# Contraction and Capillary Flow of a Carbon Black Filled Rubber Compound.

INTRODUCTION

Industrially produced rubber compounds consist of one or more synthetic or natural-based polymers (gum), reinforcing fillers (silica, carbon black), and several additives such as plasticizers, processing aids, and a curing system listing only the most important ones. This complex and heterogeneous composition leads to a unique rheological behavior, caused by the interactions of fillers with the polymer and each other: Compared to an unfilled polymer, the strain dependency of several rheological properties (for instance, storage modulus) increases, inducing a highly nonlinear viscoelastic material behavior . These characteristics are often referred to as the "Payne effect" in literature . Studies [3, 4] show that the level of nonlinearity is strongly related to the filler content. Exceeding a percolation threshold, a filler network is formed . A filler network prohibits a distinctive linear viscoelastic region, which corresponds to a constant viscosity at low shear rates ("Newtonian plateau") . Rubber compounds exhibit a shear thinning ("power law") behavior even at shear rates lower than [10-.sup.6] [s-.sup.1] . Consequently, a yield (shear) stress [[tau].sub.0] instead of a Newtonian plateau is observed; [[tau].sub.0] reflects the minimum stress necessary to initiate flow . A schematic representation of the velocity distribution v(r) of a fluid exhibiting yield stress is illustrated in Fig. 1. In dependence of the wall shear stress [[tau].sub.w], a plug flow region with a constant velocity is assumed, while shearing occurs only in regions in which the yield stress is exceeded. A rheological model that is able to describe the flow behavior of such fluids was introduced by Herschel and Bulkley . Like all generalized Newtonian fluid flow models, however, it omits the viscoelastic nature of the polymer matrix.

At high filler loadings, moreover, the Cox-Merz rule  [eta]* ([omega]) = [eta]([??]) is inapplicable. This relationship was proposed first in 1958 and it empirically states that the complex shear viscosity [eta]*, measured in dynamic tests, matches the steady-state shear viscosity [eta], if the angular frequency [omega] equals the steady-state shear rate [??]. This was found to be true for a large variety of unfilled polymers , but it fails for rubber compounds  and most filled polymer systems . Properties obtained from high pressure capillary rheometry (HPCR) always reflect a specific deformation history. In highly filled polymer systems, the deformation history changes the microstructure of the material. The exposure to strain causes a breakdown of the agglomerated filler network. A breakdown of the filler network may be achieved in dynamic testing under fixed preconditions ("preshearing") as well. Fasching  outlined a procedure, where the tested rubber compound is exposed 4 min to preshearing conditions (strain amplitude 42%, frequency 31.42rad/s) before starting the actual dynamic measurement. He found the listed setting leading to a maximum breakdown of the filler network with minimized damage to the macromolecular chains.

No slippage at the wall is a boundary condition (BC) assumed for most rheological test procedures and typically applied in computational fluid dynamic (CFD) simulations. However, fillers and lubricants in a polymer matrix increase the probability of slippage . Consequently, this flow phenomenon was previously observed for several mbber compounds [15, 16]. A popular method to identify slippage at the wall was proposed by Mooney . Applied to capillary flow experiments, Mooney's test varies the die geometry but keeps the length to diameter (L/D) ratio constant. In a second step, the (Bagley corrected) wall shear stress is plotted as a function of the apparent shear rate [[??].sub.a]. Overlapping curves prove according to Mooney no slippage at the wall. Geiger  reported difficulties when quantifying the wall slip velocity of an EPDM rubber compound, as Mooney's approach gave negative values. He improved the original evaluation proposed by Mooney for a slit die rheometer system. Making the wall slip velocity not only dependent on the wall shear stress but on the slit height allowed to quantify plausible wall slip velocity values also for rubber compounds. Furthermore, pressure fluctuations found in the raw data as well as a stick-slip shape of the extruded strands indicate slip flow [19, 20]. Another popular method to identify and quantify wall slip may be applied using rheological tools with different surface textures. Friesenbichler  and Chauffoureaux et al.  proposed geometries designed for slit die rheometer systems, which are able of suppressing slippage at the wall and allow the quantification of a slip velocity for smooth surfaces. This is an approach used in oscillatory and rotational rheometry as well . Kukla et al.  found serrated plates to be most effective suppressing the slip behavior of feedstocks.

