Modelling of the processing of incompatible polymer blends

Citation for published version (APA):Elemans, P. H. M. (1989). Modelling of the processing of incompatible polymer blends. [Phd Thesis 1 (ResearchTU/e / Graduation TU/e), Chemical Engineering and Chemistry]. Technische Universiteit Eindhoven.https://doi.org/10.6100/IR316189

DOI:10.6100/IR316189

Document status and date:Published: 01/01/1989

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MODELLING . OF THE

PROCESSING OF INCOMPATIBLE

POLYMER BLENDS

P.H.M. Elemans

MODELLING OF THE PROCESSING OF

INCOMPATIBLE POLYMER BLENDS

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de

Technische Universiteit Eindhoven, op gezag van

de Rector Magnificus, prof. ir. M. Tels, voor

een commissie aangewezen door het College van

Dekanen in het openbaar te verdedigen op

dinsdag 5 september 1989 te 14.00 uur

door

Petrus Henricus Maria Elemans

geboren te Oss

Dit proefschrift is goedgekeurd door

de promotoren

prof. dr. ir. H.E.H . Meijer

en

prof. dr. P.J. Lemstra

Dit onderzoek werd mogelijk gemaakt door financiële steun

van DSM .

voor mijn ouders

voor Alice

CONTENTS

1. INTRODUCTION

1.1. Structured blends

1.2. Modelling of mixing equipment

1.3. Distributive mixing

1.4. Dispersive mixing

l.S. Survey of the thesis

1.6. References

1

1

6

8

10

13

14

2. APPROACHES TO THE MODELLING OF MIXING EQUIPMENT 17

2.1. Mixing equipment 17

2.2. Modelling of mixing equipment 22

2.2.1. Liquid-liquid mixing 23

2.2.2. Solid-liquid mixing 28

2.3. References 32

3. MODELLING OF COROTATING TWIN-SCREW EXTRUDERS 35

3.1. Introduetion 36

3.2. Screw geometry

3.3. Analysis of simplified geometry

3.3.1. Relative lengtbs

37

41

42

3.3.2. Specific energy 43

3.4. Improved analysis 47

3.4.1. Leakage flows 47

3 . 4.2. Power consumption over the flights 49

3.4.3. Mixing elements 51

3.4.4. Sequence of screw elements 55

3.4.5. Nonisothermal powerlaw calculations 57

3.5. Calculated results 61

3.5.1. Specific energy

3.5.2. Combination of parts band c

3.5.3. End temperature

61

63

64

3.6. Experimental verification of the newtonian,

isothermal analysis

3.6.1. Throughput versus screw speed

characteristic

3.6.2. Pressure gradients

3.6.3. Filled lengths

3.6.3.1. Experimental setup

3.6.3.2. Results

3.7. Residence time distribution

3.8. Discussion

3.9. References

4. MODELLING OF THE CO-KNEADER

4.1. Introduetion

4.2. Screw geometry and working principle

4.3. Summary of the Newtonian, isothermal

analysis

4.4. Mixing

4.5. Experimental

4.5.1. Throughput versus pressure

64

65

68

73

73

75

77

79

80

82

83

84

87

91

93

characteristic 95

4.5.2. Filled length 100

4.5.3. Pressure gradients 103

4.6. Nonisothermal, non-Newtonian analysis 106

4.7. Residence time distribution 108

4.8. Discussion 111

4.9. References 111

5. SCALING

5.1. Sealing laws

5.2. Geometrical sealing

5.3. Thermal sealing

5.3.1. Laminar flows

5.3. 2. Ideally mixed annular

L/D = a constant

5. 3. 3. Ideally mixed annular

H/D = a constant

flow,

flow,

113

113

114

1Hi

117

118

120

5.4. Sealing laws

5.5. Example: Glass-fibre reinforcement

5.6. Conclusion

5.7. References

6. TIME EFFECTS IN THE DISPERSIVE MIXING OF

INCOMPATIBLE LIQUIDS

6.1. Introduetion

6.2. Affine deformation of droplets in

simple shear flow

6.3. Breakup of threads

6.4. Breakup of droplets

6.5. Experimental

6.5.1. Experimental setup

6.5.2. Model fluids

6.5.3. Results

6.5.3.1. Affine deformation

6.5.3.2. Breakup of threads in

120

121

122

123

124

124

125

127

129

134

135

135

136

137

simple shear flow 138

6.5.3.3. Stable deformation and

relaxation of droplets 139

6.6. Conclusion 141

6.7. References 142

7. MORPHOLOGY OF THE MODEL SYSTEM

POLYSTYRENE/POLYETHYLENE 144

7.1. Introduetion 145

7.2. Phase inversion 147

7.2.1. Materials 147

7.2.2. Blend preparatien 148

7.2.3. Phase inversion diagram 148

7.2.4. Influence of compression moulding 153

7.3. Film thinning 153

7.4. Influence of block copolymers 156

7.5. References

8. STABILITY OF MORPHOLOGIES, OR THE EXPERIMENTAL

DETERMINATION OF INTERFACIAL TENSION 161

8.1. Measurement of interfacial tension via

breakup of threads 161

8.2. Interfacial tension between two

homopolymers

8.2.1. Materials

8.2.2. Experimental procedure

8.2.3. Results

8.3. Influence of block copolymers on

interfacial tension

8.3.1. Materials

8.3.2. Results

8.4. Contact angle measurements

8.4.1. Experimental procedure

8.4.2. Results

8.5. Breakup of molten polymerie layers

162

162

163

166

168

169

171

173

175

176

- an illustration 177

8.6. Conclusion 179

8.7. References 179

9. COUPLING OF DETAILED AND OVERALL MODELLING 181

9.1. Examples of calculations on

dispersive mixing 181

9.1.1. Combined affine deformation

and reorientation 182

9 . 1.2. Dispersion of Rn isolated droplet

in a screw extruder 183

9.1.3 . Tot al shear in a corotating

twin-screw extruder 186

9.2. Breakup of threads 188

9.3. Experimental 190

9.4. Results 190

9.5. Conclusions 193

9.6. Reierences 1 94

SUMMARY

SAMENVATTING

NOMENCLATURE

CURRICULUM VITAE

196

199

202

207

-1-

CHAPTER 1

INTRODUCTION

Various morphologies can be realized via processing of

incompatible polymer blends, for instanee droplets or fibers

in a matrix and stratified or cocontinuous structures. The

structures induced are usually intrinsically unstable.

Modelling of extrusion processes and continuous mixers

yields expressions not only for the shear rate and shear

stress, but also for the limited residence time and the

number of reorientations. These results can be combined with

detailed knowledge of respectively distributive and

dispersive mixing processes to predict the development of

specific morphologies, i.e. structured blends, as a tunetion

of time.

1.1. STRUCTURED BLENDS

Similar to rubbers and thermosets, thermoplastic polymers

are hardly used in their pure form. Additives are needed to

improve for example processability and lifetime (lubricants,

stabilizers), modulus and strength (mineral fillers like

glass beads, chalk, clay, mica or glass- fiber

reinforcement), appearance and colour (pigments),

conductivity (conductive fillers like steelwire, aluminium

Reprinted partly from: H.E.H. Meijer, P.J . Lemstra and P.H.M . Elemans,

Makromol. Chem., Macromol. Symp., ~' 113 (1988), by permission of

Hüthig & Wepf Verlag, Basel.

-2-

flakes or carbon) or flammability (flame retardants) .

Despite of the continuous development of new polymers, a

large number of properties can only be obtained when

different polymers are combined. Well known examples are the

impact modified, (rubber) toughened polymers, where polymers

with different glass transition temperatures are blended,

and the group of barrier polymers for packaging, where

specific polar and apolar polymers are combined in order to

increase the resistance against water- and gas- (oxygen,

carbondioxide) transport simultaneously.

Of course there are various routes to combine polymers in

order to achieve optimum properties. Polymer blends can be

made directly on a microscopie scale in the reactor. The

other extreme, on a macroscopie scale, is co-extrusion to

produce multi-layered structures via casting, blowing, blow

moulding and injection-moulding. Extrusion (melt) blending

is a route in between and in principle a rather flexible

one. The limited miscibility of polymers (1,2) complicates

this processing route however.

Unless specific interactions exist, phase separation usually

occurs (3,4). Of course, processing of miscible polymer

systems is of interest since tailor made properties can be

obtained by just changing the volume fractions. Although

over 300 pairs of miscible polymers are known (2) only a few

systems have been commercialized. Well known is the

successful blend PPE/PS. Other systems of commercial

interest are PC/PET and PC/PBT (5) .

In general, however, we have to deal with incompatible

polymers and depending on the processing conditions various

morphologies can be obtained. Figure 1.1 displays some

characteristic morphologies as obtained by extruding the

incompatible blend of Polystyrene/High Density Polyethylene

(PS/HDPE). Figures 1.1a, 1.1b, and 1.1c display extrudates

obtained from a corotating twin-screw extruder. Figure 1.1d

shows a PS/PE composition made via the Multiflux static

mixer (6,7).

-3-

All these morpbologies were realized by extruding the model

system Polystyrene/High Density Polyethylene (PS/HDPE), by

changing the volume fractions, viscosity ratio or processing

route (8).

These structures have been classified befare (9) and have

been found in practice, for example, with SBS block

copolymers with different percentages of polybutadiene

(10,11). As one can imagine, there are one ortwoorders of

magnitude difference in the length scale between the

incompatible system PS/PE and the blockcopolymers of SBS.

Figure 1.1a . Figure 1.1b.

Figure 1.1c. Figure 1.1d.

Figure 1.1a. Scanning electron micrograph of the microtomed surface of a

85/15 PS/HDPE blend (viscosity ratio 1).

Figure 1.1b . As Figure 1.1a, of the edge of a microtomed surface of a

75/25 PS/HDPE blend (viscosit y ratio 2) . HOPE (in black)

still forms the continuous phase .

Figure 1.1c. As Figure 1.1a, of a 55/45 PS/HDPE blend.

Figure 1.1d. As Figure 1.1a, of 50/50 PS/HDPE Multiflux blend.

-4-

A lot of attention has been paid to the morphology shown in

Figure 1.1a, and especially to routes to obtain a small

partiele size. Experimental results have been reported by

Borggreve (12) and Wu (13) for the system PA/EPDM, where it

is clearly demonstrated that the tough-brittle transition

temperature is not only depending on the amount of rubber

(Figure 1.2) but also, at the same volume fraction, on the

partiele size of the dispersed phase (Figure 1.3). To obtain

this small partiele size of the dispersed rubbery phase

maleic anhydride modified EPDM had to be used (14).

1100 100 ---, .>!:

L 80 80 g, L ..-> ., g'

'-1il 60 ~ 60 ..-> u t);;' ~ ~-€ -~ 40 E --, 40 "8 ·- "'

"8~ ~ ~ u 20

~ 20 .,

L L ~ 15 0 0 c c

20 -40 -20 0 20 40 60 80 -20 0 T('C) T('C)

Figure 1-2- Figure 1-3

Figure 1.2. Brittle-tough transition in Nylon/rubber blends_

Effect of rubber concentration_ Data from Ref. 12.

40 60

(• 0;\/ 2.6;• 6.4;0 10.5;0 13.0;(', 19-6;e 26.1 voL%)

Figure 1.3. As Figure 1.2. Effect of partiele size_ Volume fraction

of the rubber is 26%. Data from ReL 12_ (• PA-6;

(j 1.59~;· 1.2~;\/ 1.14~;0 0.94~;0 0.57~;· 0.48~)

80

Structures as displayed in Figure 1.1b, i.e. flbrils in a

matrix are aimed for as reinforcement and in the fabrication

of synthetic paper or artificial leather (15,16,17).

-5-

Cocontinuous structures, Figure 1.1c, are usually obtained

if a 50/50 blend is extruded or, if the viscosity ratios of

matrix and dispersion differ from one, at other mixing

ratios as well (18). The morphology of the cocontinuous

structures is tosome e xtent similar to IPN's basedon

direct chemistry, although the scale is two orders of

magnitude higher. (19).

Layered structures, see Figure 1.ld, can be made rather

easily with specially designed static mixers like the Ross

mixer or even better with the Multiflux mi xer (6,7). The use

of layered structures is important for instanee in the area

of pac kaging. Figure 1.4 shows the barrier properties,

expressed in the effective diffusion coefficient, of a cast

or blown film as a function of the composition, with the

morphology as parameter.

-en ~ E 0 -

<D > -0 <D --<D

c

10- 6 ~-----------------------------------------------,

10-7

10-8

10-9

parallel

·~ EVOH spheres

lamel la;---.:=·~ -- ~ --- · -...........:::--:--.._ EVOH cylinders

~---- ~"':"--- ·~ ------- ·~

PP cylinders

-----~

---···---... ..__ ',<~', , --... ....._ ',\ ···-- \

PP spheres

10-104---------~--------r-------~r--------,--------~ 0 0.2 0.4 0.6 0.8

vol. fractlon EVOH

Figure 1.4 . Effective diffusion coefficient as a function o f composition

for a polypropylene/ethylene-vinyl alcohol copolymer

(PP/EVOH) blend, with different morphologies. After Ref. 20.

The two limiting curves correspond with the two extremes:

-6-

layers of the barrier material oriented either parallel or

perpendicular to the plane of the film. Note that a

logarithmic scale is used on the vertical axis, indicating

that the upper curve obeys the additive rule of mixtures.

As can be seen from the examples mentioned, these so-called

structured blends (21) all exhibit a distinct morphology.

It is important to understand how different morphologies in

a blend of two incompatible polymers can be obtained, and

guaranteed during subsequent processing. Detailed knowledge

of the processing equipment is necessary as well as an

understanding of the mixing process itself. Therefore, the

rest of this chapter will consist of a brief review of these

topics.

1.2. MODELLING OF MIXING EQUIPMENT

From simplified flow analysis inside extruders, important

overall parameters for mixing, such as residence time t,

shear stress "' shear rate y, total shear y and the nuffiber of

reorientations nr can be deduced, at least locally.

Especially if only melt-fed equipment is considered, all

geometries such as extruder channels and clearances, and

also converging flows with one or two moving boundaries,

e.g. the two roll mill, have been analysed (22,23,24,25). As

a consequence the local conditions present for mixing are

known even in typical compounding equipment like

batchmixers, counterrotating twin-screw extruders, Farrel

Continuous Mixers, corotating twin-screw extruders and

reciprocating pin extruders like the Buss Co-kneader.

Of course more elaborate calculations can be performed,

yielding the complete two- or three dimensional flow field

in the complex geometries of the mixing sections in a

corotating twin-screw extruder (26) and in the Co-kneader

(27).

-7-

However, it has to be postulated a priori that these mixing

sections are completely filled with melt and all

calculations are still isothermal. Here a more overall

investigation of these continuous mixers is developed, part

of which has been published already (28,29). By simplifying

the geometry again, the lengths which are completely filled

with melt are determined depending on the screw geometry

used and on processing conditions like screw speed and

(independently) metered throughput. Moreover, via an

averaged local heat balance, the temperature rise during the

compounding process and the specific energy, depending on

processing conditions, can be calculated .

If combined with criteria originating from a more complete

model of the dispersion process itself, this would be

sufficient to predict the morphology of an, as-processed,

blend. However, ignoring complicating factors like

coalescence of droplets, even the time effects af the

dispersion process are not well understood. For the more

simple dispersive mixing of carbon black in rubber in an

extremely simplified geometry of a completely filled

batchmixer (no time effects involved in breaking, no

influence of partiele size), see Figure 5, an interesting

analysis exist& (30,31), which is later extended

V

Q

HIGH SHEAR ZONE

Figure 1.5. High shear section in series with an infinitely well mixed

section. The fluid is continuously pumped from one section

into the other. From Ref. 30.

-8-

to two roll mills (32) . Provided that dispersive mixing of

blends is better investigated, these examples may be

extended to the rnadelling of the blending process in

continuous mixers, since the mathematical tools, necessary

for this kind of calculations, already exist from continuurn

mechanics, see for example (33) and (34).

1.3. DISTRIBUTIVE MIXING

For distributive mixing total shear y and number of

reorientations during the shear history are the only

determining factors. This has been clearly illustrated by Ng

and Erwin (35) who performed a c lassical experiment by

placing coloured slices of a polymer between two concentric

cylinders and rotate one of them. Either the number of

layers formed (measured radially) or the total interface,

bath being a measure for distributive mixing, is directly

proportional to the total shear.

A (A interfacial area) (1.1)

Since y = yt, shear rate and -time are interchangeable. If

the already formed layers are reoriented relative to the

direction of flow, mixing becomes much more effective. This

is illustrated by stopping the rotation, freezing, cutting

slices which are turned over 90°, an ideal reorientation,

heating up again and further shearing. If this procedure is

repeated n-1 times, Eq. 1.1 reacts (see Figure 1.6),

A = Ao ( 1 In y) n (1. 2)

A much more effective way of distributive mixing, because

reorientation does not cast energy (shear rate or -time)

Static mixers are the prime exponents of mixing by

reorientation rather than by total s hear y, but also in

corotating twin-screw extruders material is continuously

reoriented relative to the shearing motion of

-9-

Figure 1.6. Shearing and reorientation during shear of black and white

segments. From Ref. 36.

Figure 1.7. Mixing and

reorientation

in corotating

twin-screw

extruders.

the surfaces, when one screw scrapes the fluid from the

other one (Figure 1.7).

The number of reorientations can be estimated and forms,

tagether with the expressions for shear rate, time and total

shear, the basis for sealing rules for distributive mixing

in corotating twin-screw extruders (29) . Although the pins

of a Buss Co-kneader reorient the flow as well, see Ref. 27

and 37, the distributive mixing is better understood by

consictering the local weaving action of the pins (the screw,

as usual, is thought to be stationary and the barrel and

pins rotating and reciprocating, yielding sinuscictal

trajectories through the screw channel) .

Combined with an overall model of the continuous mixer (29)

-10-

this analysis directly provides insight in the distributive

mixing of additives, pigments, fillers and already dispersed

masterbatches in a matrix.

1.4. DISPERSIVE MIXING

If two incompatible polymers have to be blended, the

interfacial tension, which is directly related to the mutual

miscibility, becomes during the mixing process of the same

order of magnitude as the shear stress applied and will

dominate the resulting morphology. An order of magnitude for

the interfacial tension o is typically 10-2 [N/m], while the

shear stress ~ for polymer roelts is of the order of 10 4

(N/m2]. Consequently, if local radii are in the order of

10-6 [m], yielding o/R = 10 4 (N/m2 ], both stresses are

equal. Starting at the first stages of mixing, the droplets

undergo affine deformation according to Eq. 1.2 describing

the distributive mixing. The resulting long slender bocties

become instable due to the interfacial tension-driven

Rayleigh disturbances (38,39), see Figure 1.8.

Figure 1.8. Sinusoidal distortions on a PA-6 thread (diameter 55 ~)

embedded in a PS matrix at 230 ° C.

-11-

The draplets formed are again subjected to shear stresses

counterbalanced by the interfacial tension resisting the

deformation process. This process has extensively been

studied in the literature (38-45). Especially the work of

Grace (43) is worthwhile reading because of the large number

of experiments performed in shear and elongational flow with

liquids with a large range of viscosity ratios. The

stability of draplets turns out to be strongly dependent on

this viscosity ratio:

(1.3)

and of the ratio of the applied shear stress ~ = DeY and the

pressure due to the interfacial tension a/R, usually

referred to as the capillary number Ca:

Ca ~R/a (1. 4)

Quite a large difference exists between the (efficiency of)

shear- and elongational flows, especially if p ~ 1. This

difference can only partly be explained by the difference in

shear- and elongational viscosity (43). It is mainly due to

the difference in type of flow: weak vs. strong respectively

(46). See Figure 1.9.

1(0)

100 T.R/a

10

1

0.1 10-7 10-6 10-5 10-4 10-3 10-2 10-1 10° 101 102 103

Viscosity Ratio, p

Figure 1.9. Comparison of effect of viscosity ratio on critical shear

in rotational and irrotational shear fields. From Ref. 43.

-12-

Although all of these studies are performed with individual

draplets of model liquids at room temperature, they

emphasize the non-equilibrium state of the morphologies

given in Figures 1.1a-1.1d. The fibrous structures found in

PE/PS mixed on a corotating twin-screw extruder are half-way

the dispersive mixing process and are typically formed in

the strong elongational flow field between screw tips and

die and in the filament between the die and water quench.

This is clearly illustrated in Figures 1.10 and 1.11 showing

two different spots in the same filament. In one case

(Figure 1.10) some fibrils (the smaller ones, of course)

start to break up exhibiting Rayleigh disturbances while in

the second case (Figure 1.11) more fibers have broken up.

These effects, including coalescence, which is also found in

Figure 1.9, always occur during the mixing process. The

morphology will continuously change and adapt itself to

local situations.

Figure 1.10. Figure 1.11.

Figure 1.10. Scanning Electron micrographof a fracture surface

parallel to the direction of extrusion of an extrudate of a

45/55 PS/HDPE blend (viscosity ratio 1) . Fibreus PS is shown

in different stages of breakup and coalescence.

Figure 1.11. As Figure 1.10, but with more fibers breken up.

-13-

1.5. SURVEY OF THE THESIS

The main field of interest, as outlined in the previous

sections, can be further explored now. The final objective is

to combine the knowledge of specific areas into a more

complete model that can be of use in the processing of

polymer blends. The content of each chapter wi11 be briefly

indicated below.

Chapter 2 gives a review of the different approaches to the

rnadelling of compounding equipment.

Chapter 3 deals with the modelling of melt-fed corotating

twin-screw extruders. It concentrates on the calculation of

locally fi1led lengths, power, specific energy and

temperature rise. With regard to mixing, the shear rate,

shear stress, residence time and the number of reorientations

can be determined. Experiments supporting the analysis are

presented.

In chapter 4, the computational model for the corotating

twin-screw extruder is applied to the Cokneader. The course

of this chapter is analogous to that of chapter 3.

In chapter 5, methods of sealing will be introduced.

Chapter 6 discusses the processes involved in the mixing , of

two (incompatible) liquids, with emphasis on the time effects

of deformation and breakup of dispersed particles in

well-defined fields of flow.

Chapter 7 gives, as an example of an incompatible blend, the

morphology of the model system Polystyrene-Polyethylene, as

processed on a corotating twin-screw extruder. Parameters

varied are blend composition and viscosity ratio.

Chapter 8 describes model experiments concerning the

stabilization of specific, non-equilibrium morphologies.

Chapter 9 attempts to couple the knowledge of the previous

chapters. The origin and the development of blend

morphologies, made on a corotating twin-screw extruder, is

followed.

-14-

1.6. REFERENCES

1. 0. Olabisi, L.M. Robeson, M.T. Shaw, Polymer/Polymer

Miscibility, Academie Press, New York (1979)

2. L.M. Robeson, Polymer Compatibility and Incompatibility,

NMI Press (1982)

3. J.M. Barlow, D.R. Paul "Polyblends '87 ", Boucherville

Canada, Polym. Eng. Sci. Zl_, 1482, (1987)

4. W.H. Stockmayer, R. Koningsveld, E. Nies in 'EQuilibrium

Thermodynamics of Polymer Systems' Vol . .l: Polymer Phase

Diagrams, Oxford Univ. (1988)

5. NN, Plastics Eng. 2, 24, (1986)

6. R. Sluijters" De Ingenieur TI, 15 ,33 (1965)

7. D. S'cbil'O, I<. Ostertag, Verfahrenstechnik .6., 2, 45 (1972)

8. P.H.M. Elemans, J.G.M. van Gisbergen, H.E.H. Meijer in

'Integration of Polymer Science and Technology', Elsevier

(1988)

9. M. Matsuo, S • .Sagaye in 'Colloidal and Morphological

Behaviour of Block anà Graft Polymers', G.E. Molau, Plenum

(1971)

10. M. Matsuo, Japan Plastics, .2., 6, (1968)

11. J.A. Manson, L.H. Sperling, Polymer Blends aod Composites,

Plenum (1976)

12. R.J.M. Borggreve, R.J. Gaymans, J. Schuijer, J.F. Ingen

Housz, Polymer ~' 1489-1496 (1987)

13. S. Wu, Polym. Eng. Sci. 27, 335-343 (1987)

14.Du Pont, US Patent 580513

15. G.V. Vinogradov, N.P. Krasnikova, V.E. Dreval, E.V.

Kotova, E.P. Plotnikova, Int. J. Polym. Mat. ~, 187 (1982)

16.M.P.Zabugina, E.P. Plotnikova, G.V. Vinogradov, V.E.

Dreval, Int. J. Polym. Mat. lQ, 1 (1983)

17.M.V. Tsebrenko, Int. J. Polym. Mat. lQ, 83 (1983)

18. G.N. Avgeropoulos, F.C. Weissert, G.G.A. Böhm, P.H.

Biddison, ACS Rubber Division Meeting, Paper 3, New

Orleans (1975)

19.K.C. Frisch, D. Klempner, H.L. Frisch, Polym. Eng. Sci.

22, 17 (1982)

-15-

20. J. Sax, J.M. Ottino, Polym. Eng. Sci. ~' 165 (1983)

21.H.E.H. Meijer, P.J. Lemstra, P.H.M. Elemans, Makromol.

Chem., Macromol. Symp. 1Q, 113 (1988)

22. J.M. McKelvey, Polymer Processing, J. Wiley (1962)

23. J.M. Funt, Mixing of Rubbers, RAPRA (1977)

24. z. Tadmor, I. Klein, Engineering Principles of

Plasticating Extrusion, Reinhold (1971)

25. Z. Tadmor, C.G. Gogos, Principles of Polymer Processing,

J. Wiley (1979)

26.W. Szydlowski, K. Brzoskowski, J.L. White, Int. Polym.

Proc. l_, 207 (1987)

27.M.L. Booy, Y.K. Kafka, Submitted to Polym. Eng. Sci.

28.H.E.H. Meijer, P.H.M. Elemans, H.H.M. Lardinoye, G. Kremer

in: 'Wärmeübertragung bei der Kunststoffverarbeitung', VDI

Düsseldorf (1986)

29.H.E.H. Meijer, P.H.M. Elemans, Polym. Eng. Sci. ~'

275-290 (1988)

30. I. Manas-Zloczower, A. Nir, Z. Tadmor, Rubber Chem. Techn .

.5..5_, 1250 (1982)

31. I. Manas-Zloczower, A. Nir, Z. Tadmor, Rubber Chem. Techn.

;u, 583 (1984)

32. I. Manas-Zloczower, A. Nir, Z. Tadmor, Dispersive mixing

in two roll mills (submitted)

33. J.M. Ottino; Polym. Eng. Sci. ~' 7 (1983)

34.D.V. Khakhar, J.M. Ottino, Int. J. Multiphase Flow~' 7

(1987)

35. K.Y. Ng, L. Erwin, Polym. Eng. Sci. 21, 4, (1981)

36. G.M. Gale, Rapra Merobers Report No. 46 (1980)

37. L.Erwin, F. Mokhtarian, Polym. Eng. Sci. ~' 49-60 (1983)

38. Lord Rayleigh, Proc. Roy. Soc. (London) 2..2_, 71 (1879)

39. S. Tomotika, Proc. Roy. Soc. Al.2Q, 322 (1935)

40. G.I. Taylor, Proc.Roy. Soc. (London) A146, 501 (1934)

41. R.G. Cox, J. Fluid Mech. n, 601 (1969)

42. E.J. Hinch, A. Acrivos, J. Fluid Mech. ~' 305 (1980)

43. H.P. Grace, Chem. Eng. Comm. ll, 225 (1983)

44. J.J. Elmendorp, Ph.D Thesis Delft University of Technology

(1986)

-16-

45. C.D. Han, Multiphase Flow in Polymer Processing, Academie

Press ( 1981)

46.W.L. Olbricht, J.M. Rallison, L.G. Leal, J. Non-Newt.F.M .

.l.Q., 291 (1982)

-17-

CHAPTER 2

APPROACHES TO THE MODELLING OF MIXING EQUIPMENT

Mixing equipment, used in the compounding and blending of

polymers, is briefly reviewed, as well as the different

approaches used in the rnadelling of mixers.

