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Influence of wall boundary conditions on sharkskin
Journal of Non-Newtonian Fluid Mechanics 228 (2016) 55–63
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Influence of wall boundary conditions on sharkskin Hiroshi Mizunuma∗, Yusuke Ohata Department of Mechanical Engineering, Tokyo Metropolitan University, Hachioji-shi, Tokyo, Japan
a r t i c l e
i n f o
Article history: Received 29 June 2015 Revised 24 November 2015 Accepted 19 December 2015 Available online 30 December 2015 Keywords: Melt fracture Sharkskin instability Wall boundary condition Polymer extrusion
a b s t r a c t Extrudate polymer flow is unstable at high throughputs, and some polymers cause sharkskin roughness on the extrudate surface. Two rotating rollers were installed in a planar die exit to give a moving die wall, and optimal control of the rollers eliminated sharkskin. Extrusion instability for polydimethylsiloxane was investigated as a function of the die wall velocity and extrudate velocities. The minimum die wall velocity VwL required to eliminate sharkskin increased with an increase in the extrudate velocity Vf , and this relationship between VwL and Vf was investigated over the range of Vf = 3.2 to 28.0 mm/s. The apparent reconfiguration rate was introduced for the flow on a moving wall of a viscous power law fluid, and was found to be useful as the sharkskin onset criterion over the Vf range investigated. A new formula was derived for VwL and reasonably agreed with the measured results. The roller die and the introduced methodology allowed extrudate instability to be investigated under extended exit conditions. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Polymer extrusion is one of the most important industrial polymer processes. Melt fracture occurs in the extrusion when a critical flow rate is exceeded, and this has been investigated by many researchers, including in an early study by Howells and Benbow [7]. Petrie and Denn [19] comprehensively reviewed the studies undertaken on melt fracturing at the time of their review. Melt fracture is caused by the instability in polymer flow, and the incipient instability for some polymers induces cyclic fine roughness known as sharkskin. Sharkskin is closely related to the abrupt change in polymer flow at the die exit, where the flow is released from the stick wall boundary inside the die. Cyclic cohesion failure [8] or cyclic stick–slip flow [16] has been observed at a narrow region of the die exit, and these anomalous phenomena are thought to be closely related to sharkskin. Denn [5] performed a detailed review of extrusion instability and polymer slip. It should be noted that instability causing sharkskin can be suppressed by enhancing wall slip. A polymer processing additive (PPA) can be added to enhance polymer slip and eliminate melt fracture. Migler et al. [14] developed a microscopic velocimetry technique for measuring the velocity profile near a die wall, and they found that slip flow was induced by the presence of a PPA. Kharchenko et al. [10] found that polymer slip was induced by absorbed layer of PPA droplets at the die exit. Migler et al. [15] introduced a reconfiguration rate T˙ at
∗
Corresponding author. Tel.: 81 42 677 2713; fax: 81 42 677 2701. E-mail address: [email protected] (H. Mizunuma).
http://dx.doi.org/10.1016/j.jnnfm.2015.12.008 0377-0257/© 2015 Elsevier B.V. All rights reserved.
the die exit and showed for a polymer with a PPA that the onset of sharkskin could be predicted by assuming a constant critical T˙ . Effective wall slip and the elimination of melt fracture have been achieved by grafting silane [11] or adding a silicon rubber coating [13] to the die wall. Sharkskin is closely related to polymer characteristics, the influence of which has been investigated in detail by Burghelea et al. [3]. They used two polymers which had different stability characteristics and investigated the sharkskin instability from various viewpoints. Another approach to sharkskin is to investigate the sharkskin onset under different boundary conditions by changing the die scale or the wall slip velocity. However, for example, slip velocity induced by adding a PPA or coating the die wall is not directly controllable. The ability to investigate sharkskin instability under arbitrary wall slip conditions enables to modify the extrudate flow in various levels and would allow more effective investigations on sharkskin onset to be performed. No-slip boundary conditions occur on the normal wall inside the die, as shown in Fig. 1a. However, sharkskin can be suppressed by enhancing wall slip, as mentioned above. A velocity profile for which wall slip occurs is shown in Fig. 1b. The change in the flow between the inside and outside of the die is decreased by using a non-zero wall slip velocity in the die, and thus extrudate instability can be eliminated. The velocity profile with wall slip resembles the velocity profile on a moving wall without wall slip, as shown in Fig. 1c. If a uniform velocity profile (v(y) = constant) is achieved, the polymer would pass through the die exit without suffering significant strain. Therefore, a moving wall may suppress sharkskin to a degree similar to that achieved using additives and wall coatings. This idea was
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Fig. 1. Extrudate flow and the boundary conditions at the die wall. (a) The no-slip flow that occurred without PPA added showed sharkskin fracture, and (b) the slip flow that occurred with PPA added eliminated sharkskin fracture [14]. (c) The no-slip flow at the moving wall had a similar velocity profile similar to the (b) slip flow, and thus the same suppression of sharkskin fracture was expected.
studied by Benkreira [2], but the detailed results have not been published. In the present study we investigated the relationship between moving wall velocity and sharkskin instability. Two counterrotating rollers were installed in the die to move the wall at the exit at a controllable velocity. We varied the extrudate velocity and the die wall velocity independently, and systematically investigated the effect of changing the wall velocity on sharkskin. The flow model introduced by Cogswell [4] was extended to the flow on a moving wall of a viscous power law fluid and was combined with the continuity condition between the die inside and outside. Based on the obtained relationship, the apparent reconfiguration rate was introduced instead of the reconfiguration rate proposed by Migler et al. [15] and was correlated with the criterion leading to the onset of sharkskin. 2. Method The polymeric fluid used in this study was the linear polydimethylsiloxane (PDMS) Silicone X-21-3043 (Shin-Etsu Chemical Co., Tokyo, Japan), which had a molecular weight of 2.28 × 105 Da and a density of 983 kg/m3 . The same PDMS of a different lot was used in a previous paper [16]. The velocity profiles were measured in an acrylic Couette flow cell using laser Doppler velocimetry, and macroscopic slip was not detected inside the cell for this PDMS. Thus the macroscopic slip between the PDMS and wall is negligible in the present study. The flow curve and the first normal stress difference N1 for the PDMS are shown in Fig. 2. The measurements were performed using a cone-and-plate rheometer (Haake RS600; Thermo Fisher Scientific, Waltham, MA, USA). The cone had a diameter of 20 mm and a gap angle of 1°. The rheological characteristics of the PDMS were similar to those used by Kissi and Piau [12] and Kalyon and Gevgilili [9] to investigate melt fracture. All the
Fig. 2. Shear stress and first normal stress difference as a function of shear rate.
measurements given in Section III were performed at room temperature (approximately 20 °C). The extrudate flow was investigated using two different dies. One was a conventional die with a 20.0 mm wide slit with a 1.0 mm gap, as shown in Fig. 3a. Polymer in the container was extruded through the slit at a constant speed using a plunger driven by a linear motor. The other die was a combined roller die, which had two counter-rotating rollers at the die exit, as shown in Fig. 3b. A typical roller die is used in the calendaring process and is separate from the extrusion die [6,18]. However, the roller die used in this study was combined with the polymer container, and the roller itself acted as the extrusion die. The rollers were covered with fluorocarbon polymer (FEP) tubes of 5 μm thickness to prevent the extrudate sticking to the rollers at a low extrudate velocity. The rollers had diameters of 20.0 mm including the FEP tubes. Both rollers were connected to a DC motor via gears, and the rollers rotated in counter directions (Fig. 4). The gap between the rollers was 1.0 mm, which was the same as the gap in the conventional die. The wall velocity Vw at the die exit was equivalent to the circumferential wall velocity of the rollers, and it was varied by adjusting the speed at which the rollers rotated. The polymer was extruded from the die using a plunger (as for the conventional die), and the average extrudate velocity Vf was controlled by the plunger speed independently of the roller wall velocity Vw . The pressure inside the polymer container was measured using a flush-mounted pressure sensor (PS-5KD; Kyowa Electronic Instruments, Tokyo, Japan) and the thrust force from the motor to the plunger was measured using an in-line load cell (LUR-A-200; Kyowa Electronic Instruments). The surface roughness of the extrudate was recorded using a digital camera (α 65; Sony, Tokyo, Japan), and measured using a laser microscope (VK-X100; Keyence, Osaka, Japan).
Fig. 3. Schematics of (a) a conventional die and (b) a roller die combined with a polymer container.
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3. Results
Fig. 4. Roller die. The polymer was extruded through the slit between two rollers. The static pressure in the polymer container was measured using a flush-mounted pressure sensor.
