Experimental study of the volume defects in polystyrene extrusion

J. Non-Newtonian Fluid Mech. 121 (2004) 175–185

Experimental study of the volume defects in polystyrene extrusion Christelle Combeaud, Yves Demay, Bruno Vergnes∗ CEMEF, Ecole des Mines de Paris, UMR CNRS 7635, BP 207, 06904 Sophia Antipolis Cedex, France Received 2 April 2004; received in revised form 8 June 2004

Abstract This paper presents an experimental characterization of the volume defects in polystyrene extrusion. In a first step, these defects are quantified in capillary rheometry. In a second step, a transparent slit die, fed by an extruder, is used to perform birefringence experiments and to visualize stress transients during the instability. Isochromatic fringe oscillations are observed in the reservoir at the onset of the defect and during its development. Their frequency has been estimated and appears to be very sensitive to flow conditions (temperature, convergent geometry), but not to apparent shear rate. Using small entry angles allows to postpone the defect occurrence. Furthermore, the use of a smooth convergent leads to the suppression of the instability, at least until an apparent shear rate of 390 s−1 . The possible origin of the instability is discussed. © 2004 Elsevier B.V. All rights reserved. Keywords: Polystyrene; Instability; Volume defects; Capillary; Slit die; Flow birefringence

1. Introduction When extruding a molten polymer, the flow emerging from the die presents extrusion defects above critical conditions. Depending on the molecular structure of the polymer, these defects exhibit different aspects, from small surface irregularities called sharkskin until completely chaotic volume distortions, called gross melt fracture. All these defects, which obviously lead to strong industrial processing limitations, have been subject to several reviews over the last few years [1,2]. Among the numerous extrusion defects, a typical behavior concerns the occurrence and development of regular volume distortions affecting the extrudate. It is classically observed for branched polymers (low density polyethylene), but also for highly elastic ones, i.e. with high normal stresses and long relaxation times (polystyrene, polypropylene). Above a critical flow rate, the extrudate does present a perfect helical shape when emerging from a circular die. In similar conditions, the extrudate would also exhibit periodic distortions when emerging from a slit die, but of course, for such flow conditions, no helical shape can be observed any more. ∗ Corresponding author. Tel.: +33 4 93 95 74 63; fax: +33 4 93 65 43 04. E-mail address: [email protected] (B. Vergnes).

0377-0257/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2004.06.007

As a result, whatever the flow geometry, we will prefer in the following the term of “periodic volume defect” instead of “helical defect”. As the flow rate increases, the extrudate progressively loses its periodicity, until it presents a chaotic aspect (“gross melt fracture”) [3]. It is worth noting that “periodic volume defect” will refer to regular distortions, whereas “gross melt fracture” will describe irregular shapes. In the literature, “upstream instability” is often indistinctly used for all these phenomena. In 1956, Tordella [4] clearly described experimental setup and flow conditions leading to extrudate distortions. Contrary to linear polymers, branched or highly elastic polymers present a continuous flow curve (pressure/flow rate relationship) in capillary extrusion. The onset of upstream instability is characterized by a change of slope on this flow curve, especially pronounced for short capillary lengths [5–7]. Critical parameters usually used to quantify the onset of the upstream instability are mostly shear flow parameters, such as apparent shear rate or wall shear stress. According to Ballenger et al. [8], volume defects occur for a critical wall shear stress around 0.1 MPa. However, this critical stress depends on the die length, the temperature and also the molecular weight [9]. Moreover, upstream die flow visualizations during the