Early rnadelling started with those sections that were

accessible in terms of flow field (two-dimensional; one

direction (lubrication approximation)), fluid (Newtonian),

and geometry (the unrolled extruder channel (1,2), the nip

region of a roll mill (3)). Throughout the years, the

analyses have become more sophisticated. Non-Newtonian,

non-isethermal effects (4), a three-dimensional (two

directions) description of the flow field (5), the use of

finite-element techniques (6,7) and even chaotic motions (8)

have been incorporated. However, they only give solutions to

local problems; overall answers are still hard to find.

Also combinations of (simplified) flow analysis with local

dispersion and breakup processes of solicts (9) or liquids

(10) are scarce.

2.1. MIXING EQUIPMENT

Mixing operations of highly viscous polymers take place in

various types of machines. These can be divided into batch

mixers (a few only) and (many types of) continuous mixers.

Batch mixers

Batch mixers are usually found in the processing and

compounding of rubbers. Most common types are the internal

mixer and the roll mill, because of their dispersive and

distributive mixing qualities (11). During each revolution,

-18-

the material is forced to pass through a nip, where the

deformation and breakup of the dispersed (solid or liquid)

particles takes place. Between two passes, the material is

reoriented, either via manual cutting (roll mill), or via

the lateral motion induced by the particularly shaped rotor

wings (internal mixer). See Figure 2.1.

c

-E@3 ~ B --== D~ ~-. ~· -

Figure 2.1. Exarnples of rotor designs (arrows indicate pumping action of

rotor wings). A: Banbury two-wing; B: Banbury four-wing;

C: Shaw Intermix three-wing; D: Werner & Pf1eiderer four-wing.

From (12).

Single screw extruders

Single screw extruders consist of a conveying screw fitting

closely in a cylindrical barrel. One wall (the barrel) is

stationary, while the other wall (the screw) is moving, thus

dragging the material in the direction of the die at the

outlet. A pressure flow is generated in the reverse

direction, due to the resistance of the die. Mixing is

achieved by the motion caused by the combination of drag and

pressure flow inthescrew channel (13,14). The mixing

quality of singlescrew extruders is generally poor (15),

but can be improved when the screw is equipped with extra

mixing elements that provide for periadie reorientation of

-19-

the material, such as barrier sections (16), pineapple

heads, blisters, eccentric disks or even cavity transfer

mixers ( 17) .

Co-kneader

More flexible in screw design is the Co-kneader (18). This

is a single screw extruder with a simultanuously rotating

and oscillating screw having interrupted flights. Pins from

the barrel are inserted into the screw channel. The combined

weaving motion of pins and flights gives rise to good

dispersive as well as distributive mixing.

Twin-screw extruders

J~~~ J~ J~

J~ G--~ ~0--~

Figure 2.2a Figure 2.2b

Figure 2.2 . Fully, partly and nonintermeshing types of corotating (Fig.

2.2a) and counterrotating (Fig. 2 . 2b) twin-screw extruders.

From (21).

As Figure 2.2 shows, twin-screw extruders may be divided

into counter- and corotating types and into closely, partly

-20-

and nonintermeshing systems (19-21). Apart from the

direction of rotatien of the screws, they can be subdivided

according to their transport mechanism: positive

displacement or drag flow. This division can be made by

investigating whether the channel is closed in the axial

direction (by the flight of the opposite screw) or open

(22-24) .

Counterrotating twin-screw extruders

Counterrotating twin-screw extruders can be constructed with

small clearances, and the closely intermeshing types are

therefore often associated with positive displacement. In

practice, this does not prove to be very realistic because

apart from the typical tetrahedral gap between the sides of

the adherent screw flights and the necessary clearance

between barrel and screws, the so-called calender gap

between screw root and tip of the flight of the opposite

screw is often rather large. This gap drags material (with

two rnaving walls!) backward into the previous C-shaped

chamber. The counterrotating extruder is treated in detail

in Refs. 25 and 26 with the final result that the pumping

characteristic, throughput versus pressure buildup, is

rather easily obtained as the number of C-shaped chambers

becoming free per unit of time multiplied by the volume of

one chamber minus the sum of all leakage flows. Even with

small clearances the backflow caused by leakage can be in

the order of half of the positive displacement, depending on

the pressure at the die. Counterrotating twin-screw

extruders are almast exclusively used in poly(vinyl

chloride) processing, because of their mild treatment of the

melt. As far as mixing is concerned, they can be treated as

a continuous two-roll mill process.

Farrel Çontinuous Mixer

The Farrel Continuous Mixer (FCM) is a combination of an

-21-

internal mixer with a nonintermeshing counterrotating

twin-screw extruder (Figure 2.3). The mean residence time of

FCMs is in the order of 20 seconds, which is short.

Therefore, they are mainly used for the fast melting and

pelletizing of premixed powdered polymers (PP, ABS, or HDPE)

Figure 2.3. Typical screw geornetry of a Farrel Continuous Mixer (FCM)

Corotating twin-screw extruders

Corotating twin-screw extruders are, much more than

counterrotating twin-screw extruders or FCMs, preferred in

the processing of polymer blends. They operate almost

completely under atmospheric pressure, since their main

pumping mechanism is drag flow. Via openings or vent ports

in the barrel, material can be added to the melt (fillers,

stabilizers, pigments) or volatiles can be removed. To do

so, these extruders normally have to be underfed, and

pressure is only generated in those parts of the screw that

are completely filled, e.g. countertransporting sections.

Mixing is achieved very effectively in the intermeshing

region between the screws. The material is passed from one

screw to the other, and is thus reoriented.

Corotating Disk Processor

Though still rarely encountered as a continuous mixer, the

Corotating Disk Processor can be mentioned (27). Its basic

-22-

geometry (processing chamber) consists of two parallel disks

mounted on a shaft, and fitting in a cylindrical barrel. The

pumping action is very efficient, since it is due to two

jointly rnaving walls. Apart from channel blocks, which

separate the processing chambers from each other, mixing

pins can be inserted through the barrel wall. They provide

for reorientation by splitting the streamlines as well as

for dispersive mixing because the material is forced through

narrow clearances, see Figure 2.4. Without these pins, the

mixing action of the Corotating Disk Processor is as poor as

that of other effective pumps like a gear pump (28).

MIXING

"~~== MIXING BLOCK CVTLET DISK 2

Figure 2.4. Processing chamber with mixing pins of a Corotating Disk

Processor. Frorn (29).

2.2 MODELLING OF MIXING EQUIPMENT

In rnadelling of mixing equipment, one aften encounters

complex geometries and fluids exhibiting strongly

non-Newtonian behaviour. Different approaches have been

developed to get a better understanding of the mixing

process in batch and continuous mixers.

Usually the material is considered to be completely molten,

-23-

having a well-defined (Newtonian) viscosity. Approaches or

roodels that focus on single- or twin-screw extruders can

aften be applied to batch mixers or vice versa.

It is interesting to review the most important approaches,

thereby camparing their practical value in the field of

distributive and dispersive mixing.

2.2.1. LIQUID-LIQUID MIXING

Liquid-liquid mixing is the approach that is found in many

textbooks on polymer processing (30-33). The geometry of

most mixing equipment (extruders, internal mixers) is

locally represented as two parallel plates. The relevant

parameters for the mixing process are firstly combined into

the shear rate y and the average residence time t.

circumferential speed 11' D N y

loc al channel depth H (2 .1)

volume 11' D L H t (pressure dependent) throughput Q(P) (2. 2)

The total strain y

y = y t, (2 .3)

and the temperature T define the basic parameters for mixing.

From this simple starting point, complications can be

incorporated, such as the number of reorientations nr and

the effect of initial orientation. Due to the complex

geometries of most mixers, it is inevitable to introduce the

use of average shear rates. In addition, residence time

distribution and the weighted average total strain are

necessary to characterize the mixer performance.

The mixing process is thought to be in its initial stage

(see Se ctiens 1.3 and 1.4), with large strains imposed by

the matrix on the suspended particles of the minor phase.

The analyses are usually based on the isothermal flat plate

-24-

model of the extruder screw (34). Newtonian flow is assumed.

From the velocity field in down-channel direction as wel! as

in cross-channel direction, the average residence time, the

respective average shear rates y2 and Yx and the total shear

strain y, can be calculated.

The problem of averaging is quite complicated for the

different types of mixers. Each fluid element, starting at a

given initia! position in an extruder channel, wil!

experience a different strain history. This was quantified

by Lictor and Tactmor (35), who introduced the Strain

Distribution Function (SDF) f(y)dy. It is defined as the

fraction of the fluid in the mixer which has experienced a

shear strain from y to y+dy. The mean strain of the fluid at

the exit of the mixer is:

00

y I y f(y)cty (2 .4)

Yo

with YQ the minimum strain.

Pinto and Tactmor (36) propose the Weighted Average Total

Strain (WATS) to calculate the amount of strain experienced

by a fluid element in a single screw extruder. It is defined

by:

00

WATS I y (tl f(tl ctt (2 .5)

0

with y(t) the strain undergone by a fluid element at a

time t

{(t) the Residence Time Distribution (37)

Unfortunately, the WATS does not constitute a quantity that

can be determined experimentally. Neither can it account for

the initia! orientation or perioctic reorientations of the

material,as brought about by mixing sections of an extruder.

Its main importance is, therefore, on the theoretica! level.

-25-

Extensions of this approach can be found in the work of Bigg

and Middleman (38) . They study the evolution of the

interfacial area, which is a measure of the degree of

mixing, between two immiscible fluids by following

tracer particles in two-dimensional flow fields that are

present in extruder channels. Mixing sections in single

screw extruders greatly enhance the formation of interfacial

area (39,40). Ottino (15) presents a complete

three-dimensional description ('framework') of the flow

field in e.g. single screw extruders, with essentially the

same conclusion as Ref. 40: The initia! orientat i on and,

most of all, the number o f reorientations are the

determining factors in liquid-liquid mixing.

Although a complete description of the velocity field is

useful to understand the working principle of a mixer, it is

not sufficient. Even seemingly simple velocity fields can

give rise to a quite complex flow or 'motion' (in terros of

continuurn mechanics) of fluid particles (41,4 2 ). As stated

before (see Section 1.3), distributive mixing is usually

analysed in terros of the deformation (to a large extent) of

'blobs' or granules, schematically represented by a material

line element. Under certain conditions, these material lines

undergo e xponential stretching. In two dimensions, this is

possible for instanee in a hyperbalie flow field, and in the

flow inside a cavity that has periodically moving walls

(43). In the latter case, the mixing has become 'chaotic'.

A chaotic flow is characterized by either of the two

following (equivalent) criteria: (i) The flow has a positive

Liapunov exponent or (ii) the flow forms so-called 'Smale

horseshoes' (44).

The Liapunov exponent can be explained using the length

stretch of an infi nitesimal materia l filame nt dx undergoing

the flow

-26-

(2 .5)

with x the position of a given material point at time t

xo the initial position

~ the deformation gradient

The length stretch À is defined by (15):

À lim I dx I I I dxo I .f(~ mo mo) ldx0 l+o

with ldxol

~

the initial length

the Cauchy-Green deformation tensor: ~

the initial orientation dxo/ldxol

(2. 6)

In most flows, it is normally observed that À oc t for long

times (e.g. in a shear flow). For that reason, these flows

are called 'weak flows' (43). Strong flows, on the other

hand, show À - eet, with o the Liapunov exponent:

lim ( 1/t) ln À (2. 7)

t+o:>

The effect of o > 0 is that fluid particles, no matter how

close tagether initially, become separated exponentially

(43). This is ref1ected in the behaviour of the

two-dimensional 'blinking vortex flow' (45).

The Smale harseshoe function (44) is shown in Figure 2.5. It

involves the stretching and folding of a square with itself.

This is the only possible mechanism of increasing length for

a two-dimensional surface in a bounded flow.

The presence of harseshoes is shown by Chien et al. (46).

They study the deformation of a blob of tracer liquid in a

cavity that has periodically rnaving upper and lower walls.

An optimal value of the dimensionless frequency f exists

which produces maximum (i.e. exponential) stretching of the

-27-

initia! blob in a given time. With regard to mixing, it is

desirable that the horseshoe functions are present over a

large portion of the mixing region. This is still difficult

to predict.

I I I

I Figure 2.5. Representation of the Smale horseshoe function. After (46).

It is clear that the concept of chaotic mixing is far from

complete and limited in practical application, but still it

may give some more insight into the process of the rapidly

decreasing length scale of two initially segregated fluids.

Also, the existence of 'demixed' regions can well be

demonstrated, see e.g. Ref 47.

-28-

2.2.2. SOLID-LIQUID MIXING

A much more simple approach can be found in the work of

Manas et al. (9,48), who develope an analysis of the

dispersive mixing process in Banbury-type of mixers, which

is later extended to rol! mills (49). Although originally

proposed for the dispersive mixing of carbon black in rubber

(no time effects involved), the analysis can in principle be

applied to e.g. the blending of incompatible polymers.

The approach successfully integrates a number öf aspects

that are relevant for the modelling of the mixing process:

(i) an extreme simplification of the mixer geometry, (ii) the

influence of (initia!) orientation and reorientation of the

dispersed particles, and (iii) a criterium for breakup of

carbon black agglomerates.

During one pass through the nip region of the batch mixer

(the high shear zone in Figure 1.5), the carbon black

agglomerates are subjected to hydrodynamic forces exerted on

them by the matrix fluid (which is considered to exhibit

Newtonian or power law behaviour) . The effectiveness of

these forces depends on the instantanuous orientation of the

agglomerates, and can be expressed as

with x s

shape factor

characteristic cross-sectional area of an

agglomerate

• shear stress

e,~ instantanuous orientation angles

(2. 8)

The agglomerates are thought to consist of clusters of

aggregates, which are held together by cohesive forces (9).

In their most elementary form, these forces can be expressed

as

-29-

1-c

c ~

) d s

with c the relative void space between the aggregates

Co a constant

d diameter of the aggregate

Agglomerate breakup is assumed to occur when the

hydrodynamic forces exceed the cohesive forces:

~> F = 1

c

With the Eqs. 2.8 and 2.9, this yields

~ z sin2e sin (f/ cos (f/ Fe

8 1-c d with z 9 x 1: ( co c

(2 .10)

(2 .11)

(2.12)

Following this criterium, agglomerate rupture is independent

of agglomerate size. With the criterium, and with some

statistics concerning the distribution of passes of a given

agglomerate through the nip, this part of the problem is

essentially solved. After leaving the nip, the agglomerates

are reoriented in the mixer chamber before entering the

high-shear zone again with a random orientation distribution.

By defining the volume fraction of undispersed agglomerates

(i.e. those above a certain critica! diameter), calculated

predictions can be compared with experimental data, see

Figure 2.6a. The agreement is fairly good, given the

relative simplicity of the model and the large number of

assumptions on which the analysis is based.

Figure 2.6b shows the calculated fraction undispersed carbon

black as a function of rotor speed.

Figure 2.6c shows an interesting Qifference in machine

performance. When styrene-butadiene rubber (SBR, with

-30-

~O = 105 Pa.s) is compounded, the fraction undispersed (I

carbon black decreases monotonically with increasing rotor

tip clearance, i.e. throughput. This means: the available

hydrodynamic farces are always much larger than the cohesive

farces (compare Eq. 2.10). Upon compounding low-density

polyethylene (LDPE, with ~O = 103 Pa.s), however, the

fraction undispersed carbon black exhibits a minimum, and

increases again with larger tip clearance. Since now the

hydrodynamic farces are of the same order of magnitude as

the cohesive farces, only the more favourably oriented

agglomerates will break up.

Ul

~ ~ Q)

E 0 .8 0 0, OI

"' '0 0.6 Q) Ul

Q; a. "' '6 0.4 c: :J

0 c: 0.2 0

·;::; u "' L.t 0

0 100 200 300 400 500 600

time (s)

Figure 2.6a. Fraction (f) of undispersed carbon black as a function of

time. Parameter: diameter D (in mm) of the internal mixer.

Symbols indicate experimental data. From (9) .

In this thesis, this line will be extended to the analysis

of the dispersive mixing of two incompatible fluids. The two

most important continuous mixers are analysed in a rather

simple but overall way (yielding data on y, t, y, nr,

depending on processing conditions) . Next we will study the

time effects of the local dispersive mixing and finally an

attempt will be made to combine these two approaches in one

model.

-31-

"' ! "' (ij E 0.8 0 0> 0)

"' " 0.6 Q)

"' (ij 90 sec. a. "' 0.4 '6 c: :::l

0 c: 0.2 0

'<3 "' Lt 0

0 40 80 120 160 200

rotor speed (rpm)

Figure 2.6b. Simulated fraction of undispersed agglomerates in a Banbury

B mixer (D ~ 97 mm) as a function of rotor speed. Parameter:

mixing time. From (9).

"' ! "' (ij E 0.8 0 0> 0)

"' " 0.6 Q)

"' Îi) a. "' 0.4 '6 c: :::l -0 c: 0.2 0

·.::; SBR 0

"' Lt 0 3 4 5

gap height (mm)

Figure 2.6c. As Fig. 2.6b, as a function of rotor tip clearance,

for a mixing time of 120 seconds. Power law model

parameters: low density polyethylene (LDPE) no = 103 Pa.s,

n 0.46; styrene-butadiene rubber (SBR) no = 105 Pa.s;

n 0.3. From (9).

-32-

2.3. REFERENCES

1. Anonymous, Engineering (London), ~' 606 (1922)

2. H.S. Rowell and D. Finlayson, Engineering (London), ZQ,

249 (1928)

3. G. Ardichvili, Kautschuk Gummi Kunststoffe, ~, 23 (1938)

4. R.E. Colwell and K.R. Nickolls, Ind. Eng. Chem., ~' 841

(1959)

5. R.M. Griffith, Ind. Eng. Chem. Fund., 1, 180 (1962)

6. W. Szydlowski and J.L. White, Intern. Polym. Proc. II,

3/4, 142 (1987)

7. R. Brzoskowski, J.L. White, W. Szydlowski, N. Nakajima

and K. Min, Intern. Polym. Proc. ~, 3, 134 (1988)

8. J.M. Ottino, C.W. Leong, H. Rising and P.D. Swanson,

Nature, ~, 419 (1988)

9. I. Manas-Zloczower, A. Nir and Z. Tadmor, Rubber Chem.

Techn., ~, 1250 (1982)

10.D.V. Khakhar and J,M, Ottino, Int. J. Multiphase Flow,

u, 71 (1987)

11. J.M. Funt, Mixing of Rubbers, Rapra (1977)

12.H.Palmgren, Rubber Chem. Techn., ~' 462 (1975)

13. J.F. Carley, R.S. Mallouk and J.M. McKelvey, Ind. Eng.

Chem., ~' 974 (1953)

14. R. Chella and J.M. Ottino, Ind. Eng. Chem. Fund., ZA, 170

(1985)

15. J.M. Ottino and R. Chella, Polym. Eng. Sci., ~, 7 (1983)

16. B.H. Maddock, SPE J., ~' 23 (1967)

17. G.M. Gale, 41st SPE ANTEC, 109 (1983)

18.P. Schnottale, Ka~tschuk Gummi Kunststofte ~, 2/85, 116

(1985)

19. C.J. Rauwendaal, Polym. Eng. Sci., 21, 1092 (1981)

20.H. Herrmann and U. Burckhardt, Kunststoffe, Qa, 753 (1978)

21.H. Werner, Dissertation, Munich University of Technology

(1976)

22.K. Eise, S. Jakopin, H. Hermann, U. Burckhardt and H.

Werner, Adv. Plast Techn., 18 (April 1978)

-33-

23.U. Burckhardt, H. Herrmann, and S. Jakopin, Plast.

Compounding, Nov./Dec. 73 (1978)

24.M.L. Booy, Polym. Eng. Sci., ~, 606 (1975)

25. C.J. Rauwendaal, Polymer Extrusion, Hanser Publishers,

Munich (1986).

26.L.P.B.M. Janssen, Twin-screw extrusion, Elsevier,

Amsterdam, (1978).

27. Z. Tactmor, P. Hold and L. Valsamis, Plast. Eng. J., ~,

20 (1979), ibid., ll, 34 (1979)

28.B. David and z. Tactmor, Intern. Polym. Proc.~, 1, 39

(1988)

29. L.N. Valsamis and G.S. Donoian, Adv. Polym. Techn., ~,

131 (1984)

30. J.M. McKelvey, Polymer Processin~, J. Wiley, New York

(1962)

31. Z. Tactmor and I. Klein, Engineering Principles of

Plasticating Extrusion, Van Nostrand Reinhold (1970)

32. z. Tactmor and e.c. Gogos, Principles of Polymer

Processing, Wiley-Interscience, New York (1979)

33. S. Middleman, Fundamentals of Polymer Processing,

McGraw-Hill, New York (1977)

34. w.o. Mohr, R.L. Saxton and C. H. Jepson, Ind. Eng.

ti, 1857 (1957)

35. G. Lidor and z. Tactmor, Polym. Eng. Sci., .l.Q, 450

36. G. Pinto and z. Tactmor, Polym. Eng. Sc i., 10, 279

Chem.,

(1976)

( 1970)

37.P.V. Danckwerts, Chem. Eng. Sci., ~, 1 (1953), ibid.,~,

93 (1958)

38. D. Bigg and S. Middleman, Ind. Eng. Chem. Fund., ~, 184

(1974)

39.L. Erwin, Polym. Eng. Sci., ~, 572 (1978)

40. L. Erwin and F. Mokhtarian, Polym. Eng. Sci., ~, 49

(1983)

41. H. Aref, J. Fluid Mech . , ~, 1 (1984)

42.D.V. Khakhar, H. Rising and J.M. Ottino, J. Fluid Mech.,

.1..12_, 419 (1986)

43.D.V. Khakhar and J.M. Ottino, Phys. Fluids, ~, 3503

(1986)

-34-

44. S. Smale, Bull. Amer. Math. Soc., 21, 747 (1967)

45.M.F. Doherty and J.M. Ottino, Chem. Eng. Sci., ~' 139

(1988)

46.W.L. Chien, H. Rising and J.M. Ottino, J. Fluid Mech.,

l..l..Q., 355 (1986)

47.D.V. Khakhar, J.G. Franjione and J.M. Ottino, Chem. Eng.

Sci., .1.2_, 2909 (1987)

48. I. Manas-Zloczower, A. Nir and Z. Tadmor, Rubber Chem.

'i'echn. ;u, 583 ( 1984)

49. I. Manas-Zloczower, A. Nir and Z. Tadmor, private

communication (1983)

-35-

CHAPTER 3

MODELLING OF COROTATING TWIN-SCREW EXTRUDERS

In many operations in polymer processing, such as polymer

blending, devolatilization or incorporation of fillers in a

polymerie matrix continuous mixers are used, e.g. corotating

twin-screw extruders (ZSK), Buss Cokneaders and Farrel

Continuous Mixers.

Theoretical analysis of these machines tencts to emphasize

the flow in complex geometries rather than generate results

which can be directly used (1-5).

In this chapter a simple model is developed for the hot melt

closely intermeshing corotating twin-screw extruder,

analogous to the analysis of the single screw extruder

carried out in 1922 and 1928 (6,7).

With this model, and more specifically with its extension to

the complete nonisotherrnal, non-Newtonian situation, it is

possible to understand the extrusion process and to

calculate the power consumption, specific energy consumption

and temperature rise during the process. These calculations

can be performed not only with respect to the viscosity of

the melt, but also dependent on the screw speed and the

screw geometry (location and number of transport elements,

kneading sections and blister3, pitch (positive or

negative), screw clearance and flight width).

To support the theoretical analysis, model experiments with

a Plexiglas walled twin-screw extruder are performed, in

actdition to practical experiments with melts on small and

large scale extruders, with very reasonable results.

Reprinted partly frorn: H.E.H. Meijer and P.H.M. Elemans, Polym. Eng.

Sci., ~, 275 (1988) by parmission of the Society of Plastics Engineers.

-36-

3.1 INTRODUCTION

Corotating twin-screw extruders are very often encountered

in the compounding and blending processes of polymers,

although some important drawbacks exist. The maximum torque

of corotating twin-screw extruders is still relatively low

in comparison with single-screw extruders and (conical)

counterrotating twin-screw extruders. This is inherent to

the flexible screw design and self-wiping action, necessary

to perform the primary task of corotating twin-screw

extruders, which is either to remove volatiles from the melt

(water, solvents, monomers) or to add fillers (glass, chalk,

talc, mica) via openings in the barrel. As a consequence,

all corotating twin-screw extruders must be underfed, which

means that the throughput locally, at least in the open

barrel sections is only a fraction of the maximum

theoretical throughput. Correct metering of the individual

components is often stated to be 80% of the compounding job.

In the modelling, attention will be focused on the hot melt

extruder, which in practice is used for devolatilization or

which is present after the melting section in each

compounding extruder. Solicts conveying-, transition- and

melting sections are difficult to analyse because no

distinguished melting mechanism can be recognized as in

single-screw extruders (8-11). Rather a mixture of solicts

and melt exists as in the dissipative mix melting mechanism

(12). Nevertheless, incorporation of the rnadelling of the

melting section will be important because during compounding

most of the limited torque is used in this stage of the

process. Sametimes even an important part of the dispersive

mixing is already achieved bere, because of the high

viscosity (low temperature) of the mixture (13).

In practice, corotating twin-screw extruders are usually

constructed with narrow flights. An exception to this is the

feed section, where broad flights are sametimes used to solve

-37-

problems with difficult-to-transport powders. However their

tetrahedral gap is from the geometrical point of view always

much larger than the one in counterrotating twin-screw

extruders. Moreover, they are constructed with closely

intermeshing screws in order to promate the self-wiping. As

a consequence the flights leave a completely open 8-shaped

chamber. Therefore, the transport mechanism is drag flow.

The analysis of the corotating twin-screw extruder can be

found in Ref. 4, but as in Refs. 14 and 15 too much effort

is paid to a detailed treatment of the complex geometry and

the reader easily gets lost. Rauwendaal (14) pays some

attention to the rnadelling of corotating twin-screw

extruders, but the analysis is incomplete and therefore of

little practical use. Of course, the more important extruder

manufacturers have developed their own computer programs to

predict the performance of their extruders depending on

screw design and to scale up the results from laboratory

measurements to production size (16), but for obvious

reasans they do not always present their know-how to the

competitors in the open literature.

Here, the corotating twin-screw extruder will be dealt with

as a single-screw extruder using the theories developed in

1922 and 1928. This is allowed with respect to the transport

characteristics of the melt-filled sections because of the

completely open channel.

3.2 SCREW GEOMETRY

Different screw elements exist: Single-, double- or

triple-flighted screws with different pitch, even with

negative pitch, mixing and kneading elements. Screw

contiguration is extremely flexible, and can be fitted to

the job, this being one of the major advantages of this kind

of extruders. At present mostly double- flighted screw

elements are used because of the larger useful volume.

Single-flighted screws are less popular because mixing

performance increases with the number of flights.

-38-

The most elementary screw geometry is given in Figure 3.1

and consists of a sequence of transport elements with

positive pitch combined with an element with negative pitch.

The principle of the analysis will be explained using this

geometry.

Figure 3.1. Elementary screw geometry.

In this elementary screwpart three functional zones can be

distinguished: part 'a', partially filled having a degree of

fill f (0 < f < 1); part 'b', completely filled, pressure

generating; part 'c', completely filled, pressure consuming.

In principle, every screw can be thought to consist of parts

'a','b' and 'c'. Some examples are given schematically in

Figure 3.2.

Local pressure gradients (following from metered throughput

compared with theoretically maximum throughput) and lengths

determine their relative dimensions.

For the moment, only one screw is considered, thereby

neglecting the presence of the intermeshing region. The

screw is thought to be stationary and the barrel rotating in

opposite direction as usual. Furthermore curvatures are

neglected by either looking very locally or by unrolling

-39-

the screw channel. In first approximation, the screw channel

is thought to have a rectangular cross-sectional shape, with

average height H and width W.