In typical CFD simulations, the viscoelastic nature of rubber compounds is omitted. Instead, generalized Newtonian fluid flow models are used to solve the conservation equations of mass, momentum and energy assuming a viscous, isotropic and incompressible fluid behavior. These simplifications may lead to considerable deviations between experiment and simulation . However, a precise numerical description of viscoelastic flow phenomena is fundamentally important for designing extrusion dies, cold and hot runners, as well as complex injection molding tools. For unfilled polymers, various research groups had great success when using the Kaye-Bernstein-Kearsley-Zapas (K-BKZ) model [26-29]. Recently, Mitsoulis et al.  proved the applicability of the K-BKZ model to a wall slipping SBR rubber compound as well.

The hydrogenated acrylonitrile-butadiene rubber (HNBR) rubber compound in this study exhibits no pronounced slippage at the wall, but a shear sensitive plug flow at the centerline. For the first time a viscoelastic integral-type fluid flow model is used to describe and predict this kind of flow behavior. Furthermore, a new approach is presented in using presheared material data for linear viscoelastic modeling. This procedure is particularly important for highly filled polymer systems, where rheological properties are affected by filler-filler and filler-matrix interactions resulting in an invalidity of the Cox-Merz rule.

EXPERIMENTAL

Material

In the present study, a carbon black filled HNBR compound was tested. The level of processing aids and lubricants was kept for this composition at a minimum. Our company partner SKF Sealing Solutions Austria GmbH, Judenburg, Austria provided the rubber compound. It contains of a peroxide based crosslinking system and exhibits a hardness of ShoreA 85. However, the exact composition remains confidential, since it is currently used for industrial applications. At room temperature, the density is 1.22 g/[cm.sup.3]. All presented experiments are carried out in an uncured state.

Oscillatory Rheology

Small amplitude oscillatory shear tests were performed using a Modular Compact Rheometer MCR 501 (equipment manufacturer Anton Paar GmbH, Graz, Austria) in parallel plate configuration. At the reference temperature of 100[degrees]C, the influence of surface geometry (Fig. 2) on the complex viscosity was tested. Any other oscillatory tests were done using the serrated plate geometry (Fig. 2b).

The HNBR rubber compound was roller milled to a sheet, allowing the extraction of samples with a diameter of 25 mm and a thickness of approximately 1.5 mm. In order to gain information on an acceptable strain level, first, tests with constant angular frequency (10.49 rad/s) and step-wise amplitude increase of the sinusoidal strain ("amplitude sweeps") were done. Second, tests with fixed amplitude level and step-wise increase of the angular frequency (0.6 - 628 rad/s) were performed ("frequency sweeps").

High Pressure Capillary Rheology

The steady-state shear viscosity was determined using an HPCR of the type Rheograph 50 (equipment manufacturer GOTTFERT Werkstoff-Prufmaschinen GmbH, Buchen, Germany). Mooney tests were carried out using two capillaries (L/D = 10/1; 20/2) with the same L/D ratio. The entrance pressure loss was detected in both cases using tapered orifice dies (L/D =0.2/1; 0.2/2). Moreover, pressure drops were measured in two additional capillaries (L/D = 5/1 ; 20/1) at the reference temperature of 100[degrees]C. In order to calculate the steady-state shear viscosity, the well-established Bagley and Weibenberg-Rabinowitsch corrections were applied . Using the entrance pressure losses, the steady-state extensional viscosity may be estimated as well. Cogswell , Obendrauf , and Binding  proposed well-established analytical models. Perko et al.  listed numerous assumptions, which may lead to critical errors for each of these models. However, data obtained according to Binding's method correlated best for an SBR rubber compound to experimental data using Sentmanat's  extensional rheometer. Consequently, Binding's model was used to calculate the steady-state extensional viscosity at the reference temperature.

