Introduction

Polymer melt extrusion is at the core of plastics processing. Extruders are used for compounding as well as for modifying and/or blending polymers, often by reactive extrusion. Extruders are also used to convert compounds into sheets, films, pipes and tubing, profiles, etc. Melt extrusion requires specific melt viscoelastic properties to ensure good quality extrudates, with smooth surface and proper dimensions, at high production rates. The rheological characterization and modelling of polymer melts at large shear rates in the nonlinear viscoelastic regime is thus of high practical relevance, namely for assessing their processability and predicting extrusion instabilities (see, e.g., Dealy 2005; Cyriac et al. 2013).

Rotational rheometry is a well-established technique for measuring the steady-state rheological functions of interest, which are essentially shear viscosity and normal stress differences. However, at high shear rates, secondary flow instabilities develop due to large first and second normal stress differences which drive surface distortion (Kulicke et al. 1979) and edge fracture (Tanner and Keentok 1989), respectively. The cone-partitioned plate (CPP) geometry has been introduced to mitigate the effects of these instabilities (Meissner et al. 1989; Schweizer 2002; Snijkers and Vlassopoulos 2011), but the large torques generated at very high shear rates reach the limits of torque transducers of commercial rotational rheometers. Thus, in practice, measurements at shear rates exceeding 100 s−1 are challenging for highly elastic melts (Schweizer 2002, 2003; Snijkers and Vlassopoulos 2011; Costanzo et al. 2018; You and Yu 2021) such as extrusion grade polymers.

The hole-pressure method (Broadbent et al. 1968) is an alternative route to assess both shear viscosity and normal stresses and can be used in in-process rheometers. The concept is illustrated in Fig. 1 for the specific case of slit rheometry. For each flow rate in the slit with height H, the shear viscosity η is inferred from the difference between the pressures P3 and P1 measured by two flush-mounted transducers on the slit, whereas normal stresses are acquired through the difference between P1 and the pressure P2, which is measured by a transducer mounted in the recess opposite to P1. A rectangular recess with depth d and width b gives access to the first normal stress difference N1, whereas the second normal stress difference N2 is measured with a circular recess, using the relationships between P3, P1, and P2 given by the Higashitani and Pritchard (1972) and Baird (1975) equations. With the correct choice of the slit and recess geometries, both shear viscosity and first normal stress difference could be accurately measured for commercial polyethylene and polystyrene melts at shear rates in the range of 10 to 300 s−1 (Teixeira et al. 2013, 2015a). However, N2 measurements showed up to be practically unfeasible for these melts (Teixeira et al. 2013), whereas less elastic materials such as liquid crystalline polymers in the nematic phase (Teixeira et al. 2015a) do not generate sufficient hole-pressure at P2 to give a measureable N1.

Fig. 1
figure 1

Schematic representation of the hole-pressure method (a) and design of slit dies for the simultaneous measurement of η, N1 (b, with rectangular recess) or N2 (c, with circular recess (4)). The thick black arrow in (b) and (c) indicates the flow of material from the extruder outlet to the slit die inlet, whereas optical windows (5) and aperture for optical measurements (6) are indicated in (c)

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With the conventional slit die design displayed in Fig. 1, only a single point of the flow curve (η or N1 vs. shear rate) is obtained at a specific operating condition (Baird 2008). The screw speed (single screw extruder) or the feed rate (twin screw extruder) must be changed to generate different shear rates in the measuring slit. Simultaneously, the residence time and stresses experienced by the material in the extruder will be also altered. In the case of multiphase materials such as polymer blends (Teixeira et al. 2015b), polymer nanocomposites, and thermo-mechanically sensitive materials such as bio-based and biodegradable polymers, this will cause changes in the material characteristics, i.e., a different material may be pumped to the measuring slit at each different shear rate. To bypass this problem, a double-slit design has been developed, whereby the melt exiting the extruder enters a measuring channel and a production channel (see Fig. 2). The flow in each channel can be regulated by a vertical valve at its entrance. Hence, by manipulating simultaneously the two valves in opposite directions, different flow rates can be generated in the two channels while maintaining the same operating conditions in the extruder (Teixeira et al. 2018).