11 1 I I 11 11 11115\\M\\\\~1>

\\\\\\\\\\\\\\\\\V '• ~\\\\\\\\\\\\\\111~

\SS'\'\ '\'\S:SS'\S:S: '\'\SS'\\\\\\\\\\\\\\\\\\\\\\\\\\\//\\\\\)

\\\\\\ \\\\\\\\\\S\Jnt!BU!Vl\S\ \\\\\\ \\\\'1\\\\ll!f\\\\IJI[\\\\\\\\\\\\\\\\\\\\>

~\\\\\\IIIIIIIIIIIIIIIIIIIIIIK\91111111mmlliiiiiK\9111111111miiiiiiiiiK\~IIIIIIIIIIIIIIIIIIIIIKS9mllllllllmiiiiiiiK\9111111111111111111111K\~IIIIIIIIIIIIIIIIIIIIIK\\'llllllllllllllllllllll@~lllllllllllllllllllijK\JIIIIIIIIIIIIIIIIIIIII@ll>

Figure 3.2. Examples of screw design.

The barrelwall moves with a velocity, V, over the screw

channel under an angle ~, the pitch angle, and drags the

liquid into the direction of the die (cos~ - 0.95 neglected

here for the time being) .

V 11'DN

with

N revolutions per second

D diameter

-40-

The maximum drag flow capacity iu parts 'a' is (17):

(3 .1)

Qd = "2VHW ( 3 . 2 )

The real throughput Q is always smaller than Qd

(3. 3)

with f the (local) degree of fill, (see Figure 3.3a), Q is

approximated to be half of the real throughput because only

one of the screws is considered here.

a

Figure 3.3. Transport in parts a,b and c.

-41-

Parts 'b' are completely filled; therefore Qd = ~VHW. They

are generating pressure, consequently (see Figure 3.3b):

Because the metered throughput Qm = Q

everywhere, the pressure flow equals

Qp = -(1-f)Qd

(3 .4)

f.Qd is constant

(3 .5)

Parts 'c' are completely filled and consume pressure because

the drag flow is in the negative transport direction: Qd

-~VHW (see Figure 3.3c). This must be overcompensated by a

pressure flow to yield a net transport. Because of

continuity of throughput, it follows that:

(3. 6)

In rectangular ducts of width W and height H (the influence

of side walls is neglected here) the expression for pressure

flow of a Newtonian fluid reacts (7,17,18)

1 dp 3 Q -- H W p = - 12~ dz (3. 7)

By substituting in Eqs. 3.2 and 3.7, Eq. 3.4 can now be

written as

( 3. 8)

3.3 ANALYSIS OF SIMPLIFIED GEOMETRY

Provided that the leads of the forward and reverse zone are

numerically equal, that the number of flights in all zones

is equal, and that the pressure generated in zone 'b' is

dissipated in zone 'c', the simplified geometry now can be

analysed.

-42-

3.3.1 RELATIVE LENGTHS

First, attention will be paid to the calculation of the

lengthof parts 'a', 'b' and 'c' from the local pressure

gradients.

, a, Q =

dp

dz

, b' Q =

dp hence,­

dz

'c' Q =

dp hence,

Fr om

dz

Eqs.

dp dz I/

b

f.l,2VHW

0

1 dp 3 l,2VHW -

12}1 dz H W f.l,2VHW

-l,2VHW -

= -(1 +

3.12 and

dp

dz c

1 dp

12}1 dz

6}-lV f)-

H2

3.14 it

1 - f

1 + f

3 H W f.l,2VHW

follows that:

(3. 9)

(3 .10)

(3 .11)

(3.12)

(3 .13)

(3 .14)

(3.15)

(The minus sign occurs because of different signs of the

pressure gradients in parts 'b' and 'c' .) In case of an

isothermal, Newtonian fluid the pressure gradient is

constant:

dp ~p

dz L (3 .16)

It is now possible to determine the length ratio with Eqs.

3.15 and 3.16:

1 + f

1 - f (3 .17)

-43-

(minus sign omitted) . Let the total length of the screw be

L; then the lengthof parts 'a', 'b' and 'c' are given by:

With Eqs. 3.17 and 3.18:

2 La = L - 1 - f Lc

If relative lengths are defined:

L ___a L

L .

~ and ll.c

and ll.c is given, this yields:

2 ll.a 1 1 f JLC -

1 + f ll.b 1 - f JLC

JLC = JLC

Example: if f 0.3 and JLC 0.2

0.37

3.3.2 SPECIFIC ENERGY

L __c_ L

then ll.a

The shear stress (N/m2 ) at the wall equals

du 'tw = ~ dy I

y=H

The shear ra te at the wall reacts (7,17,18)

du V 1 dp

dy I H + - - H

y=H 2~ dz

(3 .18)

(3 .19)

(3 .20)

(3. 21)

0.43 and ll.b

(3.22)

(3.23)

-44-

So, for the three parts 'a' ,'b', and 'c' this yields:

'a' Eqs. 3.10, 3.22 and 3.23,

y_ 'tw = }l H

'b' Eqs. 3.12, 3.22 and 3.23,

'tw = }l ~ (4 - 3f)

'c' Eqs. 3.14, 3.22 and 3.23,

'tw = }l ~ (-4 - 3f)

(3.24)

(3. 25)

(3. 26)

The force (N) acting on the wall equals the product of

shear stress and surface area where the stress is active.

'a' Eqs. 3.21, 3.24 and 3.27,

V 2 F = f }l trDL(1 - !!.cl H 1 - f

, b' Eqs. 3.21, 3.25 and 3.27,

V 1 + f F = }l trDL(4 - 3f) l!.c H 1 - f

, c, Eqs. 3. 21, 3.26 and 3.27,

V

F = }l H trDL(4 + 3f)l!.c

(minus sign in 1:w should be omitted here)

The total force is the sum of these three forces:

V }l H trDLf(1 +

2(3f + 4)

f

(3.27)

(3 .28)

(3. 29)

(3 .30)

(3. 31)

-45-

The torque (Nm) equals the force times the radius:

T (3.32)

The power consumption (Nm/s) is torque times screw speed

(rad. /s):

p T.N.211' (3.33)

From Eqs. 3.31, 3.32 and 3.33 it follows that

V 2(3f + 4) p = \A H

11' 2D2LNf(1 + f l!.c) (3.34)

Or (with COS<p - 1 still)

Q f l,(zVHW' V 'Tl'DN; w 'Tl'Dsin<p (3 .35)

21.l11'DLNQ 2(3f + 4) p

H2 sin<p (1 + l!.c)

f (3. 36)

Without screw elements 'c' with negative pitch l!.c = 0 so

the term between the brackets equals 1 and the usual

expression for the power consumption of a single screw

extruder is found (19). With the elements 'c', the term

between the brackets yields the relative importance of the

presence of elements with negative pitch, see Table 3.1.

The lower the value of f, the more important is the

contribution of the completely filled part to the total

power consumption. In absolute sense, its contribution is

less, because the filled length Lb decreases with

increasing pressure gradient in part 'b'.

Another way to illustrate the relative importance of the

channel part which is completely filled (total relative

length l!.b+l!.c) by the presence of 'c' elements is given by

the ratio

~ p

a

l+f (4-3f) 1-f + 4 + 3 f

2

f (1-f)

-46-

4-3f2

f (3.37)

Thus: f = (0.4; 0.3; 0.2) yields Pb+c/Pa (9; 13; 20).

In words, for f = 0.3 a part with negative pitch involves

the consumption of 13 times the power relative to a partly

filled channel 'a' of the same length. These simple

calculations illustrate the relative importance of negative

pitch parts, yet it must be kept in mind that until now the

power consumption in the clearance between flight and

cylinder has not been taken into account .

Table 3 . 1. The relative importance of elements with negative pitch:

power consumption in a completely filled channel (%) .

Q,c 0.01 0.05 0.1 0.2

f = 0.4 26l1,c/(1 + 26l1,c)•lOO% 21 56 72 84

f = 0.3 33l1,c/(1 + 33l1,c)•100% 25 62 77 87

f - 0.2 46l1,c/(1 + 46l1,c)•100% 32 70 82 90

The specific energy (kWh/kg) is power (kW) divided by

throughput (kg/h). From Eq. 3.36 and with Q = (m3/s):

Qm = Q.3600.p

This yields:

E sp.

J.nr DLN (1 +

2(3f + 4)

f Q, )

c

0.4

91

93

95

(3.38)

(3 .39)

Again Eq. 3.39 yields, for Q,c = 0, the identical result as

derived for single-screw extruders .

-47-

3.4 IMPROVED ANALYSIS

Until now the analysis was relatively naive and only

qualitative. Nevertheless, some understanding of the basic

working principles has been raised. Next, some improvements

will now be incorporated in the model.

3.4.1 LEAKAGE FLOWS

In corotating twin-screw extruders leakage flows are more

important than in single-screw extruders, because they are

relatively large with respect to the throughput, which is

only a part 'f' of the maximum theoretical throughput by

drag flow. Moreover, the screw speed is higher than that of

single-screw extruders because of its desired significant

contribution to the mixing process.

So far, we have not considered the influence of the flights

since we have tried to simplify the treatment as much as

possible. Improvements however can, and will, be easily

incorporated.

To compensate for the area occupied by flights (double

flighted screws in most cases), the channel width should be

reduced and replaced in all equations by:

w ~Dsin~ (1-ne) (3.40)

where e is the relative width of the flights, given by

~ e = w ~Dsin~ (3.41)

with Wf the flight width measured (as W) perpendicular to

the flight, and n is the number of flights. A typical value

of e is 0.1.

-48-

For the present analysis of double- or triple-flighted

screws, all flights can be taken together; mixing, which

indeed is dependent on the absolute number of flights, is

not being analysed bere. The corrected value of the channel

width must be introduced into all the equations derived so

far; among other things, this correction influences

directly the value of f, the degree of fill, in Eqs. 3.2

and 3.3.

Figure 3.4. Unrolled screw channel.

The local throughput that must be transported to the die is

larger than the metered throughput Qm, because of backflow

through clearances. Simple drag flow is enough to estimate

the backflow because, in general, pressure flows are

negligible in narrow channels. Therefore Qm should be

replaced by (see Figure 3.4):

(3.42)

with

~v&~Dsin~cos~(l-ne) (3.43)

where & flight clearance

-49-

The degree of fill, f, is influenced once more by this

correction. Having corrected for f, the cos~ which was

neglected so far also can be brought in, starting with Eq.

3 .1.

3.4.2 POWER CONSUMPTION OVER THE FLIGHTS

Over the flights power is consumed. The shear stress equals

V

}. = Jlf ó (3. 44)

with Jlf the viscosity above the flight and & the flight

clearance.

The surface area A equals

Af = nelTOL (3.45)

Therefore, force, torque and power are respectively

V Ff Jlf ó nelTOL (3. 46)

V Tf ~Jlf ó

ne1T02L (3. 4 7)

V pf = Jlf ó ne1T2o 2LN ( 3. 48}

A comparison of the power consumed above the flights with

that consumed in the channels can be made now if Eq. 3 .34

is corrected according to Eqs. 3.40, 3.42 and 3.43 and cos~

is incorporated:

V 2(3f + 4} Pc Jlc H

1T202LNf(l + f !te) (1-ne} (3. 49} and

Qm Qm + ~VólTOsin~cos~(l-ne) f

~Vcos~HW ~VH1TOsin~cos~(l-ne} (3.50}

-50-

and, rewriting this expression,

&

f = + ~2D2HNcos~sin~(l-ne) H

(3.51)

yielding

~f H ne 1 (3 .52) Pc ~c & (1- ne) (1 + 2(3f + 4)2-c)

In Eqs. 3.44 throügh 3.49 and Eq. 3.52, ~f and ~c are

introduced to allow one to be able to choose a somewhat

lower Newtonian viscosity above the flights. This is

necessary because of pronounced shear thinning and

temperature rise due to local dissipation here.

With the typical data

H

& 40; n 2; e 0.1;

\lf

~c = 0.15

Table 3.2 follows. As is clearly illustrated in thi s Table,

power consumption between the screw flights and the barrel

is large compared to the power consumption in the channel.

This is typical for these (usually) low values of degree of

fill, f.

Table 3 . 2. Power consumption above the flights in comparison with

that in channels (Ef/Ecl. Parameter: relative length of

parts c (l/, c ).

ll,c 0.01 0.05 0.1 0.2 0.4

f - 0.4 3.0 1.6 1.0 0.6 0 . 33

f ~ 0. 3 3.8 1.9 1. 2 0.66 0.36

f - 0. 2 5.2 2 .3 1. 3 0.74 0 . 39

-51-

3.4.3 MIXING ELEMENTS

So far we have only dealt with the simplest combination of

screw elements with positive and negative pitch; moreover,

only one screw has been analysed. The geometry of two

combined screws is more complex and also many other

elements exist, as is illustrated in Figure 3.5.

fffiH1ft ma· .. -m~ ...