The surfaces of extrudates produced using the conventional die are shown in Fig. 5. Sharkskin was observed when the average extrudate velocity Vf was 0.8 mm/s, which was the lowest velocity achievable using the linear motor. The critical Vf at which sharkskin first appeared was therefore less than 0.8 mm/s using the conventional die. The roller die had a wall velocity caused by the rotation of the rollers, as mentioned above. Using an appropriate die wall velocity eliminated melt instability and led to the production of an extrudate with a smooth surface. Fig. 6 shows the extrudates produced for the roller wall velocities Vw = of 5.0, 10.0, and 15.0 mm/s with Vf keeping a constant 15.0 mm/s. The extrudate had a smooth transparent surface like a glass plate when Vw was 10.0 mm/s, as shown in Fig. 6b. However, a rough sharkskin-like extrudate surface was found when a lower velocity of Vw = 5.0 mm/s was used, and the illuminated surface had cyclic reflections indicating that it possessed two-dimensional roughness, as shown in Fig. 6a. A Vw of 15.0 mm/s caused the surface to have irregular three-dimensional roughness, as shown in Fig. 6c. The preliminary experiments showed that a smooth extrudate was continuously produced if the wall velocity Vw increased in line with Vf . Various combinations of Vw and Vf were therefore tested to allow us to determine the conditions under which extrudates with smooth surfaces were produced. The occurrences of exudates with smooth surfaces (◦) and with rough surfaces (×) for various combinations of Vw and Vf are shown in Fig. 7. This stability diagram can be divided into a stable region II and unstable regions I and III. When Vf was increased using the stationary roller die (Vw = 0) a smooth extrudate was produced up to Vf = 2.0 mm/s,
Fig. 5. Sharkskin produced using the conventional die (Fig. 3a). The extrudate velocity Vf was (a) 0.8 mm/s and (b) 2.0 mm/s.
Fig. 6. Change in extrudate roughness caused by varying the wall velocity Vw at a constant extrudate velocity Vf of 15.0 mm/s. (a) Vw = 5.0 mm/s, (b) Vw = 10.0 mm/s, and (c) Vw = 15.0 mm/s.
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Fig. 7. Stability as a function of roller wall velocity Vw and average extrudate velocity Vf .
and sharkskin started to be observed at Vf = 3.2 mm/s. When the rollers rotated to give a Vw of 18.0 mm/s, however, a smooth extrudate was produced up to a Vf of 28.0 mm/s, which was the maximum Vf that could be achieved using the system used in this study. The critical Vf was 3.2 mm/s for the stationary roller die, which was a factor of four lower (0.8 mm/s) for the conventional die. The difference between these critical Vf values may have been partly caused by the difference between the surface materials of the dies. The conventional die was made of acrylic resin without any covering or coating, but the roller die surface was covered with a fluorocarbon polymer (FEP). In order to investigate the influence of the die material, the conventional die wall was covered with a FEP adhesive tape similarly as the roller die. The FEP die surface did not influenced the extrudate roughness. One of the other possible causes is the curvature of the die exit. The conventional die was a 90° exit, and the roller die had the curvature radius of 10 mm. Arda and Mackley [1] used the planar dies of 1.2 mm gap and showed that the curvature of a die exit reduced sharkskin. That is, the die with a curvature radius of 2 mm postponed sharkskin up to 21% higher shear stress than the die with a 90° exit. Although the onset Vf values of sharkskin are different between the conventional die and the roller die, our purpose is to investigate the effect of moving wall. Thus the results for moving wall are compared with those for the roller die of Vw = 0 hereafter. The three-dimensional surface roughness was investigated quantitatively using a laser microscope. The roughness Ra was defined as
Ra =
1 L
0
L
|Rn |dx,
(1)
where x is the extrusion direction coordinate, L is the measurement length, and Rn is the deviation of the local surface height from the average surface height. The extrudates produced at an average extrudate velocity Vf of 22.0 mm/s are shown in Fig. 8. At a Vf of 22.0 mm/s, a Vw of 5.8 mm/s or 11.0 mm/s was not high enough to suppress melt instability, as shown in Fig. 7, and the extrudate produced had a roughness Ra of 32 μm at a Vw of 5.8 mm/s (Fig. 8a) and an Ra of 7 μm at a Vw of 11.0 mm/s (Fig. 8b). When Vw was 5.8 mm/s (Fig. 8a), the surface roughness was high and approximately two-dimensional. A Vw of 14.0 mm/s eliminated sharkskin at a Vf of 22.0 mm/s, as shown in Fig. 7, and the extrudate surface was flat at this optimized Vw of 14.0 mm/s, as shown in Fig. 8c. The Ra was 0 μm for these velocities. Tests were also performed for Vw higher than critical, as shown in Fig. 7. The Ra values 5 μm at a Vw of 18.0 mm/s (Fig. 8d), and 14 μm at
a Vw of 21.5 mm/s (Fig. 8e). The surfaces produced at the lower and upper boundaries of the stable region II are shown in Fig. 8b and 8c, respectively. The surfaces produced at these critical conditions were visibly rough and irregular compared with the surface shown in Fig. 8a. The irregular three-dimensional roughness was remarkable when Vw was 21.5 mm/s (Fig. 8d), and contrasted with the regular two-dimensional roughness shown in Fig. 8a. The extrudate kept the surface roughness apparently constant along the upper and lower boundaries of the stable region II by adjusting the wall velocity Vw . Along these boundaries, a shear stress τ w and a ratio of (N1 /τ ) w on the die wall were calculated using the wall shear rate γ˙ w = (2n + 1 )/n · (V f − Vw )/H for a plane Poiseuille flow of a power law fluid τ = K γ˙ n . The wall shear rate γ˙ w calculated was in the range of 34 to 139 s−1 on the lower boundary and 32 to 75 s−1 on the upper boundary. Although these ranges are higher than the range plotted in Fig. 2, the PDMS (a molecular weight of 4.28 × 105 gmol−1 ) used by Kissi and Piau [12] showed a constant n (∼ =1/3) in the range of 10−1 s−1 to 3 −1 10 s . Similarly, Fig. 2 showed n = 0.3 in the high shear range of 0.8 s−1 to 8 s−1 . Thus we used n = 0.3 and K = 104 Pa sn in Fig. 2 to calculate τ w and (N1 /τ )w on each boundary. On the lower boundary, an increase in Vf increased τ w from 29 kPa to 44 kPa and increased (N1 /τ )w from 6.1 to 11, as shown in Fig. 9. Meulenbroek et al. [17] analyzed an instability of viscoelastic Poisueille flow, and showed that the instability emerged at the critical ratio (N1 /τ )w ∼ = 5. Although this critical ratio is in good agreement with that for the stationary roller die in the present study, the ratio (N1 /τ )w plotted in Fig. 9 increases to the double along the lower boundary. Thus it is difficult to use τ w or (N1 /τ )w as the onset criterion of instability in the present study. Since the surface roughness resembles sharkskin near the lower boundary of region II, as shown in Fig. 8a, another approach to predict sharkskin instability is tested to give the onset criterion in the next section. Along the upper boundary, an increase in Vf changes the signs of τ w and (N1 /τ )w from minus to plus. That is, the wall velocity Vw is higher than Vf for Vf < 10 mm/s, and the flow is mainly driven by the moving wall at the exit. An increase in Vf turns the exit flow to a pressure-driven flow for Vf > 10 mm/s (Vf > Vw ) on the upper boundary. Vf = Vw suggests a uniform velocity profile and thus a strainless stable flow. However, even if Vf = Vw , the stable flow could not be achieved for Vf > 10 mm/s. The velocity profile on a die wall is influenced by the upstream flow uniformity. In the present setup, the moving wall drags flow from the upstream location. If the drag flow is too much to pass through the narrow die exit, the excess drag flow would disturb the exrudate flow. Thus it is speculated that the present setup could not achieve a uniform velocity profile under the influence of the upstream flow. Not only the die exit but also the upstream flow need to be considered to investigate the transition between region II and III. A very high pressure is applied to a very viscous polymer being extruded through a conventional narrow die. On the other hand, moving the die wall is expected to decrease the pressure inside the polymer container, and this would be beneficial to the extrusion process. The pressure was therefore measured inside the polymer container using a flush-mounted pressure sensor. Since the pressure sensor capacity (500 kPa) was not high enough to apply a very high pressure to the polymer, Vf was limited to 4.0 mm/s, which is rather low. When Vw was increased from 2.0 mm/s to 4.0 mm/s, the pressure decreased from 536 kPa to 158 kPa. In addition to the pressure sensor, a load cell was inserted between the linear motor and the piston, and the thrust force F was measured when Vf was 16 mm/s. The dimensionless force F/F∗ and the dimensionless pressure P/P∗ are shown as functions of Vw /Vf in Fig. 10 (F∗ and P∗ are the thrust force F and the static pressure P, respectively, when Vw /Vf = 1). Only a limited range of Vw and Vf values could be tested because of the low capacities of the
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Fig. 8. Three-dimensional surface roughness. The extrudate velocity Vf was 22.0 mm/s. (a) Cyclic sharkskin fracture. Ra = 32 μm. (b) Lower boundary of the stable region II. Ra = 7 μm. (c) Elimination of melt fracture. Ra = 0 μm. (d) Upper boundary of the stable region II. Ra = 5 μm. (e) Irregular melt fracture. Ra = 14 μm.