Numerical Setup

All CFD simulations in the present study were performed using the commercial software package POLYFLOW (developed by ANS YS Inc., Canonsburg, PA) in the version 19.2. The geometries including all BC are illustrated in Fig. 3. A two-dimensional axisymmetric finite element model of the tapered orifice die and three abrupt capillaries was built. The governing equations for incompressible fluids under steady-state, isothermal flow conditions may be written as:

[nabla]*[bar.v] = 0 (1)

-[nabla]p + [nabla]*[??] = 0 (2)

where [bar.v] is the velocity vector, p is the pressure, and [??] is the stress tensor. Any influence of inertia and gravity was neglected due to the high viscosity of the HNBR rubber compound. The flow resistance (viscosity [eta]) of purely viscous fluids depends only on the rate of deformation [??], but not on the magnitude of the imposed deformation (strain). The viscous stresses of non-Newtonian, incompressible, and inelastic fluids are given as:

[??] = 2[eta]([??],T,p,c)[??] (3)

There are numerous mathematical models available, which are able to reflect the dependency of the viscosity on the shear rate [??], the temperature T, the pressure level p and even the level of conversion c. In the present study, only shear-rate dependency was considered. Reasons for this decision will be listed in the following chapter. First, the simple power-law model was used:

[eta]([??]) = K[([lambda][??]).sup.n-1] (4)

where K is the consistency index, [lambda] is the natural time (reciprocal of the reference shear rate), and n is the flow exponent. Second, an analytic expression of the viscoplastic Herschel-Bulkley model was selected.

[mathematical expression not reproducible] (5)

where [[tau].sub.0] is the yield stress, [[??].sub.c] is the critical shear rate, K is the consistency index, and n is the flow exponent.

Third, the viscoelastic nature of the tested rubber compound was considered using the integral and time dependent K-BKZ model:

[mathematical expression not reproducible] (6)

with the material constant [theta], the present time t, the past time t', the memory function M(t - t'), the damping function H([mathematical expression not reproducible]), the Finger strain tensor [[??].sup.-1.sub.t](t'), and the Cauchy-Green tensor [[??].sub.t](t'). The material constant [theta] is given by:

[N.sub.2]/[N.sub.1] = [theta]/1-[theta] (7)

where [N.sub.1] and [N.sub.2] are the first and second normal stress differences. Due to difficulties in measuring these characteristics, especially for highly filled polymer systems, the material constant [theta] was set to zero for the HNBR rubber compound.

The memory function describes the linear viscoelastic material behavior. One exponential function is not able to reflect the complex relaxation mechanisms of polymers. Consequently, a series of eight Maxwell elements (prony series) was used (=discretization of continuous relaxation spectrum).

M(t-t') = [N.summation over (i=1)] [g.sub.i]/[[lambda].sub.i]exp(-t-t'/[[lambda].sub.i]) (8)

with i representing the ith mode of the total number of modes N, the relaxation moduli [g.sub.i], and relaxation times [[lambda].sub.i].

Wagner's  damping function reflects the dependency of rheological properties on the magnitude of the imposed deformation (strain):

[mathematical expression not reproducible] (9)

where [mathematical expression not reproducible] and [I.sub.c] are the first invariants of the Finger strain tensor and Cauchy-Green tensor, respectively, [alpha] and [beta] are the material constants.

All BCs used in CFD simulations are listed in Table 1 with corresponding surfaces illustrated in Fig. 3. At the inflow (BC 1), a volumetric flow rate Q was imposed, which can be calculated as:

Q = [pi][D.sup.2.sub.b][v.sub.p]/4 (12)

where [D.sub.b] is the diameter of the HPCR barrel and [v.sub.p] is the set velocity of the piston. The apparent shear rate (no Weissenberg-Rabinowitsch correction) at the capillary wall may be written as:

[[??].sub.a] = 32Q/[pi][D.sup.3] (13)

with the diameter D of the capillary die (smallest diameter in case of orifice die).

At the exit a distinction was made between viscous/ viscoplastic and viscoelastic models. Using viscous and viscoplastic models, a vanishing normal force together with a zero tangential velocity was applied. Dealing with a viscoelastic fluid flow model a zero normal force BC results in nonzero normal stresses and therefore in a nonzero pressure level (dependent on the elastic memory) at the outlet. The K-BKZ/Wagner model is not able to distinguish between filler elasticity and the elastic memory of the polymer matrix. This possible error was minimized by preshearing the material. Nevertheless, the pressure level of the HNBR rubber compound is overestimated considerably at the exit using a zero-force outflow BC (normal and tangential force are set to zero). One possible explanation may be found in the high carbon black loading. Adding carbon black to a polymer matrix decreases the elastic memory of the polymer. Some macromolecular chains are trapped in the agglomerated filler structure and consequently immobilized ("trapped entanglements") , POLYFLOW offers a second outflow BC for viscoelastic flows consisting of a zero tangential velocity together with a fully developed normal velocity profile (same volumetric flow rate as inflow). Additionally, a pressure value of zero was assigned at the node of the exit barrel wall. This BC gave pressure predictions much closer to experimental data. Thus, it was exclusively applied in this study. In order to ensure mesh independency of all presented simulation results, a sensitivity analysis on the mesh quality was performed first. All utilized meshes in this study are listed in Fig. 4.

