Slip and friction of polymer melt flows

Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) 9 1996 Elsevier Science B.V. All rights reserved.

357

Slip and friction of polymer melt flows N. E1 Kissi, J-M. Piau Laboratoire de Rh~ologie, Domaine Universitaire, BP 53X, 38041 Grenoble Cedex (France)*

1. I N T R O D U C T I O N The appearance of extrusion defects and instabilities [1-3] in polymer melts flowing through sudden contractions is usually related to fluid slip at the wall [4, 5]. It is therefore understandable t h a t this phenomenon has been widely investigated in m a n y studies. However, it is important to s t a r t with, to note the existence of various types of slip. In the field of polymer melt rheology, the following are referred to: Slip with additives containing ~ e . This shows the role of the additives, the wall material and the surface state of the wall. It has been studied by various authors [6-8] who have shown that the additive migrates towards the wall, forming a thin lubricating film and/or a thin film of low viscosity between the wall and the polymer. The polymer may then slip on this fine lubricating layer. Linear slip between polymer and wall. The main characteristic of this type of slip is that it occurs at all stress levels. It has been modelled by de Gennes in 1979 [9], but is difficult to observe experimentally since it supposes the use of ideal surfaces: perfectly smooth and developing no interaction with the flowing polymer. Cohesive slip with weak interacti.o.ns. Considering t h a t the occurrence of slip is highly dependent on the interaction developing between the polymer and the wall, Brochard and de Gennes in 1992 [10] described the case where there is only few sites of adsorption of the polymer chains at the surface of the dies. The initial situation, prevailing before flow, is called the mushroom regime and the friction law controlling slip has been deduced theoretically: - For low flow regimes, the absorbed chains at the wall are entangled with the bulk polymer. Slippage is then very small and wall stress increases with slip velocity. - When flow regimes are increased, the marginal state could be observed. It corresponds to a transition zone, with chains which are still u n a b l e to disentangle, because the balance between elastic and friction forces tends to m a i n t a i n a coiled structure. This regime is characterized by i m p o r t a n t slippage, during which the wall stress is no more dependent on the slip velocity. Variations in extrapolation length [10] as a function of slip velocity then follow a power law. * Grenoble University (UJF and INPG) - UMR 5520 of CNRS.

358 Finally, for higher flow regimes attached chains are completely elongated and disentangled from the bulk polymer. Slippage is then easy, wall stress is again proportional to slip velocity and friction is similar to that which could be obtained with ideal surface. It should be underlined t h a t it was possible to obtain slip at low stress values, by considering the flow of a polydimethylsiloxane (PDMS) through a silica die with walls grafted by a fluorinated monolayer [11]. It can also be found in the e x p e r i m e n t a l s t u d y published in 1993 by Migler et al. [12], which moreover validates the model prediction in term of extrapolation length, for sufficiently high slip velocities. Macroscopic slip at the wall with strong interactions. It is observed when considering high surface energy dies (stainless steel die ...). It is comparable to that which would occur between a soft solid polymer and another strongly interacting solid. This was observed by Benbow and Lamb in 1963 [13], with a silicon of high molecular weight. It has also been demonstrated by means of p r e s s u r e loss m e a s u r e m e n t s a n d p h o t o g r a p h s [14, 15]. E x p e r i m e n t a l l y , macroscopic slip occurred at the wall for the flow of sufficiently entangled polymers, when a critical stress value has been exceeded [14, 16]. In spite of the experimental results mentioned above, there is at present no accepted theory for macroscopic slip with polymer fluids. Analytical modelling results have indeed already been published [9-10, 17-20]. Generally, the laws assume a linear relation between stress at the wall and slip velocity [9], which can then be extended by a non-linear relation to high stress values [10, 17-20]. These often have the advantage of being simple to use, but nevertheless have serious drawbacks. The m a i n feature is that they consider the existence of slip for any stress value 1;R at the wall. As a consequence, a polymer of fixed molecular structure slips for any stress level, which is in contradiction with the experimental observations referred to [13, 14, 16]. Recently, in 1989, Atwood and Schowalter [21] studied the slip of a high-density polyethylene. In a g r e e m e n t with the works mentioned above [13, 14, 16], they demonstrate the existence of a threshold stress, below which the fluid adheres to the wall. Beyond, variations in stress at the wall as a function of slip velocity would appear to follow a strictly increasing linear relation. For their part, Hatzikiriakos and Dealy in 1992 [22] also take into account the existence of a static friction stress. They propose a friction relation t h a t takes into account various p a r a m e t e r s and in p a r t i c u l a r the existence of a normal stress. The results obtained for the flow of a polyethylene show that, beyond the critical shear stress, variations in slip velocity as a function of shear stress at the wall follow a strictly i n c r e a s i n g power law. Thus these slip relations t h a t take into account the existence of a critical stress [21, 22], exclude the possibility of the shear stress at the wall decreasing when the slip velocity increases, though this is a common situation in tribology. It is obvious that these relations are presented as being consistent with the experiments to which they refer. However, because of the difficulties i n h e r e n t in this type of experiment (choice of fluids, limitation of transducers, etc.) they usually cover only a small range of slip velocities, and are unable to offer an overall view of friction with slip for polymer melts [23, 24]. -

E x p e r i m e n t a l and theoretical studies propose a physical explanation of the mechanisms governing slippage phenomenon, based on the interaction likely to

359

develop between the flowing polymer and the wall of the die. Thus, in the case where no polymer chain is adsorbed at the wall, the flow of a highly entangled polymer takes place with a positive slip velocity whatever the wall stress value considered: this is the linear slip [9]. In practice, such a flow configuration needs the use of an ideal surface, perfectly smooth and developing no interaction with the fluid. It is clear that this situation is not possible to obtain experimentally, and a theoretical study shows that the combined effects of roughness and polymer-wall interactions reduces slippage by promoting the adsorption of polymer chains at the wall [10]. This result has been validated by experimental studies [11, 12, 25] considering the flow of highly entangled polymers through dies characterized by small roughness and/or small surface energy. When the polymer is strongly attached to the die wall, adhesion prevails at low flow regimes, and slippage occur only above a critical wall stress value [10, 12, 14]. It should be underlined that it is this situation which is usually encountered in industrial polymer processes, where the dies used are machined in classical materials (stainless steel ...) and with no special control of the wall roughness. In the present study, after having described the experimental means used, this later flow configuration will be described and characterized first. To do this, pressure loss measurements and appropriate visualization will be analyzed in section 3. They permit the determination of the friction law of polymer melts flowing and slipping along a solid wall. Secondly, the effect of wall material on the flow properties will be examined. This will be done using fluorinated dies which are characterized by their particularly low surface energy. 2. M E A N S U S E D 2.1. F l u i d s u s e d