sensors, but it was clear that F/F∗ and P/P∗ decreased as the Vw /Vf increased. The rotating roller die exerted a drag force on the polymer, and the contraction flow was driven by this drag force rather than by the pressure force, and the pressure required to extrude the polymer was decreased to some degree. Using the roller die at an appropriate Vw therefore allows sharkskin fracture to be suppressed and at the same time decreases the driving pressure required to extrude the polymer. The sharkskin elimination obtained here is not limited to a planar die. The same elimination has been observed in our preliminary experiment on a circular die, which is composed of counterrotating semicircular grooves on two rollers. Although the die swell was not measured for the present planar die because of the difficulty of observation, the die swell has been observed to decrease in the stability region II for the circular die. If the installation of a roller die is possible, the instability modification by a moving wall would appear independently of a planar or a circular die. 4. Discussion The moving wall used in this study gave a similar near-wall flow to that achieved using a PPA to enhance wall slip, and allowed sharkskin to be eliminated. Migler et al. [14,15] measured the slip velocity induced by a PPA and investigated its effect on the elimi-
nation of sharkskin, and the results presented by Migler et al. [15] are plotted along with our results in Fig. 11. The Vf range of 10.5 to 105 mm/s found using a PPA was higher than the range of 2.0 to 28.0 mm/s found using the moving wall. The non-dimensionalized velocities Vw /Vf (this study) and Vs /Vf [15] are plotted as a function of Vf /Vf0 in Fig. 11 so that the results can be compared in the same range. Vs is the slip velocity experimentally determined by Migler et al. [15] and presented in their Fig. 9. Vf0 is the critical Vf at which sharkskin appears when a stationary roller die or a conventional die is used without a PPA being added. The Vf 0 was 3.2 mm/s for our stationary roller die and 3.8 mm/s (a throughput of 0.4 g/min) for the polymer without the PPA added in the study performed by Migler et al. [15]. The stable extrudates appeared in region II when the roller die was used, and the stable range of Vw /Vf became narrower as Vf /Vf0 increased. Region II using the roller die covered the Vs /Vf points that were stabilized by adding the PPA. On the other side, when Vf /Vf0 was 16 or 28, the unstable region I covered the Vs /Vf points that were not stabilized by adding the PPA. This consistency suggests that melt instability could be eliminated through the same mechanism in both the moving wall and the polymer wall slip techniques. The surface layer of the polymer is stretched rapidly outside the die, and Cogswell [4] proposed that the instability induced by this stretching causes sharkskin. Cogswell proposed that
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sured the flow velocity directly using tracer particles, and they calculated T˙ from these measurements. However, the local flow velocity was not measured in our study. Thus we used a viscous power law model τ = K γ˙ n , and assumed a Poiseuille flow inside the die and a uniform jet outside the die, as shown in Fig. 12. This flow model was introduced by Cogswell [4] for a Newtonian fluid without wall slip. A Poiseuille flow on a moving wall has the velocity profile for a viscous power-law fluid described by Eq. (3),
υ (y ) = Vw +
Fig. 9. Changes in wall shear stress τ w and wall stress ratio (N1 /τ )w along the lower and upper boundaries of the stable region II. τ w and (N1 /τ )w were calculated using τ w = 1.0 × 104 γ˙ w 0.3 and (N1 )w = 1.5 × 104 γ˙ w 0.7 in Fig. 2.
(2k − 1 )n + 1 n+1
(V f − Vw ) 1 − (y/H )(n+1)/n ,
(3)
where k is 3/2 for a plane Poiseuille flow and 2 for an axisymmetric Poiseuille flow, y is the distance from the center line, and H is the half-channel height for a plane Poiseuille flow and the radius for an axisymmetric Poiseuille flow. The PDMS used here did not show macroscopic slip on the acrylic wall, as mention in Section II, and sharkskin appeared similarly for both the acrylic and FEP walls, as mention in Section III. Thus the slip between polymer and wall is not considered here. A thin layer adjacent to the die wall is stretched outside, and the thickness is reduced from δ H to h, as shown in Fig. 12. The total extensional deformation is proportional to Vf /Vave , where Vave is the near-wall average velocity for Eq. (3) from y = H − δ H to H, calculated as shown in Eq. (4).
Vw 31−n Vave = + k+ Vf Vf 21+n
Vw 1− Vf
δH H
+ O((δ H/H ) ) 2
(4)
The average velocity Vave of the thin layer inside the die is related to the final velocity Vf outside the die by the continuity equation (Eq. (5)).
δ H · Vave = h · V f
(5)
The higher order terms of δ H/H can almost be neglected in Eq. (4). Removing Vave from Eqs. (4) and (5) gives Eq. (6).
δH H
=
⎧ ⎨ ⎩
2 k+ Fig. 10. Dimensionless static pressure P/P∗in the polymer container and the dimensionless thrust force F/F∗ as functions of Vw /Vf .
extensional strain rate ε˙ controls the onset of instability, and this was used by Migler et al. [15] to explain how sharkskin was eliminated using a PPA. However, the extensional strain rate could not be used as the instability criterion for a polymer with a PPA added. Cogswell analyzed the extrudate flow under no-slip conditions, and only a narrow range of flow velocities could be investigated. However, the polymer with a PPA added underwent wall slip, and a stable extrudate was produced up to a much higher critical extensional strain rate than was produced by the polymer without a PPA. Rather than the extensional strain rate, Migler et al. [15] proposed that the reconfiguration rate T˙ (the product of the maximum extensional strain rate ε˙ max at the die exit and the extensional deformation T from the inside of the die to the final extrudate, as shown in Eq. (2)) controls sharkskin instability.
T˙ = ε˙ max (T + 1 )
(2)
Microscopic measurements showed that sharkskin started at approximately the same reconfiguration rate T˙ ∼ 700/s for a polymer with and without a PPA added, despite their ε˙ max values that differed by a factor of five. Here we investigate what kind of parameters determine the transition at the lower boundary of region II. Migler et al. mea-
Vw Vf
2
31−n +4 k+ 21+n
31−n 21+n
1−
Vw Vf
Vw 1− Vf
⎫ ⎬
Vw h − H Vf ⎭
(6)
Then the relationship between h and δ H is shown in Eq. (7).
δH H
=
h H
k+
31−n for Vw ∼ = h for Vw ∼ = 0, and δ H ∼ = Vf 21+n (7)
Since h/H ∼ 0.01 at the lower boundary of region II in our study, the higher order terms of h/H and δ H/H are almost negligible, and thus these higher order terms will be neglected from now on. The die wall velocity Vw and the extrudate velocity Vf are determined by the operating conditions, and if h is given, the other parameters δ H and Vave can be obtained from Eqs. (6) and (5). In addition, the continuity Eq. (5) between the inside and outside velocities enables to estimate the extensional deformation of the wall layer. That is, Eqs. (4) and (5) gives Vf /Vave as a monotone decreasing function of Vw /Vf , and this Vf /Vave is used as the total extensional deformation Ta (≡ Vf /Vave ) in place of (T + 1) in Eq. (2). The only remaining unknown is the maximum extensional strain rate ε˙ max , which is discussed below. The extrudate flow is characterized by very low Reynolds numbers, much less than unity. The flow boundary condition changes discontinuously at the die exit, and the free surface is accelerated rapidly outside the die exit. Richardson [20] analyzed this inertialess ‘stick–slip’ flow and found that v ∼ 1.162x1/2 is the asymptotic velocity along a jet surface, where x is the streamwise distance from the die exit. This asymptotic velocity suggests that there will
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61
Fig. 11. Stability as a function of the dimensionless wall velocity Vw /Vf and the dimensionless average extrudate velocity Vf /Vf0 . The dimensionless slip velocities Vs /Vf (● and ∗) are measurements made by Migler et al. [15], in which a circular die with a diameter of 1.6 mm was used. hc in Eq. (13) was assumed to be 7 μm for both this study and the study by Migler et al. [15].
Fig. 12. Die exit flow and moving wall velocity Vw . The inside near-wall layer with a thickness of δ H is stretched and reduced to a thickness h. The velocities of these layers are Vave and Vf , and the total extensional deformation is reduced to Vf /Vave .
be a very large maximum extensional strain rate ε˙ max at the die exit x = 0. Tanner [21] numerically simulated capillary die swell for a Newtonian fluid, and showed that the extensional stress reached approximately 20ηVf /H at the die lip, where η is the viscosity. Therefore, ε˙ max is estimated to be 20Vf /H. We define the apparent extensional strain rate ε˙ a (≡ (Vf − Vw )/H) rather than ε˙ max , and we include the effect of the non-zero wall velocity through Vw . Migler et al. [15] used the reconfiguration rate T˙ as the onset criterion for sharkskin instability, as mentioned above. Here we define the apparent reconfiguration rate T˙a as shown in Eq. (8).