I

~~~ ~ ~ ~

Figure 3 . 5 . Various mixing and kneading elements.

Performing calculations on these elements is now rather

straightforward. First, we have to evaluate in more detail

the effective channel width to be used in the calculations.

Bath screws have to be visually investigated

simultaneously, preferably in the unrolled state (see Refs.

1, 4, 5 and 14). For the throughput, it follows:

2n-1

n (3.53)

the total metered throughput, corrected for leakage flow

over the flights. The factor (2n-1) represents the number

of independent channels in a multiple flighted twin-screw

-52-

extruder, with n the number of flights (4). E.g. n=2 yields

(2n-1) = 3 channels; n=3 yields (2n-1) = 5 channels. The

actual channel width through which the material is being

transported amounts to:

2n-1 w n nDsin~(1-ne) (3.54)

Second, for the mixing elements a main pitch angle ~,

positive (e.g.~= 45°), neutral (~ = 0°) or 'negative'

(e.g. ~ = 135°) must be recognized. Finally, we must sum up

the number of holes in the flight leaving room for positive

or negative drag- and pressure flows. If the lengthof the

sum of the remaining screw flight is defined as being a

factor r of the totalscrew flight length (Lf rnD/cos~),

the sum of the holes equals the factor (1-r).

A leakage by drag flow back into the previous channel

exists, caused by the presence of the holes. Therefore the

throughput brought about by the extruder should be

increased to compensate for this leakage flow:

2n-1 Q n

with

2n-1 where FN =--n--(1-ne)

(3.55)

(3.56)

F0 is a correction factor, like Fp, that compensates for

the influence of the side walls (the flights) on the drag

flow in the case of the ratio of channel depth to channel

width not being negligible anymore (see Refs. 6, 18 and

20) . Not only is the throughput influenced

-53-

by the presence of the holes, but also is the pressure

gradient in axial direction dominating the one in

down-channel direction. One has to consicter the parallel

combination of two flow resistances, represented by their

characteristic cross-sections: The rectangular screw

channel and the passage through the holes (For more

details: See Chapter 4). This is reflected in the Eqs. 3.57

and 3.58.

k

Qd - J.l Ap

The factor k can be calculated:

1 k = 12 !!. H

3'11'DFPC(sin

2q> Fifp + (1-r) Fp')

(3.57)

(3.58)

In a neutral kneading element, for instance, neither flight

nor pitch angle can be distinguished (r = 0, Cf>= 0), so the

leakage flow (Eq. 3.55) completely compensates the drag

flow (Eq. 3.2), withall the correctionsas indicated. In

this case axial pressure gradients are the only transport

mechanism, compare Eq. 3.7.

The flow in the intermeshing region results in an extra

contribution to the conveying capacity and can be treated

using the description given by Booy (5) . The average

contribution, Qa, per N revolutions to the drag flow is

given by:

(3. 59)

with the displacement factor ka a tabulated function (see

Ref. 5)

-54-

This term can be introduced in the equations used so far,

Q (3. 60)

The analysis is summarized in Table 3.3, yielding the

expressions for the pressure gradients and volume flows

with all corrections brought in until now. Figure 3.6 shows

the throughput versus pressure characteristic for the

various screw elements mentioned above.

Q (g /mln)

IV

2

111

100 2 0 ~p/di.10E·3 (Palm)

11

Figure 3 . 6. Throughput versus pressure characteristics for various screw

geometries. I transport e1ements, ~ = 17.66 • ; II transport

e1ements, ~ = 180 ° -17.66 • ; III kneading e1ements, ~ = 45 ° ;

IV kneading e1ements, ~ = 135°.

All kinds of screw geometries can be modelled now, also

screws where no negative pitch elements are present or

screws which consist of any combination of transport,

mixing or kneading elements. Before dealing with that, we

have to analyse in yet more detail what happens if several

different screw elements are combined sequentially.

-55-

Table 3.3. Expressions fordrag and pressure flow in screw elements,

based on the total metered throughput, Qm•

Definition: Q = Qd + Qa + Qp

Q Qm + QL + QLD

(1)+(2) -Qp = (Qd + Qa- QL- QLD) - Qm

with

pressure flow

drag flow

flow in the

intermeshing

region

leakage flow

over flights

leakage flow

k - t.p 1.1

(1)

(2)

(3)

(4)

(5)

(6)

(7)

through flights QLD 1/2 v cos~ H(1-r) nu sin~ FN F0 F0c (8)

V nDN; Lf r = _...",.,. __

nD/cos<p

2n-1

n (1-ne);

k 1

3 ( . 2 F .,. + (1 ) .,. ') 12 ~ H nD FPC s1n ~ N"P -r "p

ka, F0, Fp, F0 ç, Fpc (see Refs. 5, 14, 17);

transport element: r = 1; 0 < ~ < 180°

kneading element: 0 < r < 1 ; 0 < ~ < 180°

3.4.4 SEQUENCE OF SCREW ELEMENTS

(9)

(10)

In the Newtonian, isothermal case with constant viscosity, a

combination of different screw elements can be investigated

independently of the preceding and following elements. To be

more precise: all elements of the same kind cao eventually

be packed together in order to calculate energy and specific

energy.

-56-

In this case, a complete screw like the one in Figure 3.7

can be dealt with as a pure a, b, d, c screw with a sequence

of partly filled parts 'a' (with total length equal to the

sum of the individualparts 'a'), completely filled

pressure-generating parts 'b', completely filled pressure­

generating or pressure-consuming parts 'd' (mixing

elements), and completely filled pressure-consuming parts

'c' (all with total length equal to the sum of their

individual components).

When, however, more detailed information is required than

only (specific) energy - for instance, if one wants to know

where the screw is partly filled or where the viscosity

changes during extrusion (the non-isothermal, non-Newtonian

case) - then only a repetition of the same calculations for

the individual sequential components and combinations can

give the answers. In all cases, the influence of one element

(pressure-consuming) can be noticeable in the preceding

sections (in a complex way, because many leakage flows are

introduced) .

If a combination (a,b,d,c,a,b,d,c) is used (see Fig. 3.7),

then the lengthof the second partly filled part 'a' might

become negative with increasing throughput, Q. This means

that the pressure buildup capacity of the second part 'b' is

not enough to compensate for the pressure consumption in

parts 'd' and 'c'. As a consequence, the absolute pressure

at the end of the first part 'c' has a positive value, the

first part 'b' increases in length to generate this extra

pressure and the first part 'a' decreases in length. If even

the lengthof the first part 'a' is negative, the transport

of this throughput is no longer possible with this screw

design: The critical screw speed has been reached.

-57-

p Q » a >>

Q »

a b de a b de a b

Figure 3.7. Pressure profiles and filled lengths. Parameter: the metered

throughput Om·

3.4.5 NONISOTHERMAL POWER LAW CALCULATIONS

The isothermal Newtonian treatment of the flow in corotating

twin-screw extruders given above is very successful in

providing qualitative information about working principles

and consequences of different screw design in the melt

section. The best understanding originates from the most

simple model. However, quantitative answers cannot be given

because the influence of temperature and shear rate on the

viscosity (and on local pressure generating capacity) must

be included.

It is beyond the scope of this Chapter to give a detailed

analysis of the nonisothermal power law rnadelling (see Ref.

21), butsome general remarkscan be made. In principle, the

treatment is completely analogous to the Newtonian

modelling, but incremental steps over the length of the

extruder must be incorporated in order to be able to define

a local viscosity depending on local

-58-

shear rates, local temperature and, in the case of solvent

containing melts, local concentrations.

The viscosity of a power law fluid depends in general on the

temperature and on the second invariant of the rate of

strain tensor. The components thereof are calculated from

the individual drag- and pressure flows. The influences of

the main drag component V/H = ~DN/H in the channel and

V/& = ~DN/& in the clearance between flight and barrel are

dominant at first sight.

With the local viscosity,it is possible to compute the local

velocity, the velocity gradient, and the pressure gradient.

Given the local throughput Q and the conservation laws of

momenturn and mass, it follows that

dp d -b(T - T0 J du n

(J.Io e (-) (3. 61) dz dy dy

H

Q w f udy (3.62) 0

These equation:s finally result .in a set of two implicit

equations for the two unknowns (the pressure gradient and an

integration constant) . Successful salution of this system

(e.g., with a Newton-Raphson procedure) strongly depends

upon the first estimate of the values of the two unknowns. A

much more convenient way to solve the problem is to write

Eq. 3.61 in a different way:

dp d -b(T - T0 J du n-1 du (-)

dz dy (J.to e dy dy) (3.63)

which means:

dp d du

dz dy (J.I (z, y) dy) (3.64)

-59-

The tridiagonal matrix resulting from this expression

combined with a last row completely filled with the digits

1, 4 and 2, resulting from the discretisation of mass

continuity (Eq. 3.62) according to a Simpson-like

integration rule results in a straightforward solution. Very

fast recursion formulas exist for the composition and

salution of tridiagonal matrices (22), while the last row of

the matrix can be brushed by Gaussian eliminatien already

during the formation of the tridiagonal matrix. Besides, an

extension to the threedimensional flow field now easily can

be incorporated :

ap a avz

az ay (}.I (z, y) ay-> (3. 65)

ap a avx

a x ay (}.I (z, y) ay-> (3. 66)

-b(T - T0 > av av 2 (n-1) /2

l.l (z, y) = llo e ( (___z) 2

+ (______A) ) (3. 67) ay ay

For this (coupled) system the same tridiagonal matrix must

be solved for bath x- and z- directions; the only difference

being the boundary condition at the barrel:

V2 = Vcos~; Vx = Vsin~ (3. 68)

and the throughput Q2 and Qx

Coup1ing of the equations via the viscosity is not only due

to the shear rate dependency but also via the temperature

(see Eq. 3. 67).

The next step, therefore, is ca1cu1ating the 1ocal

temperature. This can be performed re1atively simp1y because

the flow in the channel is considered to be perfectly mixed

(one screw taking over the material from the other while

completely reorientating it) . Therefore a 1ocal heat balance

(Fig. 3.8) dictates that the heat stored in the local volume

(resulting in a temperature rise) equa1s the dissipative

heat generated by the velocity gradients plus the heat

exchanged with the wal1s (positive or negative) .

-60-

The last term is calculated with an average heat transfer

coefficient <a> multiplied by the temperature difference.

This engineering <a> is not known a priori, and experiments

must yield its value. After the new temperature has been

calculated (Tout in Fig. 3.8) the average value of the

viscosity is adapted to the average temperature in the

volume and calculation of pressure gradients and average

temperature is repeated. If the results are not improved

anymore (only a few iterations are needed), the next

incremental step is taken. See Table 3.4.

m l'q, Tin D~ m PCp Tout

dx

Figure 3.8. Heat balance over an incremental step dx.

Table 3.4. Nonisothermal power law calculation. Numerical scheme.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Newtonian analysis

Estimation of lengths

Step dz

Determination of local pressure gradients, Eqs.

3.59, 3.60 and 3.61.

Determination of local mean temperature, Fig. 3.8

Analysis of leakage flow, average temperature and

power consumption

Corrections on throughput and mean temperature in

channel because of leakage flow(s).

Return to 4, until no changes in mean temperature

and pressure gradients occur anymore.

Return to 3, until checkpoint on absolute pressure

is reached.

If not correct: improved estimation of lengths,

return to 2.

If correct, next screw section or end.

-61-

It is recommended not to make the incremental steps too

small (a good cernpromise is one third of the diameter) as

the elements are coupled via leakage flows and the

calculation must be iterated because of the inflow of

material (with an other mean temperature) from leakage gaps

and following channels.

Moreover, when the calculation is started, only analytica!

estimations of the lengths of parts 'a' and 'b' are known.

Therefore, testing at parts where absolute (atmospheric)

pressures are known (degassing houses; feed ports; in the

worst case at the end of the die) is necessary to check

whether the calculated pressure equals the known pressure.

If not, a new guess of the lengthof element 'a' and 'b'

must be made and the calculations repeated.

3.5 CALCULATED RESULTS

As a demonstratien only a limited nuffiber of results will be

presented here. At first, a comparison of simple isothermal

Newtonian calculations with those for the nonisothermal

power law case will be made.

3.5.1 SPECIFIC ENERGY

In Figure 3.9 the specific energy for a hot melt extruder is

plotted as a function of throughput with as parameters the

screw speed and the degree of fill. Figure 3.10 shows the

same plot for the nonisothermal power law calculations. The

conclusion of this comparison is that the Newtonian model

provides qualitatively the right information: minimum

specific energy only at maximum degree of fill. Quantitative

predictions, however, can only be made with the more

complicated calculations shown in Figure 3.10. Especially

when the influence of a screw modific ation is t o be

predicted only the nonisothermal calculations are important

(as experienced in practice), because changes in viscosity

through local heat dissipation in one part of the screw have

great influence on the heat dissipation in the following

parts.

ëi -" ....

0.5

~ 0.4 =. >­OI êi; c Ql 0.3 0 Ql Q.

f/)

' •· ' I I

0.2

0.1

:-62-

Parameter 1: screw speed [i'P~ Parameter 2: filling ratio [ -]

160

140

120

100

80

60

40

0~---r----.----.----.----.----.-----.----.----.----, 0 2000 4000 6000 8000 10000

--.- .,. Output(kg/hr]

Figure 3.9. Specific energy as a function of throughput.

OI .5

.:ti. ...... -'= 3: .4 ~ >-Cl Qj .3 c::: Ql

u Gl D. .2 (IJ

.1

0

Parameter: screw speed, degree of fill.

Isothermal, Newtonian model. From Ref. 23.

0

40 60 ' 80

' ~_oo 120

2000

f=0.1

4000

Parameter 1: screw speed [rpm] Parameter 2: filling ratio [-]

6000 8000 10000 --.;,. Output [kg/hr]

Figure 3.10. Specific energy as a function of throughput.

Parameter: screw speed, degree of fill.

Nonisothermal, power law model. From Ref. 23.

-63-

3.5.2 COMBINATION OF PARTS BAND C

As a second example, a (local) combination of transport- and

mixing elements has been investigated: a positive, neutral

and negative mixing element as is shown in Fig. 3.11. The

difference in pressure gradient in the mixing elements is

clearly demonstrated by drawing horizontal lines of 'zero

pressure' from the end of the mixing element towards the

left; the filled length can be determined graphically. The

difference in pressure gradient between positive and neutral

mixing elements is not very large. However, because of the

positive transport, screws with positive kneading elements

remain (almost) empty. This is why those kneading blocks are

in practice always followed by a screw element (or kneading

block) with a negative pitch.

50,--------------------r------------------~ ZSK 40

N = 150 rpm

40 Q = 40 kg/hr

"L' 30 0

_Q .....__"

Q) 20 L

:J (/) (/) Q) L 0... 10 -- ------------- - --- ---- ---------- --- 111 .

0~------------.-----~-------.------,-----_, 0 40 20 20 40

transport sectlon x ( m) kneading sectlon -10J-------------------~------------------~

Figure 3.11. Pressure gradient in a combination of a transport

section (~b = 27.4 ° ) and kneading section with different

pitch angle: I ~c = 36.5°; II ~c = 0 ° ; III ~c = 180° -36.5°.

From Ref. 23.

-64-

3.5.3 END TEMPERATURE

The third example refers again to devolatilization extruders

and reveals the end temperature and specific energy in

dependenee of processing conditions and materials, see Table

3.5.

Table 3.5. Comparison between final results from model calculations and

those from measurements for a melt extruder (ZSK300) under

several processing conditions.

measured values calculated values

viscosity powerlaw

at y=l s-1 index

(Pa. s)

12,500 0.43

11,300 0.43

10,500 0.43

9,000 0.43

6,250 0.43

485 0.82

296

292

284

294

286

228

0.17 297 0.18

0.15 294 0.156

0.137 280 0.148

0.126 293 0.13

0.123 287 0.125

0.052 224 0.05

From these and many other simulations it can be stated that

the more exact predictions about extrusion behaviour can

only be obtained when a temperature (great influence) and

screw speed (smaller influence) dependent heat transfer

coefficient <a> is used. It is only the heat transfer

coefficient on the polymer side that is responsible for this

dependency because a thicker isolating polymer layer is

formed with lower wall temperatures, screw speeds, and with

larger leakage gaps (25,26). A procedure for calculation of

<a> is presented in Ref. 21.

3.6 EXPERIMENTAL VERIFICATION OF THE NEWTONIAN,

ISOTHERMAL ANALYSIS

The correlation between theoretica! predictions and

practical values of Section 3.5.3 is fairly good. However,

-65-

one has to realize that an automatic feed-back exists:

Because of the temperature dependency of the viscosity a

more or less limiting value for the melt temperature and the

(specific) energy is present.

Therefore, to provide for a more direct check on the

validity of the Newtonian, isothermal analysis, experiments

were performed with model liquids (paraffinic oil, ~1 23 oc 0.2 Pa.s) on a Berstorff ZE25 laboratory corotating

twin-screw extruder.

It was possible to investigate the drag flow capacity of the

two combined screws, including the extra conveying action of

the intermeshing region between the screws.

To determine the pressure gradients and (locally) filled

lengths, the extruder could be fitted with a Plexiglas

cylinder.

3.6.1. THROUGHPUT VERSUS SCREW SPEED CHARACTERISTIC

The purpose of this experiment was to determine the maximum

drag flow capacity of different screw geometries. The

maximum theoretical drag flow can be obtained by letting

Op = 0 in Eq. 3 of Table 3.3:

This yields: Om (3. 70)

Two distinct screw geometries were investigated:

(i) positive transporting elements, having no leakage

through the flights; Eq. 3.70 reduces to

Om = Od + Oa - OL (3. 71)

and

(Ü) positive kneading elements

By substituting Eqs. 5 through 9 of Table 3.3, the

throughput 0 is obtained as a function of N, the screw speed.

-66-

(i) positive transporting elements (3.72a)

ó 2n-l ~ 3 Qm = ~VCOSfP H ~Dsinff' (1 - H) ( -n-(1-ne)) JD EDe+ 4 D Ntanff'ka

(li) positive kneading elements (3.72b)

ó 2n-1 ~ 3 Qm = ~VcosfP H ~Ds inff' r ( 1 - H ) ( -n-( 1-ne) ) JD EDe + 4 D Ntanff'ka

Table 3.6 summarizes the relevant geometrical parameters

used in the calculations. In this Table, H represents the

nominal channel depth. The real cross-channel profile can be

calculated from Eq. 10-7 of Ref.27, which gives the

cross-channel profile of the extruder screw as a function of

the circumferential angle e:

H(9) (3.73)

with Lc = the centerline distance between the screws

In the evaluation of the experiments, this real

cross-channel profile will be approximated by a rectangular

cross-section, with depth Heff· See also Figure 3.12.

E' s I

1.0

0.0

1.0

2.0

3.0

4.0

5.0 +---,----.-,----,----,.-.,--l--,----,--,--,----,--c-----1 -6.5-5.5-4.5-3.5-2.5-1.5-0.5 0.5 1.5 2.5 3.5 4.5 5.5 6.5

x (mm)

Figure 3.13. Cross-channel profile of the self-wiping screw

geornetry of a corotating twin-screw extruder

calculated frorn Eq. 3.73. The dashed line indicates Heff·

-67-

Table 3.6. Numerical values for the geometrical parameters plus

correction factors of the Berstorff ZE25 corotating

twin-screw extruder. (i) positive transporting elements;

(ii) positive kneading elements

ge ometry (i) (ii)

diameter D (mm) 25 25

channel depth H (mm) 4 4

flight clearance & (mm) 0.15 0.15

flight width Wf (mm) 1.4 6.6

Heffective Heff (mm) 3.1 3.1

pitch angle Cf> 17.66 ° 45 °

number of flights n 2 2

relative remaining

screw flight r 1 0.23

correction factors (14) H Heff H Heff

Fo 0.75 0.81 0.76 0.84

Foc 1.02 1.02 0.88 0.9

Fp 0.73 0.79 0.74 1.0

Fpc 1.15 1.11 0.98 1.0

Fp, 0.85 0 . 85

displacement factor ka 0.17 0.17

(Eq. A-10 in Ref. 5)

The maximum (theoretica!) drag flow capacity, obtained from

Eqs. 3.72a+b and from the data in Table 3.6, can be compared

with experiments. The extruder was completely filled (f = 1)

with paraffinic oil (SHELL Ondina 68; Pi 23 oç = 872 kg/m3,

Dl23oc = 0.2 Pa.s). No die was present, to prevent pressure

generation. Care was taken that the oil level in the feed

port remained constant. The throughput was measured at

several screw speeds and corrected for gravity.-induced

leakage flow.

Figure 3.13 shows the measured throughput versus screw speed

characteristics for different geometries, as compared with

those calculated from Eqs. 3.72 a+b.

-68-

50

....... 40 ..: .c: ....... 0) ~

conveylng screw - 30 -::J Q. .c: 0) ::J 20 0 ... .c: -

10 knaadlng sorew

0 0 50 100 150 200 250

screw speed (rpm)

Figure 3.13. Comparison of measured and calculated throughput versus

screw speed characteristics for transport elements

(~ = 17.66° ) and positive kneading elements (~ = 45 ° ).

Transporting elements show a large deviation (+25%) from the

predicted throughput, whereas in the case of positive

kneading elements, the agreement between experimental and

calc ulated results is relatively good (-10%), consictering

the assumptions in the analysis. Fitting the measured

throughput data of the transporting elements with a channel

depth H = 4 mm gives better agreement. Obviously, the

transporting screw elements are not completely self-wiping,

so l arger leakage gaps e xi s t. The main differenc e, howeve r,

betwee n transporting and kneading element s is t heir ability

to generate pressure, which will be discussed in the next

Section.

3 .6. 2 PRESSURE GRADIENTS

As stated in Section 2 .1, corotating twin-screw e xtruders

normally operate almast c ompletely under atmospheric

-69-

pressure. The screw is partially filled, having a degree of

fill f < 1 (parts 'a'). Pressure is generated only in those

parts of the screw where a combination of elements with

positive pitch (parts 'b') and elements with negative or

neutral pitch (parts 'c' and 'd') is present. Parts 'b', 'c'

and 'd' are completely filled.

It is possible to determine the axial pressure gradient in

parts 'b', 'c' and 'd' visually via the setup shown in

Figure 3.14. The Plexiglas cylinder is fitted with a series

of open tubes, which are directly connected with the

extruder channel. By measuring the oil level in the tubes,

the axial pressure gradient can be obtained via regression

analysis and compared with results from calculations.

Figure 3.14 Visualization of the axial pressure gradient in a

corotating twin-screw extruder.

(a) Combination of transport elements ha ving positive pitch

('{'b 17.66°) with transport elements ha ving negative pitch

('{'c 180°-qlb; Lc = 50 mm). Om= 36 g/min.; N = 130 rpm.

(b) Combination of transport elements having positive pitch

('{'b 17.66°) with kneading elements having negative pitch

(~c = 180°-45°; Lc = 150 mm). Om= 26 g/min.; N = 190 rpm.

-70-

Figures 3.15 and 3.16 compare the calculated and

experimental pressure gradients for different screw

geometries and processing conditions. The pressure gradients

in parts 'b' and 'c' are calculated from the Eqs. 3 through

10 of Table 3.3, plus the geometrical data in Table 3.6.

They can also be determined graphically from the pumping

characteristic (Figure 3.6), by drawing horizontal lines,

according to the present throughput followed by

interpolation towards the desired screw speed.

The correspondence between calculated and experimental

pressure .gradients in the kneading elements having negative

pitch (Figure 3.16) is quite good. For the pressure

gradients in transporting elements, however, errors as large

as -25% (Figure 3.15) or even -60% (Figure 3.16) are found.

This can be explained by consictering the (axial) pressure

gradient as a function of the channel depth, as shown in

Figure 3.17. Transporting elements exhibita strong decrease

in pressure gradient as the channel depth increases from

Heff = 3 mm to H = 4 mm. In kneading elements, however, the

pressure gradient proves to be less sensitive to changes in

channel depth.

{ !:;,

~

JOOOOO,.--,..------,---------------,

200000

100000

0

transport

knea~~ -------·-·-2

H (mm) 4

Figure 3.17. Axial pressure gradient as a function of channel depth as

calculated frorn the equations of Table 3.3 and the data of

Table 3.6. Qrn = 150 g/rnin; N = 200 rprn.

"' ,. 0 "l

P !Palm) P (Palm) PI f-'· ·o t-' · colculated vol'u~s H rt §. 'ê cclculofed valuts 0=15 N=188

~ ::r f-'· H

CD rt ::l CD 3000 rt H PI CD PI rt w H ::l t-'·

c.o 0 ..... '0 ::l (J1

-;; 0 2000 H 0

rt Hl () t-'· PI

c.o ::l rt ..... 0 IQ H 0 H PI .: CD CD ::l ..... ,. ..... c.o PI

CD '0 rt c.o ~ 0 CD '0 H p.

CD ::l rt CD rt f-'· PI p. c.o ::l ::l ' IQ p. 1 0 140 120 100 80 60 40 ::r

1 (mm) I ex. !mml H PI CD CD IX . '0 ~ i-' x negatlve tranaport (l0 - 26 mm) pos. trensport negative kneeding tLc- 150 mm) pos. tranaport ë f-'· CD '0

::l i3 CD IQ CD H

::l ~-::l rt 0" CD c.o CD

P (Palm) P (Palm) IQ ::l f?~Perlmentol vcftJes experfmentol valuts PI ::r rt

rt rt PI PI ( ::r t-'· ~ i-' .._] H ~ t-'· 3000 3000 I-' 0 CD ::l '0 ( .: IQ H

IQ '0 CD ::r f-'· '0 c.o '0 rt 0 c.o .: 0 c.o .:

2000 rt ::r f-'· H rt CD

iä .:ë" t-'· ~ IQ ...... CD H

i3 PI t-' · '0 p. 2. ..... f-'· f-'·

"' rt CD 0 0 ::l

0 ::r rt I c.o

..... .:ë" ..J f-'·

::l 0 140 120 100 80 60 40 CS\

1 ex. (mm) 1 ex . !mml" CS\ PI 0 ..... negative tranaport (lc- 25 mm) pos. transport negative kneadlng (lc - 160 mm) poa. transport ..J

CS\ m

0

(a) (b)

'0 ::r '"'l P (Po/m) P (Po/ml 1-'· ll> 1-'· colculoted volut;ts N=188 colculoled volues N=235 ("t ~ <Q 0 1-'· c ::r ::;, t1

<Q (1) 3000 3000

~ '0 w 0 co .... 1-'· "' .... ("t

2000 w 1-'· U1 ~ )>

(1) co '0 '"'l 1-'· 1-'·

"' ("t <Q ll> 0 c t1 ::r t1

~ (1)

(1) ~ w ("t (1) .... t1 (J1 1 0 140 120 100 BO 60 40 ....

I ax. lmml I ax. lmml _, ("t Hl nagative transport llc • 25 mml pos. transport negative k.neading llc- 150 mml pos. transport ::r "' 0 t1 "' t1 0 0

c ll> <Q ::r :( 0

P (Pa/m) P (Palm) '0 1-'· 0 experimentol ..-olues experlmentol voNes c ("t g. I ("t ::r

1-'· -..1

~ )I" ::;, 3000 3000 [\.) ::;, ll> I

'- (1) ("t

s ll> 1-'· 1-'· 0. 0 2. 1-'· ::;,

::;, <Q 0 2000

Hl (1) .... ("t (1) t1 s ll> (1) ::;, ::;, co ("t '0 co 0

t1 ::r ("t

ll> 1-'· ~ ::;, 1-'· <Q 0 140 120 100 BO 60 40 ::;,

Iu. lmm) lex. (mml <Q (1) .... negative transport tl.c- 26 mm) pos. transport negative k.neading (Lc • 160 mm) pos. trensport ::;, ~ (1)

<Q (1) ll> ::;, ("t ("t

1-'· co ~ (1)

(a) (b)

-73-

By combining Eqs. 3.15 through 3.17, it is possible to

calculate the filled length Lb from the measured pressure

gradients:

(dp/d!!.l I L - L • ( ) bax.- cax. (dp/d!!.llb (3. 74)

The filled length thus obtained is only a crude estimate,

because of the errors in the pressure gradient. In the next

Section, a more accurate method will be discussed.

3.6.3 FILLED LENGTHS

Via the method discussed in the previous Section, the

axially filled lengths can be estimated. Comparison of these

lengths with calculated values inevitably leads to large

deviations. The calculations (using the equations of Table

3.3) yield results that are valid in down-channel direction.

Of course, these results can be converted into the

corresponding values for the axial direction, but the best

evaluation can only be expected with filled lengths that are

measured in down-channel direction.

3.6.3.1 EXPERIMENTAL SETUP

Experiments were performed with paraffinic oil (SHELL Ondina

68, ~1 23 oc = 0.2 Pa.s) in the Plexiglas-walled laboratory

corotating twin-screw extruder. With the setup shown in

Figure 3.18, the combination of a part c (transport elements

having negative pitch or kneading elements - neutral or with

negative pitch) with transport elements can be investigated

directly, campare Sectien 3.5.2.

-74-

Figure 3.18. Setup of Plexiglas-walled laboratory corotating twin-screw

extruder with High Speed camera.

To measure the length of the channel that is completely

filled preceding a part c at different screw speed and/or

throughput, a High Speed camera was used. Figure 3.19

represents one image of the film, showing the transition of

a partly filled channel into a completely filled channel, as

can be seen with a Motion Analyzer.

Figure 3.19. Combination of a transport element having a pitch angle

~b = 17.66 ° with a negative transport element

(~c = 180°-~b). The screws rotate clockwise. N

Qm = 100 g/min.

300 rpm,

-75-

3.6.3.2 RESULTS

A large number of screw geometries was investigated using

the High Speed Camera. Same representative results will be

discussed here. For details, the reader is referred to the

report by J.H. Truijen (28).

Figure 3.20 compares the predicted and experimental filled

lengtbs in screws with different combinations of parts 'b'

and 'c'. The sections 'c' may consist either of kneading

elements with neutral or negative pitch, or of transporting

elements having negative pitch. It is important to notice

that all sections 'c' are of the same axial length, and that

the filled lengtbs are measured in down-channel direction,