RESULTS AND DISCUSSION

Experimental Results

A popular hypothesis in literature dealing with slippage of rubber compounds predicts the presence of a thin "slip layer" at the wall. This layer consists of additives that migrated to the surface. Wiegreffe  verified these assumptions observing higher concentrations of zinc stearate at extruded EPDM strands near the surface. Applying Mooney's test, this slippage mechanism results in higher wall shear stresses, when the diameter of the capillary is increased (constant L/D ratio). The influence of the thin slip layer gets less and less important, as the surface area is decreased compared to the volume ratio. An increasing shear stress level should enhance slip behavior at the wall. Mourniac et al.  published corresponding flow curves for an SBR rubber compound.

Flow curves observed for the HNBR rubber compound (Fig. 5) exhibit a very different behavior. At same apparent shear rate levels, the wall shear stress (Bagley corrected) of the L/D =10/1 capillary exceeds the one of the L/D = 20/2 capillary. An influence of geometry is proven, since they do not form a master curve. However, slippage at the wall should result in contrary flow curves. Moreover, the impact of geometry decreases with increasing shear stress level. No deformation (for instance, stick-slip) was observed at the surface of extruded strands even at higher shear rate levels (Fig. 5). Analyzing the measured raw data, no pressure fluctuations were present in any capillary experiments, which would indicate slippage/stick-slip as well. These results indicate no pronounced slippage at the wall but a plug flow region at the centerline. The plug area is enhanced with increasing flow gap. Similar to the schematic representations of the velocity profile illustrated in Fig. 1, an increasing shear stress level suppresses the plug section and consequently the geometrical influence is reduced as well.

A Bagley plot including pressure data of the orifice die and three capillaries is illustrated in Fig. 6. Apparent shear rate levels range from 44 to 7,975 [s.sup.-1] covering the most important shear rate regions of rubber processing (injection molding and extrusion). All analyzed shear rate levels exhibit almost perfect linearity. However, some important conclusions may be drawn analyzing these results.

First, another slip mechanism proposed by Uhland  is ruled out for the HNBR rubber compound. Polymers obeying this slip mechanism exhibit nonlinear pressure profiles along the capillary length. Moreover, pressure and temperature dependency of the viscosity effectively cancel each other out. The dominance of one effect results either in an increase of the pressure level as a function of the capillary length (pressure) or in a decrease (temperature). There is no rheological model available in POLYFLOW reflecting pressure dependency of the viscosity, so the second best option was to consider neither of them. Thus, only shear rate dependency of the viscosity was considered in CFD simulations.

Comparing the complex viscosity to the steady-state shear viscosity proves the inapplicability of the Cox-Merz rule to the tested HNBR rubber compound (Fig. 7). Preshearing the HNBR rubber compound following Fasching's  guideline reduces the gap between steady-state and complex viscosity but fails closing it. The observed deviations between steady-state and dynamic testing primarily prove a sensitivity of rheological properties to the history of deformation. It seems impossible to identify preshearing conditions valid for any material and any steady-state conditions. Therefore, the best option was to further obtain linear viscoelastic properties following Fasching's guideline.

Performing frequency sweeps at 100[degrees]C, the influence of two surface geometries was tested. Comparing the complex viscosity obtained with smooth (Fig. 2a) and serrated (Fig. 2b) parallel plates, no significant influence was observed (Fig. 8). These results are in good agreement with HPCR data indicating no slippage at the wall. Consequently, a no-slip BC was applied in CFD simulations. However, slippage was observed for several other rubber compounds [15, 16, 18, 38, 39]. Thus, the serrated (Fig. 2b) plate geometry was used in further dynamic testing excluding any influence of possible slippage mechanisms, which may occur at other temperatures.