V f − Vw V f · T˙a = ε˙ a Ta = H Vave
(8)
Vf / Vave was calculated by substituting the experimental results Vw /Vf and hc /H on the lower boundary into Eqs. (5) and (6), and by removing δ H from those equations. Then T˙a was obtained from Eq. (8) along the lower boundary and plotted in Fig. 13. The power law constant n = 0.3 and the surface roughness hc = 7 μm were assumed in Eqs. (5) and (6). Along the lower boundary, an increase in Vf increased the apparent extensional strain rate ε˙ a = (Vf − Vw )/H and by contrast, decresed the total extensional deformation Ta = Vf /Vave (= δ H/hc ). Their product T˙a was approximately kept constant along the lower boundary. These results suggest that a constant T˙a along the lower boundary can be used as the onset criterion T˙a crit of sharkskin. T˙a calculated for n = 1 is also plotted in Fig. 13. The smaller n corresponds the steeper velocity gradient
Fig. 13. Changes in T˙a , (Vf − Vw )/H, and Vf /Vave along the lower boundary of the stable region II. Vf and Vw were the experimental values on the lower boundary, from which Vave was calculated using Eqs. (5) and (6). The surface roughness was assumed to be hc = 7 μm. T˙a was calculated using Eq. (8).
in the thin layer of a thickness δ H. However, δ H decreases with an increase in Vf , and Vave is approximately reduced to Vw . Thus the influence of n is negligible for large Vf , as shown in Fig. 13. Next, we show that the lower boundary can be obtained as a function of hc /H and Vf /Vf0 instead of T˙a crit , because these can easily be found experimentally. Using Eq. (8), Vave can be removed from Eqs. (4) and (5) to give Eqs. (9) and (10).
1−
δH H
Vw Vf
−1 =1+
=
h Vw 1− H Vf
Vf 31−n − k+ 21+n H T˙a
−1
H T˙a Vf
δH H
(9)
(10)
Assuming that VwL is Vw at the lower boundary, and removing δ H/H from Eqs. (9) and (10), VwL is given as a function of Vf as
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shown in Eq. (11).
5. Conclusions
3 1 − n V f /H T˙a crit hc Vw L = − k+ Vf 21+n 1 + V f /H T˙a crit H
Vf Vf 1+ H T˙a crit H T˙a crit
(11) Assuming that Vf0 is the critical extrudate velocity for a stationary die (VwL = 0), the point Vf = Vf0 crosses the horizontal axis (Vw = 0) and the lower boundary (Vw = VwL ) of region II in Fig. 7. At this point, Eq. (11) can be reduced to Eq. (12),
Vf 0 = H T˙a crit
31−n hc k+ , H 21+n
(12)
which defines the relationship between Vf0 , hc , andT˙a crit . A polymer with a higher T˙a crit (i.e., a more stable polymer) gives a higher Vf0 for the same roughness height hc . Defining ξ as the right hand side of Eq. (12), Eq. (11) can be reduced to Eq. (13).
ξ · V f /V f 0 hc 31−n Vw L = − k+ Vf 1 + ξ · V f /V f 0 H 21+n
ξ·
Vf Vf 1+ξ · Vf 0 Vf 0
Two counter-rotating rollers were installed in a die exit, and the effect of Vw on sharkskin instability of PDMS was investigated. Sharkskin appeared at a Vf of 3.2 mm/s when the roller die was used to extrude PDMS without the rollers rotating. In contrast, sharkskin did not appear up to a Vf of 28.0 mm/s when the rollers were rotated. A stability diagram was plotted for the various combinations of Vw and Vf that were tested, and a stable region was found for a range of Vw values. The apparent reconfiguration rate T˙a was defined as the product of the apparent extensional strain rate ε˙ a and the total extensional deformation Ta for the flow on a moving wall of a viscous power law fluid. The experimental results showed that Ta decreased with an increase in ε˙ a , and that their product T˙a kept constant along the lower boundary of the stable region II. This constant T˙a was found out to be the onset criterion of sharkskin over the range of Vf = 3.2 to 28.0 mm/s and was defined as the critical apparent reconfiguration rate T˙a crit . T˙a crit can be correlated with Vf0 , and the relationship between these onset criterions and polymer characteristics is a next important issue . Acknowledgments
(13) Therefore, if Vf0 is given from the sharkskin instability for Vw = 0, the lower boundary of region II can be obtained as a function of Vf /Vf0 for a constant hc /H . Although T˙a crit does not explicitly appear in Eq. (13), it can be calculated from experimental results for Vf0 and hc /H using Eq. (12). If Vf is sufficiently high, Eq. (13) can be reduced to approximately VwL /Vf = 1, which means that the polymer passes through the die exit maintaining a uniform velocity profile v(y) = VwL . In other words, any weak deviation from a uniform velocity gives T˙a higher than T˙a crit under such a high Vf . It is therefore necessary to use a carefully designed system, including the parts upstream of the die, to achieve a uniform velocity profile at the die exit. As shown in Fig. 8, the sharkskin roughness was 7 μm for the lower boundary, and this was in the same range as the sharkskin roughness found by Venet and Vergnes [22]. Assuming hc to be 7 μm, the plotted curve of Eq. (13) agrees qualitatively with the experimental lower boundary of region II, as shown in Fig. 11. Although Eq. (13) was plotted for n = 0.3 and 1, the influence of n on VwL /Vf was not remarkable, similarly as shown on T˙a in Fig. 13. When we calculated Eq. (13) for the result of Migler et al. [15], we assumed n = 1 because of unknown n. Howells and Benbow [7] and Cogswell [4] found that sharkskin appeared at a constant extrudate velocity Vf0 , independent of the die diameter. Therefore, Vf 0 is closely related to the polymer characteristics when sharkskin occurs. Assuming that Vw = 0, Eq. (9) can be reduced to V f 0 /T˙a crit = (k + 3(1 − n )/2(1 + n ))δ H. Migler et al. [15] considered δ H to be independent of the die diameter, and suggested that the choice of T˙a crit is consistent with the use of Vf 0 as the onset criterion for sharkskin instability. For the moving wall die, Eq. (13) suggests that the transition between regions I and II is dependent on the die geometry H through hc /H. However, the influence of H is rather modest. We extended the flow model developed by Cogswell [4] to the flow on a moving wall of a viscous power law fluid, and its combination with the continuity condition enabled to correlate the apparent reconfiguration rate T˙a with Vw and Vf . A constant T˙a crit corresponded to the onset criterion of sharkskin and could be used to predict the instability transition across the lower boundary of the stable region II. The polymer tested here is still limited to a PDMS. However, since sharkskin can be controlled by adjusting Vw for the moving wall, a more detailed study on the moving wall effect would allow the melt instability mechanism to be described in more detail.
The first author (H. M.) stayed at the laboratory of Prof. M. M. Denn, who suggested H. M. to turn his attention to a die exits. H. M. would like to thank Prof. M. M. Denn for this and other insightful suggestions. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.jnnfm.2015.12.008. References [1] D.R. Arda, M.R. Mackley, The effect of die exit curvature, die surface roughness and a fluoropolymer additive on sharkskin extrusion instabilities in polyethylene processing, J. Non-Newton. Fluid Mech. 126 (2005) 47–61. [2] H. Benkreira, The control of extrudate swell and instabilities using the rotating roller die, in: Proceeding of Polymer Processing Society 24th Annual Meeting, II, 2008, p. 12. [3] T.I. Burghelea, H.J. Griess, H. Münstedt, Comparative investigations of surface instabilities (“sharkskin”) of a linear and a long-chain branched polyethylene, J. Non-Newton. Fluid Mech. 165 (2010) 1093–1104. [4] F.N. Cogswell, Stretching flow instabilities at the exits of extrusion dies, J. NonNewton. Fluid Mech. 2 (1977) 37–47. [5] M.M. Denn, Extrusion instabilities and wall slip, Annu. Rev. Fluid Mech. 33 (2001) 265–287. [6] M.P. Groover, Fundamentals of Modern Manufacturing: Materials, Processes, and Systems, Prentice Hall, New Jersey, 1996. [7] E.R. Howells, J.J. Benbow, Flow defects in polymer melts, Trans. Plast. Inst. 30 (1962) 240–253. [8] Y.W. Inn, R.J. Fischer, M.T. Shaw, Visual observation of development of sharkskin melt fracture in polybutadiene extrusion, Rheol. Acta 37 (1998) 573–582. [9] D.M. Kalyon, H. Gevgilili, Wall slip and extrudate distortion of three polymer melts, J. Rheol. 47 (2003) 683–699. [10] S.B. Kharchenko, P.M. McGuiggan, K.B. Migler, Flow induced coating of fluoropolymer additives: development of frustrated total internal reflection imaging, J. Rheol. 47 (2003) 1523–1545. [11] N. EL Kissi, L. Leger, J.M. Piau, A. Mezghani, Effect of surface properties on polymer melt slip and extrusion defects, J. Non-Newton. Fluid Mech. 52 (1994) 249–261. [12] N. EL Kissi, J.M. Piau, The different capillary flow regimes of entangled polydimethylsiloxane polymers: macroscopic slip at the wall, hysteresis and cork flow, J. Non-Newton. Fluid Mech. 37 (1990) 55–94. [13] O. Kulikov, K. Hornung, A simple way to suppress defects in the processing of polyethylene, J. Non-Newton. Fluid Mech. 124 (2004) 103–114. [14] K.B. Migler, C. Lavallée, M.P. Dillon, S.S. Woods, C.L. Gettinger, Visualizing the elimination of sharkskin through fluoropolymer additives: coating and polymer-polymer slippage, J. Rheol. 45 (2001) 565–581. [15] K.B. Migler, Y. Son, F. Qiau, K. Flynn, Extensional deformation, cohesive failure, and boundary conditions during sharkskin melt fracture, J. Rheol. 46 (2002) 383–400. [16] H. Mizunuma, H. Takagi, Cyclic generation of wall slip at the exit of plane Couette flow, J. Rheol. 47 (2003) 737–757.