Fig. 2
figure 2

Schematic representation of the double-slit rheological die (left) and photo of the actual die coupled to a prototype mini-scale twin-screw co-rotating extruder (right) taken during the compounding of a carbon-based polymer nanocomposite (Infurna et al. 2020). P0 indicates the location of the pressure transducer used to monitor the pressure at the output of the extruder, which is kept constant by balancing the flows in the measuring (A) and the production (B) channels using vertical valves located at both channels entrances. The measuring channel (A) reproduces the design displayed in Fig. 1 for the hole-pressure method with optical access for light scattering, microscopy, or turbidity measurement

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It is worth noting that the set-up illustrated in Fig. 2 is unique in establishing relationships between extrusion conditions and flow curves. In short, the advantage of the double-slit die over the single-slit rheometer is that it can produce a flow curve without changing the extrusion conditions (screw speed or feed rate) to generate different shear rates in the slit. Indeed, when using off-line rotational rheometry to characterize the extrudates, the extra thermal cycles associated to disk preparation of the samples and loading in the rheometer might change the morphology of the material due to possible coalescence of the dispersed phase, degradation, or chemical reactions between the ingredients (Teixeira et al. 2020, 2021).

The in-line rheometers presented in Figs. 1 and 2 were validated by a direct comparison with the viscosity flow curves measured by off-line capillary rheometry performed on rheologically simple systems such as industrial polystyrene and polyethylene melts (Teixeira et al. 2013, 2018). However, these materials were too elastic to allow for normal stress measurement with rotational rheometry at shear rates overlapping those achieved with the in-line slit or double-slit rheometers. So far, the confidence in the validity of the measured N1 data at high shear rates could only be based on comparison to Laun’s empirical relationship with the frequency dependence of the storage modulus G′(ω) (Laun 1986). In this paper, a comparison of the experimental data obtained by off-line (rotational) rheometers in the low shear-rate regime and by the in-line rheometers in the high shear-rate regime with predictions of tube models for entangled polymer melts, the Doi-Edwards model (Doi and Edwards 1978), and the Hierarchical Multi-mode Molecular Stress Function (HMMSF) model (Narimissa et al. 2015, 2016; Narimissa and Wagner 2016a, b, c, 2018, 2019) is presented. The HMMSF is based on a few well-defined rheological assumptions and has been shown to model accurately the rheology of polydisperse linear and branched polymer melts in extensional and shear flows with only one or two nonlinear material parameters. However, the comparison of experimental data and HMMSF model in shear flow was so far restricted to linear (HDPE) and long-chain branched (LDPE) polyethylenes at shear rates up to 10 s−1 (Narimissa and Wagner 2016a, c).

The objective of this paper is thus twofold: (i) To compare the rotational, slit, and double-slit rheometer data with the predictions of the Doi-Edwards and the HMMSF model, and (ii) to test the model predictions for shear viscosity and normal stresses with experimental data of commercial polymers processed under industrial (lab-scale) conditions at shear rates up to 3000 s−1 and 300 s−1, respectively.

Materials and methods

Materials

For experiments with the set-up displayed in Fig. 1, a polystyrene (PS) Edistir N2560 from Polimeri Europa and a low-density polyethylene (LDPE) Lupolen 1840H from BASF were used. A polystyrene (PS) RXP 3002 Natural from Resinex was used to validate the double-slit in-line rheometer presented in Fig. 2. This commercial plastic has a melt flow index of 11 g/10 min (200 °C/5.0 kg).

Rheometry

Detail about the calibration of the pressure transducers and their mounting on the measuring slit was given elsewhere (Teixeira et al. 2013). The slit rheometer displayed in Fig. 1 was coupled to a capillary rheometer (Rosand RH10) in view to feed the polymer melts into the measuring channel under steady flow conditions. The method for producing flow curves with the slit rheometer coupled to the capillary rheometer has been detailed elsewhere (Teixeira et al. 2013). The Rosand RH10 capillary rheometer was also used to produce the shear viscosity data for PS Edistir N2560 at 190 °C and at shear rates overlapping and exceeding those achieved with the double-slit rheometer.

In-extruder experiments with the double-slit rheometer were conducted at constant extrusion conditions (screw speed or feed rate), each point in the flow curve being obtained with a given combination of the positions of the valves used to set the flow rates in the corresponding channels. The experimental set-up and methodology to produce flow curves were described in detail elsewhere (Teixeira et al. 2018).