A broad frequency range of storage and loss moduli is of great importance fitting viscoelastic fluid flow models. Thus oscillatory tests were performed over a temperature range of 80 K (40[degrees]C-120[degrees]C). The definition of the dissipation factor tan ([delta]) results in an invariance to vertical shifting. Rubber compounds are complex thermorheological systems, where vertical shifting has to be considered. Therefore, a good option to start time-temperature superposition  is by constructing a master curve for the dissipation factor first.

tan([delta]) = [b.sub.t]G"([a.sub.t][omega])/[b.sub.t]G'([a.sub.t][omega]) (17)

where [b.sub.t] is the vertical and [a.sub.t] is the horizontal shifting factor. As a result, a frequency range of more than five decades ([10.sup.0]-[10.sup.5] rad/s) was covered including the range of deformation rates of the capillary flow experiments (Fig. 9). In order to check plausibility, the obtained horizontal shifting factors were plotted as a function of the inverse temperature. The coefficient of determination ([[R.sup.2)] proves a linear correlation exceeding a value of 99%. In order to construct a master curve for the desired linear viscoelastic moduli, the obtained horizontal shifting factors were maintained at the exact same values. Additionally, vertical shifting was applied minimizing any deviations to storage and loss moduli at the reference temperature.

Exposing polymers to oscillatory deformation, the following relationships are valid:

G'([omega]) = [n.summation over (i=1)][g.sub.i][[omega].sup.2][[lambda].sub.i.sup.2]/1 + [[omega].sub.2][[lambda].sub.i.sup.2] (18)

G"([omega]) = [n.summation over (i=1)][g.sub.i] [omega][[lambda].sub.i]/1+[[omega].sup.2] [[lambda].sub.i.sup.2] (19)

The continuous relaxation spectrum was discretized using eight modes, which is the maximum number available in POLYFLOW. Comparing predictions to measured data (Fig. 10a) validates the obtained model coefficients describing the linear viscoelastic material behavior of the HNBR rubber compound (Table 2). The remaining parameters [alpha] and [beta] were fitted using steady state shear viscosity and extensional viscosity data, respectively. Comparing model predictions to measured data (Fig. 10b), proves that the steady-state shear viscosity is described well. However, the extensional viscosity is overpredicted at lower strain rates and under-predicted at higher ones. Luo and Tanner  proposed a multiple beta ([[beta].sub.i)] approach that is able to reflect the true extensional behavior of polymers. It is not possible to consider multiple betas in POLYFLOW. Thus, the listed [beta] value (Table 2) gives the best fit for the used model in this study.

Dynamic and steady-state rheological data show no Newtonian plateau for the tested HNBR rubber compound. This is a characteristic related to the high filler loading. The observed flow curves (Fig. 5) indicate a shear sensitive plug flow with no pronounced slippage at the wall. Therefore, a viscoplastic model able to reflect these characteristics was used in the study (Herschel-Bulkley). Additionally, the purely viscous power-law model was selected, since it is able to reflect the shear thinning behavior of rubber compounds at low shear rates. A comparison of model predictions to measured data proves the ability of both models to describe the steady-state shear viscosity of the HNBR rubber compound well (Fig. 11).

Numerical Results

Starting CFD simulations of the orifice die a sensitivity analysis on the mesh quality was performed first. The mesh design is based on previous studies [30, 43], who used a grid getting denser moving towards the stress singularity at the entrance of the investigated dies. The first mesh utilized in the sensitivity analysis consists of 2,030 elements (Fig. 4). Aiming to find a mesh giving quick results without impacting the obtained pressure levels, the number of elements were progressively decreased to 936 (Mesh 2). The predicted pressure drops for a set volumetric flow rate of 38 [mm.sup.3/s]/s ([[??].sub.a]=391 [s.sup.-1)] are listed in Table 3. Any obtained deviations are considered to be minimal. Consequently, Mesh 2 was used in further CFD simulations of the orifice die.