C. Combeaud et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 175–185

instability, using either flow birefringence [10–13], tracers [6,13–18], or laser Doppler velocimetry [17,18] proved that the region upstream the contraction was the site of the defect initiation. Nigen et al. [19] have recently carried out investigations using particle image velocimetry and confirmed the link between entry flow destabilization and gross melt fracture. It is almost clear that the capillary inlet is a region where the fluid is submitted to high elongational deformations. This extensional flow may play an important role in upstream instabilities. However, in spite of many experimental and theoretical works, no totally satisfactory explanation has been proposed for the initiating mechanism. Some authors [5,10,20] claimed that the instability is due to the presence of large recirculation zones in the reservoir for dies with abrupt contraction. The vortices would oscillate, above a critical size, and then burst into the main convergent flow, destabilizing it. Vortex instabilities have been clearly demonstrated in recent works on Boger fluids polymer solutions [16,17]. However, applications to molten polymers remain less common. Kim and Dealy [21] have recently suggested a critical tensile stress as a criterion for the onset of gross melt fracture. They argue that gross melt fracture would involve the true fracture (rupture) of the melt, specifying that it is not clear whether this is the result of chain scission or sudden disentanglement and chain pull-out. More generally, previous studies have clearly pointed out that both viscoelastic fluid properties and flow geometry are deeply involved in the occurrence of upstream defects. Consequently, the aim of the present work is to focus on crucial and necessary flow conditions for which the volume instability of a molten polymer is expected to occur. This study highlights the influence of flow conditions on the defect occurrence and may also lead to a better understanding of the dynamic response of a viscoelastic fluid forced into a convergent. The flow behavior of polystyrene is investigated, using both capillary and transparent slit die experiments. Influences of both temperature and convergent geometry on the periodic volume instability occurrence are studied. The development of the defect is also investigated, especially regarding the instability frequency.

Table 1 Characteristic data of the polystyrene Density at 23 ◦ C (g cm−3 ) Density at 200 ◦ C (g cm−3 ) Mn (g mol−1 ) Mw (g mol−1 ) Mz (g mol−1 ) Mw /Mn

We =

˙ N1 (γ) , ˙ γ˙ 2η(γ)

(1)

where N1 is the first normal stress difference. Unfortunately, the range of measurements for N1 (between 0.2 and 3 s−1 ) is very far away from the region where the instabilities develop (above 100 s−1 ). Consequently, an estimation of a Weissenberg number should be made by extrapolation from the low shear rate range. Depending on the type of extrapolation, and with the same value of regression parameter, values between 2 and 10 can be estimated at 100 s−1 . Consequently, and despite the interest of such approach, no elastic characteristic number will be used in the following. Fig. 1b presents the elongational behavior measured in uniaxial extension using a Meissner-type RME rheometer, at an elongation rate of 1 s−1 . In these conditions, the polymer 10

6

10

5

10

4

10

3

10

2

The polymer used is a commercial polystyrene supplied by Dow Benelux (Terneuzen, The Netherlands). Its physical and molecular characteristics are listed in Table 1. The complex moduli (G and G ) and the complex viscosity at 200 ◦ C are presented in Fig. 1a. The polymer shows a shear-thinning behavior, with a zero-shear viscosity η0 of 33 800 Pa s. Its activation energy is around 146 kJ/mol. In order to evaluate the elasticity of the polymer, we have to estimate a relaxation time. From the horizontal tangent of the Cole–Cole plot [20], we can deduce an average relaxation time of 10.85 s. How-

Elongational viscosity (Pa.s)

2.1. Polymer

(b)

G'

η∗ G"

1

10 -3 -2 -1 0 1 2 3 4 10 10 10 10 10 10 10 10

(a)

2. Material and methods

1.047 0.973 136100 295600 461500 2.17

ever, to characterize the elastic behavior in different flow conditions, the better would be to use a Weissenberg or a Deborah number. From continuous shear measurements, the Weissenberg number is defined as:

Complex viscosity (Pa.s) Storage and loss modulus (Pa)

176

-1

Frequency (rad s )

10

6

10

5

4

10 -1 10

0

10 Deformation

10

1

Fig. 1. (a) Complex shear viscosity at 200 ◦ C and (b) uniaxial extensional viscosity at 190 ◦ C and 1 s−1 (the full line represents the linear viscoelastic behavior).

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177

profiles used.

3. Capillary rheometer results 3.1. Apparent flow curves and extrudate distortions

Fig. 2. Slit die geometry.