Rotational rheometry was performed with a stress-controlled ARG2 rheometer (TA instruments) equipped with a 25-mm plate-plate geometry, and with two Advanced Rheometric Expansion System rheometers (ARES, TA Instruments) coupled to a cone-and-plate geometry (25-mm diameter and 0.1 rad cone angle) or a 25-mm plate-plate geometry. These were used in order to produce the linear viscoelastic data needed as input for the HMMSF model, and to compare η, N1, and N1 – N2 values with model predictions at lower shear rates of the flow curves. Additional flow curves were obtained from two Advanced Rheometric Expansion System rheometers (ARES, TA Instruments) in two different laboratories (Polymeric Materials Group at the Karlsruhe Institute of Technology, Germany, indicated as “KIT” in the figures, and Applied Rheology and Processing Section at the Katholieke University of Leuven, Belgium, indicated as “KUL”). For experimental details, see Teixeira et al. (2013, 2018).

Modelling

The Hierarchical Multi-Mode Molecular Stress Function (HMMSF) model was developed by Narimissa and Wagner for the prediction of uniaxial (Narimissa et al. 2015), multiaxial (Narimissa et al. 2016), and shear (Narimissa and Wagner 2016a) rheological behaviours of polydisperse long-chain branched (LCB) polymer melts as well as polydisperse linear melts (Narimissa and Wagner 2016b, c). A summary of development of the HMMSF model was presented by Narimissa and Wagner (2018), a comparison between this model and other prominent tube-models for polydisperse linear and long-chain branched polymer melts was given by Narimissa and Wagner (2019), and a recent extension for elongational flow of LCB polymers by Wagner et al. (2022).

We shortly summarize the basic equations of the HMMSF model:

The extra stress tensor of the HMMSF is given by,

$$\sigma \left(t\right)=\sum_{i}\underset{-\infty }{\overset{+\infty }{\int }}\frac{\partial {G}_{i}\left(t-{t}^{^{\prime}}\right)}{{\partial t}^{^{\prime}}}{f}_{i}^{2}\left(t,{t}^{^{\prime}}\right){\mathbf{S}}_{DE}^{IA}\left(t,{t}^{^{\prime}}\right){dt}^{^{\prime}}$$
(1)

The extra stress is a sum over all stress contributions from discrete Maxwell modes making up the relaxation modulus,

$$G\left(t\right)=\sum_{i}{G}_{i}\left(t\right)=\sum_{i}{\mathrm{g}}_{i}\mathrm{exp}\left(t/{\tau }_{i}\right)$$
(2)

with partial relaxation moduli \({\mathrm{g}}_{i}\) and relaxation times \({\tau }_{i}\).

\({\mathbf{S}}_{DE}^{IA}\) is the Doi and Edwards orientation tensor with the independent alignment (IA) assumption (Doi and Edwards 1978), which is equal to 5 times the second-order orientation tensor S.

The molecular stress functions \({f}_{i}={f}_{i}\left(t,{t}^{^{\prime}}\right)\) are functions of observation time t and the time of creation of tube segments by reptation at time t′, and are obtained by integration of the evolution equations,

$$\frac{{\partial f}_{i}}{\partial t}={f}_{i}\left(\mathbf{K};\mathbf{S}\right)-\frac{1}{\alpha }\left(\frac{1}{{\tau }_{i}}+\beta CR\right)\left[\left({f}_{i}-1\right)\left(1-\frac{2}{3}{w}_{i}^{2}\right)+\frac{2}{9}{f}_{i}^{2}\left({f}_{i}^{3}-1\right){w}_{i}^{2}\right]$$
(3)

K is the velocity gradient tensor and \(\alpha\) a topological parameter which depends on the architecture of the chain,

$$\begin{array}{c}\begin{array}{cc}\alpha=1&\mathrm{for}\;\mathrm{LCB}\;\mathrm{melts}\end{array}\\\begin{array}{cc}\alpha=1/3&\mathrm{for}\;\mathrm{Linear}\;\mathrm{melts}\end{array}\end{array}$$
(4)

CR is a dissipative Constraint Release term which has nonzero values only in shear flow (CR = 0 in extensional flow) and can be expressed in terms of shear rate \(\dot{\gamma }\) and the normal components of the orientation tensor S as \(\mathrm{CR}=\frac{1}{2}\sqrt{\left.{\dot{\gamma }}^{2}\right|{S}_{11}}-{S}_{22}\). β is a numerical parameter which is determined experimentally.