The results reported here concern highly entangled polydimethylsiloxanes (PDMSs), polybutadienes (PBs) and polyethylenes (PEs) containing no additives [3, 14, 15]. Their main rheological characteristics are set out in Table 1. Two linear PDMSs (LG2 and LG3) provided by Rh6ne-Poulenc were used. The advantage of these fluids is that they are molten at ambient temperature, which means that the results of flow experiments can be interpreted without having to worry about problems of heat regulation. They are also reputed for their thermal, chemical and mechanical stability. In addition, they are transparent, which is vital for observing results and, lastly, they have low surface energy. Two PBs of different molecular weight provided by Michelin were also studied. One is linear (PB) and the other star-branched (PBb). These fluids are also molten at ambient temperature. Moreover they are reputed to be well-defined polymers. Finally, they are highly birefringent, making it possible to study the local stress field during flow, thanks to the use of flow birefringence methods. Lastly, two linear polyethylenes were considered. The LLDPE provided by Enimont is a linear low-density fluid, and the HDPE provided by Du Pont Canada a highdensity one. As these are commercial polymers that melt at high temperatures, they enabled the study to be performed under quasi-industrial conditions. Being linear or branched and melting at ambient or high temperatures, these fluids provide a comprehensive view of the stable or unstable phenomena likely to occur during the extrusion of polymer melts with slip.

360

Table 1 Fluids used Fluid PDMS LG2 23~ PDMS LG3 23~ PB 23~ PBb 23~ LLDPE 190~ HDPE 185~

Mw (g/mol) 758 000

Mw/Mn 3.9

Tl0(Pas) 540 000

1 670 000

7.9

?

220 000

1.05

170 000

1.38

143 000

3.9

216 700

18.09

1 260 000

9 800

All the experiments were performed in an air-conditioned room. In the case of the PDMSs and PBs, the t e m p e r a t u r e was 23~ and in the case of the LLDPE and HDPE, 190~ and 185~ respectively.

2.2. E x p e r i m e n t a l a p p a r a t u s Two types of installation were used for the experiments, depending on whether the upstream pressure or upstream mean flow rate were controlled. In the first experimental installation, the fluid was placed initially in a reservoir and then forced out under pressure exerted by nitrogen into an axisymmetrical or two-dimensional capillary. In this system, the total pressure loss is monitored [11, 14]. This is determined by using a Bourdon-type pressure gauge. The mass flow is measured by weighing and timing. The second installation considered is a capillary rheometer (GSttfert 2001). In this apparatus, the fluid is contained initially in a reservoir, into which a piston slides. It is then forced into an axisymmetrical capillary, with the piston moving at a controlled speed. In this system, the mean flow rate is monitored, with the pressure being measured by sensors with a measuring capacity ranging from 50 to 2000 bar. Two types of die were used, namely axisymmetrical capillaries and narrow slits. The dimension of the capillaries used are shown in Table 2. Most are made of metal (tungsten carbide, stainless steel) and received no particular surface treatment. However, in order to study the effect of the wall material on polymer friction, capillaries coated with an industrial fluoride t r e a t m e n t by Isoflon were also considered. Two axisymmetrical narrow orifices of stainless steel, 0.5 and 2 mm in diameter, were also used. The ratio of their length to diameter is less than 1. For a given diameter, pressure m e a s u r e m e n t s for a fluid flowing through one of these orifices enables entrance effects to be estimated for the same fluid flowing into a capillary [15, 26]. It should be noted t h a t another and more precise way of determining the order of m a g n i t u d e of entrance effects is to use the CouetteBagley correction. This is the method that was used for the PEs.

361 The aim of using the narrow slits was to produce quasi-2D extrusion. The slits have a gap (2e) of 2 ram, are 45 mm wide and 20 mm long. Here again, flow is considered in both stainless steel and fluorinated dies. Fluorination was obtained by coating the wall of the dies with strips of commercial PTFE, which is normally used in plumbing for sealing pipes. An industrial t r e a t m e n t process (by Isoflon) was also used. Lastly, low-roughness silica surfaces could be treated in the laboratory by grafting fluorinated trichlorosilanes to the wall [11]. A narrow orifice die m a d e of stainless steel and characterized by its short length (0.05 mm) in comparison with the gap (2e = 2 ram) will also be considered. As with flow in the axisymmetrical capillary, flow through this type of orifice can be used to evaluate pressure losses for a given fluid flowing into a slit with the same gap and width (45 mm). 3. F L O W IN H I G H S U R F A C E E N E R G Y D I E S

These feature essentially metal walls (stainless steel, t u n g s t e n carbide). The results reported may also apply to dies made of Plexiglas [14] or silica [11]. 3.1. V i s u a l i z a t i o n a n d f l o w c u r v e s For each of the fluids tested, variations in pressure loss d u r i n g flow were represented as a function of flow rate. The general run of the flow curves obtained [14, 15, 27] is shown in Fig. 1. With low flow rates, the flow is stable and takes place without any slip occurring [14]. Above a certain flow regime, which varies depending on the dimensions of the dies and the degree of entanglement of the polymer considered, macroscopic slip appears at the walls. This was demonstrated by observing the streamlines within a t r a n s p a r e n t axisymmetrical capillary. To do this, a tracer was injected n e a r the wall, perpendicular to the direction of flow, and its movement observed. With sufficiently high regimes, Fig.2 shows t h a t the tracer injected into the flow moves very distinctly near the wall, thus clearly demonstrating the existence of a definite slip velocity in this area.

On t h e flow curves, this slip is accompanied by a d i s c o n t i n u i t y in the instantaneous flow values (Fig. 1). Its occurrence differs depending on w h e t h e r or not the installation involves any storage of elastic energy connected w i t h the compressibility of the polymer upstream of the die. In the case of controlled-pressure flow, once a critical pressure value AP* is exceeded, it was shown that slip was seen on the flow curves in the form of a j u m p in flow rate, reaching a factor of as much as 100 (portion (BC) of the flow curve) [15]. Moreover, by carrying out tests with decreasing controlled pressure, it can be seen t h a t the slip persists with pressures of less t h a n AP* (portion (FD) of the flow curve). As the pressure continues to decrease, Fig. 1 shows t h a t there is a sudden decrease in flow rate (portion (DA) of the flow curve) and there is once again the portion of curve corresponding to flow with adherence to the wall. Thus, in this flow configuration, macroscopic slip is accompanied by a hysteresis. The h i g h e r the molecular weight of the polymer, the greater the hysteresis value [14, 16]. In the case of flows with a controlled mean flow rate, the fluid compressibility plays an important role and the appearance of slip is accompanied by oscillations in the instantaneous pressure and flow rate as a function of time.