Fig. 3. Convergent profiles for the slit die experiments (H = 1 mm and L = 20.75 mm).

does not present any strain-hardening. 2.2. Experimental set-up

The main goal of this part of the work was to characterize the helical instability, i.e. to describe the onset and the development of the periodic volume defect along the flow curve. The occurrence of the instability was determined by visual inspection of the extrudate shape. This polymer does not present any surface defect. Above a critical apparent shear rate, the smooth extrudate becomes suddenly affected by typical helical distortions, which lose their periodicity for higher flow rates. Fig. 4 presents apparent flow curves obtained at a temperature of 200 ◦ C with different L/D ratios for a diameter of 1.39 mm. For each apparent flow rate, a sample of extrudate has been cut off and carefully cooled at ambient temperature. The flow curves (pressure as a function of the apparent shear rate) are continuous and show different zones, with various extrudate aspects (separated by dashed lines on the graph). The pictures illustrate the extrudate aspect for the case L/D ≈ 0: at low shear rates (until 100–150 s−1 ), the extrudate is perfectly smooth (Fig. 4a). Above a critical shear rate, a helix begins to affect the extrudate surface (Fig. 4b). When flow rate increases, the helix becomes more and more

2.2.1. Capillary rheometer Preliminary investigations have been carried out with a capillary rheometer (Rheoplast© , Courbon, France). The circular dies have a half entry angle of 45◦ . The available die diameters D are 0.93, 1.39, 2.0 and 3.0 mm. Capillary lengths L correspond to L/D ratios of 0, 4, 8, 16 or 32. The pressure is measured in the reservoir just before the die convergent. 2.2.2. Slit die The experimental set-up has already been described in detail elsewhere [22]. It will only be described here briefly. It consists of a single screw extruder, equipped with a gear pump and a transparent modular slit die, where flow-induced birefringence (FIB) measurements are possible. Fig. 2 describes the flow channel, which consists of a reservoir with a cross section Hr × W = 18 × 10.1 mm and a slit die of gap H = 1 mm and length L = 25 mm (the contraction ratio Hr /H is 18:1). The pressure is measured in the reservoir, 20 mm upstream the contraction. The temperature has been set at 180, 190 or 200 ◦ C. Different entrance angles have been tested, to check their influence on the periodic volume defects. Half-angle was varied from 90◦ (abrupt contraction) to 45 and 30◦ , while the gap H and the effective die length L remained constant. Another smooth entry shape has also been studied (trumpet-shaped profile). For practical reasons, the die length was reduced to 20.75 mm for all these dies. Fig. 3 summarizes the different convergent

Fig. 4. Apparent flow curves at 200 ◦ C, for different L/D (D = 1.39 mm) and extrudate morphologies for L/D ≈ 0.

178

C. Combeaud et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 175–185 3

Helical periodicity volume (mm )

pronounced (Fig. 4c–f), affecting all the extrudate volume (helical pitch and diameter increase with flow rate). For apparent shear rates higher than 400 s−1 , we first observe a degeneration of the helix (with still a more or less periodic organization, Fig. 4g), followed by the development towards the chaotic defect (Fig. 4h–i). No strictly periodic organization is observable any more. As described in the literature [5–7], the onset of the helical defect is characterized by a change in the slope of the flow curve, well pronounced in the case of small L/D ratios (Fig. 4). The helix direction (right or left) does not seem to depend on the die geometry (D or L/D). As soon as the helical instability has appeared (in a direction which is randomly defined), this direction does not evolve any more with the flow rate. This clearly appears on the pictures sequence.

