Rheologica Acta

, Volume 46, Issue 6, pp 877–888

Instability of entangled polymers in cone and plate rheometry

Authors

  • Changping Sui
    • Department of Chemical EngineeringTexas Tech University
    • Department of Chemical EngineeringTexas Tech University
Original Contribution

DOI: 10.1007/s00397-007-0169-8

Cite this article as:
Sui, C. & McKenna, G.B. Rheol Acta (2007) 46: 877. doi:10.1007/s00397-007-0169-8

Abstract

Flow instability in three entangled polymer systems including a 10 wt% 1,4-polybutadiene (PBD) solution, an 11.4 wt% polyisobutylene (PIB) solution, and a long chain branched polyethylene melt (LD 146) was investigated in both stress-controlled and rate-controlled experiments in the cone–plate geometry. It was found that flow instability occurred for experiments in both rate- and stress-controlled modes. The effects of cone angle or rim gap and shearing time on flow instability were studied. The smaller cone angle and shorter shearing time delay (in terms of stress or shear rate) the occurrence of severe instability and mass loss of the PBD solution but not for the PIB. Our data are consistent with the dramatic shear rate jump for the flow curve constructed from the stress-controlled experiments being associated with mass loss after the severe instabilities. We also find that the Cox–Merz representation gives a powerful tool for investigation of flow instability. Finally, another interesting result in this work is that it seems that the stress overshoot can be related to the onset of flow instability in the present system.

Keywords

Flow instabilityEdge fractureMass lossStress controlledRate controlledStress overshootCox–Merz rule

Introduction

“Flow anomalies” have been reported by Wang’s group recently (Tapadia and Wang 2003, 2004, 2006). Tapadia and Wang (2003, 2004) observed a dramatic shear rate jump in the plateau region of a steady flow curve for a 10 wt% polybutadiene (PBD) solution. They ascribed this discontinuity to a “yield-like entanglement–disentanglement transition (EDT).” However, Inn et al. (2005) argued that this discontinuity is associated with mass loss after the appearance of the instability in the measured polymer system. Later, Tapadia and Wang (2006) observed strain inhomogeneity through the gap of their measuring system using a particle-tracking technique, and they asserted this observation is consistent with the concept of EDT. We remark here that whether the discontinuity was caused by mass loss or strain inhomogeneity, it is not a constitutive property of the material when it occurs.

To add further information related to the dispute on the origin of the discontinuity in the flow curves of entangled PBD solutions, we performed both stress- and rate-controlled experiments on the same material as that investigated by Wang’s group. We monitored the edge condition with a charge-coupled device (CCD) camera during the tests and observed edge instability and mass loss. Results from our stress-controlled experiments agree more with Inn et al. (2005) than with Wang’s group, although it is possible that the EDT is a precursor to the mass loss. In addition, we performed the same experiments on a poly(isobutylene) (PIB) solution (SRM 2490), which has a viscosity more than two orders of magnitude lower than that of the PBD solution. The results of this simple PIB solution in the high shear rate region further confirm the conclusions made from the PBD solution work.

The data representation of figure 1 in Tapadia and Wang (2003) is that corresponding to a modified form of the Cox–Merz rule proposed by Gleissle and Hoshestein (2003). We find that the Cox–Merz rule and this representation provide powerful tools in the investigation of flow instabilities. Particularly, we add to the discussions new data for a branched polyethylene that is not consistent with either view of the reported flow behaviors.

Flow instabilities have also been observed at very low shear rates in rate-controlled experiments. Although it was reported that smaller cone angle or rim gap (Keentok and Xue 1999; Macosko 1994; Pattamaprom and Larson 2001) and shortened time of shearing (Pattamaprom and Larson 2001) can significantly delay the onset of the instabilities, we did not find this to be the case in our experiments. Interestingly, we also find that the stress overshoot in rate-controlled, start-up experiments seems to relate to the onset of flow instability.

In the current work, both stress-controlled and rate-controlled (start-up) steady shearing experiments were conducted on a stress-controlled rheometer (Paar Physica MCR 501). Two cone–plate geometries with cone angles of 1° (CP25-1) and 6° (CP25-6) were used to study the effects of cone angle or rim gap on the onset of instabilities. To check the accuracy of rate-controlled data measured from the stress-controlled MCR 501 rheometer, we also performed the same rate-controlled experiments on a rate-controlled rheometer (ARES) with cone angle of 5.7° (CP25-5.7). The materials investigated here include a PBD solution with high elasticity (nominally the same material as that studied by Wang’s group) and a National Institute of Standards and Technology (NIST) SRM 2490 (PIB solution) with relatively low elasticity. This work is organized into five sections: “Introduction”; “Background”; “Experiments”; “Results and discussion,” which is sub-divided into three parts: “Stress-controlled experiments and the cause of the discontinuity in the plateau region of a steady flow curve,” “Deviation from the Cox–Merz rule and its interpretation for the disagreement between Inn et al. (2005) and Tapadia and Wang (2003, 2004),” “Rate-controlled experiments and stress overshoot”; and “Conclusions.” The effects of cone angle or rim gap and shearing time on the instabilities are discussed in “Stress-controlled experiments and the cause of the discontinuity in plateau region of a steady flow curve.”