The mass fractions \({w}_{i}\) of chain segments in the evolution Eq. (3) take into account hierarchical relaxation and are determined by,

$$\begin{array}{c}\begin{array}{cc}{w}_{i}^{2}=\frac{G\left(t={\tau }_{i}\right)}{{G}_{D}}=\frac{1}{{G}_{D}}\sum_{j=1}^{n}{\mathrm{g}}_{j}\mathrm{exp}\left(-{\tau }_{i}/{\tau }_{j}\right)& \mathrm{for} {\tau }_{i}>{\tau }_{D}\end{array}\\ \begin{array}{cc}{w}_{i}^{2}=1& \mathrm{for }{\tau }_{i}\le {\tau }_{D}\end{array}\end{array};$$
(5)

The mass fractions \(w_{i}\) of dynamically diluted segments with relaxation time \(\tau_{i}\) longer than the dilution relaxation time \({\tau }_{D}\) are smaller than 1, while the mass fractions of permanently diluted segments with \({\tau }_{i}\le {\tau }_{D}\) are set equal to 1. \({G}_{D}\) is called the “dilution modulus” and is a free parameter of the model.

The HMMSF model requires only a single fitting parameter for extensional flow (i.e., \(G_{D}\)) and an additional CR parameter (β) for shear flow.

The Doi-Edwards model with the independent alignment assumption (DEIA) is recovered from Eq. (1) by setting \({f}_{i}^{2}\equiv 1\) (Doi and Edwards 1978). We will also test a combination of the HMMSF model and the DEIA model by use of Eq. (1), but assuming that polymer chain segments with relaxation times \({\tau }_{i}\) below a certain threshold value \({\tau }_{c}\) are so short that they are only oriented, but not stretched, i.e., that

$$\begin{array}{cc}{f}_{i}^{2}\equiv 1& if{\tau }_{i}<{\tau }_{c}\end{array}$$
(6)

We call this the “HMMSF-IA” model in the following.

Results and discussion

Linear-viscoelastic characterization

From small amplitude oscillatory shear (SAOS) experiments (Fig. 3), the relaxation spectra of PS Edistir N2560 at 190 °C, PS RXP 3002 at 210 °C, and LDPE 1840H at 150 °C were determined by the IRIS software (Winter and Mours 2006; Poh et al. 2022), and are summarized in Table 1.

Fig. 3
figure 3

Storage (G′) and loss (G″) modulus of (a) PS Edistir N2560 at 190 °C, (b) PS RXP 3002 at 210 °C, and (c) LDPE 1840H at 150 °C. Lines are fit by parsimonious spectra of Table 1

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Table 1 Relaxation spectra of PS Edistir N2560 at 190 °C, PS RXP 3002 at 210 °C, and LDPE 1840H at 150 °C
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Shear viscosity

The shear viscosity of the two polystyrene melts and the low-density polyethylene melt measured by rotational, capillary, and slit die rheometry up to shear rates of 1000–3000 s−1 are presented by symbols in Figs. 4 and 5. As far as the shear rate regimes of the different measurement methods overlap, general agreement of the experimental data was obtained. Predictions of the DEIA and the HMMSF models are indicated in the figures by lines. For the two polystyrene melts (Fig. 4), good agreement between experimental data and predictions of the DEIA model is found. This is consistent with a recent analysis of the steady-state shear viscosity of several monodisperse PS melts (Narimissa et al. 2020). Concerning the HMMSF model, we use a dilution modulus of \({G}_{D}=\mathrm{7,000}Pa\) for both polystyrenes, which has been shown earlier (Narimissa and Wagner 2016c) to give excellent agreement with uniaxial and equibiaxial extensional flow data of a commercial PS melt (PS158K of BASF AG). We recall that the dissipative Constraint Release (CR) for shear flow of linear polymers such as polystyrene is large and use a CR parameter of β = 1 here. Predictions of the HMMSF model are therefore only slightly larger than those of the DEIA model, and are within experimental accuracy in good agreement with the experimental data.

