experimental and theoretical investigation of extrudate swell of polymer melts from small...

Experimental and Theoretical Investigation of Extrudate Swell of Polymer Melts from Small

(Length)/(Cross-Section) Ratio Slit and Capillary Dies DAVID HUANG*

Polymer Engineering University of Tennessee

Knoxville, Tennessee 3791 6

and

JAMES L. WHITE

Polymer Engineering University of Tennessee

Knoxville, 3791 6

and

lnstitut fur Kunststoffverarbeitung RWTH Aachen

Aachen, West Germany, (BRD)

An experimental and theoretical study is presented of extrudate swell from short capillary and slit dies. The polymer melts studied were polystyrene and polypropylene. The swell from slit dies is greater than the swell from capillaries. Decreasing die entry angle for capillary dies decreases swell. The argu- ment is made that elongational How existing in the die entry region and for short dies determines extrudate swell. Dimen- sional analysis arguments are used to relate extrudate swell to a Weissenberg number based on elongational flow at the die entrance and the detailed die geometry. Correlations are developed. The theoretical study is based on unconstrained elastic recovery following elongational How through the die entrance region.

INTRODUCTION enerally extrusion dies are used to develop desired G shapes and profiles of the emerging polymer extru-

dates. Die lengths are produced long enough to achieve this desired goal but no longer than necessary as this is costly in pressure. On the other hand, most basic studies of extrudate characteristics treat long dies (usually capillary dies). There is an obvious reason for this, as extrudate swell is obviously acomplex problem and it can only be hoped to be understood in terms or well-defined flow histories such as exist in long dies of simple and constant cross-section. This justification however does not re- solve the problem of the necessity of understanding swell from short dies. It is with this problem that this paper is concerned.

It is useful to concisely review pertinent studies of extrudate swell here. First experimental studies of swell from capillaries have been extensive (l-ll), while there have been few studies from slit dies (12116). Swell from long slit dies is generally greater than that from capil-

* Present address: Chemplex, Rolling Meadows, Illinois

h i e s (16). At any extrusion rate, swell decreases to an asymptotic value with increasing die length. Swell in- creases with extrusion rate and at a critical characteristic shear rate usually equivalent to a die wall shear stress of 106 dynes/cm2 (lo5 pascals). The emerging extrudates become distorted (3, 4, 14, 16-19). It is clear from flow visualization studies (18-23) that the instabilities giving rise to these distorted extrudates arise in the die entry flow region. Quantitative experimental studies of the magnitude of extrudate swell have been reported by various investigators for long capillary (7-11, 16, 24) and slit dies (16). The detailed extrudate swell-shear rate or shear stress relationship varies from polymer to polymer and is especially sensitive to molecular weight distribu- tion, which also results in increased delayed recovery (often found by annealing extrudates (8, 11, 24)). More general correlations have been predicted from viscoelastic fluid theory (11, 16, 24, 25) which relate swell to normal stress-shear stress ratio at the die wall. This seems to reasonably well, though not perfectly, correlate data. Numerous theoretical studies of swell from long dies have appeared. The most useful of these would

182 POLYMER ENGINEERING AND SCIENCE, FEBRUARY, 1980, Yo/. 20, No. 3

Experimental and Theoretical Investigation of Extrudate Swell of Polymer Melts

seem to be the formulation of Tanner (25) based on unconstrained recovery from long duration Poiseuille flow in a capillary. This has since been extended by the authors and their colleagues to the cases of the swell of extrudates held in uniaxial tension (11) and from slit dies

The characteristics of die entrance flow are obviously of importance to swell from short dies. The flow patterns in this region are complex and often involve vortex motions (18-23). Cogswell (26) and Metzner, Uebler, and Chan Man Fong (27) were the first to note the elongational flow character of the die entrance region. The former author suggested it could be used for measurements of elongational viscosity of polymer melts. More detailed discussions about the kinematics of entrance flows were published shortly thereafter by Metz- ner and Metzner (28), Cogswell (29) and White and Kondo (23). Both Cogswell and White and Kondo argue that it is the elongational flow characteristic of polymer melts which gives rise to the vortex-like die entrance flows.

In the present paper we carry out an experimental and theoretical study of swell from short capillary and slit dies. The magnitude of swell is studied as a function extrusion rate (die wall shear rate) and die entrance geometry. A theory of swell based upon unconstrained elastic recovery considerations is developed and com- pared with experiment.