Helical instability frequency (Hz)

This characterization was focused on the defect frequency. The volume V of each helical periodicity (i.e. the volume of a portion of extrudate whose axial length is equal to the helix pitch) has been estimated by: M , ρn

(2)

where M is the mass of the extrudate, ρ the density of the polymer at 200 ◦ C and n is the number of periodicities encountered on the sample. As we work at constant volume flow rate Q, the instability frequency f is calculated as follows: f =

Qρn Q = . V M

L/D = 16

500 L/D = 4

400 300

L/D = 0

200 100 0 100

(a)

3.2. Characterization of the helical defect

V =

600

150 200 250 300 -1 Apparent shear rate (s )

350

3,5 3

L/D = 0

2,5 L/D = 4

2 1,5

(b)

L/D = 16

1 50

100 150 200 250 300 350 -1 Apparent shear rate (s )

Fig. 5. Variation of (a) the helical volume and (b) the helical frequency as a function of the apparent shear rate (T = 200 ◦ C, D = 3 mm and L/D = 0, 4 and 16).

[3,7,8,20] could be a visual artifact: as the pitch is higher, the extrudate looks much less distorted, even if the volume affected by the instability is larger.

(3)

Fig. 5 shows the influence of the apparent shear rate on the instability development: these results are in close agreement with the sequence of pictures presented before. The helical volume increases linearly with the flow rate, which means that the helix grows up in a very organized way. Consequently, the defect frequency is not very sensitive to the apparent shear rate (Fig. 5b). To summarize, for a fixed geometry, the instability develops with a constant frequency and with growing organized helical volume distortions. These results are consistent with the findings of Nigen et al. [19] obtained by particle image velocimetry. If we consider the length of the capillary, it can be also shown that the frequency increases as the length decreases [23]. For a diameter of 3 mm, f increases from 1.45 to 2.1 and 2.8 Hz while L/D decreases from 16 to 4 and 0. However, the helical volume increases with the die length (or L/D): at 300 s−1 , it increases from 280 to 360 and 500 mm3 , when L/D increases from 0 to 4 and 16 (Fig. 5a). The pitch of the helix increases also linearly with L/D at constant shear rate. In other words, it seems that there is no attenuation of the instability along the capillary, what was first reported by Bergem [24] using flow visualizations. Consequently, we may imagine that the attenuation of the instability for long capillaries, largely cited in the literature

4. Study of the flow-induced birefringence in the slit die 4.1. Apparent flow curves and associated flow-induced birefringence patterns For each geometry, the pressure in the reservoir is measured as a function of the apparent shear rate, calculated by: γ˙ app =

6Qm , ρWH 2

(4)

where Qm is the mass flow rate. Contrarily to the case of capillary extrusion, where extrudates have perfect helical shapes, the onset of volume defect in a slit die is much more difficult to observe. Indeed, slit die extrudates are affected by periodic volume distortions, which can be due to volume defects but also to side wall effects. As the presence of these side effects prevents one from observing clearly the defect occurrence at the die exit, it is necessary to use upstream visualization, for example by flow-induced birefringence. Fig. 6 presents the apparent flow curve at a temperature of 180 ◦ C and the associated FIB patterns in the case of an abrupt contraction (as explained in

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Fig. 6. Apparent flow curve and associated FIB patterns: (a) 30 s−1 , (b) 87 s−1 and (c) 263 s−1 (abrupt contraction, T = 180 ◦ C, L = 25 mm and H = 1 mm).

Section 5, the increase in principal stress difference with each fringe corresponds to 0.015 MPa). The isochromatic fringes corresponding to the point (a) are stationary: during time, they remain identical and symmetrical about the reservoir axis. As the apparent shear rate reaches a critical value around 90 s−1 (point (b)), they start to oscillate regularly, perpendicularly to the flow direction. They periodically move into asymmetrical positions, from one side of the reservoir to the other. The times t1 and t2 in Fig. 6 correspond to the two extreme fringe positions in the reservoir. In this situation, the melt flow coming from the reservoir enters the contraction asymmetrically. This phenomenon clearly depicts the volume defect as a consequence of the flow destabilization at the die inlet, and not as a destabilization of the flow inside the channel, as recently proposed by Meulenbroek et al. [25]. Moreover, as the volume instability develops, the fringes oscillation amplitude also increases, as it can be seen by comparing fringes positions from Fig. 6b and c1 .