362

0 0

0 0 0 r~

o~,,~

r~

0 0 ~ 0

0

0

0 0 0

c

"c (I c~ ~

c c~ .p-

C

c

~2

~

,

E E

c

r~

.r,,l

"~.N

363

Log Pu

H B

9f . . . .

9

/

4

. . . . . . . .

j

D

Log q y v

Figure 1 9Typical flow curve for highly entangled polymer melt extrusion through two-dimensional or axisymmetric dies. During the compression phase, section (AB) of the flow curve applies, and the fluid adheres to the wall. During the relaxation phase, portion (CD) of the flow curve is found and t h e r e is slip [14, 22, 28]. When the controlled m e a n flow r a t e is sufficiently high to correspond to a rising branch of the flow curve, there is once again a steady pressure regime with p e r m a n e n t slip. When sufficiently high flow regimes with slip can be reached, there is a further discontinuity in the slope of the flow rate-pressure curve, followed by a slight drop in the pressure (Fig. 1). In fact, in a g r e e m e n t with other works [29], this discontinuity is connected with the appearance of a second area of oscillation in the instantaneous pressure, indicating the existence of a second h y s t e r e s i s loop between two p r e s s u r e values. In comparison with the first loop, the second zone of oscillations occurs in a slip regime, during both the compression phase (portion (EF) of the flow curve) and relaxation phase (portion (GH)). Indeed, the minimum pressure reached in this loop (points E and H of Fig.l) remains sufficiently high to be p e r m a n e n t l y outside the zone corresponding to stable flow with adherence. After this second zone of oscillations, there is once again a steady pressure regime and the flow curve represented in Fig. 1 once again has a slope discontinuity. The flow curves characterizing the extrusion of highly entangled polymer melts in conventional dies (metal, with any reasonable degree of roughness and having no special prior treatment) thus show that such flows are governed by: - adherence to the wall at the lowest flow regimes; this property must obviously be taken into account for modelling this type of flow, - the existence of a critical stress beyond which flow takes place with slip. This is therefore a type of slip which occurs under high stress and which has been identified as macroscopic slip. Visualization of the velocity field shows t h a t it occurs very close to the wall (less t h a n 10 ~m) in the case of the flows with heterogeneous shear rates and stress fields considered here.

364

c~

~-,-~

~

.i-4

0

C~1 .,-G

~

~

"~

~ U

~

s~

~

v.-.4

9

~

c~

~.~ ~

o~

oo

365

3.2. Friction

curves

Let us consider the flow of a highly entangled polymer in an axisymmetrical die of d i a m e t e r D and length L. In the case of polymers, when the s h e a r rates are sufficiently high or within a certain shear rate domain, variaff~ns in viscosity as a function of shear rate m a y be represented by a power law [30]: T! = k ~n-1.

(1)

If it is a s s u m e d that flow is established in the capillary, the exact equation taking into account entrance effects and enabling slip velocity to be calculated is written as follows [26]" n -

UR

=

3n + 1

R[R 2--~

(z~Pt-

/kpe)]l/n,

(2)

where ~r is the mean velocity through the section, UR the slip velocity at the wall, R the radius of the capillary, APt the total pressure loss and APe the pressure loss at the entrance to the capillary. The a p p a r e n t shear stress at the wall Xa and the real shear stress at the wall XR are thus defined by: xa =

APt 4L / D

and

XR =

A P t - APe . 4L / D

(3)

To determine the rate of slip, three different methods may thus be used: Local m e a s u r e m e n t s by using laser velocimetry, m a r k i n g the flowing polymer or by optical techniques [12]. A direct method. According to relations (1) to (3), this requires knowledge of the following:. - the rheometric behavior of the fluid during shear, in order to determine the power law parameters, - the flow curves, giving variations in total pressure loss and entrance pressure loss as a function of flow rate. This method requires great precision in determining "n", otherwise it can lead to contradictory results [5, 27]. In particular, the present authors have shown t h a t it m a y be n e c e s s a r y to r e s o r t to t i m e - t e m p e r a t u r e superposition to obtain variations in viscosity over a sufficiently wide range of shear rate [15]. It is also obvious t h a t when pressures reach sufficiently high levels, the dependence of viscosity on pressure must also be taken into account [27]. A method based on Moon ey's diagram. This involves representing the variations in wall shear gradient defined by

i,=

3 n + 1 r162 n R'

356 as a function of l/R, with a constant real s h e a r stress at the wall. According to relations (2) and (3), the curve obtained is then a straight line of slope 3n+ 1

UR.

This method requires the use of dies of various diameters and precise knowledge of the flow curves for each one. R a m a m u r t h y [4] applied this m e t h o d by plotting variations in apparent shear rate at the wall as a function of I]R, with Xa constant. Relation (2) shows that the plot is only equivalent to t h a t with c o n s t a n t x'R in two cases: - the dies considered are very long (L/R-> ~). - entrance effects are very weak and can be neglected. If this is not the case, neglecting entrance and exit effects can lead to erroneous interpretations [27]. The direct method was used here to d e t e r m i n e the friction curves for a series of polymers in axisymmetrical capillaries of various dimensions. The characteristic flow curves for the flow considered here are r e p r e s e n t e d by the curve in Fig. 1. It can be seen t h a t this curve, on log-log scale, consists of a series of s t r a i g h t segments, with a discontinuity in slope and flow rate once a certain flow regime is reached. Let q* be the flow rate corresponding to the first discontinuity in flow rate and q** the flow r a t e for which oscillating plug flow d i s a p p e a r s . The curve s h o w n schematically in Fig. 1 can then be represented by the following equations: q < q*,

ZXP t

=

Ktc q ntc,

(4a)

q > q** > q * ,

APt

-

Ktd q ntd,

(4b)

where q is the m e a n flow rate, APt the total p r e s s u r e loss in the capillary, "Ktc", "Ktd", "ntc" and "ntd" coefficients t h a t depend on the fluid and capillary u n d e r consideration (cf. Fig. 3a). It should be noted that, in the case of flow involving two zones of instability, these will be distinguished. With a given diameter D, and for each flow rate value considered, the flow curves for an orifice die or the Couette-Bagley correction give an indication of the entrance pressure losses APe, for flow in a capillary of the same diameter D. On log-log scale, allowing for m e a s u r e m e n t errors, these curves are segments of straight lines. They all display a discontinuity in slope, corresponding to the a p p e a r a n c e of unstable p u l s a t i o n s u p s t r e a m of the die and to the p h e n o m e n o n of m e l t f r a c t u r e d o w n s t r e a m [3]. It was established that, for all the polymers considered in this study and within the accuracy of the m e a s u r e m e n t s , the flow rate corresponding to this b r e a k in the slope of entrance pressure loss curves is equal to the flow rate q* characteristic of the discontinuity in flow rate occurring on the total pressure loss curves for the same polymer and with the same d i a m e t e r [14]. The curves relating to the entrance pressure losses can thus be represented by the equations 9

m

~ m

~D mmm

m

mm

mm

,mm

.m

mmmu o~

F~

m

m

~,q

,.-,i c~

c~

o

o .,=..4 m~

0r2 o

o

.,=.4

~

o oP.4

o

o

.,.~

o

.~

g

367

368

n

q _< q* ,

AP e = Kec q ec,

(5a)

q > q*,

AP e = Ked q ned ,

(Sb)

where q is the m e a n flow rate, APe the entrance p r e s s u r e loss, "Kec", "Ked", "nec" and "ned" coefficients depending on the fluid and orifice considered (cf. Fig. 3b). With a given d i a m e t e r and for each polymer, the above r e s u l t s can be used for each flow r a t e value considered to calculate the pressure loss APc in the capillary itself, defined by: /~Lp c