Analyzing numerically obtained results, the use of a viscous or viscoplastic fluid flow model leads to a distinctive underestimation of measured pressure values (Fig. 12). This is an outcome expected and in good agreement with published data [26, 27, 30]. The pressure drop in the orifice die is dominated by the contraction flow from the barrel to the short (L = 0.2 mm) die. In the contraction area, the rubber compound is primarily exposed to extensional deformation. Any viscous fluid flow model assumes by definition (Eq. 3) a ratio of three between simple shear and extensional deformation. However, calculated extensional viscosity data (Fig. 10b) indicate a ratio clearly exceeding three, especially at higher deformation rates ([[??].sub.a] = 3,906 and 7,975 [s.sup.-1]). As a result, viscous and viscoplastic models are not expected to give an accurate pressure prediction of contraction flow dominated geometries. Applying the viscoelastic K-BKZAVagner model all measured pressure drops are matched well.

The corresponding velocity profiles reveal the influence of the set viscoelastic outflow BC. At the Position A-A (Fig. 3a), the outlet BC does not affect the viscoelastic flow (Fig. 13a). Thus, the use of the K-BKZAVagner model in POLYFLOW seems to lead in general to a parabolic-shaped velocity profile. However, at the outlet, the velocity profile is similar to the one predicted by the viscous and viscoplastic models (Fig. 13b). Performed capillary rheometer experiments indicate (Fig. 5), a velocity profile similar to this shape. Due to the short length of the orifice die, the whole area crucial for pressure prediction is influenced by the viscoelastic outflow BC. As a result, measured pressure values are matched well. Nevertheless, in CFD simulations, a fluid model is expected to predict the true flow conditions by itself. This is a requirement not fulfilled by the K-BKZAVagner model for the investigated HNBR rubber compound.

CFD simulations of capillary dies were pursued, carrying out a sensitivity analysis on the mesh quality first. Starting with a mesh (Fig. 4) consisting of 2,180 elements (Mesh 3), the number of elements was reduced to 1,250 (Mesh 4) and finally to 450 (Mesh 5). No significant influence on the predicted pressure levels was obtained for any model (Table 3). Thus, Mesh 5 was used in further simulations.

In contrast to the first flow problem, which is dominated by contraction, both the viscous and the viscoplastic models predict pressure drops in the short capillary die with an L/D ratio of 5/1 better (Fig. 14). Only for the two highest apparent shear rates ([[??].sub.a] =3,906 and 7,975 [s.sup.-1)] significant deviations are observed. Comparing predicted and measured values of the orifice die (Fig. 12), these deviations may be assigned to the inability of these two models to predict the true extensional viscosity of the HNBR rubber compound, especially at higher deformation rates. However, measured pressure data are overestimated by the KBKZ/Wagner model. The K-BKZ/Wagner model predicts in general, but contrary to experimental observations, a parabolic-shaped velocity profile. For this geometry, the whole area crucial for pressure prediction (capillary die and contraction region) is not affected by the outflow BC any longer. Consequently, pressure drops are overestimated.

Performing CFD simulations for the second capillary (L/D = 10/1), a mesh (Fig. 4) similar to the one of the shorter die consisting of 600 elements and 666 nodes (Mesh 6) was used. Predicted and measured pressure data are in good agreement with previous results. Both the viscous and viscoplastic models are able to describe and predict the shear dominated capillary flow of the HNBR rubber compound well (Fig. 15). As predictions of the steady-state viscosity (Fig. 11) and observed velocity profiles (Fig. 13) are similar, minor deviations are expected and found to be true for these models. However, the applied viscoelastic fluid flow model overestimates measured pressure drops, since it is unable to reflect the assumed plug-shaped velocity profile.

CFD simulations of the longest die (L/D =20/1) were carried out using Mesh 7 (Fig. 4) consisting of 600 elements and 666 nodes. The influence of the contraction flow to the overall pressure drop is minimized for this geometry. As a result both, the viscous and the viscoplastic models match measured data almost perfectly (Fig. 16). Comparing pressure predictions of the K-BKZAVagner model to experimental data proves that mean deviations do not stay at a similar level for the three different capillary flow geometries (L/D = 5/10/20). These results indicate that temperature and pressure dependency of the viscosity do not completely cancel each other out. For the shorter die (L/D = 5), the temperature dependency is dominant. Thus pressure drops are overestimated by isothermal simulations. For the longest die (L/D = 20), the pressure dependency is dominant, causing an underestimated pressure drop. This effect is missing when comparing viscous/ viscoplastic CFD simulation results. The contraction flow, which is not correctly predicted by these models, impacts each geometry differently.