1 The video films of all the FIB experiments presented in this paper can be seen on the following web site: http://eve.cma.fr/presentation films-eve.htm.

The onset of volume defect is indicated in Fig. 6. As in capillary experiments, the apparent flow curve clearly presents a change of slope at this point, which is in good agreement with the literature [5–7]. If we call t the time interval between two successive fringe movements, 1/t would represent the defect frequency when these movements are periodic. Fig. 7 presents the evolutions of 1/t with time, for different apparent shear rates. For the critical apparent shear rate of 87 s−1 (volume instability onset), the fringes movement is strictly periodic. When increasing the flow rate (110 s−1 ), it remains periodic and the frequency (0.46 Hz) is quite independent of the imposed flow rate, as observed in capillary experiments. For larger flow rates (215 s−1 ), 1/t increases (around 0.59 Hz), but the movement loses its periodicity: 1/t tends to fluctuate with time. For even larger flow rates (328 s−1 ), we observe an amplification of this phenomenon. The fringes displacements are getting more rapid and more complex: it seems that several successive modes occur and are superimposed to the first one. These results are qualitatively similar to those reported by Yesilata et al. [16] concerning pressure drop fluctuations in an abrupt contraction during elastic instabilities.

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328 s

0,8 1/∆t (Hz)

-1

215 s

0,6

-1

110 s 0,4

-1

87 s 0,2 0 0

2

4

6 8 Time (s)

10

12

Fig. 7. Time evolution of 1/t (fringe oscillations movement) () 87 s−1 , (䊉) 110 s−1 , (
) 215 s−1 and () 328 s−1 (abrupt contraction, T = 180 ◦ C, L = 25 mm and H = 1 mm).

It is also interesting to investigate the time evolution of the instability frequency along the die channel. As it is impossible, regarding the classical experimental set-up, to quantify the instability frequency at the die exit because of side effects, an axisymmetric channel is used. The flow emerging from the parallelepiped reservoir is forced into an axisymmetric capillary die that allows observing extrudate distortions, at the defect occurrence and during its development. The contraction ratio is similar to previous ones and the capillary length L was imposed to be large (L = 35 mm). In the upstream region, we clearly observe isochromatic fringe destabilization, whose kinematics is exactly the same as previously described for the slit die. The upstream instability frequency is characterized in a similar way as presented before, especially during the “periodic regime” corresponding to the “periodic volume instability”. At the die exit, helical distortions do affect the extrudate as soon as fringe oscillations are detected in the upstream region. The instability frequency at the die exit is characterized with the same procedure as in capillary rheometry (Section 3.2). Fig. 8 presents a comparison of the experimental upstream and exit frequencies. It shows that the frequency is not damped down during the flow along the die channel [24].

If we compare the characterization of the volume defect, regarding both extrudate aspect from capillary rheometry and FIB fringes displacements in the reservoir from slit die experiments, it appears clearly that the initiation and the development of the defect can be summarized as follows: the instability occurs for a critical apparent shear rate (in fact, rather a critical Weissenberg or Deborah number), with a fixed frequency. As the flow rate increases, the frequency does not evolve any more, whereas the amplitude of both volume distortions at the die exit and fringes displacements at the die inlet increases. Furthermore, we observe a lost of periodicity for larger flow rates. In capillary rheometry, it corresponds to a degeneration of the helix shape and the transition towards the chaotic regime. Similarly, in the slit die, the fringe displacements also lose their perfect periodicity and symmetry: it is probably also the transition towards a chaotic defect. We will only focus in the following on the first regime, i.e. helical or periodic volume instability. 4.2. Influence of the temperature Apparent flow curves and FIB patterns were characterized for die temperatures of 180, 190 and 200 ◦ C in the case of the abrupt contraction. Fig. 9a presents the apparent flow curves at the three temperatures, and Fig. 9b shows the reduced flow curves obtained by applying the time–temperature superposition principle at a reference temperature of 190 ◦ C. 100 80 60

Pressure (bar)

180

180˚C

40

190˚C Onset of the volume defect

20 200˚C

10

1

(a)

Pressure (bar)

100 80

2

10 -1 Apparent shear rate (s )

10

3

Reference temperature: 190˚C

60 40 Onset of the volume defect

20

10

(b) Fig. 8. Volume instability frequency in the reservoir (䊉) and at the die exit (), as function of apparent shear rate (capillary channel, T = 180 ◦ C, D = 4 mm and L/D = 8.75).