=

/~kp t

-

Ape

(6)

.

Now: '~R

=

AP c 4L/D

=

rl,it"

It is therefore possible to deduce the following:

aPe

= Kcq~ with Kc = 4_.LLk F 3 n + 1 D L 4n

32 ]~ D

3

(7) ,

where n is the power law coefficient defined in (1). It should be noted that, to determine APc using relation (7), it is necessary for: q < q*. In this case the flow is stable and it is possible to m e a s u r e variations in viscosity coefficient, q to be sufficiently high to consider the variations in viscosity coefficient as a power law of the shear rate. In the field of flow r a t e variations defined in this way, i.e., outside t h e a r e a of Newton]an behaviour and the transition zone between N e w t o n i a n a n d power law behavior, Kc a n d n m a y also be d e t e r m i n e d from the flow curves by c o m p a r i n g relations (6) and (7) (cf. Fig. 3c). Hence, in o r d e r to m e a s u r e the friction of flowing p o l y m e r s w i t h slip in axisymmetrical capillaries, it is simply a question of m e a s u r i n g total and e n t r a n c e p r e s s u r e losses. The curves obtained in this way and relations (4a) and (5a) can then be used to d e t e r m i n e the values of Kec, Ktc, nec and ntc. It is t h e n possible to deduce the values of Kc and n by applying relations (6) and (7). In addition, these s a m e curves a n d relations (4b) and (5b) can be used to d e t e r m i n e the values of Ked, Ktd, ned and ntd. Let us consider a value q of the m e a n flow rate corresponding to a flow with slip, i.e., such t h a t UR > 0, or again q > q*. It is t h e n possible to d e t e r m i n e t h e corresponding values of: - APt (cf. Fig. 3a) by means of relation (4b) ; - APe (cf. Fig. 3b) by means of relation (5b) ; - APc (cf. Fig. 3c) by m e a n s of relation (7).

369 By introducing the pressure loss APc obtained in this way into relation (3), it is possible to calculate the stress I;R associated with the flow rate q considered. Relation (2) can then be used to determine the value of UR corresponding to this same flow rate, thus indicating variations in TR as a function of UR, and hence the friction relation for the fluid under the flow conditions considered. This method was applied to two PDMSs (LG2 and LG3), to the PB and the LLDPE (cf. Table 1). The values of"Kec", "Ked", "Ktc", "Ktd", "Kc", "nec", "ned", "ntc", "ntd", "n", and the value of q*, determined from the experimental curves obtained for these fluids are given in Table 3. This table also contains the value of the shear stress at the wall for flow rate q*, denoted x*. By choosing these fluids, it was possible to obtain a comprehensive view of the friction curves for polymer melts with slip, over a significant r a n g e of slip rate at the walls. Figs. 4a to 4d demonstrate: - The existence of a static friction stress, corresponding in fact to the flow rate q* on flow curves. Thus in the conditions used in this study, the polymer melts only slip above a critical stress threshold. They adhere when at rest. 1,o

5 R

(10

Pa)

0,8

A

0,6

A

j

0,4

---"i,.5-J

-'-

Gum_LG2

9

A L/D = 10/0.5 Q L / D = 20/0.5 0,2

9

L / D = 20/2 R e l a t i o n (10)

0,0, 10 4

9

'

'

'

'

1 ~ 1 -

10

'

-3

"

'

' ' " 1

1 0 -2

V R (m/s) '

'

'

'

' ' ' 1

10

,

"1

,

1-

,

,,

,,

10 ~

Figure 4a " Friction curve of PDMS LG2. x* is the static friction stress (Table 3). - At low slip velocities, the stress at the wall decreases when the slip velocity increases, which is a current situation in tribology. This decrease corresponds to the flow regimes preceeding the m i n i m u m obtained on the flow curve. Consequently, it is not possible to deduce shear stress value in the decreasing p a r t of the friction curve from the flow curves, since they correspond to the oscillatory flow regimes. Following this decrease, the shear stress at the wall increases again with slip velocity for flow regimes corresponding to portion

370 (DF) on the flow curves. Note that in the case where the flow curves exhibit two instability zones, the corresponding friction curve exhibits two minima, the last one being followed by a rising branch with a high slip velocity corresponding to portion (HG)on the flow curves. The friction curve depends slightly on the degree of e n t a n g l e m e n t in a given family of polymers (comparison of Figs. 4a and 4b). In addition, in the case of the LLDPE, the friction curve does not appear to depend on t e m p e r a t u r e (Fig. 4d).

9

- -

_

.

(105 Pa) R 0,8

~, (5/0.5)

LG3"

Gum

o I A ) = 5/0.5 9 I./D = 20/2

0,6 x, (20/2)

00 0,4

ll E l l

mo

ooOOO

9

o

0,2

U R (m/s) 0,6 10 -4

9

'

,

,

,'

,

9

10 -3

"v

,"

,

9

9

,

i

'1

10 -2

10 -1

Figure 4b 9Friction curve of PDMS LG3. ~* is the static friction stress (Table 3). Thus it appears that, generally speaking, the run of the friction curves obtained for the various polymer melts is characterized by significant non-linearity. On a semilog scale, these curves have in particular a bell shape, which is reminiscent of the work carried out by Moore in 1972 [31] on friction using slightly r e t i c u l a t e d elastomers. These also show t h a t adherence and h y s t e r e s i s are the m a i n mechanisms governing slip phenomena. This may be explained by the fact t h a t the polymer melts used in this study are highly entangled. In addition, these curves have two minima corresponding to the two oscillation regimes in the instantaneous pressure. It should be noted that, in the case of the PDMSs, the small quantities of fluid available for the s t u d y were insufficient to r e a c h flow r a t e v a l u e s corresponding to the second minimum, if it exists. It is also possible to characterize the polymer-wall slip by representing variations in the extrapolation length b as a function of the slip velocity. It should be recalled t h a t b is defined as the ratio of the slip rate to the shear rate at the wall [9]. With the assumptions made in this study (power law, etc.) relation (2) can be used to determine the shear rate at the wall for flow regimes including slip, i.e.:

371

3,0

5 R

Pa)

(10

O

2,5

O

O $

O

O

9 8o

2,0

@0

0 o~

9

1,5

1,0" o

L / D = 5/0.5

9

][.JD = 20/2

0,5

U 0,0

R

...... . '

'

(m/s) 9

'

'

"

9

'

1 0 -3

I

'

9

9

.=,

.