indicated by the coordinate z. The coordinates z and ~are

related:

z = ~/sin~ (3.75)

In Figure 3.20, three levels of filled lengtbs can be

distinguished, each of which corresponds to the type of

section 'c'. Within each level, filled lengtbs are measured

at a different throughput. The general agreement between the

predicted values and those from the experiments is fairly

good. If the results are examined in more detail, however, a

few remarks have to be made.

At a screw speed below N = 200 rpm, the predicted and

experimental results start to deviate. This is observed for

all combinations and throughputs. It is mainly due to the

fact that gravity causes the fluid to prolapse. To our

regret, no accurate measurements of the filled length can be

expected for these screw speeds, when using a fluid of low

viscosity (~ = 0.2 Pa.s)*.

* A fluid of higher viscosity causes nonisothermal effects

and might exhibit elasticity, due to the presence of trapped

tiny air bubbles.

-76-

It is quite an interesting region of screw speeds, however,

where a large differentiation in filled length is predicted

by the model.

The geometrical description of kneading elements having

negative pitch is rather poor. Still, the assumption of a

rectangular channel with effective channel depth Heff, and a

relatively remaining flight length of about 25% yields

acceptable results.

800,---------~-------------------------------,

700 lc • 112.5 mm

neg. tranaporllng .. otlon

600

0

500

400 -E E -.a

300 ...1 .s::. -CD c Q)

200

100 neg. kneadlng eecllon

150 200 250 300 350 400 450

screw speed (rpm)

-77-

previous page:

Figure 3.20 Filled channel length Lb as a function of screw

speed for combinations of transport elements with various

parts 'c'. The axial lengthof parts'c', Lc = 112.5 mm in all

cases. Parameter: Om (0 100; <) 140; 0 170; e 240 g/min).

Drawn lines are values calculated from the equations of

Table 3.3. The symbols represent experimental results.

3.7 RESIDENCE TIME DISTRIBUTION

Apart from their local pressure generating or pressure

consuming capacity, the different screw geometries of the

corotating twin-screw extruder can be characterized by their

residence time distribution RTD (29,30).

At a given moment during the compounding process (in the

experiments, LDPE Stamylan 2300G, DSM was used), titanium

dioxide (Ti02 ) tracer particles are added as a pulse at the

entrance of the e xtruder. At the exit, samples are taken

from the strand at regularly spaeed intervals in time (~t

7.5 s). From the amount of tracer material that can be

detected in the sample via X-ray fluorescence (with an

accuracy of 1 ppm), the response is determined.

The results are presented in dimensionless farm. The value

of the cumulative response F at a time t represents the

fraction of the material with a residence time less than t.

It is clear that F ~ 1 as t approaches infinity. It is

camman toplot F(t) versus the dimensionless time t/t, where

t is the average residence time.

Two extreme screw geometries (see Figure 3.21) are compared

here. The one shown in Figure 3.21a is a conventional screw

with a moderate amount of kneading sections. Figure 3 .21b

shows a s c rew geometry which consists of a series of

alternating Couette elements and kneading elements. In its

most simple farm, the average residence time can be

estimated for bath screws by calculating

-78-

V 'II'DLH t

Q Q (3. 75)

with V the volume that is completely filled with melt,

i.e. corresponding more or less to the length of

the kneading section.

Q the volumetrie throughput

(a)

(b)

Figure 3.21 Screw geometries of a Werner & Pfleiderer

ZSK30 laboratory corotating twin-screw extruder

(L/D = 42D) . (a) Standard kneading screw;

(b) Couette-kneading screw.

The experiments were now performed on a Werner & Pfleiderer

ZSK30 laboratory corotating twin-screw extruder. Typical

data are: The length of the kneading sections (L z 900 mm

and 140 mm) . The channel depth H = 5 mm, and the diameter of

the screw D = 30 mm. The density of the LDPE z 800 kg.m-3.

Substituting these values in Eq. 3.75 yields forthescrew

geometries of Figures 3.21 a and b: tr z 50 s and 300 s,

respectively. In spite of this simplistic way of

calculating, these results correspond quite well to the t

from the RTD measurements, as is shown in Figure 3.22. In

this Figure, the Couette-kneading screw geometry has the

stronger tendency towards plug flow.

-79-

0 .8 i(/ I

- 0.6

l -Ir

0 .4

screw type (t) ' 0.2 /; ZSK 02, (306 s)

ZSKOB_,_ (~ __ ___ .

~lug flow

0 0 2 3 4

t/t

Figure 3.22 Residence time distribution in a ZSK30 corotating

twin-screw extruder. Screw speed = 250 rpm;

metered thoughput Q = 6 kg . hr-1 ; barrel temperature

Tw = 200 •c. Screw geometries as in Figure 3.19.

3.8 DISCUSSION

It has been shown that a simplified model for a corotating

twin-screw extruder is able to predict the correct energy

consumption, specific energy, and temperature rise, not only

over the extruder as a whole, but also locally during the

processing (depending on local screw geometry, processing

conditions, and material properties) . This is of great

practical importance in polymer processing because an

understanding of the process is within reach and gives a

perspective for solving problems in s c ale up. Despite o f

this, the mo re dire ct measuring o f transport c apac ity, loc al

pressure gradients and filled lengths with model liquids in

Plexiglas-walled extruders yields larger deviations, as

expected more or less. However, some of the differences

between calculated and measured values can be attributed t o

-80-

the difficulties in the experiments caused by the low

viscosity of the fluid used. Gravity-induced leakage flow

apparently can not always be neglected. Other deviations

originate from the geometrical approximation of the channel

depth. Small improvements of the model, such as a local wall

and screw temperature dependent leakage gap (important in

large extruders) can be brought in. An independent

determination of the (scale-up rules for the) heat transfer

coefficient is necessary, but most useful will be an

incorporation in the modelling of the pressure-generating

capacity of the feed section combined with a model of the

melting section. Only if these eonditions are fulfilled may

a complete model exist (which in its turn could be combined

with the existing theory of the dispersion process (see for

example, Ref. 31) in order to predict the compounding

performance for the blending operation.

3.9 REFERENCES

1.W. Szydlowski, R. Brzoskowski and J.L. White, Int. Pol.

Proc., 1., 207 (1987)

2. J.L. White and K. Min, Paperpresentedat the European

Symposium on Polymer Blends, Strasbourg, France (May

1987) .

3.M.L. Booy, Pol. Eng. Sci., .la, 973 (1978).

4. H. Werner, Ph.D Thesis, Munich University of Technology,

West Germany (1976).

5.M.L. Booy, Pol. Eng. Sci., ZQ, 1220 (1980)

6.Anonymous, Engineering (London), 114, 606 (1922).

7. H.S. Rowell and D. Finlayson, Engineering (London), llQ,

249,385 (1928).

8.B.H. Maddock, SPE ANTEC (1959),383

9. Z. Tadmor, Pol. Eng. Sci., .Q., 185, (1966)

10. J.F. Ingen Housz, H.E.H. Meijer, Pol. Eng. Sci., 21

(1981), 352

-81-

11. J.F. Ingen Housz, H.E.H Meijer, Pol. Eng. Sci., ~

(1981)' 1156

12. Z. Tadmor, P. Hold, L. Valsamis, SPE ANTEC (1979),193.

13. K. Nordmeier, Ph.D. Thesis, Aachen University of

Technology, West Germany (1986)

14.C.J. Rauwendaal, Polymer Extrusion, Hanser Publishers,

Munich (1986).

15. L.P.B.M. Jansen, Twin-screw extrusion, Elsevier, '

Amsterdam, (1978).

16. H. Gosler, (Werner & Pfleiderer), private communications.

17. J.F. Carley, R.S. Malleuk and J.M. McKelvey, Ind. Eng.

Chem., ü, 974 (1953)

18. J.F. Carley and R.A. Strub, Ind. Eng. Chem., ~, 970

(1953)

19. R.S. Malleuk and J.M. McKelvey, Ind. Eng. Chem., ~, 987

(1953)

20. C.D. Denson and B.K. Hwang, Pol. Eng. Sci., ZQ, 965 (1980)

21. C.P.J.M. Verbraak, H.H.M. Lardino ye and H.E.H. Meijer,

Polym. Eng. Sci. (in progress)

22. J.R.A. Pearson and S.M. Richardson (ed.), Computational

Analysis of Polymer Processing, Appl. Sci. Londen (1983)

23. H.E.H. Meijer, P.H.M. Elemans, H.H.M. Lardinoye and G.

Kremer in: 'Warmeobertragung bei der

Kunststoffverarbeitung', VDI Dosseldorf (1986)

24. H.H.M. Lardinoye and G. Kremer, internal report DSM (1988)

25. C.H. Jepson, Ind. Eng. Chem.,~, 992 (1953)

26. H. Janeschitz-Kriegl and J. Schijf, Plastics & Polymers,

Dec., 523 (1969)

27. M. L. Booy, Polym . Eng. Sci.,.l.5_, 606, (1975)

28. J.H. Truijen, internal report Eindhoven University of

Technology (1989)

29.P.V. Danckwerts, Chem. Eng. Sci., ~' 1 (1953), ibid. a, 93 (1958)

30. G. Pinto and Z. Tadmor, Polym. Eng. Sci., ~, 279 (1970)

3i. J.J. Elmendorp, Ph.D. Thesis, Delft University of

Technology (1986)

-82-

CHAPTER 4

MODELLING OF THE CO-KNEADER

The Buss Co-kneader is a single screw extruder with

interrupted flights. Pins from the barrel are inserted into

the screw channel. The screw is both rotating and

oscillating. Due to this action screw flights are

continuously wiped by the pins. During one passage of the

pin, the material is not only subjected to high shear

stress, but it is reoriented as well, thus promoting the

distributive mixing process by the local weaving action of

the pins and screw flights. Attempts to model the Co-kneader

tend to focus on a single pin passing through the hole in a

screw flight (1,2). However, a more overall model can start

with the same equations that apply to the corotating

twin-screw extruder (3). Because now not only the effect of

leakage flows on the local pressure gradient has to be

considered but also that due to the local dragging action of

the pins (neglecting the oscillatory action), experiments

with model liquids have been performed to evaluate the

overall model. Additional experiments with a

Plexiglas-walled Co-kneader support the calculations

concerning filled lengths in various screw geometries. These

results, and those of model calculations, which are extended

to the nonisothermal, non-Newtonian situation, will be

presented.

Reprinted partly from: P.H.M. Elemans and H.E.H. Meijer, Polym. Eng. Sci.

(submitted) by permission of the Society of Plastics Engineers.

-83-

4.1. INTRODUCTION

The Buss Co-kneader is a continuous mixer which is widely

used in many operations in the processing of polymers and

foodstock. It offers possibilities for homogenization,

colouring, reinforcing, and (highly) filling of critical and

difficult-to-handle compounds of thermoplastics, rubbers or

thermosets (4-7). Provided the roetering of the individual

components is correct, torque and temperature rise can well

be kept under control, even for thermally instable materials

and/or critical additives partly because both screw and

barrel are oil thermostated.

Invented in 1945 (8), practical application of the

Co-kneader has been far ahead of theoretical understanding.

According to the design specifications, the Co-kneader is

said to operate fully adiabatic and scale-up should be no

problem since every next size in the series has ten fold the

throughput of the previous size (PR46, PR100, PR200).

Although the more modern machines have not altered their

main werking principle, their length has been increased from

7D to 11D (for special purposes even 15D can be chosen) and

screw- and barrel design is flexible now, also on the

laboratory kneader. This flexibility now requires for the

development of roodels that can give predictive answers with

regard to energy, specific energy, temperature rise and

residence time, dependent on metered throughput, chosen

screw design and eperating conditions.

Recently, one of the first more serieus studies concerning

the rnadelling of flow in pin-barrel (rubber) extruders

appeared (9). Like Ref. 1, it is confined to calculations of

one type of screw element of invariable geometry only (no

sequence of different screw elements as used in the

Co-kneader) . The flow is said to be Newtonian and

isothermal. However, it can be expected that processing of

highly viscous model fluids will give rise to a considerable

increase in melt temperature. Moreover, the mixing of pin

-84-

screw extruders used in rubber processing is much less

effective than that of the weaving action of the pins and

remaining screw flights in the Co-kneader due to its

rotating and oscillating movements (see Figure 4.3).

Besides, apart from improved mixing performance, see Ref.

10, one of the most important effects of the pins in rubber

extruders could be the impravement of the transport

characteristics by breaking through slip layers in the

beginning of the process where an extremely high viscous,

plastic material is present. Kneading is normally promoted

in these types of extruders by the choice of the screw:

multichannel, multicompression and decompression with many

local barrier flights.

Still, it would be interesting to cernpare the results of

Ref. 7 with experiments in those sections of rubber

extruders where only melt is present and the roodels apply.

Here it is tried to model the Buss Co-kneader in a way

analogous to that of the corotating twin-screw extruder (3).

First we will only deal with the simplest case of a

Newtonian liquid under isothermal conditions. This analysis

can be extended, as in Ref. 3, to the non-Newtonian,

nonisothermal situation of a melt-fed Co-kneader.

4.2. SCREW GEOMETRY AND WORKING PRINCIPLE

The screw of a Co-kneader consists of interchangeable

elements. Figure 4.1 shows three common screw elements:

conveying elements (single flighted), having two rows of two

pins per lead inserted in the barrel, closed channel

kneading elements, double flighted, with three rows of one

pin per lead, and kneading elements, also double flighted

and having three rows of two pins per lead.

In contrast with the corotating twin-screw extruder, no real

negative transporting elements exist, except for the element

shown in Figure 4.6e, which has a locally negative pitch

angle. Staggering elements, however, are used. The most

-85-

common ones are buffle rings, used preceding degassing

sections in the barrel, and the die that connects the

kneader to the discharge extruder if present. See Figures.

4.2a and 4.2b.

An alternative setup, which replaces the discharge extruder,

to assure regular material outflow, as proposed in Ref. 11,

is discussed in the Section 'Experimental'.

Figure 4.1. Common screw e1ements a) conveying (2 rows of 2 pins);

b) closed channel kneading (3 rows of 1 pin);

c) kneading (3 rows of 2 pins) .

4.2a 4.2b

Figure 4.2. Staggering e1ements (schematica11y). After Ref.3.

a) buff1e ring; b) die connecting kneader and discharge

extruder.

-86-

Figure 4.3 illustrates the combined (self-wiping) action of

the interrupted flights of a kneading element with the

corresponding kneading bolt in the barrel wall (the kneading

bolts are usually diamond-shaped, except in the PR46, which

has cylindrical bolts) . It is due to this weaving pattern

that the Co-kneader possesses an excellent distributive

mixing quality. The unrolled "standard screw" geometry

(Figure 4.4) clearly shows the variety in mixing intensity

along the screw.

!A A•B

n]c c~~ ~c c

~ r ~ ~ ~J 0 18 0 A

~u ~. ~~ .~ ~ A

JU ~. ~ ~ ~ c I ~ ~ ~

~ I A AB

Figure 4.3. Trajectories of (stationary) pins relative to the shaded

flight (Figs. 4.3(a) - 4.3(d)). Fig. 4.3(e) shows the weaving

pattern as a result from the motion of the pins.

0 0 0 L(l

~ ~ ~ ~ 0 0 0 L(l . 0 0

....: r.: ....: r--: • N N N N Ltl

N N N UJ UJ UJ

0 0 0

r-: ....: ....: N N N

0

<Ti IC

Figure 4.4 . Unrolled "standard screw" geometry (D = 46 mm, L/ D = 11),

showing the sinuscictal trajectories of the pins relative to

the screw. The circles indicate the feed ports.

Explanation of screw elements' abbreviations: EZ: conveying ;

KE: kneading; GS: closed channel kneading; OS: buffle ring.

-87-

4.3. SUMMARY OF THE NEWTONIAN, ISOTHERMAL ANALYSIS

Like in the case of a corotating twin-screw extruder, the

analysis of the Co-kneader will depart from the combination

of a screw element (conveying or kneading) with a staggering

element.

The screw elements of the Co-kneader all have interrupted

flights, leaving room for the pins to pass during each

revolution. Thus, two main influences on the usual

drag-and-pressure flow can be distinguished: (i) the extra

conveying action due to the pins. And (ii): The leakage flow

through the flights' holes, back into the previous channel.

The remaining screw flight is considered to be a factor 'r'

of the total length of the screw flight (Lf = r • ~D/cos ~).

The sum of the holes' length equals the factor (1-r). The

pins are thought to cause a plug flow right in front of

them. This is expressed by the factor a: The aspect ratio of

the pins' surface over the channel surface, which equals

(dpin)2/(L•W/n). We will investigate these two influences

separately in order to understand the transport

characteristics of the kneader.

Although the kneader is rotating as well as asciilating in

axial direction, we will neglect the influence of the latter

in the analysis of the transport characteristics. The

equations for Newtonian, isothermal pressure and drag flow

in an unrolled screw channel will be applied, with the usual

assumptions, including the negleetien of pressure flow over

the flight clearance (3). Table 4.1 gives the resulting

equations that apply to the Co-kneader. Because of the large

number of relatively big holes in the interrupted screw

flights of the Co-kneader, the flow rates due to local

pressure gradients are important nat only in down-channel

direction, but also in axial direction. In Figure 4.5, the

influence of a hole on these pressure gradients is

visualized in the unrolled screw. The pressure difference ~P

between the two points A and B in the screw channel can be

-88-

generated in two ways. Either by going a distance z in

positive z-direction: AP = (dp/dz)•z, or by going a distance

x in negative x direction: ~ = (dp/dx)•x. In each direction,

the pressure flow equals the pressure gradient times a

geometrical resistance factor k for that particular

direction, see Eq. 5 of Table 4.1. The coordinates x and z

are related to the axial coordinate, ~:

z = ~/sinq>

x= -~/cosq>

(4 .1)

(4 .2)

Figure 4.5. Pressure generation in down-channel and in cross-channel

direct ion.

-89-

The total pressure flow equals:

1 dp Qp =- 12}.1 H31fD FPC(sin<p Fr/p dz +

(1-r) ---F'

COS(j) P ( 4. 3)

Combining Eq. 4.3 with the Eqs. 4.1 and 4.2 , the pressure

flow is given by Eqs. 5 and 11 o f Table 4.1.

Table 4.1. Expressions for drag and pressure flow in screw elements,

based on the total metered throughput, Q.

Qc hannel

Qchannel

(~in> 2 I (L W/n)

with

pressure flow

drag flow

leakage flow

over flight

leakage flow

through flight

c = efficiency

pararoeter

extra drag flow

due to pins n V H ~in (if f = l)

V 'TfDN Lf

r = ---:-"'---'TfD / costf

z = _2._ sin<p

F0 , Fp, F0 c, Fpc (see Re fs. 12,13 , and 14)

1-ne

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(9)

(10)

(11)

-90-

In Table 4.1, Foc and Fpc are correction factors that

account for the influence of curvature on the flow in the

kneaders' screw channels. The kneader has an H/D of about

0.15, which is rather large compared to most corotating

twin-screw extruders, where the H/D < 0.1.

It is important to know what value of r, the relative

remaining flight length, should be used in the calculations.

In principle, there are two ways to determine r, as is

demonstrated in Figures 4.6a and 4.6b. The results are shown

in Table 4.2. The fact that two sets of 'r' values can be

obtained wil! turn out to be important in the later

evaluation of the experiments.

(a) (b) (c) (d)

Fig. 4.6. Detail of unrolled screw channel.

a) conveying element, Lh/Lf = (1-r)

b) conveying elemerit, Lh/Lf = (1-r)

c) closed channel kneading element

d) kneading element

e) as Fig. 4.6c, with locally negative pitch angle

(e)

-91-

Tab1e 4.2. Geometrica1 estimation of r, the re1ative

remaining f1ight 1ength and ~ (aspect ratio

(dpin)2/(L W/n)) for severa1 screw e1ements.

r ~

according to

Figure

4.6a 4.6b

conveying 0 . 92 0.8 0 . 005

e.c. kneading 0.78 0.55 0.011

kneading 0. 71 0 . 4 0.011

4.4. MIXING

The mixing qualities of the Co-kneader, which have been

somewhat in the background during the analysis, will now be

discussed. The number of reorientations can be calculated by

starting with the analysis by Booy et al. (1), who consicter

the elementary passage of a single pin through the opening

between two adjacent flights of a kneading element. Their

assumptions are: (i) the element is completely filled with

melt, and (ii) after the passage of the pin, the material is

- at least locally - reoriented, whi c h can be expected from

the simulations in Ref. 15.

The average residence time in a screw element with length L

approximately amounts to:

t

where

11'DLH(1 - ne)

Q

D diameter

H c hannel depth

n = the number o f f lights

e relative flight width

Q throughput

( 4. 4)

-92-

The average residence time required for the passage of a pin

between two flights (the length of one flight is about 1/3

times the circumference) is given by:

1/3'11'0 2

1/2'1l'DN 3N

h d (s -1) w ere N = screw spee

The number of reorientations, nr, is given by:

nr N

3/2'1l'DLH ( 1-ne) Q

(4. 5)

(4. 6)

This analysis is valid if only one pin is present in the

screw channel. However, in a real Co-kneader screw element,

a total amount of p pins can be recognized. For each

element, p can be obtained by multiplying the number of pins

per lead of the screw by the number of pin rows on the

barrel circumference. (See Sectien 4.2). Thus, for closed

channel kneading elements, p = 3, whereas for kneading

elements, p = 6.

For the number of reorientations in a completely filled

screw element, this yields the final expression (with L

'll'Dtan~, the screw lead) :

nr 2 2 p.3/2'1!' D Htan~ (1-ne) N

Q (4. 7)

In the case of partially filled screw elements, the number

of reorientations is evaluated with respect t o the filled

length Lf.

nr N

p.3/2'1l'D L~tan~ (1-ne) Q ( 4. 8)

-93-

Example: consicter the completely filled kneading element

(L = 27 mm) of a Buss PR46 laboratory Co-kneader, with

typical data:

~ 10.5°

H 7.10-3 m

D 46.10-3 m

n 2

e = 0.133

N 240 rpm = 4 s-1

Q 20 kg.hr- 1 z 5.5.10-6 m3.s-1

With these data, the number of reorientations, calculated

from Eq. 4.8, becomes nr = 128. The amount of shear between

two reorientations, Yn• is estimated to be y = y.t = 2/3 ~D/H ~ 14.

4.5. EXPERIMENTAL

The goal of the experiments was threefold: (i) to provide a

check for the geometrically estimated values of the relative

remaining flight length; (ii) to determine the influence of

the pins; (iii) to check the complete Newtonian, isothermal

analysis of the kneader. To achieve this, extrusion

experiments were performed using silicon oil (~23oc

1 Pa.s) and paraffinic oil (~23 oc = 0.2 Pa.s), under

isothermal conditions. Throughput versus screw speed,

throughput versus pressure generation as well as locally

filled lengtbs were measured, using severa1 screw types

(conveying, closed channel kneading and kneading), different

number of pins mounted in the barrel wall (from 0 to max.)

and processing conditions (screw speed, throughput) .

Finally, attempts were made to measure the pressure gradient

directly, i.e. from the oil level in a series of open tubes

along the cylinder. All experiments were performed on a

PR 46-llD laboratory kneader, which could be fitted with a

Plexiglas cylinder,

-94-

~igure 4.7. The Co-kneader fitted with a Plexiglas cylinder. Via a system

of small glass tubes that act as open manometers, the

pressure gradient along the cylinder can be visualized.

see Figure 4.7. A problem arises when a constant kneader

outflow is required, for instanee for measuring the pumping

characteristics (Q - ~p) of the Co-kneader. The oscillatory

motion of the kneader screw relatively to the barrel causes

severe pressure fluctuations. Usually a discharge extruder

is needed to guarantee constant pressure at the die. This

can be avoided by using a special die (11). The cylinder is

fitted with a flange having the same boring as the kneader.

A ring fitting closely inside this flange is mounted to the

kneader shaft. The die opening now yields a constant

pressure and consequently gives smooth filaments of constant

diameter which can be caoled and cut. For measuring the

pumping characteristic of the kneader the resistance of the

die is made adjustable and the pressure is measured using a

manometer mounted at the end of the screw. See Figure 4.8.

The pumping characteristics of the screws as well as the

filled lengths were measured using this special die.

-95-

conneet ion to manome ter

conneetion to adjustab~e 'die

Figure 4.8. Setup used to equalize pressure fluctuations due to the

oscillatory motion of the screw.

4.5.1. THROUGHPUT VERSUS PRESSURE CHARACTERISTIC

Figures 4.9a through 4.9c show the throughput versus

pressure characteristic for various screw elements. Note the

large difference between conveying and kneading elements.

The intersections with the vertical axis represent the net

maximum outflow of the (completely filled) kneader. In this

case, no die was present. This was done to prevent pressure

generation due to the oscillation of the screw. Because

relatively low viscous fluids were used, the experiments had

to be corrected for gravity-induced flow.

The intersections with the horizontal axis yield the maximum

pressure generating capacity of the screws. Here, the axial

oscillatory motion was compensated for with the special die

mentioned in the previous Section.

For conveying elements - single-flighted, i.e. having a much

wider channel compared with the other, double-flighted, ones

- no distinct influence of pins on either drag or pressure

flow was observed.

-96-

1500·-,---------------------,

N • 180 rpm ~

è e ..... 1000-~

'S a_

"§, ~conveylng ~ 500~ -~. :5 '' ' k d ' ""' """'- ~.c. neo 109 · ""'

"""-~ .

f~kneoding """- , ~ ' 0~~~-,_--~_,~---,----,---_,

è e

0.00 0 .05 0.10 0.15 0.20 0.25

Ap (bar)

1500,---------------------,

N • 200 rpm

..... 1000

""-~. ~

··~.

""".conveying

""" """ """ """ -~ ~(jl

0.00 0.05 0. 10 0.15 0.20 0.25

i\p (bar)

Figure 4.9. Throughput versus pressure characteristics for conveying,

closed channel kneading and kneading screws in a 46 rnm

Co-kneader. Model liquid s: silicon oil Rhodorsil V1000

1 Pa.s) and paraffinic oil SUNPAR 2280

1 Pa.s). Parameters: Screw speed N (rpm) and

number of pin rows.

Open resp. closed symbols: 0 • conveying; 0 e c. c. kneading ;

a•kneading screw with resp . without pins.

-97-

With the equations of Table 4.1 and the data presented in

Figure 4.9, the two important parameters r (relative

remaining flight length) and c (efficiency parameter) can be

evaluated.

Overall equation: Q + QL + QLD (1-a.) •Qct + c•Q · + Q p1n p ( 4. 9)

no pins: a. 0, c 0

( 4 .10)

The vertical intersection is calculated from the condition:

Q = 0 p

Q

Qd (1 - r•ó/H - (1-r))

The horizontal intersection is calculated from the

condition: Q = 0

-Q p Qd r• (1 - &/H)

r J.l Qd (1 - ó/H)

k

with k a function of r: k

see last Eq. of Table 4.1.

C •r -1 ) '

(4 .11)

(4 .12)

(4 .13)

(4 .14)

-98-

with pins: a. ~ 0, c ~ 0

The vertical intersectien (Qp 0) is given by

Q r•Qd(1 - ó/H) - a.•Qd + c•Qpin ----

(4 . 15)

(4 .16)

The horizontal intersectien (Q

4.14:

0) yields, analogous to Eq.

Ap (4 .17)

------------Apla.=O

Table 4.3 shows the values of rand c that follow frorn the

respective horizontal and vertical intersections.

Table 4.3. Values for r (relative remaining flight length) and c

(efficiency of pins) as determined from horizontal and

vertical intersections in Figures 4.9(a) - 4.9(c),

r c

intersectien (hor.) (vert.) (hor . ) (vert.)

conveying 0 . 93 0 . 75 0.01 0.01

closed channel

kneading 0.90 0 . 6 0.04 0.15

kneading 0.52 0.18 0.09 0.21

-99-

In the case of conveying and closed channel kneading

elements, the values of the experimentally determined

relative remaining flight length (using Eqs 4.9 till 4.17)

as shown in Table 4.3 correspond reasonably well to those

that can be determined from geometrical considerations. The

results from the horizontal intersections correspond to the

values in the first column of Table 4.2, whereas the results

from the vertical intersections correspond well to the

values in the second column of Table 4.2. This can be

explained by realizing that the pressure gradients in the

Co-kneader are almost exclusively in axial, rather than in

down-channel direction, because of the large number of

holes. The values of the relative remaining flight length,

calculated from the horizontal intersection will be close to

the ones determined according to Figure 4.6a. In the absence

of pressure flow, the value of the relative remaining flight

length is determined from drag flow, which is assumed to be

present along the length of the flight in down-channel

direction. Consequently, the values of r calculated from the

vertical intersectien correspond to the ones determined

according to Figure 4.6b. In the case of kneading elements,

however, the holes' length is so large in comparison with

the remaining flight length that the model equations hardly

apply.

With regard to the evaluation of the influence of pins as

expressed in the efficiency parameter c we can conclude, as

could be expected, that no impravement in transport

properties in single-flighted elements is found. There is,

however, an effect in the double-flighted elements