1

10

2

10

3

-1

Reduced apparent shear rate (s )

Fig. 9. (a) Apparent flow curves at different temperatures and (b) mastercurve at 190 ◦ C obtained by time–temperature superposition (abrupt contraction, L = 25 mm and H = 1 mm).

As presented before, the defect occurrence is detected by the upstream flow destabilization. The three arrows in Fig. 9a show that the defect onset is largely postponed at higher shear rates as the temperature increases. The critical shear rate increases from 90 to 150 and 300 s−1 when the temperature increases from 180 to 190 and 200 ◦ C. By applying time–temperature superposition (Fig. 9b), we observe that the instability appears for approximately the same critical pressure. This is in good agreement with capillary rheometer results [23] and confirms the strong sensitivity of the instability initiation to the pressure (or stress) level of the fluid forced in the die convergent. From FIB pattern oscillations, we measured the characteristic “frequency” of the instability (1/t, see previous paragraph), for the three temperatures. Fig. 10a shows that the instability frequency increases markedly with the temperature (0.5, 0.8 and 1.8 Hz for 180, 190 and 200 ◦ C, respectively). It is worth noting that we observe once again the low dependency of the frequency with the apparent shear rate. If we apply a time–temperature superposition, a mastercurve is obtained (Fig. 10b). The frequency corresponding to each temperature can then be deduced from one single experiment. The flow instability at die entrance appears to be very sensitive to the temperature. As both instability occurrence and frequency do obey the time–temperature superposition principle, there is evidence that the volume instability is directly linked to intrinsic viscoelastic properties of the polymer.

Instability frequency (Hz)

2,5 2 200˚C

1,5 1

190˚C

0,5 0

180˚C

0

100

200

300

400

500

-1

(a)

Apparent shear rate (s )

Pressure (bar)

C. Combeaud et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 175–185

10

181

2

30˚ 45˚ 90˚

10

1

10

1

10

2

10

3

-1

Apparent shear rate (s )

Fig. 11. Apparent flow curves for the different half entry angles, at a temperature of 180 ◦ C. Arrows indicate the onset of volume instability (L = 20.75 mm, () 30◦ , (
) 45◦ , ( ) 90◦ and ( ) smooth convergent).

4.3. Influence of the die converging angle Different entry geometries were used, with half entry angles of, respectively, 30, 45 and 90◦ , as well as a smooth convergent (Fig. 3). Apparent flow curves are presented in Fig. 11. The entry effects on the total measured pressure seem to be very low, as the flow curves are quite well superimposed. It appears that the volume instability is all the more postponed as the entry angle is low. As shown in Fig. 12, the critical apparent shear rate increases from 70 to 80 and 160 s−1 when the converging angle decreases from 90 to 45 and 30◦ . It is worth noting that, for half-entry angles higher than 45◦ , the influence on the critical shear rate is very low. These observations are consistent with literature [5,8,26,27]. Fig. 13 compares the FIB patterns corresponding to the converging angles of 30, 45 and 90◦ , at the same apparent shear rate of 270 s−1 (for which the instability is already well developed). As in Fig. 6, the time values t1 and t2 correspond to extreme positions regarding the fringes oscillations. The time values are very different from one geometry to another, as the defect frequency depends on the entry angle (Fig. 14). It appears in Fig. 13 that the defect is all the more developed as the entry angle is important: we can see that the fringes oscillations have a far more important amplitude in the case of the flat entry (90◦ ). Moreover, Fig. 14 shows that the entry geometry controls the defect frequency: it increases when

Reference temperature: 190˚C

500

2 -1

1,5 1 0,5 0 0

(b)

Critical shear rate (s )

Reduced frequency (Hz)

2,5

100

200

300

400

500 -1

Reduced apparent shear rate (s )

Slow convergent

400 300 200 100 0 0

Fig. 10. (a) Instability frequency as a function of the apparent shear rate, for three temperatures and (b) mastercurve at 190 ◦ C obtained by time–temperature superposition (abrupt contraction, L = 25 mm and H = 1 mm).