Background

It is well known that viscoelastic materials under shearing flows are unstable at high shear rates or shear stresses. An elaborate review of instabilities from both theoretical and practical points of view was given by Larson (1992). In this paper, we give a brief review of the instabilities in cone–plate and parallel-plate geometries, which are characterized by the appearance of distortions of the meniscus (free surface) in flow visualization studies (Hutton 1969; Kulicke and Porter 1979; Pearson and Rochefort 1982; Inn et al. 2005). Instabilities are also revealed by the gradual reduction in the shear stress as a function of time for non-Newtonian fluids (Hutton 1963) or a dramatic change of shear stress and first normal stress difference for a Boger fluid (Magda and Larson 1988). The types of surface distortions in cone–plate and parallel-plate geometries reported include (1) axisymmetric indentation in the free surface (Hutton 1969; Crawley and Graessley 1977; Tanner and Keentok 1983; Keentok and Xue 1999) and has generally been referred to as edge fracture, (2) irregular, non-axisymmetric surface distortions described as vortices (Kulicke and Porter 1979; Kulicke et al. 1979; Kulicke and Wallbaum 1985), (3) gross meniscus distortions (Pearson and Rochefort 1982), and (4) elastic spiral instability (Magda and Larson 1988; Kocherov et al. 1973; McKinley et al. 1991, 1995). Several criteria have been developed to predict the onset of the flow instabilities. Hutton (1963, 1965, 1969) proposed a critical first normal stress difference (N1c) for the first type of instability (edge fracture) based on a concept of limit of elastic energy for a liquid. Subsequently, Tanner and Keentok (1983) proposed that edge fracture is controlled by the second normal stress difference based upon computations of the stress and velocity in the parallel-plate geometry for a second-order fluid. Edge fracture occurs when the second normal stress difference exceeds a critical value, N2c, which is correlated to the surface tension of the liquid and the size of the edge fracture. This criterion was confirmed in later studies (Lee et al. 1992; Huilgol et al. 1993, 1994).

Compared to edge fracture, the criterion for the instabilities with irregularities on the free surface is quite different. Kulicke and Porter (1979) observed that the onset of surface irregularities was sensitive to the ratio of N1/σ12. At the point where the surface irregularities occur, the slope of a plot of N1/σ12 vs shear stress jumps dramatically. −N2/N1 is found to be another sensitive indicator for the onset of the surface irregularities by Kulicke and Wallbaum (1985). A critical Deborah number is proposed as a criterion for the onset of instabilities by a number of authors (McKinley et al. 1995; Phan-Thien 1985; Olagunju and Cook 1993). Importantly, the critical Deborah number developed by McKinley et al. (1995) for the elastic spiral instability takes into account the effect of cone angle as well as the nonlinear features of the viscoelastic properties.

The occurrence of instabilities limits the ability to obtain reliable data for the properties of polymer melts and solutions in cone–plate and parallel-plate rheometers at high shear rates (Kulicke and Wallbaum 1985; Macosko 1994; Rosen 1993). Especially for the more elastic fluids, the shear rate is limited to very low values. Kulicke and Wallbaum (1985) reported the limiting shear rate for two very viscoelastic systems due to the occurrence of instabilities. For a 5 wt% polystyrene (PS)/toluene solution, the severe instability of this system occurred at a shear rate of 16 s−1 and was followed by sample loss at a shear rate of 25 s−1. In the case of a strong polyelectrolyte system, instability and mass loss occurred virtually simultaneously at 0.4 s−1. Inaccurate data due to the occurrence of instabilities and the mass loss may lead one to make incorrect measurements of the constitutive properties of materials. One example of such issues is the importance of flow instability or unknown deformation field and whether the kinked-type behavior for the relaxation modulus at large deformations (Fukuda et al. 1975) is a real material behavior. As discussed in a previous work (submitted to Journal of Rheology), this type of behavior was interpreted as the consequence of the non-uniform deformation through the sample (Marrucci 1983; Larson et al. 1988) and/or wall slip (Archer et al. 2002; Sanchez-Reyes et al. 2002; Venerus and Ritesh 2006) rather than a real material behavior. Another example is the controversy between Wang’s group (Tapadia and Wang 2003, 2004) and Denn’s group (Inn et al. 2005) as introduced above.