All velocity profiles observed in capillary flow (Fig. 17) are in good agreement with previously discussed data (Fig. 13). Both the viscous and the viscoplastic models predict the assumed plug-shaped velocity profile. However, applying the K-BKZ/Wagner model leads to a parabolic-shaped velocity profile, resulting in an overestimation of the predicted pressure. The set viscoelastic outflow BC affects only areas near the outlet, but imposes a plug flow similar to the one predicted by the viscous and viscoplastic models. Nevertheless, with increasing length of the die, the effect of the BC to the overall pressure prediction decreases. As a result, it provided only a valid workaround solution for the short orifice die.

Critically reviewing the applicability of the K-BKZ/Wagner model to highly filled rubber compounds, three key areas for improvement were found:

1. Need to distinguish between filler elasticity and elastic memory of the polymer matrix.

2. Need to take filler-polymer-related mechanisms into account (trapped entanglements).

3. Need to reflect plug-shaped velocity profiles.

A differential viscoelastic fluid flow model that is able to consider Points 1 and 2 was proposed first by Leonov  and modified by his PhD student Simhambhatla . Testing the Leonov-Simhambhatla model exceeds the scope of this study. Thus, the authors cannot comment on its applicability in CFD simulations.

CONCLUSIONS

The pressure driven contraction and capillary flow of a carbon black filled HNBR rubber compound was studied both experimentally and numerically. Rheological testing indicates no pronounced slippage at the wall but a shear sensitive plug flow at the centerline. As rheological properties of rubber compounds are affected by filler-filler and filler-matrix interactions, the influence of the deformation history to linear viscoelastic modeling and to the invalidity of the Cox-Merz rule is investigated and discussed. As a result, this article provides a guideline on how to determine and to best fit linear and nonlinear viscoelastic data with the K-BKZ/Wagner model for rubber compounds. Carrying out CFD simulations, only the K-BKZ/ Wagner model was able to predict pressure drops in a contraction flow dominated orifice die correctly. The pressure drops in all tested capillaries were overestimated, since the K-BKZ/ Wagner model fails to reflect the assumed plug-shaped velocity profile. However, the tested viscous (power law) and viscoplastic models (Herschel-Bulkley) were able to predict pressure drops in shear dominated regions well.

ACKNOWLEDGMENTS

This research work was supported by the Austrian Research Promotion Agency (FFG) as part of the "RubExject H" project (corresponding project number 855873) and the company partners SKF Sealing Solutions Austria GmbH, Judenburg, Austria; IB Steiner, Spielberg, Austria; and ELMET Elastomere Produktions- und Dienstleistungs-GmbH, Offering, Austria. The authors thank DI Dr. Ivica Duretek, DI Stephan Schuschnigg, and DI Dr. Matthias Haselmann for their respective contributions.

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Sebastian Stieger (iD), (1) Roman Christopher Kerschbaumer, (2) Evan Mitsoulis, (3) Michael Fasching, (4) Gerald Roman Berger-Weber, (1) Walter Friesenbichler, (1) Joachim Sunder (5)

(1.) Institute of Injection Moulding of Polymers, Montanuniversitaet Leoben, Leoben, 8700, Austria

(2.) Polymer Competence Center Leoben GmbH, Leoben, 8700, Austria

(3.) School of Mining Engineering and Metallurgy, National Technical University of Athens, Zografou, 157 80, Greece

(4.) SKF Sealing Solutions Austria GmbH, Judenburg, 8750, Austria

(5.) GOTTFERT Werkstoff-Prufmaschinen GmbH, Buchen, 74722, Germany

Correspondence to: S. Stieger; e-mail: sebastian.stieger@unileoben.ac.at Fractions of this work were presented at the Polymer Processing Society (PPS) regional conference in Boston, MA, November 2018 and at the PPS-35 International Conference in Cesme, Turkey, May 2019. Contract grant sponsor: ELMET Elastomere Produktions- und Dienstleistungs-GmbH. contract grant sponsor: IB Steiner. contract grant sponsor: SKF Sealing Solutions Austria GmbH. contract grant sponsor: Austrian Research Promotion Agency; contract grant number: 855873. DOI 10.1002/pen.25256

Published online in Wiley Online Library (wileyonlinelibrary.com).