'

1 0 "1

1 0 .2

F i g u r e 4c" F r i c t i o n c u r v e of PB.

5 I; R ( 1 0

Pa)

05


|

]] D~

ii

i~

I

I

i

LLDPE I:* ( D = 2 m m )

LLDPE + x 9

9 9 9 o A o

; T = 160~

I / D = 30/2 I / D = 20/2 I.JD = 10/2

; T = 190~ I/D I/D I/D L/D I/D L/D

= 2O/O.5 - 15/0.5 - 10/0.5 = 30/2 = 2O/2 -- 10/2

U R (m]s) i

10

-2

9

,

9

,

|

II

I

10

I

"]

,

I

9

9

I

'9

10

9

0

'

9

'

|

"

| l l

10

1

10

F i g u r e 4d" F r i c t i o n c u r v e of L L D P E . x* is t h e static friction s t r e s s (Table 3).

372

~/S = 3 n + 1 n

~r

R

UR

=

[ R ~

]l/n ( A P t - APe) ,

(8)

and hence: b = UR

(o)

The r e s u l t s obtained are presented in Fig. 5. It can be seen that, w i t h a given polymer, the extrapolation length hardly depends on the geometric characteristics of the dies. In addition, the results obtained with the PDMSs show that, with the same family of polymer, b increases with the molecular weight. Lastly, with the polymer-wall pairs envisaged here and the range of slip velocities achieved in this study, b appears to be an increasing function of the slip velocity. F u r t h e r m o r e , on log-log scale, variations in b m a y be represented by a portion of a straight line with a slope of about 0.9 for all the experiments carried out, in the case of flow in high surface energy dies.

b (m)

Gum LG3 Stainless steel die L/D = 20/2 ,.,.~~ El

I]D=5/0.5

'

E!

P__BB Stainless steel die o L / D = 20/2 = L/D=5/0.5

Q

&

9 PBb 99. o 9 cod Slippery_ die 4"

.& r

e~

f Gum LG2 Stainless steel die 9 L/D = 20/2 & L/D = 10/0.5 9 L/D = 20/0.5

~11

LLDPE & Stainless steel die A j "~~9- a IJD=10/2 9 I.]D = 20/2 i L/D = 30/2 L / D = 10/0,5 9 L/D = 15/0,5 -t I_]D= 20/0,5

=

x

U R (m/s) 10

._4

......

10,

..~

......

...... 10, '-5' . . . . . . 10, . . -4

10, ~3 . . . . . . 10, -'2". . . . . . 1 0, : 1

......

10, "0' . . . . . . 10, "1. . . . . 10 2

Figure 5 9Extrapolation length as a function of slip velocity at the wall for different polymer-wall pair.

4. F L O W I N D I E S W I T H L O W S U R F A C E E N E R G Y The effect of polymer-wall interactions is an essential p a r a m e t e r t h a t h a s to be studied in order to understand the conditions under which slip is triggered. To do so,

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if:

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373

374 numerous studies have analyzed polymer flow in capillaries made of different materials or by using polymers containing various types of additive [4, 5, 8, 16, 32]. The particular effect of fluorine is emphasized in all cases, but the results obtained for similar polymers may be contradictory [32]. Indeed, not all commercially available fluorinated materials are equivalent [25]. The flow of various polymers was analyzed using axisymmetrical or twodimensional fluorinated dies (cf. Tables 1 and 2). Several types of fluorine were tested [11, 25, 33]: solid PTFE in various forms (strips, massive die, etc.), industrial surface treatment (Isoflon), or treatment in the laboratory enabling fluorinated trichlorosilanes to be grafted on to the surface of silica objects of low roughness [34, 35]. The characteristics of the wall layer (chemical composition, surface energy, etc.) of fluorinated surfaces obtained by grafting on trichlorosilanes could be more thoroughly controlled in comparison with the surfaces obtained using commercial fluorine. The flow of a given polymer in such dies is thus a valuable way of studying the effect of wall material when extruding polymer melts with slip. Flow curves were plotted for all the polymer-wall pairs envisaged, and the influence of the wall material on the appearance of the extrudate was observed. In the case of the PBb flowing in the 2D dies treated by Isoflon, the results were supplemented by stress charts obtained by birefringence and by measurements of the velocity profile determined by laser velochnetry [25]. 4.1. G e n e r a l r e s u l t s In the case of all the polymer-wall pairs considered, analysis of the section of the flow curves corresponding to stable flow shows that flow rate at a given pressure is greater when the fluid flows in fluorinated dies. This can be seen in Fig. 6 for the particular case of the HDPE [11, 25, 36]. This increase, which may reach a factor of 5, demonstrates the combined effect of roughness and the wall material. It may be explained by the occurrence of polymer slip in the fluorinated dies even at low flow rates. In the case of regimes corresponding to unstable flow, the flow curves obtained with steel dies and fluorinated dies are virtually the same (Fig. 6). Therefore, they appear to be independent of the characteristics of the die wall. In fact, these relatively high flow regimes are governed by bulk instability upstream of the contraction [3, 33]: volume phenomena appear, therefore, to play a leading part in comparison with interface phenomena.

Wall material also appears to have a decisive effect on the appearance of the extrudate. Indeed, the classic succession of the various extrusion regimes can be observed in the steel dies [11, 25]. When the same polymers flow through fluorinated dies, the occurrence of sharkskin cracks or plug flow could be significantly delayed or completely eliminated [11, 33]. Thus, depending on the characteristics of the fluorine available (in particular in terms of surface energy), and the flow configuration considered, it was possible to multiply extrusion rates by 16 in comparison with situations where flow takes place in a conventional die made of stainless steel, while at the same time keeping the same qualities of transparency in the extrudate [11, 25, 33]. Consequently, by provoking slip at the wall, fluorination of the dies is also likely to prevent sharkskin defects (Fig. 7). These surfaces, characterized by their low energy and referred to as slippery surfaces [33], produce a different type of slip from that occurring with a high stress level along the high surface energy and relatively rough walls described in section

375 2.2. An attempt was made to identify and characterize this type of slip by studying the flow of the PBb in detail. 10 3

AP (10 9

5

Pa)

Stainless steel capillary Isoflon capillary 9

A

1 0 2.