Fig. 4
figure 4

Viscosity versus shear rate for (a) PS Edistir N2560 and (b) PS RXP 3002

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Fig. 5
figure 5

Viscosity versus shear rate for LDPE 1840H. Lines indicate (a) predictions of the DEIA and HMMSF models; (b) predictions of the DEIA and HMMSF-IA models

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Figure 5 presents the shear viscosity of LDPE 1840H measured by rotational, capillary, and slit die rheometry up to shear rates nearly 2000s−1. Considering the wide range of shear rates investigated, good agreement of the different experimental methods is found. In the shear rate range between 10−2 and 1 s−1, the DEIA model is slightly underpredicting the experimental data, while the high shear-rate predictions are in agreement with the measurements by capillary and slit die rheometry. On the other hand, the HMMSF model with a dilution modulus of \({G}_{D}=\mathrm{10,000}Pa\) and a CR parameter of β = 0.5 results in good shear viscosity predictions within experimental accuracy in the full shear-rate range investigated (Fig. 5a). A dilution modulus of \({G}_{D}=\mathrm{10,000}Pa\) has already been shown earlier to result in an excellent description of the elongational viscosity for LDPE 1840H [Wagner 2022]. Predictions of the HMMSF-IA model, which becomes important in modelling the normal stress difference data (see below), are shown in Fig. 5b. Using a cut-off relaxation time of \({\tau }_{c}={2s}^{-1}\) for polymer chain stretch according to Eq. (6), agreement with the experimental data in the low shear-rate range between 10−2 and 1 s−1, and convergence to the DEIA model predictions at high shear rates is obtained.

Normal stress differences

Figure 6 shows the normal stress differences of PS Edistir N2560 measured by rotational and slit die rheometers. While rotational rheometry is limited to shear rates up to 1 s−1, the slit die rheometers cover the shear-rate range from 10 to 300 s−1. As in the case of shear viscosity, good agreement between experimental data of the first normal stress difference and computed N1 data from the DEIA model is found in the full range of shear rates investigated (Fig. 6a), and again this is consistent with the recent analysis of the steady-state first normal stress difference of several monodisperse PS melts (Narimissa et al. 2020). The HMMSF model with a dilution modulus of \(G_{D} = 7,000Pa\) and a CR parameter of \(\beta = 1\) are slightly larger than predictions of the DEIA model for N1, but are clearly within experimental accuracy in good agreement with the experimental data.

Fig. 6
figure 6

Normal stress differences versus shear rate for PS Edistir N2560. (a) N1; (b) \({N}_{1}-{N}_{2}\)

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Concerning the difference of the first and second normal stress differences (Fig. 6b), while predictions of both DEIA and HMMSF agree with the rotational measurements, the experimental data of \({N}_{1}-{N}_{2}\) obtained by the slit-die circular slot technique at high shear rates are increasingly lower than predicted by the models. This points towards the difficulty in accurately measuring N2 with the hole-pressure method, as established elsewhere (Teixeira et al. 2013).

The normal stress differences of PS PXP 3002 measured by rotational rheometers and the double-slit die rheometer are shown in Fig. 7. While agreement with predictions of the DEIA and HMMSF models is seen for the low shear rate rotational measurements of \({N}_{1}-{N}_{2}\), the double-slit die measurements give only a very rough estimation of N1.

Fig. 7
figure 7

Normal stress differences N1 and \({N}_{1}-{N}_{2}\) versus shear rate for PS RXP 3002

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The normal stress differences of LDPE 1840H measured by rotational and slit die rheometers are presented in Fig. 8. While rotational rheometry is limited to shear rates up to 10 s−1, the slit die rheometers cover the shear rate range from 10 to 300 s−1. The DEIA model clearly underpredicts the low shear rate range, but reasonable agreement between N1 data and prediction in the range of high shear rates accessible by the slit-die rheometer is achieved (Fig. 8a). It may come as a surprise that the DEIA model gives not only a quantitative prediction of the N1 data of the linear PS melts, but also of the N1 data of the long-chain branched LDPE 1840D at high shear rates. However, this is in line with earlier findings of Ianniruberto and Marrucci (2013) that due to arm withdrawal, entangled melts of branched PS behave like linear PS in the steady state of fast elongational flows. The effect of arm withdrawal is most likely also the root cause of the agreement of the N1 data of LDPE 1840H and the prediction of the DEIA model at high shear rates. We further note that as in the case of the polystyrene melt, the experimental data of \({N}_{1}-{N}_{2}\) obtained by the slit-die circular slot technique are increasingly lower than predicted by the DEIA model (Fig. 8b).