Stress overshoot in start-up flows is a phenomenon in which shear stress and the first normal stress difference do not increase monotonically but pass through a maximum with increasing time or strain at sufficiently high shear rate (Graessley 1974; Bird et al. 1977; Menezes and Graessley 1982). It is accepted as a constitutive material behavior and has been related to the molecular chain stretch-retraction process (Ferry 1980; Osaki et al. 2000ac). Qualitatively, it is related to the strain magnitude criterion for nonlinear creep behavior (Ferry 1980). Osaki et al. (2000ac) quantitatively predicted the magnitude of shear (\(\gamma _{{\sigma {\text{m}}}} \)) corresponding to the maximum shear stress with an empirical equation for polystyrene solutions. Theoretical evidence for chain stretch effects on stress overshoot is that the modified Doi and Edwards model (Pearson et al. 1991) incorporating the chain stretch-retraction process qualitatively captures the observed N1 overshoot. Surprisingly, in the present work, it appears that the shear stress overshoot observed may be related to the onset of surface distortion or flow instability. Hence, the question arises whether the flow instability is one of the contributions to the stress overshoot. This is discussed subsequently.

The Cox–Merz rule (Cox and Merz 1958) has been proven to be valid for many polymeric systems by a number of authors (Mead and Larson 1990; Marrucci 1996; Mhetar and Archer 1999; Al-Hadithi et al. 1992). However, it does not hold for all materials (Kulicke and Porter 1980; Ferri and Lomellini 1999; Pattamaprom and Larson 2001; Wen et al. 2004), and its failure has been related to wall slip (Osaki et al. 1975). To date, there seems to have been little attempt to relate the deviations from the Cox–Merz rule to edge fracture or the flow instabilities of the polymeric fluids in steady shearing flows. Two previously reported deviations from the Cox–Merz rule \( \left[ {\eta {\left( {{\mathop \gamma \limits^ \bullet }} \right)}} \right. < {\left| {\eta * {\left( \omega \right)}} \right|} \) and \( \left. {\eta {\left( {{\mathop \gamma \limits^ \bullet }} \right)} > {\left| {\eta * {\left( \omega \right)}} \right|}} \right] \) are considered in the present work.

Experiments

Materials and methods

Two solutions are investigated in this work: (1) a 10 wt% solution of PBD (Polymer Source, Mn = 1.1 × 106 g/mol, Mw = 1.243 × 106 g/mol, polydispersity index [PDI] = 1.13) in a solvent of butadiene oligomer (Sigma-Aldrich, Mn = 1,000 g/mol), which is the same material as that examined by Tapadia and Wang (2003, 2004); (2) Standard Reference Material SRM 2490 from NIST, which is an 11.4 wt% non-Newtonian polymer solution of PIB (Mn = 1.0 × 106 g/mol) dissolved in 2,6,10,14-tetramethylpentadecane. The zero-shear rate viscosity of SRM 2490 at 25 °C is around 100 Pa·s, which is more than two orders of magnitude lower than that of the 10 wt% PBD solution (3.94 × 104 Pa·s at 30 °C). We also considered a long chain branched polyethylene melt with Mn = 1.65 × 105 g/mol, Mw = 3.9 × 104 g/mol, PDI = 4.22, and η0 = 1.79 × 104 Pa·s at 150 °C.

To prepare the PBD solution: 1.70 g PBD was dissolved in a mixture of 15.35 g butadiene oligomer and 60 ml hexane (cosolvent) in a 100 ml round-bottom flask. Most of the cosolvent was evaporated in a rotary evaporator at 30 °C. The rest of the cosolvent was removed by an oil pump (0.1 mm Hg) for 5 days at room temperature (the sample was weighed every 24 h until the weight change was less than 0.01 g).