H. Mizunuma, Y. Ohata / Journal of Non-Newtonian Fluid Mechanics 228 (2016) 55–63 [17] B. Meulenbroek, C. Storm, V. Bertola, C. Wagner, D. Bonn, W. van Saarloos, Intrinsic route to melt fracture in polymer extrusion: a weekly nonlinear subcritical instability of viscoelastic Poiseuille flow, Phy. Rev. Lett. 90 (2) (2003) 1–4 024502. [18] J.R.A. Pearson, “Mechanics of Polymer Processing,” Elsevier Applied Science Publishing, London and New York. [19] C.J.S. Petrie, M.M. Denn, Instabilities in polymer processing, AIChE J. 22 (1976) 209.
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[20] S. Richardson, A ’stick-slip’ problem related to the motion of a free jet at low Reynolds numbers, Proc. Camb. Philos. Soc. 67 (1970) 477–489. [21] R.I. Tanner, Die-swell reconsidered: some numerical solutions using a finite element program, Appl. Polym. Symp. 20 (1973) 201–208. [22] C. Venet, B. Vergnes, Experimental characterization of sharkskin in polyethylenes, J. Rheol. 41 (1997) 873–892.
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Journal of Non-Newtonian Fluid Mechanics journal homepage: www.elsevier.com/locate/jnnfm
Influence of wall boundary conditions on sharkskin Hiroshi Mizunuma∗, Yusuke Ohata Department of Mechanical Engineering, Tokyo Metropolitan University, Hachioji-shi, Tokyo, Japan
a r t i c l e
i n f o
Article history: Received 29 June 2015 Revised 24 November 2015 Accepted 19 December 2015 Available online 30 December 2015 Keywords: Melt fracture Sharkskin instability Wall boundary condition Polymer extrusion
a b s t r a c t Extrudate polymer flow is unstable at high throughputs, and some polymers cause sharkskin roughness on the extrudate surface. Two rotating rollers were installed in a planar die exit to give a moving die wall, and optimal control of the rollers eliminated sharkskin. Extrusion instability for polydimethylsiloxane was investigated as a function of the die wall velocity and extrudate velocities. The minimum die wall velocity VwL required to eliminate sharkskin increased with an increase in the extrudate velocity Vf , and this relationship between VwL and Vf was investigated over the range of Vf = 3.2 to 28.0 mm/s. The apparent reconfiguration rate was introduced for the flow on a moving wall of a viscous power law fluid, and was found to be useful as the sharkskin onset criterion over the Vf range investigated. A new formula was derived for VwL and reasonably agreed with the measured results. The roller die and the introduced methodology allowed extrudate instability to be investigated under extended exit conditions. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Polymer extrusion is one of the most important industrial polymer processes. Melt fracture occurs in the extrusion when a critical flow rate is exceeded, and this has been investigated by many researchers, including in an early study by Howells and Benbow [7]. Petrie and Denn [19] comprehensively reviewed the studies undertaken on melt fracturing at the time of their review. Melt fracture is caused by the instability in polymer flow, and the incipient instability for some polymers induces cyclic fine roughness known as sharkskin. Sharkskin is closely related to the abrupt change in polymer flow at the die exit, where the flow is released from the stick wall boundary inside the die. Cyclic cohesion failure [8] or cyclic stick–slip flow [16] has been observed at a narrow region of the die exit, and these anomalous phenomena are thought to be closely related to sharkskin. Denn [5] performed a detailed review of extrusion instability and polymer slip. It should be noted that instability causing sharkskin can be suppressed by enhancing wall slip. A polymer processing additive (PPA) can be added to enhance polymer slip and eliminate melt fracture. Migler et al. [14] developed a microscopic velocimetry technique for measuring the velocity profile near a die wall, and they found that slip flow was induced by the presence of a PPA. Kharchenko et al. [10] found that polymer slip was induced by absorbed layer of PPA droplets at the die exit. Migler et al. [15] introduced a reconfiguration rate T˙ at
∗
Corresponding author. Tel.: 81 42 677 2713; fax: 81 42 677 2701. E-mail address: [email protected] (H. Mizunuma).
http://dx.doi.org/10.1016/j.jnnfm.2015.12.008 0377-0257/© 2015 Elsevier B.V. All rights reserved.
the die exit and showed for a polymer with a PPA that the onset of sharkskin could be predicted by assuming a constant critical T˙ . Effective wall slip and the elimination of melt fracture have been achieved by grafting silane [11] or adding a silicon rubber coating [13] to the die wall. Sharkskin is closely related to polymer characteristics, the influence of which has been investigated in detail by Burghelea et al. [3]. They used two polymers which had different stability characteristics and investigated the sharkskin instability from various viewpoints. Another approach to sharkskin is to investigate the sharkskin onset under different boundary conditions by changing the die scale or the wall slip velocity. However, for example, slip velocity induced by adding a PPA or coating the die wall is not directly controllable. The ability to investigate sharkskin instability under arbitrary wall slip conditions enables to modify the extrudate flow in various levels and would allow more effective investigations on sharkskin onset to be performed. No-slip boundary conditions occur on the normal wall inside the die, as shown in Fig. 1a. However, sharkskin can be suppressed by enhancing wall slip, as mentioned above. A velocity profile for which wall slip occurs is shown in Fig. 1b. The change in the flow between the inside and outside of the die is decreased by using a non-zero wall slip velocity in the die, and thus extrudate instability can be eliminated. The velocity profile with wall slip resembles the velocity profile on a moving wall without wall slip, as shown in Fig. 1c. If a uniform velocity profile (v(y) = constant) is achieved, the polymer would pass through the die exit without suffering significant strain. Therefore, a moving wall may suppress sharkskin to a degree similar to that achieved using additives and wall coatings. This idea was
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Fig. 1. Extrudate flow and the boundary conditions at the die wall. (a) The no-slip flow that occurred without PPA added showed sharkskin fracture, and (b) the slip flow that occurred with PPA added eliminated sharkskin fracture [14]. (c) The no-slip flow at the moving wall had a similar velocity profile similar to the (b) slip flow, and thus the same suppression of sharkskin fracture was expected.
studied by Benkreira [2], but the detailed results have not been published. In the present study we investigated the relationship between moving wall velocity and sharkskin instability. Two counterrotating rollers were installed in the die to move the wall at the exit at a controllable velocity. We varied the extrudate velocity and the die wall velocity independently, and systematically investigated the effect of changing the wall velocity on sharkskin. The flow model introduced by Cogswell [4] was extended to the flow on a moving wall of a viscous power law fluid and was combined with the continuity condition between the die inside and outside. Based on the obtained relationship, the apparent reconfiguration rate was introduced instead of the reconfiguration rate proposed by Migler et al. [15] and was correlated with the criterion leading to the onset of sharkskin. 2. Method The polymeric fluid used in this study was the linear polydimethylsiloxane (PDMS) Silicone X-21-3043 (Shin-Etsu Chemical Co., Tokyo, Japan), which had a molecular weight of 2.28 × 105 Da and a density of 983 kg/m3 . The same PDMS of a different lot was used in a previous paper [16]. The velocity profiles were measured in an acrylic Couette flow cell using laser Doppler velocimetry, and macroscopic slip was not detected inside the cell for this PDMS. Thus the macroscopic slip between the PDMS and wall is negligible in the present study. The flow curve and the first normal stress difference N1 for the PDMS are shown in Fig. 2. The measurements were performed using a cone-and-plate rheometer (Haake RS600; Thermo Fisher Scientific, Waltham, MA, USA). The cone had a diameter of 20 mm and a gap angle of 1°. The rheological characteristics of the PDMS were similar to those used by Kissi and Piau [12] and Kalyon and Gevgilili [9] to investigate melt fracture. All the
Fig. 2. Shear stress and first normal stress difference as a function of shear rate.