A 9 A A A A A ~/ (S "I ) 1 10

1

9

01

. . . . . . . .

'

10 2

. . . . . . . .

9

,

,

,

, ,

'

103

Figure 6 9 Flow curves for the flow of HDPE t h r o u g h axisymmetrical capillaries, 2 ram diameter and 20 mm long. 4.2. C o m p l e t e a n a l y s i s o f P B b f l o w The PBb was chosen for various reasons. In comparison with PEs, it melts at a m b i e n t t e m p e r a t u r e and thus avoids any problems of h e a t regulation. In comparison with the PB, it is t r a n s p a r e n t , m a k i n g it possible to observe the changes t h a t occur and to carry out velocity m e a s u r e m e n t s by laser velocimetry. Lastly, in comparison with the PDMSs t h a t were available, it is properly birefringent, making it possible to analyze the local stress field. The flow of this polymer under a controlled nitrogen p r e s s u r e was examined in sudden, fiatbottomed contractions. The extrusion dies considered are two-dimensional, 45 mm wide and characterized by a gap (2e) of 2 mm. One was made of stainless steel and was 20 m m long. The second had the same geometry as the first but the walls were coated with PTFE by Isoflon. The third was a narrow stainless steel orifice characterized by its short length (only 0.05 ram) in comparison with the gap. It should be recalled t h a t flow in such an orifice gives a correct r e p r e s e n t a t i o n of entrance effects in a given fluid flowing into a capillary die with the same gap and width.

4.2.1. Flow curve- Visualization of extrudate In Fig. 8, flow in the 20 mm long stainless steel die is represented by a straight line with no slope discontinuity. This results indicates that flow is stable for all the regimes considered [3]: there is no melt fracture or macroscopic slip for the flow conditions reproduced here. The flow curve for the orifice die has the same features, though the overall energy losses are lower than in the long die. In the case of flow in the Isoflon die, Fig. 8 shows that the flow curve plotted on

376

~

~. 9

~ -,~~

~

o

m

;-Bm

~

2=

0

m c~

9

~

c~

i:l~ -~-~

~.~

. ~ ~cl o ,.~

~ ~-=

2 ~ ~ ~-~.~

377 log-log scale consists of three portions of straight line. The first portion is shined by about 30% with respect to the flow in the steel die at a pressure of 5 bar. The second portion corresponds to a transition zone before reaching high-slip regimes, which are represented by the third portion of straight line, where the difference in flow rate for a given pressure between flow in the steel die and t h a t in the Isoflon die m a y reach 500%. This difference may be a t t r i b u t e d to the triggering of slip at low stress levels along the fluorinated walls [25, 36]. F u r t h e r m o r e , at high flow rates, the pressure loss curve for the Isoflon die appears to be superimposed on the curve obtained for the steel orifice die. As this die gives a good order of magnitude for the e n t r a n c e pressure losses, this result d e m o n s t r a t e s the fact t h a t energy losses in the Isoflon die occur almost exclusively in the inlet area, and are thus considerably reduced inside the capillary due to slip along the fluorinated walls. The e x t r u d a t e in the stainless steel die is first smooth a n d t r a n s p a r e n t and t h e n s h a r k s k i n cracks gradually invade the surface as the flow regime increases. No extrusion defects were observed with flow in the Isoflon die and with any of the regimes t h a t could be obtained in this study, which are up to 24 times higher t h a n the regime in which sharkskin occurs in the stainless steel die [25]. 10 2

5

AP (10

Pa) 9

0o

9

0 o

O OO&

n

101

0

0

0 0

9 Stainless steel capillary o Fluorinated capillary 9 Orificedie

0 9

(g/h)

q Ick.l.vO 1 0 "1

. . . . . . . .

, 10 o

. . . . . . . .

m

,

101

.

"

10 2

9

,

.

.

,

.

10 3

Figure 8 9Flow curves for the flow of PBb through two dimensional dies, 2 m m gap and 20 or 0.2 mm long. The orifice die is made of stainless steel. Consequently, by considerably reducing the shear stresses in the capillary and hence stresses in the outlet area, slip modifies the conditions of flow for the polymer, which can then be extruded under low stresses and without any surface defects [36].

4.2.2. Measurement of velocity at the wall Velocity m e a s u r e m e n t s were taken by using a laser doppler velocimeter with a green laser beam [25, 36]. Fig. 9 shows the velocity profiles m e a s u r e d with this

378 system for both the steel and P T F E dies, in an axially symmetrical plane and at 10 m m from the inlet. Fig. 9a confirms t h a t flow in the steel die is similar to a fully developed Poiseuilletype flow, with adherence at the wall, even at the highest flow r a t e s studied. Indeed, no slip was detected near the wall with the apparatus used. Fig. 9b shows that, at high flow rates in the PTFE die, there is a plug flow at a velocity close to the m e a n velocity. At 50 ~tm from the wall, the velocity is practically equal to the maximum velocity in the section. These results show that, with flow in the P T F E die, energy is dissipated in a very t h i n w a l l layer. However, it h a s a l r e a d y been seen from the p r e s s u r e m e a s u r e m e n t s t h a t there is relatively little dissipation in the same experimental conditions. The results of Figs. 8 and 9b obtained at high flow rates with the PTFE die thus show that the conditions for slip with practically no friction are achieved.

4.2.3. Birefringence patterns Qualitative information concerning the distribution and zones of concentration of stresses in the various areas of flow and the different dies considered was obtained by birefringence. Like m a n y other authors [37-39], we shall a s s u m e t h a t the photo-elastic law, which expresses the relation between the optical tensor and stress tensor, is l i n e a r for the experiments performed here. Thus, at least in principle, this relation may be used to obtain an experimental m e a s u r e m e n t of stresses at all points of the flow [37, 38, 40, see also chapter III.1 in this book]. The isochromes, lines where the transmitted intensity is cancelled by interference, provide isovalue curves for the difference in the principal stresses. Along the die axis, the maximum order of the fringes in the inflow area will be denoted Ke and the extreme order in the outflow area denoted Ko. The order of the fringes at the wall before the outflow area will be denoted Kw.

-3

O AP=25bar A AP = 35 bar +

V R

o,oo t~0

- Theoreticalprofile

\\A .o,o

0.,~

,,

%% ,,, . ' , < , ,

-0,06 y (ram) -0,08

.

0

,

1

"

2

Figure 9a : Velocity profile for the flow of PBb through the 2D stainless steel die.