Figure 1 shows our dynamic data for the 10 wt% of PBD solutions as well as the data from Wang’s group and Inn et al. (Denn’s group) for the same material solution (data from the other two groups are from private communication with Wang). Clearly, the crossover point from our data is in fairly good agreement with Wang’s group but different from the result of Inn et al. This implies that the small difference of polydispersity (1.13 vs 1.2) due to the different batch of materials does not have a big effect on the experimental data, but that the Inn et al. (Denn’s group) material is somewhat different.
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Fig. 1

Comparison of crossover point of G′ and G″ among three groups as indicated in the legend (data of Wang’s and Denn’s groups are from private communication with Wang)

Rheological measurements

The rheological properties of the polymer solutions were measured using both a stress-controlled rheometer (MCR 501 [Paar Physica]) and rate-controlled rheometer (ARES [TA Instruments]). The MCR 501 is a multifunctional machine that allows one to perform both stress- and rate-controlled experiments and has high resolution for torque (0.2–2,000 g·cm) but low sensitivity for normal force (2–5,000 g). To investigate the effect of cone angle or gap at the rim on the occurrence of flow instability, two measuring systems were used with this stress-controlled machine: one is the cone–plate geometry with 1° cone angle (with a gap at the rim 0.267 mm) and 25 mm in diameter (CP25-1), the other with 6° cone angle (with a rim gap of 1.363 mm) and 25 mm in diameter (CP25-6). To check the accuracy of rate-controlled data measured from the stress-controlled MCR 501 rheometer, we also performed the same rate-controlled experiments on a rate-controlled rheometer (ARES) with cone angle of 5.7° (CP25-5.7). Furthermore, two shearing periods (102 s and 19.43 s) were applied to the sample for the stress-controlled experiments with the CP25-6 measuring system to examine the effects of shearing time on the onset of flow instabilities. We took videos for the two types of measurements with the different geometries using a CCD camera.

All the experiments for the PBD solution were carried out at 30 °C except for those done for the video recordings, in which case, the experiments were performed at room temperature. The experiments for the SRM 2490 were all performed at room temperature (note that room temperature in our laboratory is approximately 23–25 °C). Shear stress was increased in steps from 25 to 6,000 Pa with 10-min rest time between two steps for the stress-controlled tests. The range of shear rate for the start-up experiments is from 0.001 to 500 s−1, and the dynamic oscillatory measurements cover five decades of frequency from 0.001 to 100 rad/s.

Results and discussion

The onset of instability and the development of surface distortions of the two investigated polymer solutions in steady shearing flows were monitored and recorded by a digital CCD camera. It was found that the characteristics of the flow instabilities for the two polymer solutions are quite different. The instability of the 10 wt% PBD solution shows non-axisymmetric spiral ripples on the free surface, whereas in the case of the SRM 2490 is of the edge fracture type (Hutton 1969; Crawley and Graessley 1977; Tanner and Keentok 1983). The visualizations of instabilities for both solutions are shown subsequently.

Stress-controlled experiments and the cause of the discontinuity in plateau region of a steady flow curve

The stress-controlled tests were performed by increasing the shear stress from the range of 25 to 6,000 Pa with 10-min rest time for material recovery between two steps at a temperature of 30 °C. Figure 2 shows the transient data, shear rate as a function of time for part of the stress-controlled tests at different constant shear stresses for the CP25-6 measuring system on the MCR 501. One can see that the shear rate jumps by more than two decades when shear stress increases from 2,500 to 2,600 Pa. A similar jump transition has been reported by Tapadia and Wang (2003, 2004) and Inn et al. (2005), respectively. Tapadia and Wang (2003, 2004) attributed this dramatic shear rate jump to the EDT transition, whereas Inn et al. (2005) claimed this jump was caused by mass loss after the occurrence of flow instability based on monitoring the sample shape during the experiments. We have observed the same phenomena as described in Inn et al. (2005) upon taking videos for experiments in which the shear stresses were applied near the critical point where the jump occurred. As shown in Fig. 2, the shear stress of 2,600 Pa is a critical point where flow instability may have occurred. Therefore, we made video recordings of the edge condition of the sample for experiments with the shear stress ranging from 2,400 to 4,000 Pa. The shearing time was 19.43 s for each constant shear stress (this shearing period is much shorter than 102 s for the experiments in Fig. 2). The sample was permitted to rest at least 15 min between tests. For convenience, all experiments for making video recordings were conducted at room temperature. The evolution of the instabilities and subsequent mass loss can be seen in Fig. 3. The left column shows the images before applying the shear at different shear stress levels, and the right column shows the corresponding images at the point where the shearing stops. The images in the right column demonstrate that the type of flow instability for this PBD solution is elastic spiral ripples on the free surface. As seen in Fig. 3, the intensity of the instability increases with shear stress and is strongest at 2,800 Pa at which point the instability propagates throughout the entire fluid field from the edge inwards towards the apex of the cone with time. Visually, the entire fluid field changes optical features from the transparent to the opaque. At 3,000 Pa, where the shear rate jumps dramatically (Fig. 4), a large amount of mass was lost. Figure 4 depicts the corresponding transient data for experiments at different shear stresses in which videos were taken (as shown in Fig. 3). From this set of transient data, we can see that the sample is already in the unstable region at shear stresses between 2,400 and 2,700 Pa where true steady state is inaccessible due to the surface distortion observed in Fig. 3. At the shear stress of 2,800 Pa, the shear rate shows an obvious trend to jump because the flow instability propagates throughout the entire gap. At a shear stress of 3,000 Pa, where mass loss occurs, a dramatic shear rate jump is observed due to the mass loss. Qualitatively, it is understandable that the dramatic shear rate jump is associated with mass loss because the “shear stress” coming from the sample will be significantly reduced due to sample loss from the cone and plate. This, in turn, results in a dramatic shear rate increase in stress-controlled experiments and shear stress reduction in rate-controlled experiments (this is shown in data from rate-controlled experiments). Therefore, from Figs. 3 and 4, we may conclude that the long transient and dramatic jump for the shear rate response at a shear stress of 2,600 Pa in Fig. 2 correlated with the mass loss that occurs after the flow instability (edge distortion) propagates throughout the entire fluid field. This result agrees with the observations of Inn et al. (2005).
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Fig. 2