(efficiency in the order of 10-20%) . The experimental values

of c also differ when determined from either vertical or

horizontal intersectien of the throughput-pressure

characteristic. The naive statement of full effectivity of

the transport capacity of pins expressed in the

straightforward expression used: Qpin = n V H dpin is

apparently an overestimation.This would be expected not only

from reasans concerning the bow-wave type of flow in front

-100-

of a pin, but also because of the consideration that this

flow is mainly in tangential direction (straight through the

holes in the flights), thus promoting leakage as well. The

pins might seem less important, yet it must be kept in mind

that pins are important for solicts transport and that the

presence of the pins is essential for the mixing principle

of the kneader.

4.5.2. FILLED LENGTH

The second way to evaluate the parameters r and c can be

realized with the setup shown in Figure 4.7. By

photographing the length of the channel that is completely

filled, the combination of a basic screw element with a

staggering element can be closely examined, see Figure 4.10.

Figure 4.10. Photograph showing the combination of a closed channel

kneading screw element, having 3 rows of pins. with a

Couette element (gap height 0.8 mm).

Figures 4.lla through 4.1ld show the comparison of

experiments and calculations for the various processing

conditions. In Figure 4.lla, the filled length is calculated

for conveying screw elements without pins, using a value for

-101-

the relative remaining flight length r = 0.93, obtained from

the horizontal intersectien (see Table 4.3). The calculated

filled length is quite sensitive to the value of r. Figure

4.11b shows calculated and measured filled lengths for

conveying screw elements with pins. As could be expected

from Figure 4.9, no distinct influence of the pins on the

filled length was observed.

For closed channel kneading elements without any pins, the

value of r = 0.9 (see the first column of Tab1e 4.3) is

correct to predict the filled length, as shown in Figure

4.11c. If pins are mounted in closed channel kneading

elements, a significant decrease in filled length is

observed, see Figure 4.11d. This effect is, however, not

satisfactorily described by carrying out the calculations

with an efficiency parameter c = 0.15. Because the exact

value of the filled length in closed channel kneading

elements is difficult to discern from the photographs, the

value of the parameter c will not be discussed further.

400

Ê 350 .§. .a 300 ..J .s::. Öl 250 c:

~ ~ 200

~ x 150

··~ ~ '0 100 Q) u---:--------=: ;;::

50 0 0

0 :sa 200 250 300

screw speed (rpm)

Figure 4.11 a) Cornpariaon of calculated (drawn linea) and rneaaured

fi11ed length Lb for a combination of conveying acrew

elernenta (no pina) with a Couette element. (r = 0.93;

a,c = 0 uaed in the calculations). Model fluid: paraffinic

0.2 Pa.s). Q (g/min): D 75.0; • 100; 0 130;

400

Ê 350 .§. .0 300 ..J

.c Ö> 250 <:: ~

2CO (ij )( ~

150

"0 100

ê 50

150

450

Ê 400 .§. .0 350 ..J

.c 300 C) <:: 250 ~

(ij 200 )( ~ 150

"0 Cl) 100

- 50

0 ' 150

<50

Ê 4CO .§. .0 350 ..J

.c 300 C) c: 250 ~

(ij 200 )(

~ 150

"0 Cl) 100

50

0 . 150

-102-

-·.....__

200 250

screw speed (rpm)

0

0

•

~ 200 250

screw speed (rpm)

~~ • -------- 0 ~ . ~

• • :: 0 0 0

200 250

screw speed (rpm)

b) 300

c) 300

d ) 300

Figure 4.11 b) As Fig ure 4.11a, with pins (r = 0.93; ~ = 0 .005, & = 0.01) .

Q (g /mi n): 0 BO; • 100; 0 140.

c) As Figure 4.1la, for closed channel kneading element s (no

pins, r ~ 0 . 9 ; ~,& = 0). Q (g/ rnin): 0 75; • 100; 0 120.

-103-

(continued from previous page)

d) As Figure 4.11a, for closed channel kneading elements

(three rows of pins, r = 0.9; « = 0.011, e = 0.15).

4.5.3. PRESSURE GRADIENTS

The third and most accurate way to evaluate r is by visually

investigating the axia1 pressure gradients with the setup

shown in Figures 4.7 and 4.12. From the data thus obtained,

and with the equations of Table 4.1, the relative remaining

screw flight length r can be calculated. The results are

given in Table 4.4, for different screw geometries and

processing conditions. (No pressure gradients were

determined for kneading elements, because of the low

pressure generating capacity of these elements)

Table 4.4. Experimental pressure gradients and relative remaining screw

flight length r in conveying elements (2 rows of pins) and

closed channel kneading elements *) (3 rows of pins) .

Q N

(g/min) (rpm)

conveying 170 150

170 200

200 150

e.c. 170 150

kneading 170 200

170 250

270 150

270 200

270 250

~ L lexp.

(Pa/m)

5390

11590

5120

2680

4200

5340

1700

3020

4390

r

0.88

0.92

0.88

0.82

0.83

0.83

0.77

0.80

0.81

*) Efficiency parameter e 0.15 used in the calculations.

-104-

Figure 4 . 12 Visualization of the pressure gradient in a closed channel

kneading element. N = 250 rpm; Q = 270 g ./min.

To conclude the evaluation of rand c, Table 4.5 gives a

listing of the values to be used in future calculations.

Table 4.5. Concluding values for r (relative

remaining flight length) and c

(efficiency of pins) for several

screw elements.

conveying

e.c. kneading

kneading

r c

0.9

0.8

0.2

0.01

0.15

0.21

With the model derived so far, it is now possible to

calculate a sequence of screw elements with different

combinations of screw speeds and throughputs. Figure 4.13

gives some examples of calculated pressure profiles in

common screw configurations, for various processing

conditions.

I L{)

0 .-l I

.--------------------,~,_g ~

0 0 U")

0 0 '<t

Ê

I r~~ I I

0 0 N

0 0

f---r---..---.,---...---...---+o 0 ",

U") N

0 U")

N - 9 U")

(.mq) a;nssa;d <---

0

l!s::: 3> 4 "L "

5 ;sr- ~-~-

r--------------------,~~g ~

0 0 U")

0 0 '<t

.......-... :-. -----E -::::::;::: EZ 2~----

gE ~ EZ27 ",'-"'

0 0 N

0 0

_l ::::::::;::. EZ 27.1l

R~ -·--··--

l!c:~ dtL '----··d-

l!c:?> d"L "--·--

r-~--r---r-....--;------1- 0 0 U") l"l N

0 U") 0 N - -

U")

(;oq) aJnssa;d <---

0

-106-

30 'L'

0 25 ...0

'--"' (!)

20 L :J , VJ (/) 15 (!) L

A 0...

10 1\ I 5 I I

0 0 100 200 300 400 500 600

L (mm)

", ", ", ", ", ", "' "' ", ", N N N N N N N N N N

0> ,N IN N UI N N N N N N N N N N N !" I ~

,;-> 1:-> ~ .... ;:-J I:-' :-> :-> :-> :-> :-> .... ,..., ,..., !m

,. i;.. i ~ ,;.. 151 j!S 151 i "' • 151 I"' 151 , ... I"' i"' !"' I I I ! I I I I

Figure 4.13. a) Calculated pressure profile and filled lengths in a

Co-kneader (D = 46 mm), ha ving standard s c rew geometry.

Model fluid: Linear low density polyethylene.

Newtonian, isothermal model . Parameter : Screw speed

b) As Figure 4.13a, with kneading screw.

c) As Figure 4.13a, with conveying screw.

4.6. NONISOTHERMAL, NON-NEWTONIAN ANALYSIS

In analogy with the analysis in Chapter 3, the NeWtonian

isothermal analysis of the Co-kneader can be extended to the

nonisothermal, non-Newtonian case. By taking incremental

steps over the length of the kneader, more quantitative

answers concerning specific energy and temperature rise can

be given. Over each incremental step a heat balance is

solved: the heat stared in the local volume (resulting in a

temperature rise) equals the dissipative heat generated by

the velocity gradients plus the heat exchanged with the

walls (positive or negative). Figure 4.14a shows the

pressure profile s and fill e d lengths for constant thr oughput

with varying screw speed. Figure 4.14b shows the temperature

profile for the same throughput, with as parameter the

initia! melt temperature.

'L' 0

..0 ....._"

Q) I... ::::s (/) (/) Q) I... 0...

!\ I I I

,.-.... u

....._"

Q) I... ::::s

-+-0 I... Q)

0...

E Q)

+-

1\ I I I

15

12

9

6

3

0 0 100 200

300

250

200

150

-107-

Q;::; 25 kg/hr

Tin= 200 c

screw speed N = 200 rpm

;---. N = 220. a!!!!__ . 1/ _____ N =-~~.l!.!J!!!' ___

1/ I

.1? /

300 400 500

Q;::; 25 kg/hr

N;::; 240 rpm

initia! tempareture TIn= 200 C

T In = 2,:Z.Q_f_. __ ..

!...!.!)_;: 24Q__Ç ________ _

600

100;--------r-------.------~--------.-------.-------~ 0 100 200 300 400 500 600

L (mm)

Figure 4 . 14 a) Calculated pressure profile and filled lengths in a

Co-kneader (D = 46 mm). Model fluid: Linear low density

polyethylene , ~ol2oo oc = 5000 Pa.s; b = 0 . 016; n = 0.43.

Twall = 200 oe .Parameter: Screw speed.

b) Calculated end temperatures in a Co-kneader (D = 46 mm) .

Model fluid: as in Figure 4.14a, parameter: Initial melt

temperature .

-108-

4.7. RESIDENCE TIME DISTRIBUTION

Analogous to the procedure of Section 3.7, the Residence

Time Distribution of the Buss Co-kneader can be

determined*. Some representative results will be discussed

here. Details can be found in the work of A. Engbersen (14)

0 .8

0 .6 --i:L

0.4

throughput, (t) 0 .2 12.5 kgL'hr, (90.2)

25kgf~G (47.11 _____

0 0 2 3 4

tlt

Figure 4.15 Residence time distribution in a PR-46 Buss

Co-kneader. Screw speed: N = 200 rpm, barrel

temperature Tw ~ 150 °C . Kneading screw geometry

(see also Figure 4 . 13). Parameter: metered throughput Q.

* In the experiments, poly(vinylchloride) (PVC) was used.

0.8

0.6 --i:L

0.4

0.2

0 0

//

t/t

-109-

2

adhering PVC, Ï=82.7

non-qd_h_ering P_Y_CJ Ï=76.!!

3 4

Figure 4.16 As Figure 4.15. Processing conditions: Q = 12.5 kg/hr,

N = 200 rpm, Tw = 150 oe. Standard screw geometry.

Material : PVC with internal and external lubrificants,

referred to as 'adhering' and 'non-adhering' (DSM

VARLAN S 5020 and S 7120, respectively).

~--"'· · · · "'''''''' r~--

.. p·

0.8

0.6 --- (t) i:L screw type standerd (43.2)

0.4 ~n~yif}SI {2_2.§},

buffle ring_(~

0.2 ~<;,__~~~1!lLi~~-fJ(]r_all<!l_pl(]le_ITl(],j<!l ___

0 0 2 3 4

t/t

Figure 4.17 As Figure 4.15 . Processing conditions Q = 37.5 kg/hr,

N = 200 rpm, Tw = 150 oe. Parameter: screw geometry .

The dotted line represents F(t) fora single screw extruder,

as predicted by Pinto et al. (16) .

-110-

0.8

0.6 --Li:'

0.4 screw type (t) slandard (82.7)

kneadi'!9. (104.()). __

0.2 buffle ring_(~

s.c_,_~"._9~'!9.J~JL

0 0 2 3

tlt

Figure 4.18 As Figure 4.17. Processing conditions: Q

N = 100 rpm, Tw - 150 °C.

4

12.5 kg/hr,

In Figure 4.15, the influence of the throughput is

demonstrated. The shape of the F(t) curve remains almost the

same, while t is being reduced by a factor 2 *.

Figure 4.16 shows a tendency towards plug flow (i.e. a more

narrow distribution) for the non-adhering PVC grade.

Figure 4.17 shows an overall comparison between the most

common screw geometries.

* By assuming the kneading elements plus buffle ring elements

to be comp1etely fi11ed with melt, t can be estimated to be

(see Sectien 3.7): 80 s and 40 s when Q = 12.5 kg.hr-1 and

25 kg.hr-1 , respectively.

-111-

4.8. DISCUSSION

It is shown that an analysis similar to that for the

corotating twin-screw extruder can be used to model the flow

in a Buss Co-kneader. Measuring the throughput versus

pressure characteristic, the filled lengths and, most

accurately, directly the pressure gradients are effective

ways to check (and estimate) important geometrical

parameters such as the effective relative remaining flight

length and the pins' influence on drag and pressure flow.

Extension of the model to nonisothermal, power law

calculations is possible. It will be important to model in

yet more detail the dissipation of heat in the narrow gaps

between pins and screw flights as well as the increased heat

exchange between the polymer and the kneader wall. At a

local scale, a finite-element-method calculation might be a

helpful tool in understanding these phenomena. Further

experiments, such as determination of critical screw speed

and temperature profile in the kneader, are needed to

evaluate the extended model.

4.9. REFERENCES

1. M.L. Booy and F.Y. Kafka, Soc. Plast. Eng. (Techn.

Papers),~' 87 (1987)

2. J. Jeisy, Y. Trouilhet and P. Grassmann,

Verfahrenstechnik, lQLZ, 79 (1976)

3. H.E.H. Meijer and P.H.M. Elemans, Polym. Eng. Sci.,

~,275-289 (1988)

4. P. Schnottale, Kautschuk und Gummi Kunststoffe, ~' 2/85,

116-121 (1985)

5. S. Jakopin, Adv. Polym. Technology, vol.~ (4), 365-381

6. S. Jakopin and P. Franz, Paperpresentedat the AIChE

Diamond Jubilee, Washington O.C., November 3 (1983)

7. O.B. Todd and J.W. Hunt, Soc. Plast Eng. (Techn. Papers),

U, 577 (1973)

-112-

8. German patent 944727 (H. List, 1945)

9. R. Brzoskowski, J.L. White, W. Szydlowski, N. Nakajima

and K. Min, Int. Polym. Proc. III (1988) 3, 134-140.

10. E.G Harms, Dissertation RWTH Aachen, West Germany (1981)

11.US patent 3642406 (H.F. Irving, 1971)

12. C. Rauwendaal, "Polymer Extrusion", Hanser Publishers,

Munich (1986)

13. J.M. McKe1vey, "Polymer Processing", John Wiley & Sons,

New York (1962)

14. z. Tactmor and I. Klein, "Engineering Princip1es of

Plasticating Extrusion", Van Nostrand Reinhold, New York

(1971)

15.L. Erwin and F. Mokhtarian, Polym. Eng. Sci., ~, 49

(1983)

16.A. Engbersen (DSM), private communications (1986)

-113-

CHAPTER 5

SCALING

The procedure adopted in the Chapters 3 and 4 reveals that

it is well worth examining the sealing up from a small

laboratory extruder to a larger one. The scale-up strategies

that are usually developed involve all aspects of the

extrusion process: solicts conveying, melting and metering. A

review of existing scale-up theories can be found in Ref. 1.

The analysis in this chapter will mainly focus on partially

filled continuous mixers.

The parameters to be analyzed are: Torque, power

consumption, throughput, specific energy consumption (the

power consumption divided by the throughput), mean residence

time, shearrate, total shear and specific surface (the -

heat exchanging - barrel surface divided by the throughput) .

5.1. DEFINITION OF SCALING POWERS

In order to evaluate the different scale-up procedures, all

variables are expressed as a power of the diameter ratio.

This results in the definitions of Table 5.1.

Reprinted partly from: H.E.H. Meijer and P.H.M. Elemans,

Polym. Eng. Sci., 28, 275 (1988) by permission of the Society

of Plastics Engineers.

-114-

Table 5.1. Definition of sealing powers. The index , o'

refers to the reference machine.

Screw No D0 n Specific ~= D es =(-) (-)

speed No Do energy ESP·o Do

Channel Ho D0 n Residence t D t =(-) (-)

depth Ho Do time to Do

Screw Lo Do Q. Throughput Q D q

=(-) (-) length Lo Do Qo Do

Power Po Do p Shearrate y D g =(-) (-)

Po Do Yo Do

Torque To D0 m Tot al y D gs =(-) (-)

To Do shear Yo Do

Specific s D s = (-)

surface So Do

5.2. GEOMETRICAL SCALING

Usually corotating twin-screw extruders scale (60 < D < 200

mm) geometrically; this implies that

H

D c;

L

D c; N = c

where c = a constant.

Furthermore (locally)

<p = c Q.c = c ; f = c

(5.1)

(5. 2)

These practical rules are based on the principles which keep

the mixing constant.

Shear rate y:

V 'II'DN y ~

H H (5. 3)

-115-

Mean residence time:

volume t

throughput

~ f~ 'lf DLH

Q

fl!.'lf DLH

f l.2VHW

with f~ = local degree of fill*

And total shear y yt = f9_ 2L f Hsinqo

(5 .4) f 'lfDNsinqo

(5. 6)

As is well known (2) reorientation relative to the direction

of shear is extremely important for the mixing efficiency

M.E . - (___l y)nr nr ( 5. 7)

(The werking principle of static mixers is completely based

on nr rather than on the total shear y) • During each

revolution of the screw, material is passed from one screw

to the other and is assumed to be reoriented.

This may be written as

nr n.N.t

with

nr the number of reorientations

n the number of flights

N screw revolutions per second

t the mean residence time according to Eq. 5.4.

* f is the degree of fill e.g. underneath the hopper used only

(instead of Q) as a kind of dimensionless measure f o r the metered

throughput (Eq . 3.3).

(5 . 8)

f~ is the local degree of fill. It is equal to one in the case of

completely filled parts, but may differ from f, for example, by a change

of pitch.

hence

nr 2 f~~DLNH

Q

-116-

(5.9)

From Eqs. 5.1 and 5.2, it can be seen that the degree of

mixing (Eqs. 5.3, 5.5 and 5.6) remains constant: f, L/D and

~c = constant implies ~b = constant and ~a = constant (Eq.

3.21). If the samematerial is used (~, p = constant) the

specific energy, Eq. 3.39, is also constant. Furthermore, it

follows that throughput, energy and torque all scale with

the third power of the diameter in this case, see Table 5.2.

5.3. THERMAL SCALING

Geometrical sealing is, however, allowed only if both

extruders are operating under fully adiabatic conditions.

As soon as heat exchange with the barrel wall is more

important in the process, throughput can only scale with the

second power of the diameter because the barrel surface

(~DL) scales with o2 . Otherwise, the temperature development

in small test extruders will be different from that in

larger ones, with all consequences for viscosity differences

(mixing !) and thermal degradation with the more recently

developed high-melting-point polymers.

The problem that arises is that sealing according to the

mixing rules does not provide the same powers as sealing

according to the rules of equal thermal development. In

practice, this means that choices have to be made and that

experiments and eventually changes in screw design on the

larger scale extruder will always be necessary. This can be

illustrated as follows.

For laminar flows, the temperature development is given by:

-117-

pc u a x

+ \l (5.10)

whereas for ideally mixed annular flows (see Fig. 5.1):

dT Pc u dx

<a.> (T - T ) w +

H

<U> (T-Tw)

L

W=Y't

Figure 5.1. Heat balance for an ideally mixed annular flow.

5.3.1. LAMINAR FLOW

(5. 11)

The dimensionless form of these equations yields the

requirements for equal temperaturé development. For laminar

flows

Gz u*

where

Gz

Br

Graetz number

du* 2 + Br • (--)

dy*

pcVH H convective --

À L conductive

(5 .12)

heat transfer

heat transfer

(5 .13)

!lv2 viscous dissipation Brinkman number 'MT heat conduction

(5.14)

-118-

If Gz- and Br-numbers are constant, solution of Eq. 5.11

always gives the same result. Provided that the same

material is used (À,p,c and ~are constant), the same wall

temperature (Tw) is employed, and (as usual) L/D = a

constant, then these conditions yield (Eq. 5.14):

V = constant; consequently N ~ D-1 (5 .15)

And from Eqs. 5.13 and 5.15 it fellows that

H ~ D0.5 (5.16)

These very severe demands result in Q ~ D1.5.

The conditions ~ = 1, n = -1, h = 0.5 and q = 1.5 (see

Table 5.2) are met in practice only when heat exchange with

the walls is extremely important, for instanee in the

melting sectien of a single-screw extruder with a grooved

feed zone (3) .

5.3.2. IDEALLY MIXED ANNULAR FLOW; L/D A CONSTANT

For ideally mixed (annular) flows, which may be assumed for

the radially well mixed flow in corotating twin-screw

extruders, the heat balance for an infinitesimal element dx

can be written as:

dT 11'. Ddx + ~ [

dÜ ] ~ p c u dx dx 11' DH <a.> (T- - Tw) dy DH dx

(5 .17)

dT <a.> ('r - T ) du 2 p c u dx H + J.l ( dy) (5 .18)

In order to make these equations dimensionless, let

u T - T dx dy * ;;* :kl * * u V , fJ.T dx L ; dy H (5; 19)

-119-

ctöf* llT <a.> T* llT ~ v2 [ :y"-r. p c ~*v dx* + H2

DH dx L H

(5 .20)

-* ctf* ~ v2 H """* p c VHu

[ :;*] 2 dx* T* +

<a.> llTH2 <a.> c (5. 21)

ctöf* -* i*

du 2 p * + B • (-) u

dx* dy* (5.22)

with

P number pc VH

<a.> L

Péclet

Nusselt

convective heat transfer

total heat transfer

(5 .23)

B number

2 ~ V

<a.> HllT

Brinkman

Nusselt * As viseaus dissipation

total heat transfer

Nusselt

Péclet

As L

H

<a.>L

pcVH

À

total heat transfer

conductive heat transfer

convective heat transfer

conductive heat transfer

Constant temperature developments in this case require

constant P- and B-numbers; therefore, using the same

material (p,c,M,À constant), the sameheat transfer

coefficient <a.>, wall temperature (Tw) and when L/D is

constant,

hence

v "' ol/3 '

and VH

D constant

(5.24)

-120-

and therefore N ~ o-2/3 and H ~ o2/3 (5.25)

For the throughput this implies, as expected, Q ~ o 2 .

Consequently: 2. = 1, n = -2/3, h = 2/3 and q = 2, (see

Table 5.2). The sealing law for channel depth (in Eq. 5.25)

is often found in practice for many extruders (3), of

course, because channel depths are determined during the

construction of the extruders. Screw speeds and throughputs

can be changed easily at any later moment.

5.3.3. IDEALLY MIXED ANNULAR FLOW; H/D A CONSTANT

If the restrietion of L/D = c is dropped, and H - D is

introduced, (as is practice in corotating twin-screw

extruders) then the requirement of constant P- and B­

numbers yields

VD V "' oO. 5 d an L a constant, so

N ~ o- 0 · 5 , L ~ o 1 · 5 and H ~ D , hence Q ~ o2 · 5

and 2. 1. 5, h 1, n -0. 5, q 2.5, see Table 5.2:

5.4. SCALING LAWS

To summarize: the (usual) sealing laws can be derived,

either from literature (4-6) or from the following

equations of this thesis:

Eqs. 3.2, 3.35, 3.36 p 3 + 2n + 2. - h

Eqs. 3.2, 3.35 q 2 + n + h

es p - q 1 + n + 2. - 2h

Eqs. 3.2, 3.31, 3.32 m 3 + n + l1. - h

Eqs. 3.1, 3.35, 3.44 t h + 1 + l1. - q l1. - 1 - n

Eq. 3.42 g = n - h + 1

gs g + t l1. - h

s = (1 + !1.) - q n + h + l1. - 1

(5. 26)

-121-

The sealing laws result in the following numerical values,

depending on the methad of sealing (see text, Eqs. 5.1

through 5. 26)

Tab1e 5.2 Sca1ing powers, fo11owing from different methods

of sca1ing.

geometrical 1aminar idea11y mixed

n 0 -1 -0.667 -0.5

h 1 0.5 0.667 1

l!, 1 1 1 1.5

e 3 1.5 2 2.5

es 0 0 0 0

m 3 2.5 2.667 3

q 3 1.5 2 2.5

t 0 1 0.667 1

g 0 -0.5 -0.333 -0.5

gs 0 0.5 0.333 0.5

s -1 0.5 0 0

5.5. EXAMPLE: GLASS-FIBRE REINFORCEMENT

The applicability of the sealing concepts derived so far

can be illustrated in the case of the glass-fibre

reinforcement of a degradation-sensitive polymer.

Processing should be performed as close as possible to the

(high) melting point (Tm) and, consequently, the scale up

rules for equal temperature development should be fulfilled

(see Table 5.2). In experiments on two corotating

twin-screw extruders with different sizes the sealing

exponents for screw speed and throughput were evaluated.

They proved to be dependent on the requirements which were

put on the average temperature of the outcoming melt (see

Table 5.3). Via simulations of the last part of the

extruder (melt!) assuming an averagemelt temperature at

the inlet (the end of the melting section) and aiming for

-122-

the same melt end-temperature on both extruders, Tm + 25 in

this case, a theoretical value for the sealing exponent 'q'

is found which is close to the experimental one (see Table

5. 3) .

Table 5.3. Experimental (rows a and b) and theoretica!

(row c) values for the sealing exponents for

screw speed and throughput. From (7).

Average Sealing exponents

Melt n q

Temperature

a Tm + 15 -1 2.3

b Tm + 35 -0.6 2.5

c Tm + 25 -0.4 2.4

5.6. CONCLUSION

The conclusions that can be drawn from this short

investigation of the sealing laws are that geometrical

sealing yields maximum throughput and meets the

requirements of constant mixing, but results in a different

temperature development as soon as the extruders do not

operate in a completely adiabatic mode. Consequently,

mixing is not the same anymore because viscosities are

temperature dependent.

In the case of equal temperature development, screw speed

(absolutely) and throughput (relatively) are lower, with as

a consequence decreased mixing, energy consumption, and

torque. Large extruders (D > 200 mm) do not allow

geometrical sealing any more. The circumferential speed and

the wear it causes become too high.

-123-

5.7. REFERENCES

1. C. Rauwendaal, Polym. Eng. Sci., ll, 1059, (1987)

2. K.Y. Ng and L. Erwin, Polym. Eng. Sci., 21, 4 (1981)

3. P. Fischer, Ph.D. Thesis, RWTH Aachen, West Germany

(1976)

4. J.F. Ingen Housz, Intern. Zeitschrift for Lebensmittel

Technologie, 1La2, 48 (1982)

5. J.F. Ingen Housz, Polymer Extrusion II, Paper 8, London

(1982)

6. J.R.A. Pearson, Plast. Rubber Processing, Sept., 119

(1976)

7. H.E.H. Meijer and P.H.M. Elemans, Polym. Eng. Sci., ~.

275 (1988)

8. H. Potente "Auslegen von Schneckenmaschinen Baureihen"

(1981)

-124-

CHAPTER 6

TIME EFFECTS IN THE DISPERSIVE MIXING OF

INCOMPATIBLE LIQUIDS

The deformation and breakup processes of single droplets in

well-defined fields of flow have extensively been studied in

the literature. In spite of the fact that in real mixers the

conditions are far from equilibrium, most studies are

confined to (Newtonian) systems undergoing almost stationary

deformation (1-4). The time effects that occur during the

different stages of the dispersive mixing process are less

well understood.

In a Plexiglas-walled Couette-apparatus, the time-dependent

deformation of Newtonian droplets into extended threads has

been studied. When the shear rate is very slowly increased,

allowing for almost equilibrium deformation, the results of

the critical capillary number Ca as a function of viscosity

ratio, as reported in the literature (3), are reproduced.

However, in transient flows at capillary numbers Ca >>

Cacrit' droplets are deformed into long slender bocties which

can remain extended until the shear has stopped.

They then desintegrate into lines of droplets because of the

interfacial tension-driven Rayleigh disturbances. The time

for breakup can well be calculated from the existing theory

on the stability of extended liquid cylinders (5).

6.1. INTRODUCTION

When a blend of two incompatible polymers is subjected to

shearing forces, e.g. in a corotating twin-screw extruder,

droplets of the dispersed phase will deform into long,

threadlike particles (7), which can break up into smaller

droplets. The ultimate morphology - the relative

-125-

distribution of the constituents - depends on volume or

weight fraction (8), viscosity ratio of both polymers and

the type of flow (4).

The word 'polymer', deliberately used in the former

paragraph, in contrast with 'liquids', as mentioned in the

title of this chapter, suggests a similarity in the

behaviour of systems used in emu1sion rheology as well as in

polymer rheology. In the next Sections, therefore, we will

review some of the literature on time-dependent deformation

and breakup of Newtonian droplets. The emphasis will be on

shear flows. Although less efficient than elongational

flows, they can be realized easier in practice. In Chapter

9, results from experiments with these systems will be

translated to the processing conditions of a polymer blend.

6.2. AFFINE DEFORMATION OF DROPLETS IN SIMPLE SHEAR FLOW.

For large capillary numbers Ca (Ca= ~cyR/o), and for

viscosity ratios p < 1, droplets will deform affinely with

the matrix. Their deformation can be expressed in terros of

the total shear y (y = yt), imposed by the flow field (9).

In a simple shear flow, a spherical droplet having an

initia! diameter 'a' will deform into an ellipsoid with

length L and width B:

L/a ( 1 + 2 1/2

y ) (6 .1)

B/a (1 + 2 -1/4

y ) (6 .2)

hence

L/B (1 + y2) 3/4

(6.3)

-126-

or, by defining the deformation D (L - B) I (L + B)'

2 3/4 (1 + y ) - 1

D (1 + y2)3/4 + 1

Figure 6.1 shows the properties L/a, B/a and D as a

function of y, calculated from Eqs. 6.1, 6.2 and 6.4.

3,-----------~----------------------------~

2.5

2

~ _J

r5 1.5

~ 0

1\ I I I 0.5

2

D

8/a

3 4 5 6 7 8 9 10

---> 'Y

Figure 6.1. Relative length L/a, relative width B/a and

deformation D of an affinely deformed droplet as a

function of y.

(6 .4)

As stated befare (10,11), periodical reorientation of the

already deformed droplet increases the efficiency of the

mixing process. This is illustrated in Figure 6.2, which

shows the influence of reorientations on the decrease in

width of a droplet in simple shear flow.

0.8

0.6

~ m 0.4

1\ I I I 0.2

0 0

-127-

·-­-10

---> 'Y

without reorientations

--·---·-with reorientations

20 30

Figure 6.2. Effect of reorientation on the decrease of B/a,

the relative width, of a droplet undergoing affine

deformation in simple shear flow.

6.3. BREAKUP OF THREADS

Until now, the shear rate y and the time t have been

interchangeable. However, once the droplet has become

highly extended, it will exhibit initially small sinusoidal

distortions, each possessing a wavelength À along the

thread. Depending on the viscosity ratio p = ~dl~c' one

wavelength, Àmr will turn out to be dominant and will cause

breakup (5,6,12). For Newtonian threads of infinite length

in a fluid mat~ix which is at rest, the time for breakup,

tb, can well be calculated using the theory of Tomotika

(5, 6) .

where

1 o.a1 R0 ll,n( ) q a.o

R0 the initial thread radius

a.o the initial distortion amplitude

q the growth rate of a distortion

(6 .5)

-128-

Predicting the stability of a thread in a matrix which is

deforrning is rnuch more difficult, because the wavelength À

is changing with time. For instance, in a shear flow the

distortions are swept away (see Figure 6.3), whereas in an

elongational flow a potentially dominant wavelength will be

extending so fast with the matrix (6,7,13) that its effect

will be darnped out before having been able to cause breakup.

------~~~.~-----~ i~,..--. --"7 e=:==--...J

------~~~>~'-------- ~ ~.-----------i

Figure 6.3. Distortiens being swept away in simple shear flow.

From (14).

Frorn the detailed studying of the dynamics of this breakup

process in sirnple shear flow and hyperbalie extensional

flow, by Mikarni et al. (15) and later by Khakhar et

al. (16), it can be said that the presence of flow has a

stabilizing effect on the thread. In hyperbalie extensional

flow, the time for breakup increases slowly with the

capillary nurnber (16):

tb - log (Ca), (6. 6)