20 40 60 80 Half entry angle (˚)

100

Fig. 12. Critical apparent shear rate as a function of convergent half entry angle, at a temperature of 180 ◦ C.

182

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Fig. 13. FIB patterns for half entry angles of 30◦ , 45◦ and 90◦ (L = 20.75 mm, γ˙ a = 270 s−1 and T = 180 ◦ C).

the entry angle decreases. If we consider the shape of the convergent, we may imagine that the fluid destabilization is physically limited by the convergent boundaries. Intuitively, for the same apparent shear rate, the volume affected by the instability is less important with a low entry angle. Therefore, the extrudate may present more frequent but smaller distortions than with an important entry angle. For the smooth convergent, the flow behavior is different: even at the maximum flow conditions allowed by the experimental set-up (390 s−1 ), no instability is detected. The isochromatic fringes remain completely stable even for very high pressures reached in the entry region. Such a profile then allows to multiply by at least a factor of four the critical apparent shear rate relative to the instability occurrence.

Instability frequency (Hz)

0,7 0,6 30˚

0,5 0,4

45˚ 90˚

0,3 0,2 0

100

200

300

400

-1

Apparent shear rate (s )

Fig. 14. Instability frequency as a function of the apparent shear rate, for different half entry angles (L = 20.75 mm and T = 180 ◦ C).

5. Discussion It is now well admitted, and it is largely confirmed by our results, that the periodic volume instability is initiated at the die entry [10–21]. An explanation proposed in the literature concerns the destabilization of entry vortices [5,10,20]. If the dynamics of secondary flows is clearly associated to some regimes of upstream instabilities [15–18], it is mainly verified for abrupt contractions and polymer solutions. For PS melt, even if some vortices can effectively be seen in the 90◦ contraction, as also measured by laser-Doppler velocimetry in the literature [28], they totally disappear for the angles of 45 or 30◦ . By carefully considering the entry region of the abrupt contraction, we have observed that, for severe flow conditions corresponding to the transition between periodic volume and chaotic defects, existing vortices can get completely unstable. Moreover, these vortices are from time to time suddenly discharged into the main flow, destabilizing it. Consequently, vortices destabilization exists, but cannot be an explanation for the onset of volume instability, as low angle convergents also present periodic volume defects. The elongational stresses and deformations developing in the convergent have often been associated with the onset of the upstream instability [12,21,30]. Using capillary rheometry experiments, we can derive approximate values of elongational stresses from entrance pressure losses, according to the Cogswell method [29]. It is shown in Fig. 15 that the elongational stress at the onset of the helical instability depends slightly on die diameter and die length. It is

Critical extensional stress (bar)

C. Combeaud et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 175–185 10

2

10

1

10

0

0

5

10

15 20 L/D

25

30

35

Fig. 15. Variation of the critical extensional stress, at a temperature of 200 ◦ C, for various die geometries ((䊉) D = 0.93 mm, () D = 1.39 mm, () D = 2 mm and (
) D = 3 mm).

also quite independent of the temperature. Its value is in the range of 0.5–0.9 MPa, whatever the flow conditions. This is consistent with the range of 1.7–3.7 MPa, cited by Kim and Dealy [30] for the onset of gross melt fracture for a series of polyethylenes. The estimation of a critical elongational stress is obviously more difficult in the case of the slit geometry. Numerical simulations are today in progress to completely interpret these cases. Anyway, we can already explain qualitatively some results. Let us consider the FIB patterns obtained in stable flow conditions, at a low apparent shear rate (20 s−1 ), for the four geometries (Fig. 16). We may observe that, for all geometries, along the centerline, the stresses start to increase in the reservoir, reach a maximum at the slit entry