measurements given in Section III were performed at room temperature (approximately 20 °C). The extrudate flow was investigated using two different dies. One was a conventional die with a 20.0 mm wide slit with a 1.0 mm gap, as shown in Fig. 3a. Polymer in the container was extruded through the slit at a constant speed using a plunger driven by a linear motor. The other die was a combined roller die, which had two counter-rotating rollers at the die exit, as shown in Fig. 3b. A typical roller die is used in the calendaring process and is separate from the extrusion die [6,18]. However, the roller die used in this study was combined with the polymer container, and the roller itself acted as the extrusion die. The rollers were covered with fluorocarbon polymer (FEP) tubes of 5 μm thickness to prevent the extrudate sticking to the rollers at a low extrudate velocity. The rollers had diameters of 20.0 mm including the FEP tubes. Both rollers were connected to a DC motor via gears, and the rollers rotated in counter directions (Fig. 4). The gap between the rollers was 1.0 mm, which was the same as the gap in the conventional die. The wall velocity Vw at the die exit was equivalent to the circumferential wall velocity of the rollers, and it was varied by adjusting the speed at which the rollers rotated. The polymer was extruded from the die using a plunger (as for the conventional die), and the average extrudate velocity Vf was controlled by the plunger speed independently of the roller wall velocity Vw . The pressure inside the polymer container was measured using a flush-mounted pressure sensor (PS-5KD; Kyowa Electronic Instruments, Tokyo, Japan) and the thrust force from the motor to the plunger was measured using an in-line load cell (LUR-A-200; Kyowa Electronic Instruments). The surface roughness of the extrudate was recorded using a digital camera (α 65; Sony, Tokyo, Japan), and measured using a laser microscope (VK-X100; Keyence, Osaka, Japan).
Fig. 3. Schematics of (a) a conventional die and (b) a roller die combined with a polymer container.
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3. Results
Fig. 4. Roller die. The polymer was extruded through the slit between two rollers. The static pressure in the polymer container was measured using a flush-mounted pressure sensor.
The surfaces of extrudates produced using the conventional die are shown in Fig. 5. Sharkskin was observed when the average extrudate velocity Vf was 0.8 mm/s, which was the lowest velocity achievable using the linear motor. The critical Vf at which sharkskin first appeared was therefore less than 0.8 mm/s using the conventional die. The roller die had a wall velocity caused by the rotation of the rollers, as mentioned above. Using an appropriate die wall velocity eliminated melt instability and led to the production of an extrudate with a smooth surface. Fig. 6 shows the extrudates produced for the roller wall velocities Vw = of 5.0, 10.0, and 15.0 mm/s with Vf keeping a constant 15.0 mm/s. The extrudate had a smooth transparent surface like a glass plate when Vw was 10.0 mm/s, as shown in Fig. 6b. However, a rough sharkskin-like extrudate surface was found when a lower velocity of Vw = 5.0 mm/s was used, and the illuminated surface had cyclic reflections indicating that it possessed two-dimensional roughness, as shown in Fig. 6a. A Vw of 15.0 mm/s caused the surface to have irregular three-dimensional roughness, as shown in Fig. 6c. The preliminary experiments showed that a smooth extrudate was continuously produced if the wall velocity Vw increased in line with Vf . Various combinations of Vw and Vf were therefore tested to allow us to determine the conditions under which extrudates with smooth surfaces were produced. The occurrences of exudates with smooth surfaces (◦) and with rough surfaces (×) for various combinations of Vw and Vf are shown in Fig. 7. This stability diagram can be divided into a stable region II and unstable regions I and III. When Vf was increased using the stationary roller die (Vw = 0) a smooth extrudate was produced up to Vf = 2.0 mm/s,
Fig. 5. Sharkskin produced using the conventional die (Fig. 3a). The extrudate velocity Vf was (a) 0.8 mm/s and (b) 2.0 mm/s.
Fig. 6. Change in extrudate roughness caused by varying the wall velocity Vw at a constant extrudate velocity Vf of 15.0 mm/s. (a) Vw = 5.0 mm/s, (b) Vw = 10.0 mm/s, and (c) Vw = 15.0 mm/s.
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Fig. 7. Stability as a function of roller wall velocity Vw and average extrudate velocity Vf .
and sharkskin started to be observed at Vf = 3.2 mm/s. When the rollers rotated to give a Vw of 18.0 mm/s, however, a smooth extrudate was produced up to a Vf of 28.0 mm/s, which was the maximum Vf that could be achieved using the system used in this study. The critical Vf was 3.2 mm/s for the stationary roller die, which was a factor of four lower (0.8 mm/s) for the conventional die. The difference between these critical Vf values may have been partly caused by the difference between the surface materials of the dies. The conventional die was made of acrylic resin without any covering or coating, but the roller die surface was covered with a fluorocarbon polymer (FEP). In order to investigate the influence of the die material, the conventional die wall was covered with a FEP adhesive tape similarly as the roller die. The FEP die surface did not influenced the extrudate roughness. One of the other possible causes is the curvature of the die exit. The conventional die was a 90° exit, and the roller die had the curvature radius of 10 mm. Arda and Mackley [1] used the planar dies of 1.2 mm gap and showed that the curvature of a die exit reduced sharkskin. That is, the die with a curvature radius of 2 mm postponed sharkskin up to 21% higher shear stress than the die with a 90° exit. Although the onset Vf values of sharkskin are different between the conventional die and the roller die, our purpose is to investigate the effect of moving wall. Thus the results for moving wall are compared with those for the roller die of Vw = 0 hereafter. The three-dimensional surface roughness was investigated quantitatively using a laser microscope. The roughness Ra was defined as
Ra =
1 L
0
L
|Rn |dx,
(1)
where x is the extrusion direction coordinate, L is the measurement length, and Rn is the deviation of the local surface height from the average surface height. The extrudates produced at an average extrudate velocity Vf of 22.0 mm/s are shown in Fig. 8. At a Vf of 22.0 mm/s, a Vw of 5.8 mm/s or 11.0 mm/s was not high enough to suppress melt instability, as shown in Fig. 7, and the extrudate produced had a roughness Ra of 32 μm at a Vw of 5.8 mm/s (Fig. 8a) and an Ra of 7 μm at a Vw of 11.0 mm/s (Fig. 8b). When Vw was 5.8 mm/s (Fig. 8a), the surface roughness was high and approximately two-dimensional. A Vw of 14.0 mm/s eliminated sharkskin at a Vf of 22.0 mm/s, as shown in Fig. 7, and the extrudate surface was flat at this optimized Vw of 14.0 mm/s, as shown in Fig. 8c. The Ra was 0 μm for these velocities. Tests were also performed for Vw higher than critical, as shown in Fig. 7. The Ra values 5 μm at a Vw of 18.0 mm/s (Fig. 8d), and 14 μm at
a Vw of 21.5 mm/s (Fig. 8e). The surfaces produced at the lower and upper boundaries of the stable region II are shown in Fig. 8b and 8c, respectively. The surfaces produced at these critical conditions were visibly rough and irregular compared with the surface shown in Fig. 8a. The irregular three-dimensional roughness was remarkable when Vw was 21.5 mm/s (Fig. 8d), and contrasted with the regular two-dimensional roughness shown in Fig. 8a. The extrudate kept the surface roughness apparently constant along the upper and lower boundaries of the stable region II by adjusting the wall velocity Vw . Along these boundaries, a shear stress τ w and a ratio of (N1 /τ ) w on the die wall were calculated using the wall shear rate γ˙ w = (2n + 1 )/n · (V f − Vw )/H for a plane Poiseuille flow of a power law fluid τ = K γ˙ n . The wall shear rate γ˙ w calculated was in the range of 34 to 139 s−1 on the lower boundary and 32 to 75 s−1 on the upper boundary. Although these ranges are higher than the range plotted in Fig. 2, the PDMS (a molecular weight of 4.28 × 105 gmol−1 ) used by Kissi and Piau [12] showed a constant n (∼ =1/3) in the range of 10−1 s−1 to 3 −1 10 s . Similarly, Fig. 2 showed n = 0.3 in the high shear range of 0.8 s−1 to 8 s−1 . Thus we used n = 0.3 and K = 104 Pa sn in Fig. 2 to calculate τ w and (N1 /τ )w on each boundary. On the lower boundary, an increase in Vf increased τ w from 29 kPa to 44 kPa and increased (N1 /τ )w from 6.1 to 11, as shown in Fig. 9. Meulenbroek et al. [17] analyzed an instability of viscoelastic Poisueille flow, and showed that the instability emerged at the critical ratio (N1 /τ )w ∼ = 5. Although this critical ratio is in good agreement with that for the stationary roller die in the present study, the ratio (N1 /τ )w plotted in Fig. 9 increases to the double along the lower boundary. Thus it is difficult to use τ w or (N1 /τ )w as the onset criterion of instability in the present study. Since the surface roughness resembles sharkskin near the lower boundary of region II, as shown in Fig. 8a, another approach to predict sharkskin instability is tested to give the onset criterion in the next section. Along the upper boundary, an increase in Vf changes the signs of τ w and (N1 /τ )w from minus to plus. That is, the wall velocity Vw is higher than Vf for Vf < 10 mm/s, and the flow is mainly driven by the moving wall at the exit. An increase in Vf turns the exit flow to a pressure-driven flow for Vf > 10 mm/s (Vf > Vw ) on the upper boundary. Vf = Vw suggests a uniform velocity profile and thus a strainless stable flow. However, even if Vf = Vw , the stable flow could not be achieved for Vf > 10 mm/s. The velocity profile on a die wall is influenced by the upstream flow uniformity. In the present setup, the moving wall drags flow from the upstream location. If the drag flow is too much to pass through the narrow die exit, the excess drag flow would disturb the exrudate flow. Thus it is speculated that the present setup could not achieve a uniform velocity profile under the influence of the upstream flow. Not only the die exit but also the upstream flow need to be considered to investigate the transition between region II and III. A very high pressure is applied to a very viscous polymer being extruded through a conventional narrow die. On the other hand, moving the die wall is expected to decrease the pressure inside the polymer container, and this would be beneficial to the extrusion process. The pressure was therefore measured inside the polymer container using a flush-mounted pressure sensor. Since the pressure sensor capacity (500 kPa) was not high enough to apply a very high pressure to the polymer, Vf was limited to 4.0 mm/s, which is rather low. When Vw was increased from 2.0 mm/s to 4.0 mm/s, the pressure decreased from 536 kPa to 158 kPa. In addition to the pressure sensor, a load cell was inserted between the linear motor and the piston, and the thrust force F was measured when Vf was 16 mm/s. The dimensionless force F/F∗ and the dimensionless pressure P/P∗ are shown as functions of Vw /Vf in Fig. 10 (F∗ and P∗ are the thrust force F and the static pressure P, respectively, when Vw /Vf = 1). Only a limited range of Vw and Vf values could be tested because of the low capacities of the
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Fig. 8. Three-dimensional surface roughness. The extrudate velocity Vf was 22.0 mm/s. (a) Cyclic sharkskin fracture. Ra = 32 μm. (b) Lower boundary of the stable region II. Ra = 7 μm. (c) Elimination of melt fracture. Ra = 0 μm. (d) Upper boundary of the stable region II. Ra = 5 μm. (e) Irregular melt fracture. Ra = 14 μm.
