, reveals a typical behaviour. Van Gurp Palmen plots (Fig. 1) has shown a curve located between those of linear polyethylene (LM1, LM2) and those of metallocene branched polyethylene (BM1 to BM6). Whereas FTR confirms this result, the latest method locates BR1 as the more branched. Thus, FTR could be more sensitive to long branches.

The rheological methods presented in this article lead to a classification from the linear up to the branched polymers. Further investigations on polymers with well controlled structure should allow to determine more precisely the effects of molecular structure; i.e. number of cross linking points, length of branches compared to Me (molar mass between two entanglements). Moreover, rheological models and numerical simulation should also help us to get a better understanding of branching effect.

Acknowledgements

The authors wish to thank the research laboratory of Feluy (Belgium) of Total Elf Fina for the PE samples and SEC experiments.

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© Springer-Verlag 2004