Transient data for stress-controlled tests at different constant shear stress as indicated in the plots for 10 wt% PBD solution with the 6° cone angle system and at 30 °C. The period of shearing is 102 s

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Fig. 3

Images demonstrate the evolution of flow instability for the 10 wt% PBD solution. All the images were taken at the point where the measuring system is subjected to shearing for 19.43 s from the videos for tests with different shear rates with the 6° cone angle system on the MCR 501 and at room temperature. Left column represents the images before applying shearing, and the right column represents the images after shearing for 19.43 s (after Sui et al. 2006)

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Fig. 4

Corresponding transient data for stress-controlled tests for taking videos shown in Fig. 3. Period of shearing is 19.43 s (after Sui et al. 2006)

In addition, by comparing the curves in Figs. 2 and 4, the effect of shearing time on instability intensity becomes obvious. With longer shearing time (102 s, Fig. 2), the most intensive instability and mass loss occurred at a lower shear stress, 2,600 Pa, whereas for shorter shearing time (19.43 s, Fig. 4), it happened at a higher shear stress, 3,000 Pa.

As mentioned previously, reducing the cone angle or the rim gap has been reported to delay the onset of flow instability. In this paper, we carried out the same experiments as those in Fig. 2 with the 1° cone angle (CP25-1) measuring system, which has a rim gap more than six times smaller than that of the CP25-6. The transient data are plotted in Fig. 5. Comparing Fig. 5 with Fig. 2 shows that similar material behavior was observed for both cone angles. The difference is that the severe instability and mass loss occurred at a shear stress of 3,200 Pa for the CP25-1 measuring system instead of 2,600 Pa for the CP25-6. Obviously, the smaller cone angle delays the occurrence of the severe instability and mass loss as expected. However, instability and mass loss still occur at a low value of shear rate because of the high viscosity and high elasticity of the PBD solution. The effect of cone angle on the instabilities is more obvious if one looks at the magnitude of the shear rate jump or the range of shear rates inaccessible because of mass loss. Figure 6 shows the flow curves constructed from the data of the stress-controlled tests in Figs. 2 and 5 and measured with the two different geometries. In the low shear rate region, shear stress increases monotonically with the shear rate, and the data measured from the two cone angles are consistent. However, these two sets of data deviate from each other in the plateau region (moderate shear rate region). For the CP25-6 measuring system, the discontinuity of the shear rate covers about two decades instead of one decade for the CP25-1 measuring system. This again indicates that the discontinuity of the shear rate in the plateau region was associated with mass loss rather than the EDT transition only. If an EDT transition proceeds, the edge distortion and mass loss cannot be ascertained by the present work.
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Fig. 5

Transient data for stress-controlled tests at different shear stress levels as indicated in the plots for PBD solution with 1° cone angle system at 30 °C. Shearing time is 102 s

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Fig. 6

Flow curves of the PBD solution constructed from the stress-controlled experiments at 30 °C with different measuring systems as indicated in the legend. The arrows indicate the data points at which mass loss occurs