(16).

EXPERIMENTAL Materials

Two polymer melts, a polystyrene (Dow Styron 678U) denoted PS and a polypropylene (experimental polymer supplied by Diamond Shamrock now ARCO) denoted PP were used. The viscosity ~ ( u & ) and principal normal stress ILTI(uII - uZ2) for shear flow (el = 7x2) are shown in F i g . 1 at 180°C as a function of shear stress uI2. The PS data is due to V. M. Lobe and the PP results are due to W. Minoshima. The viscosity-shear rate data were determined at low shear rates in a Rheometrics mechanical spectrometer and at higher shear rates in an Instron capillary rheometer. The N , data were obtained on the mechanical spectrometer.

105, I I I ,lo5

poscol

Fig. 1 . Shear viscosity and normal stress as a function of shear stress for P S and PP at 180°C.

POLYMER ENGINEERING AND SCIENCE, FEBRUARY, 1980, Vol. 20, No. 3

Dies and Extrusion Apparatus

The various dies used in this study are tabulated in Table 1. Dies of varying length and entrance geometry have been used.

The cross-sections are of both slit and capillary type. The extrudate swell measurements were made at

180°C in an Instron capillary rheometer.

Experimental Procedures

In our earlier study (16) we found the melt streams especially from slit dies badly necked down due to grav- ity at low die wall shear rates. Therefore, all the data collected in this paper have wall shear rate higher than 1 s-l. Extrudates directly cooled in the air were measured with a micrometer to determine the diameter (for the capillary) and center thickness (for the slit).

For long slit and capillary dies we use the data from our previously reported measurements (16).

The room temperature data were normalized to 180°C by using a density factor [p(20"C)lp(180"C)]"3. (1.027 for polystyrene and 1.057 for polypropylene.)

RESULTS

Extrudate swell is plotted as a function of die length for the PS for capillary and slit dies in Fig. 2. We contrast swell dID and hIH from long (LID, LIH 4 m) and short (LID, L/H -+ 0) dies with 180" entrance angle for both PS and PP in Fig. 3 . It may be seen that swell decreases with increasing die length for both cross-sections. Swell is in all cases higher for the slit die. The swell at high extrusion rates becomes very large for the short dies reaching 3-4 for the capillary and 7-8 for the central portions of the slit.

The level of swell at constant 7, is greater for the PP relative to the PS for the short capillary dies.

The influence of die entry geometry on swell is shown in Figs. 4 and 5 . I t is found that decreasing the entrance angle at constant LID or LIH from 180" decreases extru-

Table 1. Dimensions of Dies

Slit dies:

H Aspect UH (mm) ratio 1 2 3

0.381 16.7 12.53 20.0 29.87 0.635 10.0 12.0 18.18 24.08

Capillary dies:

Diameter UD (mm) 1 2 3 4

0.762 8.0 11.95 15.97 1.702 5.02 9.98 15.04 20.02

L H

aspect ratio 8.0 entrance angle 180"

L D

Slit orifice: - = 0 H = 0.394 mm

Capillary orifice: - = 0

Entrance angle 180" 120" 90" 60" 30" Diameter (mm) 1.575 1.575 1.575 1.549 1.575

183

David Huang and James L. White

4 0 I I I I

3 5 PS (unannealed) Capillary Orif ice L I D = o Capillary Die D = 1702 mrn

3 0

2 0

1.5

(sec-'

689

232 101 45.9

r n I I I I I ."

0 10 2 0 30 4 0 5 0 L I D

(01

I I I I + 4'5 1 PS (unannealed) 180"c

H= 0.635 rnm

3.5

$ 3.0 I, - - m"

2.5

2.0

1.5

(sec-

243

112

52.4

20.0 95.3 4.53 1.59

1 .o 0 10 20 3 0 4 0

L / H (6)

Fig. 2 . Ertrudate swell B as a function of die length for P S (a) capi l lay dies, (b ) slit dies.

8 0

PP at 180" Die Entrance 180" S,,t 0 L / H = a

60 m L / H z W

Copiilary O a L /D=m

I "

1 00 1 0' 1 0 2 1 o3 y , (sec-'I

(6) Fig . 3 . Comparison of extrudate swell B f rom short (LID LIH + 0) and long (LID, LIH m) slit and capillary dies (a) P P , ( b ) P S .