sensors, but it was clear that F/F∗ and P/P∗ decreased as the Vw /Vf increased. The rotating roller die exerted a drag force on the polymer, and the contraction flow was driven by this drag force rather than by the pressure force, and the pressure required to extrude the polymer was decreased to some degree. Using the roller die at an appropriate Vw therefore allows sharkskin fracture to be suppressed and at the same time decreases the driving pressure required to extrude the polymer. The sharkskin elimination obtained here is not limited to a planar die. The same elimination has been observed in our preliminary experiment on a circular die, which is composed of counterrotating semicircular grooves on two rollers. Although the die swell was not measured for the present planar die because of the difficulty of observation, the die swell has been observed to decrease in the stability region II for the circular die. If the installation of a roller die is possible, the instability modification by a moving wall would appear independently of a planar or a circular die. 4. Discussion The moving wall used in this study gave a similar near-wall flow to that achieved using a PPA to enhance wall slip, and allowed sharkskin to be eliminated. Migler et al. [14,15] measured the slip velocity induced by a PPA and investigated its effect on the elimi-
nation of sharkskin, and the results presented by Migler et al. [15] are plotted along with our results in Fig. 11. The Vf range of 10.5 to 105 mm/s found using a PPA was higher than the range of 2.0 to 28.0 mm/s found using the moving wall. The non-dimensionalized velocities Vw /Vf (this study) and Vs /Vf [15] are plotted as a function of Vf /Vf0 in Fig. 11 so that the results can be compared in the same range. Vs is the slip velocity experimentally determined by Migler et al. [15] and presented in their Fig. 9. Vf0 is the critical Vf at which sharkskin appears when a stationary roller die or a conventional die is used without a PPA being added. The Vf 0 was 3.2 mm/s for our stationary roller die and 3.8 mm/s (a throughput of 0.4 g/min) for the polymer without the PPA added in the study performed by Migler et al. [15]. The stable extrudates appeared in region II when the roller die was used, and the stable range of Vw /Vf became narrower as Vf /Vf0 increased. Region II using the roller die covered the Vs /Vf points that were stabilized by adding the PPA. On the other side, when Vf /Vf0 was 16 or 28, the unstable region I covered the Vs /Vf points that were not stabilized by adding the PPA. This consistency suggests that melt instability could be eliminated through the same mechanism in both the moving wall and the polymer wall slip techniques. The surface layer of the polymer is stretched rapidly outside the die, and Cogswell [4] proposed that the instability induced by this stretching causes sharkskin. Cogswell proposed that
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sured the flow velocity directly using tracer particles, and they calculated T˙ from these measurements. However, the local flow velocity was not measured in our study. Thus we used a viscous power law model τ = K γ˙ n , and assumed a Poiseuille flow inside the die and a uniform jet outside the die, as shown in Fig. 12. This flow model was introduced by Cogswell [4] for a Newtonian fluid without wall slip. A Poiseuille flow on a moving wall has the velocity profile for a viscous power-law fluid described by Eq. (3),
υ (y ) = Vw +
Fig. 9. Changes in wall shear stress τ w and wall stress ratio (N1 /τ )w along the lower and upper boundaries of the stable region II. τ w and (N1 /τ )w were calculated using τ w = 1.0 × 104 γ˙ w 0.3 and (N1 )w = 1.5 × 104 γ˙ w 0.7 in Fig. 2.
(2k − 1 )n + 1 n+1
(V f − Vw ) 1 − (y/H )(n+1)/n ,
(3)
where k is 3/2 for a plane Poiseuille flow and 2 for an axisymmetric Poiseuille flow, y is the distance from the center line, and H is the half-channel height for a plane Poiseuille flow and the radius for an axisymmetric Poiseuille flow. The PDMS used here did not show macroscopic slip on the acrylic wall, as mention in Section II, and sharkskin appeared similarly for both the acrylic and FEP walls, as mention in Section III. Thus the slip between polymer and wall is not considered here. A thin layer adjacent to the die wall is stretched outside, and the thickness is reduced from δ H to h, as shown in Fig. 12. The total extensional deformation is proportional to Vf /Vave , where Vave is the near-wall average velocity for Eq. (3) from y = H − δ H to H, calculated as shown in Eq. (4).
Vw 31−n Vave = + k+ Vf Vf 21+n
Vw 1− Vf
δH H
+ O((δ H/H ) ) 2
(4)
The average velocity Vave of the thin layer inside the die is related to the final velocity Vf outside the die by the continuity equation (Eq. (5)).
δ H · Vave = h · V f
(5)
The higher order terms of δ H/H can almost be neglected in Eq. (4). Removing Vave from Eqs. (4) and (5) gives Eq. (6).
δH H
=
⎧ ⎨ ⎩
2 k+ Fig. 10. Dimensionless static pressure P/P∗in the polymer container and the dimensionless thrust force F/F∗ as functions of Vw /Vf .
extensional strain rate ε˙ controls the onset of instability, and this was used by Migler et al. [15] to explain how sharkskin was eliminated using a PPA. However, the extensional strain rate could not be used as the instability criterion for a polymer with a PPA added. Cogswell analyzed the extrudate flow under no-slip conditions, and only a narrow range of flow velocities could be investigated. However, the polymer with a PPA added underwent wall slip, and a stable extrudate was produced up to a much higher critical extensional strain rate than was produced by the polymer without a PPA. Rather than the extensional strain rate, Migler et al. [15] proposed that the reconfiguration rate T˙ (the product of the maximum extensional strain rate ε˙ max at the die exit and the extensional deformation T from the inside of the die to the final extrudate, as shown in Eq. (2)) controls sharkskin instability.
T˙ = ε˙ max (T + 1 )
(2)
Microscopic measurements showed that sharkskin started at approximately the same reconfiguration rate T˙ ∼ 700/s for a polymer with and without a PPA added, despite their ε˙ max values that differed by a factor of five. Here we investigate what kind of parameters determine the transition at the lower boundary of region II. Migler et al. mea-
Vw Vf
2
31−n +4 k+ 21+n
31−n 21+n
1−
Vw Vf
Vw 1− Vf
⎫ ⎬
Vw h − H Vf ⎭
(6)
Then the relationship between h and δ H is shown in Eq. (7).