The cone angle (rim gap) effects on the flow instabilities were also investigated in rate-controlled experiments. Figure 7 presents the flow curves measured in rate-controlled experiments with two different cone angles on the MCR 501 along with measurements from the CP25-5.7 measuring system on the ARES. The data obtained from the ARES are in very good agreement with those from the MCR 501 (CP25-6 system), and both sets of data are consistent with the measurements from the CP25-1 on the MCR 501 in the low shear rate region (for shear rate less than 0.25 s−1 as seen in Fig. 7). This demonstrates that the data obtained from the MCR 501 is reliable. Furthermore, a stress undershoot was seen in both flow curves measured with different cone angles on the MCR 501. The difference is that the stress undershoot measured in the CP25-6 occurs one decade earlier than that measured in the CP25-1 (0.25 s−1 vs 2.5 s−1, which are the points in the severe flow instabilities just before mass loss is observed). The smaller cone angle also delays the occurrence of mass loss in rate-controlled experiments. In addition, the shear stresses measured from the CP25-1 measuring system are well above those from the CP25-6 and the CP25-5.7 systems at the same shear rates. This discrepancy may be caused by greater mass loss in the CP25-6 and the CP25-5.7 measuring systems than in the CP25-1. In this paper, we need to point out that the data points measured after losing sample are not meaningful for material property determination. The reason we show them here is to evaluate the cone angle effects on the flow instabilities. Even the data points after the onset of flow instability may not provide reliable material property information, although they do exhibit smooth apparent flow curves before the occurrence of mass loss (as seen in Figs. 12, 13, and 15, which are discussed subsequently).
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Fig. 7

Flow curves for PBD solution constructed from the rate-controlled tests at 30 °C with different measuring systems and different rheometers as indicated in the legend. The arrows indicate the data points just before mass loss occurs

We remark in this context that Wang’s group obtained a smooth flow curve from stress-controlled tests for a polydispersed polymer solution by using a film technique (the free surface of the measuring system was surrounded by a thin film that prevents the mass loss). As seen in Fig. 8 from Philips and Wang (2005, private communication), the discontinuity or the “jump transition” in the plateau region of the flow curve is suppressed by the film surrounding the edge.
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Fig. 8

Flow curves for a polydispersed polymer measured in stress-controlled experiments. The diamonds are data measured by surrounding the edge of measuring system with film, and the squares are data measured in normal way (Philips and Wang, 2005, personal communication)

Tapadia and Wang (2006), while not emphasizing the EDT transition, have moved their work forward to examine flow behavior using particle-tracking methods. In this case, they observe significant deviations from flow-field homogeneity across the gap in the cone–plate geometry, and this has significant implications for nonlinear rheological measurements. In this paper, while not doing particle tracking, we are finding subtle edge effects that seem to be related to similar phenomenological behaviors. Whether the two types of event (instability vs inhomogeneity) are related remains to be investigated. Wang and his coworkers (Wang et al. 2006) have concluded that the flow inhomogeneity is related to a network breakdown, similar to the EDT, in step shearing flows.

To expand upon the results obtained from the PBD solution, we performed the same type of stress-controlled experiments on a low-viscosity material SRM 2490. Because the viscoelastic responses of this material are relatively low due to its low viscosity, we represent all steady shear data for the SRM 2490 on a linear scale, which makes the results somewhat clearer than the double logarithmic representation used above. The flow curves measured in stress-controlled experiments with both the CP25-6 and the CP25-1 geometries are plotted in Fig. 9. Figures 10 and 11 present the images taken corresponding to the data in Fig. 9 for CP25-6 and CP25-1 systems, respectively. We observe a shear rate jump at a shear stress of 700 Pa in the flow curve measured with the CP25-6 geometry. The jump is associated with the sample coming out the gap as seen in Fig. 10. On the other hand, no shear rate jump was seen in the flow curve measured with the CP25-1 system because the small cone angle suppresses mass loss (Fig. 11).
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Fig. 9

Flow curves of the SRM 2490 constructed from the stress-controlled experiments at room temperature with different measuring systems as indicated in the legend. The arrow pointing to square indicates the data point at which mass loss occurs, and another arrow pointing to circle indicates a comparison between two flow curves measured with different cone angles

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Fig. 10

Images demonstrate the evolutions of the flow instability for the SRM 2490. All the images are captured at the point where the measuring system is subjected to shearing for 5.68 s from videos for the stress-controlled tests with the 6° cone angle system on the MCR 501 and at room temperature (corresponding to the data points [squares] in Fig. 9; after Sui et al. 2006)

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Fig. 11

Images demonstrate the evolutions of the flow instability for the SRM 2490. All the images are captured at the point where the measuring system is subjected to shearing for 5.68 s from videos for the stress-controlled tests with the 1° cone angle system on the MCR 501 and at room temperature (corresponding to the data points [circles] in Fig. 9)

Deviation from the Cox–Merz rule and its interpretation for the disagreement between Inn et al. (2005) and Tapadia and Wang (2003, 2004)