Fig. 8
figure 8

Normal stress differences N1 and \({N}_{1}-{N}_{2}\) versus shear rate for LDPE 1840H. Lines indicate (a) and (b) predictions of the DEIA and HMMSF models; (c) and (d) predictions of the DEIA and HMMSF-IA models

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In contrast to the DEIA model, the HMMSF model with a dilution modulus of \(G_{D} = 10,000Pa\) and a dissipative CR parameter of β = 0.05 results in good agreement with experimental N1 (Fig. 8a) and \({N}_{1}-{N}_{2}\) data (Fig. 8b) in the low shear rate range up to a shear rate of about 1 s−1, while a considerable deviation at high shear rates is obvious. This deviation between model and data can be mitigated by assuming that polymer chain segments with relaxation times \({\tau }_{i}\) below a threshold value of \({\tau }_{c}=2s\) are so short that they are only oriented, but not stretched, as in the HMMSF-IA model, which is a cross-over between the HMMSF and the DEIA model. Note that the value of \({\tau }_{c}=2s\) corresponds to a shear rate of \({\dot{\gamma }}_{c}={0.5s}^{-1}\), which is approximately the shear rate when the slope of the normal stress curves starts to decrease. Indeed, this assumption leads to a transition between the low shear-rate HMMSF and the high shear-rate DEIA prediction. While the HMMSF-IA model results in an improved description of the experimental N1 data in the full range of shear rates investigated (Fig. 8c), the deviations between model and the \({N}_{1}-{N}_{2}\) data persist in the high shear-rate range (Fig. 8d).

Conclusions

The comparison of the experimental data obtained by off-line (rotational) rheometers in the low shear-rate regime and by the in-line rheometers in the high shear-rate regime with predictions of tube-based models for entangled polymer melts leads to a number of interesting conclusions:

For polystyrene, the DEIA and the HMMSF models give accurate descriptions of shear viscosity and first normal stress difference N1 for shear rates up to 3000 s−1 and 300 s−1, respectively. This is an important result, which on the one side validates the experimental data obtained by rotational as well as slit-die rheometry, but also demonstrates the agreement between experiments and models up to very high shear rates, which were not experimentally accessible earlier. It is equally important that modeling works for the industrial polymers investigated here. Due to the high dissipative Constraint Release in shear flow of linear polymers, predictions of the HMMSF model are only marginally larger than predictions of the DEIA model and are in good agreement with the experimental data within experimental accuracy. However, we note that the HMMSF model with the same parameter of the dilution modulus for polystyrene as used here has been shown earlier (Narimissa and Wagner 2016c) to give excellent agreement with uniaxial and equibiaxial extensional flow data, while the DEIA model fails utterly in extensional flows. Therefore, the HMMSF model can be used in 3D simulations of complex flows in polymer processing combining both extensional and shear flows such as in the case of abrupt contraction flow (Olley et al. 2022).

In the case of \({N}_{1}-{N}_{2}\), the agreement between experimental data for polystyrene and predictions of the DEIA and HMMSF models is restricted to data obtained by rotational rheometry at shear rates up to 1 s−1, while the slit-die circular slot rheometry shows larger deviations from the predictions of the models.

For long-chain branched polymer melts such as the LDPE investigated here, which are known to feature strong strain hardening in extensional flows, the DEIA model shows a small underprediction of the shear viscosity at low shear rates as well as a strong underprediction of the first normal stress difference, while the HMMSF model with suitable choices of the parameters for dilution modulus and Constraint Release results in a good fit of shear viscosity and N1 data. In contrast, the DEIA model describes an upper limit of the first normal stress difference data in the high shear-rate regime, which are largely overpredicted by the HMMSF model. We explain this by the effect of arm withdrawal at high deformation rates as reported by Ianniruberto and Marrucci (2013) for entangled melts of branched PS, which behave like linear PS in the steady state of fast flows. The HMMSF model does not take into account the effect of arm retraction. We therefore propose a combination of the HMMSF and the DEIA model by assuming that polymer chain segments with relaxation times \(\tau_{i}\) below a threshold value of \({\tau }_{c}=2s\) are so short that they are only oriented, but not stretched. This HMMSF-IA model leads to a transition between the low shear-rate HMMSF and the high shear rate DEIA prediction and results in an improved description of the experimental shear viscosity and N1 data in the full range of the shear rates investigated. However, further validation of the HMMSF-IA model will be required. We also note that deviations exist between all three models and the \({N}_{1}-{N}_{2}\) data obtained by the slit-die circular slot technique in the high shear-rate range. These deviations are associated with the experimental difficulty in assessing small N2 values with the hole-pressure method.