183

and then decrease along the slit. By counting the number of fringes and assuming the validity of the stress optical rule [10], we can deduce the profile of the principal stress difference (PSD) along the centerline: kλ |σI − σII | = |σxx − σyy | = , (5) CW where σ I and σ II are the principal stresses, σ xx and σ yy the normal stresses (x is the flow direction), k the order of the fringe, λ the wavelength of the monochromatic light, C the stress-optical coefficient and W is the die width. For C, we selected a value of 4 × 10−9 m2 N−1 (at 180 ◦ C), taken from the literature [31]. It appears clearly in Fig. 16 that the fringes number is maximum for the flat entry and decreases when the angle decreases. For the smooth convergent, the stress level is undoubtedly lower, compared to the other geometries. This is depicted in Fig. 17, where we plotted the principal stress difference as a function of the position along the centerline. The PSD is maximum for the flat entry (0.63 MPa), followed by the angles of 45◦ (0.54 MPa), 30◦ (0.36 MPa) and finally the smooth convergent (0.26 MPa). Consequently, we can assume that the shape of the convergent plays a major role in the establishment of the extensional stress field, and that elongational stresses are largely reduced with a smooth convergent. This could explain the effect observed on the onset of volume instability. For the onset of elastic instabilities, McKinley et al. [32] have proposed a criterion based on both curvature of

Fig. 16. FIB patterns in stable flow conditions for the different geometries (L = 20.75 mm, γ˙ a = 20 s−1 and T = 180 ◦ C).

C. Combeaud et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 175–185 Principal stress difference (MPa)

184

Acknowledgements

0,7 0,6

Slit

Reservoir

This work has been carried out through the frame of the 3PI project, funded by European Commission (programme GROWTH, contract G5RT-CT-2000-00238), which is gratefully acknowledged. Extensional viscosity data were provided by LPMI (Pau, France). Thanks to J.F. Agassant and R. Valette (CEMEF) for fruitful discussions concerning this work, and to R. Hainaut (CEMEF) for his help for some birefringence pictures.

0,5 0,4 0,3 0,2

90˚ 45˚

0,1 30˚ 0 -5

Smooth convergent

0 5 10 Axial position (mm)

15

Fig. 17. Variation of the principal stress difference along the centerline for the different geometries (same conditions as in Fig. 16) (L = 20.75 mm, () 30◦ , (䊉) 45◦ , (
) 90◦ and () smooth convergent).

streamlines and elastic normal stresses that give rise to a tension along each streamline:


λ1 Uτ11 Rη0 γ˙

1/2 > Mcrit ,

(6)

where λ1 is a relaxation time, U the characteristic velocity, R the radius of curvature of the streamline, τ 11 the tensile stress in flow direction, η0 the zero-shear viscosity and γ˙ is the characteristic deformation rate. If numerical simulations would be necessary to check the validity of this criterion for the cases we have characterized, it can nevertheless bring some light on some of our results. In particular, the use of small entry angles or smooth convergents will allow to increase the radius of curvature of the streamlines, and thus, together with the decrease of tensile stress, will postpone the instability.

6. Conclusion We have experimentally investigated the volume defects of a polystyrene with a capillary rheometer and an extruder equipped with a transparent slit die. We report that the instability exhibits a strict organization, developing with a constant frequency. The onset of volume defects corresponds to a periodic fluctuation of the FIB pattern at the die entrance. Amplitude and frequency of these fluctuations increase with apparent shear rate, until a progressive loss of periodicity and a transition towards a chaotic regime. It appears clearly that the defect is very sensitive to the temperature and that its characteristics are controlled by the viscoelastic properties of the polymer. We have also put in evidence the strong influence of the convergent shape on the instability occurrence: small entry angles and, much better, a smooth convergent allow to considerably postpone the defect initiation. We have shown that this effect is directly related to the intensity of the extensional stress field in the convergent.

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