379

-0,1 0

~176~

UR(10

o

-3

o

0

0 o

m/s)

o

o

o

o

o

0

0

00

0

o

AP = 25 bar A AP = 35 bar o

AA

+ AP = 40 bar - Mean velocity AA A A A A A AA A A A A A A A A A A

-e

+

, .

-0,2

o

+

+

+

+

+

+

+

+

-~-

y (mm) -0,3

9

0

!

1

.

2

Figure 9b : Velocity profile for the flow of PBb through the 2D fluorinated capillary.

a- Visualization of birefringence in the stainless steel die The results obtained are in conformity with those of m a n y other a u t h o r s [25, 37]. The birefringence p a t t e r n is t h u s s y m m e t r i c a l with respect to the flow axis. Moreover, the n u m b e r of fringes increases with pressure, i n d i c a t i n g t h a t the isochromes d e p e n d on the flow regime. Lastly, it is possible to observe the concentration of stresses around the capillary inlet. In the capillary, following a transition zone of the order of 2-3 times the die gap, the isochromes are parallel to the walls [36]. This results demonstrates the fact t h a t for a given flow rate, shear depends only on the direction perpendicular to the axis of flow. In particular, the shear stress at the wall remains constant along the capillary. As flow approaches the outlet section, the central fringes converge towards the sides of the die, showing t h a t there is a new zone of stress concentration at this point [36]. b- Visualization of birefringence in the Isoflon die At low flow rates, the appearance and development of the isochromes are similar to those described for the steel die. However, with u p s t r e a m p r e s s u r e s of between approximately 6 and 13 bar, pulsations in the n u m b e r of fringes were observed throughout the flow field. Thus, as shown in Fig. 10a, at a pressure of 6.4 bar, the order of fringe Ke is 9 with brief pulsations at 11. It should be noted t h a t with a similar m e a n flow rate, the order of fringe Ke corresponding to flow in the steel die was 9 [36]. The oscillation in the u p s t r e a m birefringence image corresponds to disturbance of the fringes (Fig. 10b) along the capillary and throughout the flow, occurring with a period of 16 s. At the wall, the order of fringe Kw is thus 9 and 8.5, depending on the side of the die, and it decreases by 0.5 of a fringe the m o m e n t Ke increases. In the case of steel, Kw was 13 with the same flow rate, and 8 bars were required to produce the same

380 flow rate (fig. 8). In the outflow area, the maximum order of fringes Ko increases from 3 to 3.5 w h e n Ke increases from 9 to 11. In the steel it was 5 for the same mean flow rate. In addition, figure 10b shows that the entrance length, aider which the fringes become parallel to the walls of the die, is distinctly longer than when the fluid flows in the steel die [36]. At 6.4 bar, this transition zone is 13 m m long, i.e. more t h a n 6 times the die gap. When the u p s t r e a m p r e s s u r e is increased, the n u m b e r of fringes oscillates throughout the field of observation in a similar way to t h a t described above, and the period of the phenomenon decreases. Simultaneously, the entrance length increases until it occupies the entire length of the capillary. With an u p s t r e a m pressure of 12 bar (fig. 11), Ke = 28, with peaks of 29 every 2 seconds, while Kw = 10 and 9.5 depending on the side of the die, and Ko = 0.5. The orders of Kw and Ko oscillate slightly. In addition, the last fringe on the axis at the outlet from the die is of order 4 and not 0 as in the case of the previous fully developed flows [36]. Thus, when the flow regime increases, the number of fringes observed between the axis of flow and the wall decreases, as shown in Figs. 10b and 11. However, the

Ke = 9

Ke = 11

Figure 10a : Birefringence p a t t e r n s for the flow of PBb at the entrance of a twodimensional fluorinated die, 2 mm gap and 20 mm long. AP = 6.4 105 Pa; qm = 1.1 g/h. order of the axial fringe is no longer 0 and it increases with the flow rate. The result is t h a t Kw, which is the sum of the order of the axial fringe and this zmmber of fringes, increases slowly. It is 11 at 5 bar, drops to 9 between 6.4 and 10 bar, rises

381 to 10 for 12 bar and 13 for 14 bar [25]. It may thus be deduced t h a t the stress at the wall varies little during the transition regime, and t h a t it even r e a c h e s a m a x i m u m and t h e n a m i n i m u m before s t a r t i n g to rise, w h e n the flow r a t e increases from zero. Hence, as suggested by the pressure loss measurements, the distribution of energy spent between the wall of the die and the two geometric singularities is v e r y different, depending on whether the fluid flows through a steel or fluorinated die. Indeed, the stress field in the upstream zone is not affected directly by the coating. In contrast, for a given flow rate, the wall and exit area are subjected to lower stresses when the fluid flows through a PTFE die. 4.2.4 F r i c t i o n curve

An a t t e m p t was made to determine the effect of wall material on friction with polymer melt slip, in terms of variations in the extrapolation length "b". To do this, the results represented in Fig. 8 were used to determine variations in pressure loss in the stainless steel and Isoflon capillaries, with the n e c e s s a r y corrections being made for entrance effects. These are obtained at a given flow rate by the equation: APc = APt - APe, where APt is the total pressure loss in the capillary and APe the pressure obtained at the same flow rate in the orifice die. Hence, for each value of APc, it is possible to determine from Fig. 8 the flow rate ql in the steel capillary and the flow r a t e q2 t h a t would occur in the Isoflon capillary. The slip rate can then be deduced by the equation: UR = q 2 - ql 2e.1 ' and the corresponding extrapolation length by: b = UR where ~/is the apparent shear rate given by: ~, =

6ql (2e)2.1

Fig. 5 shows the variations in extrapolation length "b" as a function of the slip rate UR for PBb flow in the fluorinated dies. On log paper, the increase in "b" for low velocities is a portion of straight line with a slope of about 0.8. It was 0.9 for polymers flowing in high surface energy dies. This slope increases for h i g h e r regimes, and variations in "b" occur as the square of the slip rate at the wall. Lastly, for the highest regimes t h a t could be reached, "b" continues to increase on a logarithmic graph along a straight line, though with a very low slope. Fig. 5 thus shows t h a t the fluorinated dies enable much lower slip regimes to be obtained t h a n those possible with the stainless steel dies. This demonstrates the fact t h a t slip along low surface energy walls is associated with wall friction t h a t is e x t r e m e l y weak compared with cases where the fluid flows along conventional walls.