I I I I PP (180°C)

L / D = a , 180' 0

L / D = o 180' 0

120" 90" 0

60' A 30" A

4 0 - CAPILLARY 8

3 0 - - a \ -0 - - - al 3 m

I I I 1 0 ' I loa 10' lo2 lo3

y, (sec-')

Fig. 4 . Influence of die e n t y geometry on extrudate swell forPP.

date swell. This is most pronounced for short dies. This is clearly seen in Fig. 4 for PP. In the case of PS, shown in Fig. 5, the same trend exists, except crossovers ap- pear with the 180" die being below the 120 and 90" entrance angles. The data is however close, as in the case of PP.

184 POLYMER ENGINEERING AND SCIENCE, FEBRUARY, 1980, Vol. 20, No. 3


I 1 I 1

PS (160°C)

L / D = @ 180' 0

L1D.o 100" 0 1200 900 0

60" A 30° A

capliiory e

1 O0 1 0' 1 O2

y , (sec")

Fig. 5 . lnpuence of die entry geometry on extrudate swell fo r P S .

DISCUSSION AND THEORETICAL ANALYSIS Die Entry Flow

The extrudate swell from short dies should be determined from the character of the entrance flow as this is the only flow history these melts know. It is thus ofvalue to discuss the entrance flow behavior of PS and PP. There have been numerous observations of die entry flow in melts, of which we may cite Tordella (20) on low density polyethylene (LDPE) Bagley and Birks (21) on LDPE and HDPE, Ballenger and White (22) for PS, PP, and LDPE and White and Kondo (23) for LDPE, PS, and HDPE. Vortex-like flows in the die corners are observed for the LDPE and PS which increase in size with extrusion rate. This is illustrated in Fig. 6. Bagley

r:l F i g . 6. Elongationalpow character of die entrance region.

and Schreiber (30) and White and Kondo have noted a reduction in vortex size with tapering of the die entry.

Of the polymer melts considered in the present study, the PS should have significant vortices. Indeed observations of vortices have been made by White and Kondo on this particular PS. The PP probably exhibits none. There are thus large differences in their die entry flows and histories as the melts emerge from short dies. The PS experiences a more "pure" elongational flow than the PP.

The reasons for these different entrance flow characteristics may be traced to the elongational flow characteristics of the two melts. To surmise the relative response of these two melts we may look to the studies of Ide and White (31) on similar melts. PS exhibits an elongational viscosity function which is constant and 3q0 at low deformation rates and is an increasing function of elongation rate, while PP appears to be a decreasing function. The large and rising elongational viscosity gives rise to an incompatible stress field in the converg- ing entrance flow. The vortices arise as a stress relief mechanism. This view is expressed by White and Kondo.

Dimensional Analysis

The simplest approach to the problem of extrudate swell rests on dimensional analysis arguments. Clearly swell from short dies should depend on geometry and the appropriate dimensionless groups representing rheological response. This would be the Weissenberg number, rrhVchlLrh (23, 32-34) where T,-h is a characteristic time, Vrh a characteristic velocity and Lrh a characteristic length.

lim B = B ( 7,k - ' r h , viscoelastic constitutive , e) (1) dimensionless ratios of

L c h parameters

where 8 is the tapered die entrance angle. (0 should not be confused with the a of Fig. 6.)

Theories of extrudate swell from long dies are usually expressed in terms ofweissenberg numbers ofform (11, 16, 25)

L+O

where Y,,. is the die wall shear rate. For extrudate swell from short slit and capillary dies

the ratio V r k l L c h should be given by the elongation rate hl1&xl defined as

The values ofA for slit and capillary dies are of form

Capillary: A = n- r tan - ( 9 (4b)

0 A = 2Wr tan - 2 Slit:

where r (-x,) is a radius vector pointed backward into the die entry region. From E 9 s 3 and 4

POLYMER ENGINEERING AND SCIENCE, FEBRUARY, 1980, Vol. 20, No. 3 185


Capillary:

Q 8 2Wr' tan

Slit: Y E ] s ( r ) =

(5b)

It is to be noted that these expressions for yE1 vary with position in the die entry.