while the size of the draplets is inversely proportional to

the strain rate.

In the case of simple shear flow, the results are presented

in terros of the initial orientation angle e0 of the thread

in the flow field. The time for breakup is given by (16):

(6. 7)

where

-129-

c = 2 tan eo eo initial orientation angle

v a constant > 0

The other initial condition, the minimum amplitude tt0 required for magnification of a distartion caused by the

interfacial tension to occur, is taken from the work of

Kuhn (13). Assuming the distortions to be due only to

thermal fluctuations, he proposed the following estimate

for tt0 :

21 kT ./(

8 3/2 Tl' (1

(6. 8)

with k Boltzmann constant

T absolute temperature

From the analysis presented in Ref. 16, it follows that

under similar conditions, the drops formed in simple shear

flow are larger than those in hyperbalie extensional flow.

6.4. BREAKUP OF DROPLETS

Numerous studies have been devoted to the deformation and

breakup of small draplets in a matrix undergoing simple

shear flow or hyperbalie extensional flow. Good reviews of

the relevant experimental and theoretical work reported in

the literature can be found in the workof Elmendorp (7),

Rallison (17) and Acrivos (18). To be recommended

separately is the almost 'classical' workon droplet

breakup by Grace (3,11), who used a Couette-apparatus and a

four-roll mill to generate simple shear flow and hyperbalie

extensional flow, respectively.

-130-

More recently, Bentley et al. (19,20) developed a

computer-controlled four-roll mill. The continuous

adjustment of the speed of each individual roller enabled

them to position a droplet almost anywhere in the apparatus

and to keep it there for a considerable period of time.

Moreover, they succeeded in (approximately) generating a

wide range of flows, characterized by a single parameter «,

which is correlated to the ratio of deformation rate to

vorticity.

The velocity field is then given by (21) :

-1 < « < 1 (6.9)

where x 1 and x 2 represent coordinates along the axes that

pass through the eentres of the rollers.

The limiting cases are:

« 1 hyperbolic extensional flow

« 0 simple shear flow

« -1: purely rotational flow

The conclusions of their experiments on steady droplet

deformation and breakup are in general agreement with those

of earlier authors.

Continuing Bentley's work, Stone et al. (22,23,24) study

transient effects in droplet breakup. By slowly increasing

the shear rate, droplets are brought to a certain

elongation ratio L/a. The flow is then stopped. Under the

influence of interfacial tension, the droplets may return

to their original shape or alternatively break up into

several fragments. For a broad range of viscosity ratios p

-131-

(0.01 < p < 10), the breakup mechanism via Rayleigh

disturbances is only observed for L/a values > 15. For

lower L/a values (6 < L/a < 15), the draplets exhibit

'end-pinching' (22). The ends of the draplets become

spherical, while the overall length decreases. The ends

pinch off, with the newly formed fluid thread relaxating

further towards a sphere. Obviously, the time scale for

'end-pinching' is much shorter than that for the growth of

Rayleigh disturbances.

Figures 6.4 and 6.5 show the influence of the viscosity

ratio and initial shape on relaxation and breakup of an

extended droplet. (Note that these theoretica! curves show

remarkably good correspondence with the experimental ones

in Ref. 23)

(a) C1C& #::JD

(b) cCC:w;:: >t:J)

(c) OCz >(:JJ=)

(d) c:= CIG>--g s(CJD =)

Figure 6.4. Relaxation and breakup of an initially extended

droplet in a fluid matrix which is at rest.

Parameter: viscosity ratio. From Ref. 23.

p = 0.05

p = 0.1

p = 1.0

p - 7.5

p 10.0

Reprinted by parmission of Cambridge Univarsity Press.

-132-

c ( (~) ) -::::> L/a 5.3

L/a - 6.4

L/a = 7.5

Figure 6.5. As Figure 6.4., for low viscosity ratio droplets.

Parameter: initial shape. From Ref. 23.

Reprinted by permission of Cambridge University Press.

Figure 6.6 shows the calculated shapes of an extended

droplet (L/a = 14), which is breaking up under the combined

action of 'end-pinching' and Rayleigh disturbances. In this

Figure, the time is made dimensionless with the factor

~cRo/cr, and can be compared with the dimensionless time tb*,

required for breakup solely by capillary waves (Eq. 6.5).

1 0.81 Ro tb q !1-n (

<XQ )

* tb ~eRa/a ~cRo/a

(6.10)

where q, the growth rate of a distortion, is given by ( 5' 6)

q (6.11)

-133-

with

Q(À,p) a tabulated function (4-6)

p the viscosity ratio ~dl~c

R0 the initial thread radius

2 hence (6.12)

with Àm the dominant wavelength at which breakup occurs

240

t* 100

t* = 0

Figure 6.6. Calculated evolution of capillary waves during

the cornbined relaxation and breakup of an

initially highly extended droplet in a fluid

matrix which is at rest (p = 1). Frorn Ref. 23.

Reprinted by perrnission of Cambridge University Press.

-134-

In the calculations performed in Ref. 23, the viscosity

ratio p = 1, and the relative distartion amplitude

tto/Ro 3.10-3. The value for Q(Àro,p) = 0.07 can be deduced

graphically from Figure 2.7 in Ref. 6, or from Figure 8.1

in Chapter 8 of this thesis. The dimensionless time for

breakup thus becomes:

t b

* 2

0.07

0.81 !!.n ( 3) ~ 160

3.10-(6.13)

From this calculation, and from Figure 6.6, it is clear

that the desintegration of the droplet by the mechanism of

'end-pinching' occurs on a shorter time scale than by

Rayleigh disturbances, and that the slowly but continually

decreasirig L/a increases the time for breakup (tb* = 240

VS. 160) .

6.5. EXPERIMENTAL

Breakup experiments with Newtonian systems were performed

in a Plexiglas-walled Couette-apparatus. The purpose of the

experiments was twofold:

(i) to investigate the behaviour of draplets during

deformation and breakup under non-equilibrium conditions

(ii) to check the time-dependent deformation of draplets

at low capillary numbers.

Although limited in direct practical applicability, we feel

that these experiments might give some more insight into

the time scales involved in the deformation of draplets in

real mixers, e.g. in the strongly varying flow field

between the kneading flight and the barrel of a corotating

twin-screw extruder.

-135-

6.5.1. EXPERIMENTAL SETUP

A simple shear flow is generated in a Couette-type

apparatus. It consists of two counterrotating concentric

cylinders. The speed of both the inner and outer cylinder

can be adjusted to keep a droplet at the same position

while being deformed at a shear rate y, which for a

Newtonian fluid is given by (25):

where

the radius of the inner, resp. outer

cylinder

(6.14)

the angular velocity of the inner, resp. outer

cylinder

In the present apparatus, R1 = 50 mm, R2 = 63.2 mm. The

height of the gap is 120 mm, which is enough to avoid

disturbance of the flow field due to the bottom. Shear

rates up to 20 s-1 can be obtained. The deformation of a

droplet can be studied from above by a microscope plus a

video system, connected to a high-resolution monitor.

6.5.2. MODEL FLUIDS

Tables 6.1 and 6.2 list the characteristic properties of

the Newtonian fluids used in the experiments. As continuous

phase, silicon oil (Rhodorsil, Rhone-Poulenc) was used. As

disperse phase, solutions of Corn Syrup (Globe 01138) in

water were used. Viscosities at 23° C were determined on a

Bohlin Vor dispersion rheometer. Newtonian behaviour was

observed up to approximately y = 10 s-1. The interfacial

tension between the Corn Syrup (C.S.) solutions and the

silicon oil was measured using a Du Nouy-ring tensiometer.

-136-

Table 6.1. Viscosities of the Newtonian model fluids at 23° C.

11J23 ° C (Pa.s)

continuous phase Rhod V 60000 63.3

Rhod V 30000 29.9

Rhod V 12500 12.5

Rhod V 5000 4.95

Rhod V 1000 1.04

disperse phase

(wt% C.S./wt% H20) 96/4 28.0

95/5 16.25

90/10 4.06

85/15 1.13

83/17 0.75

Table 6.2. Interfacial tension between Corn Syrup (C.S.)

solutions and silicon oil.

wt% C.S./wt% H20

96/4 40.0

95/5 40.0

90/10 41.4

85/15 46.7

83/17 47.0

6.5.3. RESULTS

We will now discuss experiments with model fluids in simple

shear flow, each representing a stage of the dispersion

process that occurs on a distinct time scale. Only the most

important results will be presented here. Details can be

found in reports by Janssen (26) and Bos (27).

-137-

6.5.3.1. AFFINE DEFORMATION

*) A supercritical simple shear flow, i.e. Ca > Cacrit is

stepwise applied to a spherical droplet (typical diameter

about 1 mm). The length L or, at deformation D > 0.8, the

width B, is measured from the monitor. The deformation, D,

can be determined and compared with the theoretical

prediction as given by Eq. 6.4. Affine deformation occurs

at Ca/Cacrit ~ 2. See Figure 6.7.

0 x 0 .9

0 .8 • 0.7 0 • • 0.6

• • p=0.135

0 0 .5

1\ I 0.4 I I 0.3

0 Co/Ca cri!= 7.1

.... Co/Co cri! = 3.2

x Co/Co cri! = 2.1

0.2 • Co/Co cri! = 1.2

0 .1

5 10 15 20

---> y

Figure 6.7 . Affine deformation at supercritical capillary number.

*)

Viscosity ratio p = 0.135. The drawn lines represent the

deformation calculated from Eq. 6.4.

Cacrit is the capillary number for breakup under steady conditions.

-138-

6.5.3.2. BREAKUP OF THREADS IN SIMPLE SHEAR FLOW

A supercritical simple shear flow, with Ca >> Cacrit' is

stepwise applied to a spherical droplet, which will deform

affinely into a thread. At a given moment, the thread will

exhibit sinusoidal distortions which will cause breakup.

The time for breakup tb is defined as the elapsed time

between the onset of the flow and the moment at which a

line of seperate draplets has been formed in the central

region of the thread. Figure 6.8 compares the results

concerning the dimensionless time for breakup with data

from similar experiments by Grace (3). Upon exceeding

Ca/Cacrit' the time for breakup does not decrease, as in

Grace's experiments. Grace possibly observed end-pinching

(see Section 6.4), which indeed yields a much lower value

for tb. The absence of a clear-cut definition of the time

for breakup in Ref. 3 might explain the large difference

shown in Figure 6.8.

1\ I I I

10

---> Co/Cocrit 100

Figure 6.8. Effect of exceeding Cacrit on dimensionless burst time.

Symbols indicate the value of p. Open symbols reprasent

data from Ref. 3. The line for p = 1 is obtained by plotting

tb* at Ca/Cacrit = 1 on the vertical axis, and assuming an

identical slope.

-139-

6.5.3.3. STABLE DEFORMATION AND RELAXATION OF DROPLETS

A spherical droplet is subjected stepwise to a subcritical

shear flow with constant shear rate. The deformation is

measured as a function of time, for viscosity ratios p =

0.09; 0.325; 0.933; 2.24. Figure 6.9 gives a typical plot

of the results (p = 0.09).

Figure 6.10 compares the final stable deformations of the

present systems with data from similar experiments by

Rumscheidt et al. (28).

The final deformation (usually reached at a dimensionless

time of about 25) is, at least for this range, almost

independent of the viscosity ratio p.

0.8

0.6

0 1\ I 0.4 I I

0.2

10 20 30

---> t*

p = 0 .09

0 0

40 50 60 70

• Ca= 0.527

• Ca= 0.463

o Ca= OA30

0 Ca= 0.393

6 Ca= 0.352

<> Ca= 0.295

'\1 Ca= 0.248

Figure 6.9. Deformation D as a function of dimensionless time

t* = t ~/(DeR), at subcritical capillary nurnber.

Upon cessation of the flow, the draplets return to their

original shape. The required dimensionless time is about

15. Figure 6.11 shows this relaxation process for a

viscosity ratio p = 0.933. The smal! deformation (D = 0.05)

that remains is due to optica! distartion by the monitor.

-140-

0 .8

• •• 0 .6

0 • D

~~· D

• o •• 0

0 0 .4 .. 0

1\ ~ • p = 0.0904

I ~ • p = 0.325

I 0 I 0.2 • p = 0.933

D p = 2.24

2 = 4.8 {R &: t.4)

0 0.0 0.1 0 .2 0.3 0 .4 0.5

---> Ca

Figure 6.10. Stable deformation as a function of the capillary nurnber.

The drawn line represents data from Ref . 27.

As stated before in Section 6.4.2, droplets that are

extended beyond a critical elangation ratio (L/alcrit will

not return to their initial shape, but will break up into

several fragments via the 'end-pinching' rnechanisrn after

the flow has stopped. Table 6.3 gives (L/alcrit for three

viscosity ratios.

0 1\ I I I

0.7-.---------------------,

0.6 p = 0 .933

0.5

0.4

0.3

0.2

0.1

0 .0 0 4 B 12

---> t*

• Co= 0.381

• Co= 0.359

0 Ca= 0.344

D Ca= 0.334

I:J. Ca= 0.309

0 Ca= 0.259

V Ca= 0.239

0 Co= 0.216

16 20 24

Figure 6.11. Relaxation of deformed droplets upon cessation of the flow.

Viscosity ratio p = 0.933.

-141-

Table 6.3. Critica! elongation ratio upon cessation of the flow.

disperse phase cont. phase p (L/alcrit

wt% C.S./wt% H20 Rhodorsil

96/4 V 30000 0.933 9.8

90/10 V 30000 0.325 6.3

83/17 V 30000 0.025 5.8

These observations are in fair agreement with those of

Stone et al. (22). Unfortunately, the critical elangation

ratio at higher viscosity ratios can not be investigated in

the present Couette device, because of the limitations

imposed by the type of flow (11).

6.6. CONCLUSION

In the case of Newtonian draplets in a Newtonian matrix,

affine deformation occurs in simple shear flow at Ca/Cacrit

~ 2. The time required for deformation can be calculated

from Eq. 6.4 (with y = yt). Upon cessation of the flow, the

time for breakup of highly extended draplets (L/a > 15) can

be calculated using Tomotikas theory (Eq. 6.5), see also

Chapter 8. For moderately extended droplets (6 < L/a < 15),

the numerical calculations of Stone et al. (23) apply, see

Figures 6.4-6.6.

If viscoelasticity is introduced, deformation and breakup

of draplets is more complicated. Some scouting experiments

with Boger fluids have been performed by Bos (27). Results

of a more systematic study can be found in Ref. 30 for

extensional flow, but especially in Ref. 32 for shear flow,

giving an increased Cacrit with increasing elasticity for

almest all values of the viscosity ratio p.

-142-

6.7. REFERENCES

1. G. I. Taylor, Proc. Roy. Soc. (London), hl.la, 41 (1932)

2. G.I. Taylor, Proc. Roy. Soc. (London), A146, 501 (1934)

3. H.P. Grace, Chem. Eng. Comm., u, 225 (1982)

4. F.D. Rumscheidt and S.G. Mason, J. Coll. and Int . Sc i.,

.1..6., 238 (1961)

5. s. Tomotika, Proc. Roy. Soc. (London), ~, 332 (1935)

6. J.J. Elmendorp, Ph.D. Thesis Delft University of

Technology (1986)

7. S. Tomotika, Proc. Roy. Soc. (London), ~, 302 (1936)

8. G.G.A. BÖhm, G.M. Avgeropoulos, C.J. Nelson and F.C.

Weissert, Rubber Chem. Tech., ~, 423 (1977)

9. J.M. Starita, Trans. Soc. Rheol., .1..6., 339 (1972)

10. K.Y. Ng and L. Erwin, Polym. Eng. Sci., ~, 4 (1981)

11. This thesis, Chapter 1, Section 1.4

12. Lord Rayleigh, Proc. Roy. Soc., (London), 2..2_, 71 (1879)

13.8. Kalb, R.G. Cox and R. St. John Manley, J. Coll. Int.

Sci., .8.2., 286 (1981)

14.W. Kuhn, Kolloid z. 132, 84 (1953)

15. T. Mikami, R.G. Cox and S.G. Mason, Int. J. Multipbase

Flow, 2, 113 (1975)

16. D.V. Khakhar and J.M. Ottino, Int. J. Multipbase Flow,

u, 71 (1987)

17. J.M. Rallison, Ann. Rev. Fluid Mech., .1..6., 45 (1984)

18.A. Acrivos, 4th Int. Conf. on Physico-chemical

Hydrodynamics, Ann. N.Y. Acad. Sci., ~, 1-11

19.B.J. Bentley and L.G. Leal, J. Fluid Mech., lQl, 219

( 1986)

20.B.J. Bentley and L.G. Leal, J. Fluid Mech., lQl, 241

(1986)

21. J.M. Ottino and R. Chella, Polym. Eng. Sci., ~, 357

(1983)

22.H.A. Stone, B.J. Bentley and L.G. Leal, J. Fluid Mech.,

m, u1 (1986)

23.H.A. Stone and L.G. Leal, J. Fluid Mech., ~, 399 (1988)

-143-

24. H.A. Stone and L.G. Lea1, submitted to J. Fluid Mech.

(1989)

25.B.J. Trevelyan and S.G. Mason, J. Colleid Sci., Q, 354

(1951)

26. J.M.H. Janssen, internal report Eindhoven University of

Technology (1989)

27.H.L. Bos, internal report DSM (1988)

28.F.D. Rumscheidtand S.G. Mason, J. Coll. Sci., ~' 238

(1961)

29. S. Wu, Polymer Interface and Adhesion, Marcel Dekker

Inc., New York and Basel (1982)

30.C. van der Reijden-Stolk, Ph.D. thesis Delft University

of Technology (1989)

31.W.J. Tjaberinga, internal report Delft University of

Technology (1988)

32.R. de Bruin, Ph.D. Thesis Eindhoven University of

Technology (1989)

-144-

CHAPTER 7

MORPHOLOGY OF THE MODEL SYSTEM

POLYSTYRENE/POLYETHYLENE

Blends of incompatible polymers often exbibit cbaracteristic

morpbologies (1). As an example, tbe morpbology of a blend

of tbe model system polystyrene/polyetbylene (PS/PE) is

sbown as a function of tbe viscosity ratio and composition.

Tbe blends are made on a corotating twin-screw extruder,

under identical processing conditions.

At low volume fractions, tbe dispersed pbase is present as

more or less spberical particles in a matrix. When tbe

volume fraction is increased, no stable dispersions are

found.

At a viscosity ratio of about unity, tbe dispersed pbase

consists of long, ultratbin fibers. Under the influence of

tbe interfacial tension, tbey eitber desintegrate into

droplets or coalesce. Co-continuous morpbologies are formed

upon blending or after compression moulding.

Wben block copolymers are introduced in tbe system, mucb

finer morpbologies are obtained. Tbe non-equilibrium

cbaracter, bowever, is still present.

Reprinted partly from: P.H.M. Elemans, J.G.M. van Gisbergen and H.E.H.

Meijer in Integration of Polymer Science and Technology. Part 2. Eds.

P.J. Lemstra and L.A. Kleintjens, Elseviers Applied Science Publishers,

Londen and New York (1966) by permission of Elseviers Applied Science

Publishers.

-145-

7.1. INTRODUCTION

As discussed in the farmer Chapter, the deformation and

breakup behaviour of isolated draplets can well be studied.

In a real blend, the final partiele size can not simp1y be

predicted from theory. As soon as the volume fraction of the

dispersed phase exceeds a critical concentration (~d z 1%),

the particles will start to collide and eventua1ly coalesce

(2,3). At volume fractions ~d > 15%, the system can no

langer be considered dilute (4). Co-continuous structures

will emerge until finally a composition is reached at which

phase inversion occurs (5) . The position of this inversion

region depends on the viscosity ratio of the components, as

is shown in Figure 7.1.

TORQUE RATIO

EPOM PBO

8 6 4

2

-~ ! ! "' ~~--• t CONT.

I ccoÖNT. ~\ I I --i- -1--i--

:~ EPDM ti .I=- .I .4 CONT ,

• 0 0.25 0.50 0.75 1.0

VOLUME FRACTION PBO

Figure 7.1. Morphological changes of blends of EPDM and polybutadiene

as a function of composition and viscosity ratio (represented

by the torque ratio). From Ref. 5.

This diagram shows that the component with the lower

viscosity or higher volume fraction will farm the continuous

phase. By assuming a similar influence of bath the viscosity

ratio and volume fraction, a semi-empirical equation (only

valid at low shear rates) expressing the location of the

mid-point of the inversion region has been proposed by

Jordhamo et al. (6):

-146-

lf'1 (7 .1)

The phenomenon of the coalescence of droplets has received a

lot of attention in the literature. With respect to polymer

blending, the theoretica! modelling is still in its initia!

stage. A few contributions are worth mentioning here.

With the aid of gravity-induced coalescence experiments,

Elmendorp (2) shows that the mobility of the interfaces in

molten polymer systems is quite high: droplets with radius R

exhibit coalescence times te ~ R, indicating nearly mobile

interfaces (whereas for droplets having fully immobile

interfaces, te ~ R5 ) . Therefore, coalescence will occur on

a rather short time scale (7). Recently, Chesters (8)

proposed a global model for partially mobile liquid-liquid

interfaces. The rate-determining step of the coalescence

process, the removal of the liquid film between the two

interfaces, is assumed to be due to a shear stress in a

boundary layer inside the droplet phase. Hence, not only the

matrix viscosity, but also the viscosity of the dispersed

phase and the viscosity ratio 'p' play an important role.

The validity of this model can be investigated by observing

the thinning rate of the film between two approaching

interfaces (9).

In the case of the gravity-induced coalescence of a droplet

approaching a flat liquid interface, the Chesters model

yields an expression for the film thickness (h) as a

function of the contact time (t), see also Ref. 8.

with

A l")d ,f (g t::.p/3)

3/2 (]

(7. 2)

(7. 3)

where

R

'ld g

!lp

(1

the

the

the

the

the

-147-

radius of the droplet

zero-shear viscosity of the droplet phase

constant of gravity

density difference between the two phases

interfacial tension

7.2. PHASE INVERSION

The schematic phase diagram of Figure 7.1 originally refers

to EPDM-butadiene rubber blends (5). However, it can also be

used to investigate the morphologies that arise upon

melt-blending two incompatible polymers. As representative

of this large group of polymer pairs, the model system

polystyrene/high-density polyethylene (PS/HDPE) is chosen.

7.2.1. MATERIALS

The properties of the materials used are listed in Table 7.1.

The flow curves (Figure 7.2) are more or less parallel,

which to some extent guarantees a constant viscosity ratio

over a wide range of shear rates.

Table 7.1. General properties of the materials used in the

extrusion experiments.

material code

HDPE (DSM)

Stamylan 8760

Stamylan 7058

Stamylan 7359

PS (DOW Chemica!)

634

638

meltindex I2

(dg/min)

0.06

5

35

3.5

25

density p123

oc (kg/m3)

956

952

954

1050

1050

-148-

7.2.2. BLEND PREPARATION

Blends of PS and HDPE were made on a Werner & Pfleiderer

laboratory corotating twin-screw extruder. The throughput

Q = 6 kg/hr, and the screw speed N = 270 rpm were kept

constant during the experiments. At the exit of the

extruder, the strand (with diameter about 3 mm) was

immediately quenched in water.

..,.. ö

.!::-

=-A I I I

1000

100

10 100

-~-> y (s-1) 1000 10000

Figure 7.2 Viscosity ~as a function of shear rate y of PS

and PE as determined on a Göttfert Rheograph

2000 capillary viscosimeter.

7.2.3. PHASE INVERSION DIAGRAM

HOPE 7359

t!.!1ITZ2~--· HOPE 8760 __

LOPE 1808 AN 99 ~H~L ____ _ PS 634

Figure 7.3 and 7.4 show the phase inversion diagram for the

system PS/HDPE. The micrographs represent the view of the

central regions of the strands. As can be inferred from

these micrographs, the sameareasas in Figure 7.1 exist.

However, with regard to the co-continuous area, there is no

symmetry around p = 1: The imaginary lines indicating the

co-continuous area are not parallel.

Some details of Figures 7.3 and 7.4, showing the different

stages of thread formation and breakup, are shown in Figures

-149-

1.1a, 1.1b, 1.10 and 1.11 of this thesis. Figure 7.5 shows

the presence of thin fibers at a viscosity ratio p z 1.

Campare the Chapters 1 and 6 for an explanation of the

formation of these extended morphologies.

Figure 7.5. Scanning electron rnicrograph of the edge of a

rnicrotorned surface of a 45/55 PS/HDPE blend (p z 1) .

Capricieus PS threads are visible in the direction of

extrusion.

next pages:

Figure 7.3. Scanning electron rnicrographs of the rnicrotorned

surface perpendicular to the direction of extrusion.

Figure 7.4. As Figure 7.3, parallel to the direction of extrusion.

Figure 7.6. Scanning electron rnicrographs of the rnicrotorned surfaces

of PS/LDPE blends (p z 1). Upper series: Befere cornpression

rnoulding, perpendicular to the direction of extrusion.

Lower series: After cornpression rnoulding.

-150'-

3dOHI..lfSdl..l

0

"' ..... 0 ClO

0 (") ..... 0 ,..._

0 ~ ..... 0 co

0 lil ..... 0 lil

0 ,..._ ..... 0 (")

0 ClO ..... 0

"'

2

UI 11. 0 :z:

s:::-..... "' 11. s:::-

0.5

0.2

10/90 20/80 30/70 40/60 50/50 60/40 70/30 80/20

%HOPE %PS

I .... (.Tl .... I

-152-

p9pJnOW UOJSS9JdWOO

0 N ..... 0 CD

0 M ..... 0 ,......

0 ~ ..... 0 CD

0 10 ..... 0 10

0 CD ..... 0 ~

0 ,...... ..... 0 M

0 CD ..... 0 N

w a.. oen J:C.. $$

-153-

7.2.4. INFLUENCE OF COMPRESSION MOULOING

Blends of the system polystyrene/low-density polyethylene

(PS 638, OOW/LOPE Stamylan 1808 AN 00, OSM) were processed

under identical conditions as listed inSection 7.2.2.

Subsequent to blending, the strands were pelletized and

compression moulded for 15 minutes at 200 °C (standard PS

cyc1e). Figure 7.6 shows the morphologies of the blends

before and after compression moulding. At higher volume

fractions ~ ~ 30%, a coarsening of the blend towards a

co-continuous structure is visible.

7.3. FILM THINNING

The experiments of Section 7.2 yield results at a rather

macroscopie scale. As a contrast, a scouting experiment will

be discussed which invo1ves the gravity-induced coalescence

of a polystyrene droplet with a horizontal interface of

molten polyethylene, at a temperature of 200 °C in an

atmosphere of nitrogen.

Typical data of the system polystyrene (PS 638, OOW

Chemical)/high-density polyethylene (HOPE Stamy1an 7058,

OSM) at 200 °C: Interfacial tension a= 5.10-3 N.m-1 (10);

density difference ap = 202 kg.m-3 (11); zero-shear viscosity

of the PS droplet ~d = 1400 Pa.s; droplet radius R = 1.5 mm.

A PS droplet is inserted in a HOPE matrix at an

approximately constant height above the interface. After a

contact time 't' has elapsed, the system is cooled down

immediately. The sample is then carefully microtomed, and

the film thickness at the droplet's plane of symmetry is

determined via scanning electron microscopy (see Figure

7.7). By varying the contact time, a first estimate for the

value of the proportionality constant 'A' in Eq. 7.3 can be

obtained.

For the present system, the Chesters model is expected to

apply at film thicknesses h ~ 40 ~ (12) . The theoretical

-154-

value of the constant A= 1.108 s.m-2. Figure 7.8 shows the

film thickness versus the reciprocal contact time. From the

slope of the line, a value A= 2.5.108 s.m-2 is found, which

is in the order of magnitude of the theoretical value.

This result can be compared, in terms of the thinning rate

(dh/dt), with systems having mobile or immobile interfaces,

respectively.

From Ref. 2 it fellows that

1 dh 8 1f (}2

h 3 dt 2 (immobile interfaces) 3 '1m Req F

1 dh 2 (}

h dt 3 '1m Req (mobile interfaces)

From Eq. 7.3, dh/dt can be expressed as:

dh

hence

where F

Req A

11c

1 dh

h 2 dt

t:.p

2R

.i R3 g 31f

1.0.108 s.m-2

1800 Pa.s

(7. 4)

(7. 5)

(7. 6)

( 7. 7)

For the present system, a global comparison between the

thinning rates predicted by the three models can be made by

substituting the parameters given in this Section. A film

thickness h = 10 pro is assumed. The calculated thinning

rates are presented below with respect to the thinning rate

predicted by the Chesters model.

-155-

dh (dh) (_)!mobile limmobile dt dt 1

:::: 20 :::: -- (7. 8) (dh) (dh) 600

dt I Chesters dt I Chesters

This means: The assumption of immobile as well as mobile

interfaces yields values for the thinning rates which differ

at least one order of magnitude from the experimental one.

Given the inaccuracy of the experiment (errors in droplet

radius are magnified to the power three), these results give

some support to the validity of the Chesters model.

I mm

Figure 7.7 Scanning electron micrographof a PS droplet in a

HDPE matrix. The film thickness (which is rather

large here) can readily be determined.

t U)

I 0

800.-----------------------~----------------,

600

• .:::::.- 400 ..c 1\ I I I 200

A=2.5•10+8 (s/m2)

)//~~~> R3/l (10-13 m3/•) Oi-----~----~.-----.------,------,-----~

0 5 10 15 20 25 30

Figure 7 . 8 Film thickness h versus reciprocal time R3/t,

of a PS droplet, approaching a horizontal PS/PE

interface.

-156-

7.4. INFLUENCE OF BLOCK COPOLYMERS

As was shown in the previous Sections, a blend of two

incompatible polymers is inherently unstable. Thread-like

particles will desintegrate into droplets, while coalescence

will destabilize the dispersion or will lead to

co-continuous structures. In principle, there are two

general routes to stabilize, at least locally, a desired

morphology (13).

(i) Functionalizing the two polymers with reactive groups.

As an example, related to the model system PS/LDPE, a

morphology of finely dispersed particles was reported for

blends of an oxazoline-modified PS (14) and a carboxylated

LDPE (15). With regard to the mechanica! properties,

however, no impravement was observed for the elongation at

break and the impact strength in PS rich blends (15,16).

(ii) Actding block copolymer to the blend.

It is well known from the literature (17-22) that block

copolymers of the type A-B may act as emulsifiers in a

system containing the immiscible polymers A and B. The block

copolymer tencts to concentrate at the interface, thereby

promoting interactions between the two phases. As a result,

the interfacial tension will be reduced. Therefore, with

regard to mixing, droplets will breakup at a smaller radius

R, yielding a finer dispersion in which coalescence is

decelerated. Finally, due to the better acthesion between the

phases, mechanica! properties will be improved * (23).

*) The only commercial use, to our knowledge, of a PS/PE blend,

is found in so-called 'easy seal/easy peel' packaging systems.

-157-

The model system PS/PE (containing up to 9% of added

(tapered) diblock copolymer) has been extensively studied by

Heikens et al. (24) and Fayt et al. (25-28). When blended on

a two roll mill, very fine dispersions were obtained.

Improvements in mechanica! properties, such as a higher

tensile strength and elangation at break were found in bath

PS rich and LDPE rich blends (25).

An additional third route may be found in the crosslinking

of the dispersed phase by gamma or electron-beam (EB)

irradiation. This can be particularly useful to fixate

fibrous or highly extended non-equilibrium morphologies (29).