δH H
=
h H
k+
31−n for Vw ∼ = h for Vw ∼ = 0, and δ H ∼ = Vf 21+n (7)
Since h/H ∼ 0.01 at the lower boundary of region II in our study, the higher order terms of h/H and δ H/H are almost negligible, and thus these higher order terms will be neglected from now on. The die wall velocity Vw and the extrudate velocity Vf are determined by the operating conditions, and if h is given, the other parameters δ H and Vave can be obtained from Eqs. (6) and (5). In addition, the continuity Eq. (5) between the inside and outside velocities enables to estimate the extensional deformation of the wall layer. That is, Eqs. (4) and (5) gives Vf /Vave as a monotone decreasing function of Vw /Vf , and this Vf /Vave is used as the total extensional deformation Ta (≡ Vf /Vave ) in place of (T + 1) in Eq. (2). The only remaining unknown is the maximum extensional strain rate ε˙ max , which is discussed below. The extrudate flow is characterized by very low Reynolds numbers, much less than unity. The flow boundary condition changes discontinuously at the die exit, and the free surface is accelerated rapidly outside the die exit. Richardson [20] analyzed this inertialess ‘stick–slip’ flow and found that v ∼ 1.162x1/2 is the asymptotic velocity along a jet surface, where x is the streamwise distance from the die exit. This asymptotic velocity suggests that there will
H. Mizunuma, Y. Ohata / Journal of Non-Newtonian Fluid Mechanics 228 (2016) 55–63
61
Fig. 11. Stability as a function of the dimensionless wall velocity Vw /Vf and the dimensionless average extrudate velocity Vf /Vf0 . The dimensionless slip velocities Vs /Vf (● and ∗) are measurements made by Migler et al. [15], in which a circular die with a diameter of 1.6 mm was used. hc in Eq. (13) was assumed to be 7 μm for both this study and the study by Migler et al. [15].
Fig. 12. Die exit flow and moving wall velocity Vw . The inside near-wall layer with a thickness of δ H is stretched and reduced to a thickness h. The velocities of these layers are Vave and Vf , and the total extensional deformation is reduced to Vf /Vave .
be a very large maximum extensional strain rate ε˙ max at the die exit x = 0. Tanner [21] numerically simulated capillary die swell for a Newtonian fluid, and showed that the extensional stress reached approximately 20ηVf /H at the die lip, where η is the viscosity. Therefore, ε˙ max is estimated to be 20Vf /H. We define the apparent extensional strain rate ε˙ a (≡ (Vf − Vw )/H) rather than ε˙ max , and we include the effect of the non-zero wall velocity through Vw . Migler et al. [15] used the reconfiguration rate T˙ as the onset criterion for sharkskin instability, as mentioned above. Here we define the apparent reconfiguration rate T˙a as shown in Eq. (8).
V f − Vw V f · T˙a = ε˙ a Ta = H Vave
(8)
Vf / Vave was calculated by substituting the experimental results Vw /Vf and hc /H on the lower boundary into Eqs. (5) and (6), and by removing δ H from those equations. Then T˙a was obtained from Eq. (8) along the lower boundary and plotted in Fig. 13. The power law constant n = 0.3 and the surface roughness hc = 7 μm were assumed in Eqs. (5) and (6). Along the lower boundary, an increase in Vf increased the apparent extensional strain rate ε˙ a = (Vf − Vw )/H and by contrast, decresed the total extensional deformation Ta = Vf /Vave (= δ H/hc ). Their product T˙a was approximately kept constant along the lower boundary. These results suggest that a constant T˙a along the lower boundary can be used as the onset criterion T˙a crit of sharkskin. T˙a calculated for n = 1 is also plotted in Fig. 13. The smaller n corresponds the steeper velocity gradient
Fig. 13. Changes in T˙a , (Vf − Vw )/H, and Vf /Vave along the lower boundary of the stable region II. Vf and Vw were the experimental values on the lower boundary, from which Vave was calculated using Eqs. (5) and (6). The surface roughness was assumed to be hc = 7 μm. T˙a was calculated using Eq. (8).
in the thin layer of a thickness δ H. However, δ H decreases with an increase in Vf , and Vave is approximately reduced to Vw . Thus the influence of n is negligible for large Vf , as shown in Fig. 13. Next, we show that the lower boundary can be obtained as a function of hc /H and Vf /Vf0 instead of T˙a crit , because these can easily be found experimentally. Using Eq. (8), Vave can be removed from Eqs. (4) and (5) to give Eqs. (9) and (10).
1−
δH H
Vw Vf
−1 =1+
=
h Vw 1− H Vf
Vf 31−n − k+ 21+n H T˙a
−1
H T˙a Vf
δH H
(9)
(10)
Assuming that VwL is Vw at the lower boundary, and removing δ H/H from Eqs. (9) and (10), VwL is given as a function of Vf as
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H. Mizunuma, Y. Ohata / Journal of Non-Newtonian Fluid Mechanics 228 (2016) 55–63
shown in Eq. (11).
5. Conclusions
3 1 − n V f /H T˙a crit hc Vw L = − k+ Vf 21+n 1 + V f /H T˙a crit H
Vf Vf 1+ H T˙a crit H T˙a crit
(11) Assuming that Vf0 is the critical extrudate velocity for a stationary die (VwL = 0), the point Vf = Vf0 crosses the horizontal axis (Vw = 0) and the lower boundary (Vw = VwL ) of region II in Fig. 7. At this point, Eq. (11) can be reduced to Eq. (12),
Vf 0 = H T˙a crit
31−n hc k+ , H 21+n
(12)
which defines the relationship between Vf0 , hc , andT˙a crit . A polymer with a higher T˙a crit (i.e., a more stable polymer) gives a higher Vf0 for the same roughness height hc . Defining ξ as the right hand side of Eq. (12), Eq. (11) can be reduced to Eq. (13).
ξ · V f /V f 0 hc 31−n Vw L = − k+ Vf 1 + ξ · V f /V f 0 H 21+n
ξ·
Vf Vf 1+ξ · Vf 0 Vf 0
Two counter-rotating rollers were installed in a die exit, and the effect of Vw on sharkskin instability of PDMS was investigated. Sharkskin appeared at a Vf of 3.2 mm/s when the roller die was used to extrude PDMS without the rollers rotating. In contrast, sharkskin did not appear up to a Vf of 28.0 mm/s when the rollers were rotated. A stability diagram was plotted for the various combinations of Vw and Vf that were tested, and a stable region was found for a range of Vw values. The apparent reconfiguration rate T˙a was defined as the product of the apparent extensional strain rate ε˙ a and the total extensional deformation Ta for the flow on a moving wall of a viscous power law fluid. The experimental results showed that Ta decreased with an increase in ε˙ a , and that their product T˙a kept constant along the lower boundary of the stable region II. This constant T˙a was found out to be the onset criterion of sharkskin over the range of Vf = 3.2 to 28.0 mm/s and was defined as the critical apparent reconfiguration rate T˙a crit . T˙a crit can be correlated with Vf0 , and the relationship between these onset criterions and polymer characteristics is a next important issue . Acknowledgments
(13) Therefore, if Vf0 is given from the sharkskin instability for Vw = 0, the lower boundary of region II can be obtained as a function of Vf /Vf0 for a constant hc /H . Although T˙a crit does not explicitly appear in Eq. (13), it can be calculated from experimental results for Vf0 and hc /H using Eq. (12). If Vf is sufficiently high, Eq. (13) can be reduced to approximately VwL /Vf = 1, which means that the polymer passes through the die exit maintaining a uniform velocity profile v(y) = VwL . In other words, any weak deviation from a uniform velocity gives T˙a higher than T˙a crit under such a high Vf . It is therefore necessary to use a carefully designed system, including the parts upstream of the die, to achieve a uniform velocity profile at the die exit. As shown in Fig. 8, the sharkskin roughness was 7 μm for the lower boundary, and this was in the same range as the sharkskin roughness found by Venet and Vergnes [22]. Assuming hc to be 7 μm, the plotted curve of Eq. (13) agrees qualitatively with the experimental lower boundary of region II, as shown in Fig. 11. Although Eq. (13) was plotted for n = 0.3 and 1, the influence of n on VwL /Vf was not remarkable, similarly as shown on T˙a in Fig. 13. When we calculated Eq. (13) for the result of Migler et al. [15], we assumed n = 1 because of unknown n. Howells and Benbow [7] and Cogswell [4] found that sharkskin appeared at a constant extrudate velocity Vf0 , independent of the die diameter. Therefore, Vf 0 is closely related to the polymer characteristics when sharkskin occurs. Assuming that Vw = 0, Eq. (9) can be reduced to V f 0 /T˙a crit = (k + 3(1 − n )/2(1 + n ))δ H. Migler et al. [15] considered δ H to be independent of the die diameter, and suggested that the choice of T˙a crit is consistent with the use of Vf 0 as the onset criterion for sharkskin instability. For the moving wall die, Eq. (13) suggests that the transition between regions I and II is dependent on the die geometry H through hc /H. However, the influence of H is rather modest. We extended the flow model developed by Cogswell [4] to the flow on a moving wall of a viscous power law fluid, and its combination with the continuity condition enabled to correlate the apparent reconfiguration rate T˙a with Vw and Vf . A constant T˙a crit corresponded to the onset criterion of sharkskin and could be used to predict the instability transition across the lower boundary of the stable region II. The polymer tested here is still limited to a PDMS. However, since sharkskin can be controlled by adjusting Vw for the moving wall, a more detailed study on the moving wall effect would allow the melt instability mechanism to be described in more detail.
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