As discussed in the “Introduction,” Tapadia and Wang (2003) represented their flow data in a modified form of the Cox–Merz rule, which compares complex modulus vs frequency along with steady shear stress vs shear rate instead of the complex viscosity vs frequency with steady shear viscosity vs shear rate in the conventional way. Clearly, the nonlinear steady shear data fall below linear dynamic data, which indicates the usual failure of the Cox–Merz rule. The question to ask here: Is the deviation from the Cox–Merz rule to be attributed to the EDT transition (Tapadia and Wang 2003, 2004) or to flow instability (Inn et al. 2005)? It becomes clear when we look at our data representation for the PBD solution and the SRM 2490 in Figs. 12 and 13, respectively. As discussed previously, the dramatic shear rate jump at a shear stress of 3,200 Pa in Fig. 12 is associated with mass loss. Instead of showing a shear stress increase, a shear stress drop at a shear rate of 2.5 s−1 occurs in the flow curve measured in the rate-controlled experiments. Again, these can be attributed to the mass loss of the sample. Therefore, we can see that the deviation from the Cox–Merz rule (at a shear rate of 0.1 s−1 for rate-controlled mode and 2,000 Pa for stress-controlled mode) occurs in a shear rate region well below where mass loss happens. From our observations, we can see that deviations from the Cox–Merz rule are related to the onset of flow instability for this PBD solution. The same case holds true for the low-viscosity SRM 2490 as seen in Fig. 13. To give explicit evidence for the above comments, the images in Figs. 10 and 14 demonstrate the evolution of flow instability for the experiments at different shear rate and shear stress levels corresponding to the data points in Fig. 13. The failure of the Cox–Merz rule and the anomalous behaviors in the plateau regime of flow curves correspond to the onset of edge instability and mass loss, respectively. This implies that the failure of the Cox–Merz rule seems related to flow instability. Interestingly, the flow curves measured from both rate- and stress-controlled experiments in Figs. 12 and 13 are smooth in the shear rate region between the onset of the flow instability and mass loss and are consistent with each other. The significance of this is that a smooth flow curve does not mean a “valid” flow curve, as there may be the occurrence of flow instability. Caution is required when taking flow curves for “unobserved” samples.
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Fig. 12

Flow curves scale for the 10% PBD solution constructed from steady shear data from stress-controlled and start-up tests at 30 °C with the 1° cone angle system on the MCR 501 rheometer along with complex modulus as a function of frequency. Data are represented double logarithmically. Shearing time is 102 s. The arrows in low shear rate region indicate the onset of flow instability for rate- and stress-controlled experiments, respectively, and these in high shear rate region exhibit mass loss

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Fig. 13

Flow curves for the SRM 2490 constructed from steady shear data from creep and start-up tests at room temperature with the 6° cone angle system on the MCR 501 rheometer along with complex modulus as a function of frequency. Axes are linear scales. The arrows have the same meanings as those in Fig. 12

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Fig. 14

Images demonstrate the evolutions of the flow instability for the SRM 2490. All the images are captured at the point where the measuring system is subjected to shearing for 5.68 s from videos for the start-up tests with the 6° cone angle system on the MCR 501 and at room temperature (after Sui et al. 2006)

Above, we have observed the usual failure of the Cox–Merz rule for two polymer solutions with different viscosities. Now, we go back to look at data for a long chain branched polyethylene melt that was investigated for other reasons. As shown in Fig. 15, contrary to the above two polymer solutions, an unusual failure of the Cox–Merz rule, i.e., \( {\left| {\eta * {\left( \omega \right)}} \right|} < \eta {\left( {{\mathop \gamma \limits^ \bullet } = \omega } \right)} \), for this branched polymer melt was observed. In addition, we see that the deviation from the Cox–Merz rule occurs at an earlier stage of the shear-thinning region and well before the onset of flow instability (the same type of flow instability as that of the PBD solution was observed for this polymer melt) as indicated in Fig. 15 at 0.1 s−1. The material behavior shown in this set of data is not consistent with the view of flow behaviors from either group (Inn et al. and Wang). Importantly, from Fig. 15, we can see that the existence of long chain branching strongly weakens the shear-thinning behavior for the branched material compared to the dynamic data. Therefore, the branching of the polymers may suppress much of what Wang’s group sees in their studies, and their results may be related primarily to linear polymers. Further investigations would be required to determine this.
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Fig. 15

Steady shear viscosity and complex viscosity vs shear rate or frequency on a double logarithmic scale for the LD 146 polyethylene. The squares, circles, and triangles represent the data of complex viscosity measured in dynamic experiments, the steady shear viscosity measured from rate- and stress-controlled tests at 150 °C, respectively. The arrow demonstrates the onset of flow instability (data from Sui and McKenna, 2007)