382

C

bi)

.,~

9 p,,,0

f,J

o

~ i,,,x

C o C

m

~

..'<1

m

o

o

~

C

~

-.-,

C

o

.C C C

cr ~

~

~

~

~r.~

9

o

o

o

t,-,-4

0

0

f::: o

O

o

~

~

!

f:::

~o

~

~

oo ~

II

.,..~ ~.~

383

384

5. D I S C U S S I O N AND C O N C L U S I O N By using polymers with various molecular characteristics and analyzing the flow curves, it has been possible to propose a comprehensive view of friction curves with polymer melt slip over a significant range of slip velocities. 5.1. H i g h s u r f a c e e n e r g y d i e s - m a c r o s c o p i c slip The slip regimes corresponding to and then succeeding the first area of oscillations may be described as cohesive, i.e., they indicate a fracture within the polymer itself [33]. Indeed, owing to the roughness and interactions likely to occur between the wall and flowing fluid, a layer of the polymer is considerably adsorbed and is thus trapped at the surface. When a sufficient level of stress is reached, the flowing polymer can become disentangled from this wall layer, and slide along it. In practice, it is t h e n possible to describe the second portion of the pressure loss curves (portion (DF) on figure 1). With higher flow regimes, it has been seen that a second zone of instability is likely to occur. This is situated entirely within the flow-with-slip regime and thermal effects should be taken into account for this zone. In addition, beyond a certain stress level and depending on wall properties, the adsorbed layer can become detached from the wall. This is another type of slip, which m a y be described as adhesive, i.e., indicating a b r e a k b e t w e e n the polymer and the wall. More experiments are clearly needed to identify the several possible slip regimes

The results set out in 3. show t h a t a friction law must m a k e allowance for the remarks made in 3.2 in order to represent friction with macroscopic slip in the case of polymer melts. An initial approach was made by Chernyak and Leonov in 1986 [18], and then Leonov in 1990 [20]. They proposed relations for modelling the bellshaped curve with its maximum and minimttm/mi_nima. It appeared worthwhile to adapt these relations to take into account the existence of a positive stress at rest and the decrease in stress at the wall when the slip velocity increases, for low slip regimes. With given t e m p e r a t u r e and pressure, these relations are written as follows:

XR(UR) = 1:s + Ae-XU* + B

1I:R(UR) = "r + Ae-~'u* + B U---K

,

/

l+m+

Ua UR

e

(lo)

- -U:-: (11)

1+ "gn-e Relation (10) is used to model friction laws t h a t have a single m i n i m u m [18] while relation (11) makes allowance for the existence of a second m i n i m u m [20]. These relations involve various parameters:

385 three stress parameters: Xs, A and B. The value of xs governs the m e a n stress level. The value of A fixes the stress level for low values of UR. The value of B governs the m a x i m u m stress level of the friction curve. In addition, when UR tends towards 0, the stress calculated by relations (10) and (11)tends towards the s u m Xs + A, which t h u s corresponds to the stress t h a t triggers macroscopic slip for the case in question; a velocity parameter: Ua. Its value fixes the position of the m a x i m u m along the slip rate axis; - ~,, an inverse velocity parameter. This governs the slope of the friction curve at low slip rate values; - two dimensionless parameters "m" and "y'. The value of "m" determines the c u r v a t u r e of the friction curve around the m a x i m u m , while t h a t of "~" determines the position of the second minimum in terms of stress and velocity at the wall. On the basis of these remarks, relations (10) and (11) could be used to model friction with slip for the polymers under consideration. -

-

For the PDMSs, only one zone of instability could be observed. Relation (10) was therefore used. Fig. 4a shows that, with the set of parameters given in table 4, the friction curves fit to within 5%. In addition, this same set of parameters can be used to give a faithfixl representation of flow regimes with oscillations [15]. In this regard, it should be underlined that the friction curves obtained experimentally and represented in Fig. 4a initially appear to be independent of the diameter and length of the capillary. It is therefore tempting to use a single set of parameters in order to model them with relation (10). However, it is impossible to represent all the results obtained in the oscillating regime with this single set of parameters [15]: the model s e t t i n g is hardly affected by the length of the capillary b u t it is necessary to choose slightly different values for the parameters, depending on the diameter of the dies used [15]. Table 4 Constants.of relation (10) for gum LG2. D (mm) 0.5 Xs (bar) 0.495 A (bar) 0.085 B (bar) 0.6 Ua (cm/s) 4.7 (cm/s)-I 6 m 0.009 .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0.47 0.08 0.48 2.8 6.5 0.01

With the flow conditions envisaged here, extrusion of the PB and LLDPE displays two zones of instability. An attempt was therefore made to model the curves of Figs. 4c and 4d using relation (11). In fact, owing to the considerable curvature of the friction curves around their respective m a x i m u m values (Figs. 4c and 4d), relation (11) cannot be applied. Indeed, the parameter governing the curvature is

386 "m" and, as "m" is assumed to be lower than unity [18], the work of Leonov in 1990 [20] showed that relation (11) would give little curvature around the maximum. This present study thus shows t h a t this relation is incompatible with the slip b e h a v i o u r of all polymer melts. In the p r e s e n t s t a t e of knowledge, only a completely empirical relation may be envisaged with a view to modelling the friction curves represented on Figs. 4c and 4d. Eventually, it will be necessary to draw up models based on physical considerations and to incorporate additional experimental results based on careful investigations. Numerical modelling could also be a valuable tool for analyzing a complex situation in which it is difficult to check all the parameters precisely, by varying them over the entire range. 5.2. L o w s u r f a c e e n e r g y d i e s The effect of polymer-wall interactions was studied by considering the flow of various polymers along fluorinated walls using various techniques. The results obtained show that, in the case of steel dies, the fluids flow while adhering to the walls, which have a high surface energy, and m a y crack as they leave the die. In the case of geometrically identical dies with fluorinated walls, the surface cracks observed at the die outlet may be delayed or even eliminated. In the particular case of the PBb flowing in two-dimensional dies, the experimental observations obtained by the various mechanical, optical and physical techniques used enable this result to be put down to the triggering of slip at the die wall when the fluid flows in the fluorinated die. In fact, two different slip regimes occur, separated by a fluctuating transition zone. This result, with a second regime at high flow rates t h a t resembles what slip might be along an ideal surface (Fig. 5) should be compared with the theoretical forecasts and m e a s u r e m e n t s recently published on chain dynamics and their coil-stretch transition near low surface energy walls [10, 12]. Friction curves for slippery surfaces differ from the curves for high energy surfaces in the low stress-low slip velocity region. Instead of a critical stress value, stress can increase progressively with UR at low values of UR, before a friction curve similar to that for a high energy surface is obtained. However, it has not been possible yet in the type of experiments described here to reach sufficiently low regimes to describe friction curves for slip velocities tending towards zero, and a full comparison with the results by Migler et al. [12] cannot be made the more so as techniques used in [12] concern velocities smaller t h a n lmm/s and did not give stress level. In addition, identification of the slip velocity measured in [12] with UR is not necessarily trivial. F u r t h e r experiments using appropriate m e a s u r e m e n t techniques are then necessary.

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