To evduate rch we again use shear flow data, i.e.,

This is plotted in Fig. 7 . Tofnterpret elongational and die entry flow we consider r to depend on the second invariant of the rate ofdeformation tensor b( trd2) . Thus for elongational flow in a capillary entrance

and for the entry to a slit

Y E z s = - Y ~ l s 7,573 = 0 IT, = 2y,571sz (7b)

while for shear flow

(74 IT,=,?' 1

2 EquatingEys 7a, 7b, and 7c gives the correspondence

of; values from shear flow to the entrance flows, i.e.,

;(Y) = ;(fi Y E , , ) = ; ( ~ E l s ) (8)

To obtain ;(YE]) it was necessary to extrapolate ~ ( 7 ) data as discussed in our earlier paper (16).

We have developed dimensional analysis correlations for the swell from tapered entry capillary dies. Based on the above discussions, we define for a capillary the characteristics dimensionless group as

i.e., we take (2) - - - t a n 2 16Q 8 cap

Similarly we find for the slit

where H is the slit thickness. In Fig. 8, we plot B as a function of rrh V c h l L r h for the

30 and 120" angle capillaries. We find that for the 30" angle the PP and PS data correlate well. This is not the case for the 120" angle (or for the 180" angle). These discrepancies probably arise from the existence of vortices in the case of PS. The question of how to remedy these problems in terms of dimensionless groups may be

I I I I

Cone-Plote PS* PPm ---Extropoloted Lines based

N on'i __ 1 = A b-1 -1 2a;2y 2 q z y

10-2 lo-' 100 10' 1 0 2 10'

T 2 0 r JTy,,,or 2y;ls(sec-li

Fig. 7. = Ni12ai2~ as u function of Y f o r P S and PP.

I I I ' 1- Capillary Orif ice 0

8 PS PP - _ _ 120- 0 0 30° A A

L /? 01 0 2 0 3 0 4 1 05 1 0 6 , 0 7

F i g . 8 . Plot of B us Weissenberg number rrhVCnlLrh f o r 30 and 1.20" angle dies. ( Y , ~ is shear rate.)

answered from the work of White and Kondo (23). They use the deformation rate dependence of the relaxation time as the additional parameter in correlating vortex size. The different dependence of 7 (U) for PS and PP is shown in Fig. 7. As pointed out by Ide and White (32) small changes in r(Y) can induce large variations in elongational flow behavior. White and Kondo find similar responses in die entrance flow.

Theoretical

We have applied the unconstrained recovery theory ofTanner (25), as reformulated in our earlier papers (11, 16) to the problem of swell from short dies. The detailed analysis is given in an Appendix. Basically we consider the flow in the die entry region to be elongational in character and shear flow contribution is small as shown in Fig. 6. The unconstrained recovery in the melt

186 POLYMER ENGINEERING AND SCIENCE, FEBRUARY, 1980, Vol. 20, No. 3

Experimental and Theoretical Investigation of Extrudate Swell of Polymer. Melts

emerging from the die entry region is for a capillary die

(10a) This equation is awkward to use because of the integrals in the exponential. Generally we must express Y E I c ( z ) as a function of z and integrate. An approximate solution should be of use. If we may express the integral in the exponent of this equation as 112 Y E l c (r = Dl2 tan 812) z , it follows that:

1 + Teff ^3EIP(r = DI2tan 812) ' I 6

(lob) B - [ 1 - 2reff yYElr(r = Di2tan 8/2) 1

For the slit die

4 0

3 0

1

0 \ D - - - W

In

2 0

( l la )

and using the same approximate as leads to E q lob 1 + .reffYEls ( r = Hl2tan 8/2) 'I4

(W z represents a dummy time variable. Here ? E l c and ? E l s

are as deficed in the previous section. reff 'may be equated to r for Maxwellian fluids.

Equations 10 and 11 predict the qualitative trengs of our experimental data with melt elasticity through .rand increasing elongation rate raising extrudate swell. Swell from slit dies is predicted to be larger than that from capillary dies.

To properly use the above equations, we should have a true elongational flow distinct from a mixed shear and elongational flow. Such elongational flows are observed in LDPE and PS as mentioned earlier, but not in PP. The PS melt used in this paper was among those investi- gated by White and Kondo (23) in their study of die entrance flow patterns. It is possible from their correla- tion to know the value of entrance angle a (as defined in Fig. 6 ) . Y E l c is taken as 8Q/rD3 tan d 2 . In F i g . 9, we compare E q lob with experiment using these data. The agreement is reasonably good.

In order to make comparison with the PP data, we should have to include combined shear and elongational flow as exists in the Harrison solution (35) for flow in a cone.