Rate-controlled experiments and stress overshoot

In the rate-controlled experiments, we find not only the flow instabilities as discussed above but also stress overshoot and subsequent undershoot in transient shear stress vs time plots as has been reported by Tapadia and Wang (2004). Figure 16 presents a set of transient data for the PBD solution measured at different shear rates with the 1° cone angle as indicated in the legend. The evolution of surface distortion or flow instability corresponding to the experiments in Fig. 16 is shown in Fig. 17. From these two figures, it seems that the occurrence of the weak stress overshoot at a shear rate of 0.5 s−1 may be related to a subtle surface distortion as shown in Fig. 17. Certainly, by a shear rate of 1.0 s−1, the surface distortion becomes obvious. The question to ask here whether this generally accepted material behavior, stress overshoot, is related to the onset of flow instability. To address this question, we performed another set of experiments with the 6° cone angle measuring system at different shear rate levels and monitored the edge conditions with a CCD camera. Figure 18 presents the images taken after shearing for 19.43 s at different shear rates as indicated beside the images and the corresponding transient data below. The incipient surface distortion or the onset of flow instability occurs at a shear rate of approximately 0.2 s−1 (upper images of Fig. 18), which is where the stress overshoot is observed (Fig. 18). This experimental evidence indicates that the occurrence of the stress overshoot is related to the onset of flow instability. The same holds true for the LD 146 melt at 150 °C based on visual observation (the high operating temperature makes it hard to take videos during experiments). Therefore, we conclude that the flow instability may be one contribution to the stress overshoot. To verify this finding, more materials subjected to start-up shearing flows need to be investigated, and more surface observations need to be made.
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Fig. 16

Transient data for start-up tests at different shear rates as indicated in the legend for the PBD solution with 1° cone angle system at room temperature. This set of data corresponds to visualization experiments shown in Fig. 17

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Fig. 17

Images illustrate the evolution of flow instability for the 10 wt% PBD solution. All the images were taken at the point where the measuring system is subjected to shearing for 19.43 s from the videos for tests with different shear rates with the 1° cone angle system on the MCR 501 and at room temperature. Left column represents the images before applying shearing, and the right column represents the images after shearing for 19.43 s

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Fig. 18

Upper: Images demonstrate the onset of surface deformation or flow instability. All the pictures were captured at the point where the measuring system is subjected to shearing for 19.43 s from the videos for the tests at different shear rate levels with the 6° cone angle system on the MCR 501 and at room temperature. Lower: corresponding transient data for the start-up tests

Conclusions

Flow instability at low shear rates was observed for two polymer solutions in both rate- and stress-controlled conditions and with cone and plate measuring systems equipped with different cone angles. A long chain branched polyethylene melt was also considered. The effect of cone angle on the flow instability was obvious for the PBD solution, but not prominent for the SRM 2490. The type of instability for the PBD solution is elastic spiral ripples in the free surface, whereas an edge fracture type of instability was observed for the less elastic SRM 2490. The onset of flow instability occurs at much higher shear rate (40 s−1) for the SRM 2490 compared to that (0.1 s−1) of the PBD solution. The SRM 2490 has two orders of magnitude lower viscosity.

The discontinuity of the flow curves in the stress-controlled mode was observed as reported by Tapadia and Wang (2003, 2004) and Inn et al. (2005). By monitoring the evolutions of flow instability at the free surface of the sample, our observations are similar to those of Inn et al. that the discontinuity was associated with the mass loss after the measuring system undergoes severe instability. The Cox–Merz rule is a powerful tool to look at the controversy between Tapadia and Wang (2003, 2004) and Inn et al. (2005). The usual failure of the Cox–Merz rule for polymer solutions in our studies is related to the onset of flow instability that occurs well before EDT transition. An unusual failure of the Cox–Merz rule for a long chain branched polymer shows the strongly weakened shear-thinning behavior. This implies the branched polymers may suppress much of what Tapadia and Wang see in their studies on linear polymers.

We also observed that the occurrence of stress overshoot in start-up experiments is related to the onset of flow instability for the materials investigated here. This implies that the flow instability may be one contribution to the stress overshoot. To verify this finding, more materials and more observations need to be investigated in start-up flows.

Acknowledgment

Thanks to the American Chemical Society–Petroleum Research Fund under Grant 40615-AC7 and the J.R. Bradford endowment at Texas Tech University for partial support of this work. The authors also thank S-Q Wang and A. Philips for highly fruitful discussions.

Copyright information

© Springer-Verlag 2007