Caption: FIG. 1. Velocity distribution v(r) of a fluid exhibiting yield stress [[tau].sub.0] at low (a) and high (b) wall shear stress levels.

Caption: FIG. 2. Smooth (a) and serrated (b) plate surfaces used in dynamic testing.

Caption: FIG. 3. CFD simulation setup of the orifice die (a) and capillaries (b).

Caption: FIG. 4. Meshes utilized in sensitivity analysis and final calculations.

Caption: FIG. 5. Flow curves including light microscope images of extruded strands at three apparent shear rate levels (44; 391 ; and 3,906 [s.sup.-1]).

Caption: FIG. 6. Bagley plot including linear approximations.

Caption: FIG. 7. Influence of preshearing conditions on the complex viscosity and comparison to HPCR data.

Caption: FIG. 8. Influence of plate surface geometry on the complex viscosil

Caption: FIG. 9. Time-temperature superposition of the phase angle o and corresponding Arrhenius plot.

Caption: FIG. 10. Experimental data and model (K-BKZ) predictions of linear viscoelastic moduli (a) and steady-state viscosities (b).

Caption: FIG. 11. Steady-state shear viscosity fitted with the power law and Herschel-Bulkley models.

Caption: FIG. 12. Experimental data and simulation results of the orifice die.

Caption: FIG. 13. Velocity profiles at the Positions A-A (a) and B-B (b) at a set volume flow rate of 38 [mm.sup.3]/s([[??].sub.a]=391 [s.sup.-1]).

Caption: FIG. 14. Experimental data and simulation results of the capillary L/D = 5/1.

Caption: FIG. 15. Experimental data and simulation results of the capillary L/D = 10/1.

Caption: FIG. 16. Experimental data and simulation results of the capillary L/D = 20/1.

Caption: FIG. 17. Velocity profiles at the Positions C-C (a) and D-D (b) at a set volume flow rate of 38 [mm.sup.3/s] ([[??].sub.a] = 391 [[s.sup.-1)].
```TABLE 1. BCs applied in CFD simulations.

BC                       Description

BC 1    Fully developed normal velocity profile
(inflow)

BC 2    Normal ([v.sub.n]) and tangential velocity
([v.sub.s]) = 0 (wall) (viscous/
viscoplastic simulations)

BC 2    Normal ([v.sub.n]) and tangential velocity
([v.sub.s]) = 0; pressure p - 0 at exit
wall node (viscoelastic simulations)

BC 3    Tangential force ([f.sub.s]) and normal
velocity ([v.sub.n]) = 0 (axis of
symmetry)

BC 4    Normal force ([f.sub.n]) and tangential
velocity ([v.sub.s]) = 0 (outflow)
(viscous/viscoplastic simulations)

BC 4    Fully developed normal velocity profile
(outflow) (viscoelastic simulations)

TABLE 2. Relaxation spectrum and material constants for the HNBR
rubber compound obeying the K-BKZAVagner model at the reference
temperature of 100[degrees]C ([theta] = 0).

Modes    [[lambda].sub.i]   [g.sub.i]   [alpha]   [beta]
(s)            (Pa)        (l)      (l)

1            1.00E-05       2.70E+06     0.49      0.4
2            7.20E-05       1.30E+06
3            5.18E-04       1.25E+06
4            3.73E-03       8.25E+05
5            2.68E-02       5.16E+05
6            1.93E-01       3.60E+05
7            1.39E+00       1.02E+05
8            1.00E+01       9.76E+04

TABLE 3. Predicted pressure drops at a set volume flow rate of 38
[mm.sup.3]/s ([[??].sub.[alpha]] = 391 [s.sup.-1]).

Mesh    Elements   Nodes   K-BKZ/Wagner   Power-law   Herschel(MPa)
(MPa)      Bulkley
(MPa)

1        2,030     2,244       9.65         5.20        3.90
2         936      1,099       9.65         5.21        3.92
3        2,180     2,314      29.70         20.88       21.19
4        1,250     1,341      29.83         20.88       21.19
5         450       501       29.74         20.88       21.19
```
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