B - [ 1 - 2reff?jEIs ( r = Hl2tan 812) 1

APPENDIX Theory of Extrudate Swell From Short Dies

Our arguments here use the unconstrained recovery theory of swell originally due to Tanner (25) which is an outgrowth of the earlier considerations of three dimensional elastic recovery by Lodge (36) and others (37). The particular formulation we use is similar to that of our earlier papers (11, 16, 37).

I ! I I I I

capillary Orifice (8=180°) o ( y w ) based on White4 Kondo (23) Polystyrene

0

0

0

10

F i g . 9. Comparison of theory and experiment for swell of P S from a 180" entrance angle die using E 9 10. { Y l 2 is shear rate.)

We use a single integral constitutive equation ofform

(A-1)

(11, 16, 23, 26, 34, 37)

= - rg + [rnl(z)~-' dz

where -' axi axj

ax, ax, ci, = - -

and

(A-2)

represents a Maxwellian relaxation function. xi is the instantaneous position and X, the past configuration in Cartesian coordinates.

We consider the die entrance flow to be of long duration and elongational in character. We write

which leads to

x l ( t ) = xI(o) 1 YElds xz(t ) = XZ(O) 1 (A-4)

Let us write E q A-1 in the form so that it represents the melt after it has exited from the die and exhibited swell. We introduce matrix Q which transforms back to the system of coordinates bLfore the swell to simplify kinematic considerations. This leads to

For such an elongational flow, we may write

Af 0 0

0 0 1 Q I - ' Q ' = 10 A; 01 (A-5)

POLYMER ENGINEERING AND SCIENCE, FEBRUARY, 1980, Vol. 20, No. 3 1 87


where

The swell has the form

Xg( t r ) = ~ 3 ( t ) (‘4-7)

There is no shear recovery. This allows us to specify Q and the left hand side of E q A-4.

-

( A 4 From the full E q A 4

1 (w11 + P )

(A-9b) Substitution of E q 9a into E y 917 gives

(A-10)

Integration across the cross-section and taking the integral of mII to be zero gives

- J o J o - i,”l,:-[.i Tefft2lPEZdzldz da

(A-12)

Generally the integration across the area may be can- celled out leaving the kernel function. This is as far as we can go with the general formulation.

We now seek to approximate E q A-12 so that we may integrate the integrals. Ifwe take kinematics to be independent of cross-section and independent of time with

this leads to

(A-14)

For a capillary

(A-l5a, b) 1 - a1 = I (2) = B6

a 2

P

(A- 15c)

This leads to

(A-16)

We might reasonably take rl as a suitable intermediate Y E I ~ , e.g.,

1 . 2 rl = - y E l r ( r = Dl2 tan $12)

This is identical to V r h l L c h of E q 9b. For a slit

7 a l = - 1 ($)=B4 ffz

Y E 2 = - Y E 1

This leads to

From E y A 5 6

v 2W12 tan 812 Y E l s =

We take

(A-17)

(A-18a, b) (A-19)

(A-20)

(A-21)

(A-22) 1 . 2 rl = - y E I ( r = HI2 tan 012)

This is identical to VrhILch of E q 9c.

ACKNOWLEDGMENT

This research has been supported in part by ;i Fellow- ship from the Phillips Petroleum Company and the Na- tional Science Foundation under grants NSF GK 43921 and ENG 76-19815. One of us (JLW) would like to thank the Humboldt Foundation for support during those parts of the research which were carried out while at IKV-RWTH.

Useful discussions on this problem were held with D. C. Bogue, P. Junk, W. Minoshima, and J. Wortberg.

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(1976), ibid., 21, 869 (1977). 12. F. Ramsteiner, Kunststoffe, 61, 943 (1971). 13. C. D. Han,AlChEJ., 19,649(1973);J.Appl. Polym.Sci., 19,

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24. R. Racin and D. C. Bogue, “Molecularweight Effects in Die Swell and In Shear Rheology,” University of Tennessee Polymer Science and Engineering Report No. 115 (March 1978). J . Rheology 23, 263 (1979).

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28. A. B. Metzner and A. P. Metzner, Rheol. Acta, 9,174 (1970). 29. F. N. Cogswell, Polym. Eng. Sci., 12, 64 (1972). 30. E. B. Bagley and H. P. Schreiber, Trans. Soc. Rheol., 5,341

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189

experimental and theoretical investigation of extrudate swell of polymer melts from small...

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