Although the results are quite encouraging, different

morphologies may result for blends having identical

composition, when blended on different machinery. The time

scale of processes during the mixing of the polymers A and B

+ block copolymer (e.g. diffusion of the block copolymer to

the interface) is less well understood. The average

residence time on a two roll mill is in the order of 15

minutes, whereas on a corotating twin-screw extruder

residence times are as short as 2 minutes.

As an illustration, the morphology is shown of PS/HDPE

blends with 5% tapered diblock copolymer added to the HOPE,

prepared on a corotating twin-screw extruder, under the

conditions listed inSection 7.2.2. The Figures 7,9 and 7.10

show that the overall length scale of the blend has

decreased considerably, yet the same non-equilibrium

fibrillar morphology (viscosity ratio p ~ 1) as in Figures

7.4 and 7.5 is found.

Figure 7.11 shows the morphology of this blend after

compression moulding. The originally extensional character

of the morphology has disappeared. Threads have been braken

up into droplets. No extreme coarsening of the blend

morphology is observed, campare Figure 7.6.

For understanding ultimate morphologies of blends, it is

important to study the influence of the presence of

-158-

compatibilizer at interfaces with regard to the

agglomeration process in yet more detail.

20/75/5 45/50/5 80/15/5 20/75/5 45/50/5 80/15/5

Figure 7.9 Figure 7.10.

HDPE/PS (reference)

+ 5% PS/PE tapered

diblock copolymer

Figure 7.9. Scanning electron micrographs of microtomed extrudate

surfaces of the system HOPE/PS with and without

compatibilizer, parallel to the direction of extrusion.

Figure 7.10 As Figure 7.9, perpendicular to the direction of extrusion.

20/75/5 30/65/5 45/50/5 80/15/5

Figure 7.11. Morpho1ogy of the system HDPE/PS + 5% tapered diblock

copolymer after compression moulding (15 min. at 200 °C).

-159-

7.5. REFERENCES

1. M. Matsuo and S. Sagaye, in Colleidal and Morphological

Behavier of Block and Graft Copolyrners, Ed. G.E. Molau,

Plenum, New York (1971)

2. J.J. Elmendorp, Ph.D. Thesis, Delft University of

Technology (1986)

3. N. Tokita, Paper 26, ACS Rubber Division Meeting, San

Francisco (1976)

4. D. Heikens and W. Barentsen, Polymer, ~, 69 (1977)

5. G.G.A. Bohm, G.M. Avgeropoulos, C.J. Nelson and F.C.

Weissert, Rubber Chem. Tech., ~, 423 (1977)

6. G.M. Jordhamo, J.A. Mansen and L.H. Sperling, Polym. Eng.

Sci., ZQ, 517 (1986)

7. A.K. van der Vegt and J.J. Elmendorp, Paperpresentedat

the Workshop on New Polymerie Materials, Leuven, Belgium,

October (1986)

8. A.K. Chesters, Proc. Conf. Turbulent two phase flow

systems, Toulouse, France (1988)

9. w.w. Jongepier, internal report Delft University of

Technology (1987)

10. S. Wu, Polymer Interface and Adhesion, Marcel Dekker

Inc., New York and Basel (1982)

11.J.H. Truijen, internal report Eindhoven University of

Technology (1989)

12. J.M.H. Janssen, internal report Eindhoven University of

Technology (1989)

13.M. Xanthos, Polym. Eng. Sci., ~, 1392 (1988)

14. R.W Hohlfeld, Plastics World, August (1985)

15.W.E. Baker and M. Saleem, Polym. Eng. Sci., 22, 1634

(1987)

16. J.G.M. van Gisbergen, internal report Eindhoven

University of Technology (1986)

17.R. Fayt, R. Jéröme and Ph. Teyssié, Polym. Eng. Sci.,22,

328 (1987)

18. s. Bywater, Polym. Eng. Sci.,~, 104 (1984)

-160-

19. G. Maglio and R. Palumbo in: Preprints of the 2nd Joint

Polish-Italian Seminar on Multicomponent Polymerie

Systems, Lodz (1981)

20. J.W. Barlew and D.R. Paul, Polym. Eng. Sci., ~, 985

(1981)

21. G. Riess, G. Hurtrez and P. Bahadur in: Encyclopedia of

Polymer Science and Technology. Vol. 2, Ed. J.I.

Kroschwitz, Wiley-Interscience, New York (1985)

22.R. Jéröme, R. Fayt and T. Ouhadi, Prog. Polym. Sci., 1Q,

87 (1984)

23.D.R. Paul, Ch. 12 in Polymer Blends. Vol. 2, Eds. D.R.

Paul and S. Newman, Academie Press Inc., New York (1978)

24.D. Heikens, N. Hoen, W.M. Barentsen, P.Piet and H. Ladan,

J. Polym. Sci., Polym. Symposia, QZ, 309 (1978)

25.R. Fayt, R. Jéröme and Ph. Teyssié, J. Polym. Sci.,

Polym. Lett. Ed., ~, 79 (1981)

26. ibid., ~. 1269 (1981)

27.R. Fayt, R. Jéröme and Ph. Teyssié, J. Polym. Sci.,

Polym. Phys. Ed., ~, 2209 (1982)

28.R. Fayt, P. Hadjiandreou and Ph. Teyssié, J. Polym. Sci.,

Polym. Chem. Ed., ~, 337 (1985)

29. J.G.M. van Gisbergen, H.E.H. Meijer and P.J. Lemstra,

accepted for publication in Polymer (1989)

-161-

CHAPTER 8

STABILITY OF MORPHOLOGIES, OR THE EXPERIMENTAL

DETERMINATION OF INTERFACIAL TENSION

This Chapter describes interfacial tension-driven

morphological transitions of systems in which molten polymer

threads or layers are present.

Extended (Newtonian) liquid threads suspended in a

(Newtonian) liquid desintegrate into lines of droplets

because of the interfacial tension driven Rayleigh

disturbances (1,2). From the growth rate of these

disturbances, which can well be predicted from Tomotikas

theory (2), the interfacial tension between the thread phase

and the matrix phase can be calculated (3) . Apart from the

influence of compatibilizers on the interfacial tension,

also the stability and breakup of embedded films will be

discussed shortly.

8.1. MEASUREMENT OF INTERFACIAL TENSION VIA BREAKUP OF

THREADS

It is well known that droplets that are highly elongated in

a flow field, exhibit sinuscictal distortions with wavelength

À which inevitably cause breakup after the flow has stopped.

For two-phase systems in which both phases are Newtonian

fluids, the growth rate q of these distortions is directly

related to the interfacial tension cr, see Eq. 6.14 (2).

By simply observing the desintegration process of a liquid

thread in a liquid matrix, this relationship can be used to

determine the interfacial tension between these phases (3).

Chappelear (4) was the first who applied this procedure to

measure the interfacial tension between some polymer melts,

such as polyethylene, polystyrene, polyethyleneterephtalate

and polyamide-6,6. The reported interfacial tensions have

-162-

errors of about 30%. This is mainly due to the fact that

Chappelear extrapolated viscosity data, obtained from

capillary viscosimetry, to the zero-shear rate region. With

the advent of more actvaneed rheological equipment, the value

of the zero-shear viscosity should not cause such

significant errors. In fact, it is the slowness of the

process that explains the successful application of

Tomotikas theory in these systems, as was shown by Elmendorp

(5,6). He found good agreement between the predicted and

experimental growth rate of the distortion, but formulated

no explicit conclusions with regard to the interfacial

tension.

The procedure for measuring interfacial tension outlined

above is generally referred to as the Breaking Thread Method

(7) . In contrast with other (statie) methods for measuring

surface and interfacial tension,such as pendent or spinning

drop techniques (7-10), the Breaking Thread Method does not

require data on the density difference between the two

phases. The method is also suitable for systems consisting

of highly viscous polymers, in which equilibrium drop shapes

are attained only after several hours (9). Finally, it can

be used to obtain information concerning the emulsifying

effect of (newly synthesized) block copolymers (11). Some

typical results wil! be discussed in the next Sections.

8.2. INTERFACIAL TENSION BETWEEN TWO HOMOPOLYMERS

To establish the validity of the method, experiments are

performed with homopolymers of which the interfacial tension

is known from other methods.

8.2.1. MATERIALS

The properties of the materials used are measured on a

Rheometrics mechanica! spectrometer, as listed in Table 8.1.

For all materials in this Table, the loss angle & is close

-163-

to 90°, implicating Newtonian behaviour during the

experiments.

Table 8.1 Zero-shear viscosity (~0 ) and loss angle (6) of the

homopolymers used in the interfacial tension measurements.

Material code T (oC) ~0 (Pa. s) &

HDPE (DSM)

Stamylan 7058 200 1800±3% 90o

230 1115±2% 88°

Stamylan 7359 200 250 90°

LDPE (BP Chem.)

BP33 190 62000 76°*

PS (DOW Chem.)

638 200 1400 90°

230 1000 90o

634 200 11000 90o

PS (BP Chem.)

HH101 190 31200 aso*

PA-6 (BASF)

B3 230 425 90°

8.2.2. EXPERIMENTAL PROCEDURE

Threads of the desired polymer are spun, e.g. from a

Melt-Index apparatus, a single screw extruder or simply from

a molten granule on a hot plate. After having been annealed

during 24 hours at 80 °C, the threads are embedded in the

matrix, which consists of two films of the second polymer,

* From: Refs. 12 and 15

-164-

each with dimensions 10 x 10 x 0.6 mm. The system is placed

under an optical microscope (Zeiss) in a Mettler FP2 hot

stage, and brought to the desired temperature.

At a given position along the thread, the distortien is

photographed at regular intervals in time. As long as the

distortien remains sinusoidal, its amplitude a can be

obtained from the expression:

a = b - a

4

where b the largest thread diameter

a the smallest thread diameter

(8 .1)

By plotting log(2a/Do) against time, the growth rate q of

the disturbance is calculated from the slope of the line

(2,4,7,13):

q = d (!!.n (2a/Dol)

dt

with Do the initial thread diameter

Substituting this value for q in Eq. 6.14 yields an

expression for the interfacial tension o:

0 =

where 11m viscosity of the matrix

llt viscosity of the thread

p viscosity ratio lltll'lm 0('\n,p) a tabulated function (3-6)' see also

Figure 8.1.

(8. 2)

(8. 3)

'\n the dominant wavelength at which breakup

occurs

-165-

A typical plot is given in Figure 8.2. The experimental

wavenumber x = ~Do/À is compared with the theoretical one

(Xm = ~Do/\n), to give an extra control on the experiment.

0.8

E :-..:

• 0.6 E

Cl

'( 0.4 I I

0.2

01-TTnm~~~~~mm~~~~~~~~~~~~ 0 0.001 0.01 0.1

---> p

10 100 1000

Figure 8.1. Growth function Q(Àro,p) and wavenumber Xm = nTiol\n as a

0

~ N

1\ I I I

function of viscosity ratio p. From Ref. 5.

10 20 30 40 50 60 ---> t (s)

Figure 8.2. The relative amplitude (2~/Do) of the distortion on a

PA-6 filament (Do = 55 ~' no = 425 Pa.s) in a PS

matrix <no - 1000 Pa.s). The measurements were

performed at 230 oe. <Xexp = 0.6; Xm- 0.61)

-166-

8.2.3. RESULTS

The interfacial tension between some incompatible

homopolymers, measured via the Breaking Thread Method, are

listed in Table 8.2. Where possible, reference data, taken

from the literature are given.

Table 8.2 Interfacial tension between some incompatible homopolymers.

The ~~ta in the last column are taken from Ref. 7, unless

otherwise stated.

Thread Matrix T

phase phase oe Pa.s Pa.s

p Xth Xexp 0.103 0.103

N.m- 1 N.m- 1

(exp) (Ref. 7)

HDPE PS 200 250 1400 0.179 0.59 0.59 4.9 4.7

PS LDPE 190 31200 62000 0.503 0.60 0.55 3.5 4.9

PA-6 PS 230 425 1000 0.425 0.61 0.60 20.0

HDPE PS 200 1800 1400 1.290 0.56 0.55 4.1 4 . 7

PA-6 HDPE 230 425 1590 0.267 0.59 0.60 13.5 10.7*

On the results shown in Table 8.2, a few remarks can be made

concerning the applicability of the Breaking Thread Method.

To avoid end-pinching (14) or relaxation, see Figure 8.3,

the thread should be highly elongated, and have a constant

diameter . In the present experiments, the ratio of length to

diameter

L/Do > 60. For the same reason, systems with a viscosity

ratio ~tl~ << 1 can be expected to give better results.

* at T - 250 oe (5)

-167-

Figure 8.3. Relaxation and breakup of a HDPE thread (diameter

Do ~ 20 ~) in a PS matrix at 200 °C.

The time needed to perfarm one experiment is dependent on

the viscosities of the polymers, the filament thickness D0 and the interfacial tension, and may range from 50 seconds

(PA-6/PS) to several hours (PS/LDPE) . In systems where bath

polymers are highly viscous, it is therefore desirable to

decrease the initia! thread diameter as much as possible, to

avoid degradation. E.g. a value a= 4.4 ± 0.3 mN.m- 1 was

measured by Elmendorp (15) for the system PS/LDPE (nps =

62000 Pa.s; nLDPE = 31200 Pa.s; T = 190 °C), when starting

with a thread of diameter Do = 5 ~- The experiment took

about 1 hour.

-168-

The optical microscope had to be fitted with a long-distance

(LD-40) objective, in order to cbserve the breakup process.

A straightforward evaluation of the tabulated function Q(À,p)

is only possible if thread breakup is generated by a

distortien having the dominant wavelength Àm· It is

therefore essential to campare the experimental wavenumber

with the theoretical ene. The values have to be the same. If

initial distortions of different wavelengths are not all of

the same amplitude, breakup might occur at some wavelength À

other than the dominant wavelength Àm· In this case, the

complete graphical representation of Q(À,p), coveringa range

of 10-3 < p < 102 and 0.1 < x < 0.9, as given in Figures 1

and 2 of Ref. 4, might be of help.

8.3. INFLUENCE OF BLOCK COPOLYMERS ON INTERFACIAL TENSION

It has been shown by several authors that bleek copolymers

lower the interfacial tension in a blend of two incompatible

polymers (16-18). This has a large influence on the blend

morphology. Rayleigh disturbances will start to grow at a

smaller fiber diameter (o/R is a constant) . Breakup occurs

at the same critical capillary number, so smaller draplets

will result. Simultaneously, coalescence of draplets is

decelerated, due to the immobilization of the interfaces by

the presence of the bleek copolymer (2), and the lower

pressure in the film between the interfaces (19).

Rumscheidt et al. (4) were the first to report on the

influence of surfactants on the breakup of liquid

(Newtonian) threads suspended in a surrounding (Newtonian)

fluid. They observed a non-uniform wavelength À along the

thread, which gave rise to considerable scatter in the

resulting values for the interfacial tension. A

well-controlled experimental procedure, however, makes it

worth while to use the Breaking Thread Methad for measuring

the effect of actding bleek copolymer to the model system

PS/HDPE.

-169-

8.3.1. MATERIALS

Gel Permeation ehromatography (GPe) was performed on

solutions of polystyrene in THF at 40 oe using Waters

~-Styragel columns 105, 104 and 103 A , and on solutions of

high-density polyethylene in 1-2-4 trichlorobenzene at

135 oe using PL-gel columns. See Table 8.3.

Table 8.3 Molecular weights (weight average) Mw and molecular weight

(number average) ~' of the homopolymers used in the

interfacial tension measurements.

Material code

HDI?E (DSM)

Stamylan 7058

PS (DOW Chem.)

638

634

kg/mole kg/mole

100.0

162.1

254.1

30.0

51.2

81.0

3.33

3.17

3.14

Diblock copolymers polystyrene-polyethylene (PS-b-PE) were

obtained via anionic polymerization of

polystyrene-polybutadiene (PS-b-PB) diblock copolymers,

followed by hydragenation of the PB block. All syntheses

were performed by W.J.J. de Vet at the Université

Sart-Tilman, Liege (Belgium) . The main properties of the

diblock copolymers thus obtained are given in Table 8.4. For

details concerning synthesis and characterization, see Ref.

20.

The block copolymer was introduced in the thread phase

(usually polyethylene) via melt-blending, either on a two

roll mill or on the extruder from which the thread was spun.

Rheological characterization was performed on a Rheometrics

Mechanica! Spectrometer, using a cone-and-plate geometry.

Table 8.5 gives the rheological properties of the

homopolymers + diblock copolymers at 200 oe. Also listed are

-170-

data on hornopolymer + SHELL KRATON G 1652. This triblock

copolymer of styrene-ethylene-butadiene-styrene (SEBS) is

cornrnercially available as cornpatibilizing agent.

-Table 8.4 Molecular weight (number average) ~ and rest percentage

Table 8.5

of homopolymer PS after anionic polymerization of the

diblock copolymers PS-b-PB.

Material code

BC-1

BC-2

BC-3

BC-6

BC-10

BC-12

~(PS)

kg/mole

39.4

37.0

44.9

8.5

66.6

78.0

Zero-shear viscosity <l'lo>

~ (PB) % (PS)

kg/mole homopolymer

46.3

96.5

5.5

11.5

46.4

143.0

11

8

19

15

and loss angle (6) at

of the combination homopolymer + block copolymer

used in the interfacial tension measurements.

Material code l'lo (Pa.s) s

HDPE (DSM)

Stamylan 7058+0 . 25% BC-6 2110±2.2% 89 0

+0.5% BC-6 2140±2.5% 89 0

+1.0% BC-6 2270±2.0% 88 °

+5 . 0% BC-6 2540±1.8% 85 °

Stamylan 7058+1.0% BC-1 2350±3.2% 86 °

+1.0% BC-2 2400±4.0% 87 °

+1.0% BC-3 2050±3.0% 85 °

+1. 0% BC-10 2430±2.8% 90 o

+1. 0% BC-12 2590±3.3% 86 °

Stamylan 7058+0.25% KRATON 2275±2.5% 87 °

+0.5% KRATON 2400±3.0% 87 °

+1.0% KRATON 2470±3.0% 87 °

PS (DOW Chem.)

638 +0.2% BC-6 2310±5.5% 88 °

+1.0% BC-6 1840±4.4% 89 0

+0.2% KRATON 1425±5.6% 89 0

+1.0% KRATON 1710±4 . 1 % 89 0

200 oe

-171-

8.3.2. RESULTS

The Figures 8.4 and 8.5 show the dependency of the

interfacial tension on the amount of block copolymer added

to the thread phase.

The wavelengths À at which breakup occurs are indeed less

uniformly distributed along the thread, as reported by

Rumscheidt et al. (3). Therefore, only those measurements 5~------------------------------------~

Ê ...... z .§. 4

c: ..2 ~ 3 Gl -QJ

ö 2 QJ -... Gl -.E A

I I I

• • • • •• ••• .. .... ··

• 0+-------~-----,,------,-------.------~

0 1 2 3 4 6

---> wt. % dlblock-copolymer Figure 8.4 . Dependency of interfacial tension between PS (matrix)

and HOPE (thread) on wt . % diblock copolymer added to

Ê ...... z .§. c: 0 ii c: Gl -QJ ö QJ -... Gl -.E A

I I I

the HOPE. (T = 200 ° C). 6,---------------------------------------ï

4

• 3 '• I • • • .. · • • • 2

i • • •

.. .. ·· . • .... • •

····· ···· ··· ·· ·····

0~------.-------.-------.-------.-------~ 0 1 2 3 4

---> wt.% trlblock-copolymer 6

Figure 8.5. As Figure 8.4, with KRATON G 1652 added to the HOPE.

-172-

are plotted in which the value of the wavelength is closest

to the dominant wavelength. The corresponding wavenumbers

x= ~Do/À are listed in Table 8.6.

Table 8.6 Influence of the actdition of block copolymer on the

interfacial tension of the system PS 638/HDPE 7058

at 200 °C.

Thread

phase

Matrix p

phase Xth. Xexp.

HDPE+0.25% BC-6 PS 638 1.5 111 0.54 0.62 3.4

HDPE+0.50% BC-6 PS 638 1.5 129 0.54 0.56 1.7

HDPE+1.0 % BC-6 PS 638 1.5 71 0.54 1.1

HDPE+2.5 % BC-6 PS 638 1.8 129 0.53 0.51 0.8

1.8 89 0.53 0.52 1.5

1.8 111 0.53 0.49 1.1

HDPE+5.0 % BC-6 PS 638 1.8 169 0.53 0.57 1.2

HDPE+1.0 % BC-1 PS 638 1.7 196

HDPE+1.0% BC-10 PS 638 1.7 116

HDPE+1.0 % BC-12 PS 638 1.4 34

HDPE+.25% KRATON PS 638 1.6 110

HDPE+.5% KRATON PS 638 1.7 90

HDPE+1. % KRATON PS 638 1.8 160

0.54 0.53

0.54 0.54

0.56 0.55

0.54 0.54

0.54 0.55

0.53 0.56

2.2

2.8

2.3

1.0

3.0

2.6

From the results it is clear that the interfacial tension is

considerably reduced by small amounts of block copolymer (~

0.50 wt.%). Actdition of more than 5 wt.% does not cause

significant changes anymore.

At concentrations which are higher than 0.50 wt.%, the block

copolymer molecules form micelles. Consequently, the

interfacial tension reaches a plateau value (Figure 8.4) or

exhibits a local minimum (Figure 8.5).

-173-

An explanation for the latter phenomenon might be that in

this case, all the interfacial impurities are concentrated

in the micelles. Due to this decrease of the concentratien

of impurities at the interface, the interfacial tension will

increase. Block copolymer molecules will be transported from

the bulk phase to the interface, until a new equilibrium is

reached.

8.4. CONTACT ANGLE MEASUREMENTS

It is obvious from the previous Sectiens that the Breaking

Thread Method does not only serve as an illustration of one

characteristic stage of the mixing process. The method is

also quite useful for measuring the interfacial tension in

systems containing two incompatible polymers plus a possible

compatibilizer. However, its dynamic character calls for a

second - preferentially static - method, to check the

validity of the results. To our regret, complications in the

experimental procedure prohibited the direct application of

methods such as spinning drop or pendent drop. Closely

related to the latter is a method in which the interfacial

tension is calculated from the contact angle between a

molten polymer droplet and a molten polymer substrate

(7,21,22). See also Figure 8.6.

;;9Ljdz ds

Figure 8.6. Contact angle of an axisymmetric droplet resting on a liquid

substrate (Pct< p9). After Ref. 7.

-174-

Assume an axisymmetric droplet resting on a liquid

substrate, as shown in Figure 8.6. The density of the

droplet is smaller than that of the substrate. At any point

(x,z) on the curved interface, the pressure P(x,z) amounts

to (7,23)

2cr p b + ilpgz (8. 4)

where er the interfacial tension

b the radius of curvature at the crigin (x,z)=(O,O)

/lp the density difference between the phases

g the constant of gravity

This pressure must be balanced by the capillary pressure

across the interface (24):

p = 0'

1 + -)

R2 (8. 5)

where R1 , 2 the local radii of curvature at (x,z),

parallel and perpendicular to the plane of

the paper

From an infinitesimal element with dimension ds 2 (ds the are

length) (Figure 8.6) it can be seen that

x ds

sine ; R2 de (8. 6)

dx ds.case dz = ds.sine (8. 7)

Substitution of Eqs. 8.4 and 8.6 into Eq. 8.5 yields the

Bashforth-Adams equation (23) :

de

ds +

sine

x

ilpg ---;- ) •z (8. 8)

-175-

This equation can be made dimensionless with respect to the

length scale (~pg/a)1/2:

~pg

c = s s.fc ; B bic z z.fc x x.fc, a

hence

cte sine 2

cts + x B + z

Combining Eqs. 8.7 and 8.10 finally gives implicit

expressions for X,Z and e:

dX co se

cte 2 sine + z -

B x

dZ sine - 2 sine cte

+ z -B x

(8. 9)

(8.10)

(8 .11)

(8.12)

Tabulated solutions of these equations can be found in Ref.

(25) . By measuring x, z and e, it is possible to determine

X, Z and B. The interfacial tension is then calculated from

the relationship (24):

x 2 a = ~pg (X)

z 2 a = tlpg ( Z)

8.4.1. EXPERIMENTAL PROCEDURE

(8.13a)

(8 .13b)

A granule of the less dense polymer is placed on a

compression moulded plate of the denser polymer. The sample

is then placed in a small oven between two hot plates

(temperature T = 200 °C, in an atmosphere of nitrogen), and

is left to itself for about 30 minutes. Upon removal from

the oven, the sample is immediately quenched in water.

-176-

The shape of the droplet at its plane of symmetry is

observed by optical or scanning electron microscopy. The

densities of the polymers at 200 oe are determined with the

aid of a pycnometer (26) . For each phase, it is assumed that

the presence of less than 1 wt.% block copolymer does not

affect its density. The interfacial tension is calculated

from the dimensions x, zand e, as shown in Figure 8.7.

Figure 8.7. The shape of the plane of symmetry of a HOPE droplet.

8.4.2. RESULTS

The results of the contact angle measurements are listed in

Table 8.7. As can be concluded from this Table, the values

correspond quite well to the ones determined by the Breaking

Thread Method, compare Tables 8.2 and 8.6 and Figures 8.4

and 8.5. E.g. consicter the system HDPE 7058+0.5% BC-6 vs. PS

638. For the interfacial tension, a value cr = 1.7 mN.m-1 is

found when using the Breaking Thread Method, whereas from

contact angle measurements cr = 1.4 mN.m- 1 . Inaccuracies are

mainly due to deviations in the measured contact angle, or

to the fact that some droplets do not attain a completely

a xisymmetric shape during the e xperiment. Fina lly, the erro r

in the density of each polymer is duplicated in the value

for the interfacial tension.

-177-

Table 8.7 Contact angle, interfacial tension and related quantities *

for the system PS/HDPE 7058+block copolymer at 200 oe. The

columns on the extreme right show values for cr calculated

from the radius (Eq. 8.13a) and the height (Eq. 8.13b) of

the droplet, respectively.

droplet substrate e x z x z cr.103 cr.1o3

phase phase mm mm N.m-1 N.m- 1

8.13<! 8.131:!

HD 7058 PS 638 73° 2.94 1.52 1.89 0.98 4.8 4.8

HD 7058 PS 638 130° 3.33 1.56 3.88 1.89 1.5 1.4

+.2% BC-6

HD 7058 PS 638 140° 3.20 1.32 3.85 1. 98 1.4 0.9

+.4% BC-6

HD 7058 PS 638 90o 3.70 1.30 4.43 1.51 1.4 1.5

+.6% BC-6

HD 7058 PS 638 125° 3.33 1. 47 4.29 1. 92 1.2 1.2

+.8% BC-6

HD 7058 PS 638 115° 2.63 1.08 4.35 1.84 0.7 0.7

+1% BC-6

8.5. BREAKUP OF MOLTEN POLYMERIC LAYERS - AN ILLUSTRATION

Polymer blends containing layered structures can be obtained

by coextrusion techniques, eventually combined with a

Multiflux or Ross static mixer (27), or via biaxial

stretching of a dispersion of the two polymers (28) . Due to

their geometry, molten polymerie layers with large aspect

ratios (width W >> height h) are expected to be more

resistant to interfacial tension driven distortions than

molten polymer threads. Their surface can only decrease by

relaxation into one or more spheres or by breaking up into a

two-dimensional pattern of draplets (29) .

*Th~ densities at 200 oe: PHDPE 7058 954 kg.m-3

-178-

The stability of thin layers in the molten state can be

studied with the setup of Section 8.2. Thin slices (diameter

about 2 mm, thickness 3 ~) are microtomed from a granule

and embedded between two sheets of the opposite polymer. The

system is then heated up above the highest melting

temperature. Figure 8.8 shows a typical example of the

breakup sequence of such a layer.

c (360 s.) d (600 s.)

Figure 8.8. Sequence of the breakup of a thin (3 ~) PA-6 film

ernbedded in a polypropylene (PP) matrix at 240 °C.

Material is transported from the centre to the edges of the

layer, where small 'fingers' start to grow, which rapidly

desintegrate into droplets via the mechanism described in

this Chapter and in Chapter 6.

-179-

8.6. CONCLUSION

The Breaking Thread Method can be used successfully for

measuring the interfacial tension between incompatible

molten polymers which exhibit Newtonian behaviour as the

shear rate approaches zero. An advantage is the fact that

density data of the polymers need not be known.

It is a dynamic, straightforward simple method which does

not require special equipment. Besides, only small

quantities of the materials are needed. Also the

determination of the interfacial tension via the static

contact angle yields good results, although there is some

scatter. The results from both methods should be compared

with these from spinning drop. Pendent drop proves to

introduce many experimental problems if used for polymer

melts (30).

The stability of (extended) molten polymer threads is

considerably increased by the introduetion of block

copolymers.

8.7. REFERENCES

1. Lord Rayleigh, Proc. Roy. Soc. (Londen),~, 71 (1879)

2. S. Tomotika, Proc. Roy. Soc. (Londen),~, 322 (1935)

3. F.D. Rumscheidt and S.G. Mason, J. Coll. Sci., ~' 260

( 1961)

4. D.C. Chappelear, Polym. Preprints, ~, 363 (1964)

5. J.J. Elmendorp, Ph.D. Thesis Delft University of

Technology (1986)

6. J.J. Elmendorp, Polym. Eng. Sci., ZQ, 418 (1986)

7. S. Wu, Polymer Interface and Adhesion, Marcel Dekker

Inc., New York and Basel (1979)

8. S. Wu, J. Coll. Int. Sci., J..l, 153 (1969)

9. J.J. Elmendorp and G. de Vos, Polym. Eng. Sci., ZQ, 415

(1986)

10. J.J. Elmendorp, M. de Wit and B. Oudhaarlem, Mikroniek,

2...8..Ll, 9 (1988) (in Dutch)

-180-

11.H.E.H. Meijer, P.J. Lemstra and P.H.M. Elemans, Makromol.

Chem., Macromol. Symp., ~, 113 (1988)

12. L.A. Utracki and P. Sammut, Poly,m. Eng. Sci., ~, 1405

(1988)

13. S. Wu, J. Macromol. Sci., Revs. Macromol. Chem., Ç1Q (1),

1 (1974)

14.H.A. Stone and L.G. Leal, J.Fluid Mech., ~, 399 (1989)

15. J.J. Elmendorp, private communication (1988)

16.P. Gaillard, M. Ossenbach-Sauter and G. Riess, Makromol.

Chem., Rapid Comm., ~' 771 (1980)

17. S.H. Anastasiadis and J.T.Koberstein, Polym. Preprints,

28, 24 (1987)

18.H.T. Patterson, K.H. Hu and T.H. Grindstaff, J. Polym.

Sci., ~' 31 (1971)

19 A.K. Chesters, Proc. Conf. Turbulent Two Phase Flow

Systems, Toulouse, France (1988)

20.W.J.J. de Vet, internal report Eindhoven University of

Technology (1986)

21.F. Neumann, Vorlesungen uber die Theorie der

Capillaritat, B.G. Teubner, Leipzig (1894)

22.H.M. Princen in : Surface and Colleid Science. vol. 2,

Ed. E. Matijevic, Wiley-Interscience, New York (1969)

23. S.B. Bashforth and J.C. Adams, An Attempt totest the

Theory of Capillary Action, Cambridge University Press

and Deighton, Bell & Co., Londen (1892)

24. P.S. de Laplace, Mécanique celeste, Suppl. au Xième

Livre, Courier, Paris (1805)

25. S. Hartland and R. Hartley, Axisymmetric Fluid-Liquid

Interfaces, Elsevier, Amsterdam, (1976)

26. J.H. Truijen, internal report Eindhoven University of

Technology (1989)

27.H.H.T.M. Ketels, Ph.D. Thesis Eindhoven University of

Technology (1989)

28.P.J. Lemstra and R. Kirschbaum, Polymer, ZQ, 1372 (1985)

29.N. Dombrowski in : An Album of Fluid Motion, Ed. M. van

Dyke, Stanford Parabalie Press (1982)

30.F. Trompetter (DSM), private comm. (1988)

-181-

CHAPTER 9

COUPLING OF DETAILED AND OVERALL MODELLING

In this Chapter, an attempt is made to coup1e some of the

knowledge from detailed studies on droplet breakup with the

overall rnadelling of continuous mixers. The development

during processing on a corotating twin-screw extruder of the

morphology of a blend is examined. The number of

reorientations in the extruder is of importance. The

expressions describing the time for breakup of molten

polymer threads can be used to predict the resulting

morphology. Besides, the influence of the contraction flow

through the die and the cooling conditions on the final

blend morphology can not be neglected.

9.1. EXAMPLES OF CALCULATIONS ON DISPERSIVE MIXING

When incompatible polymers are blended on a corotating

twin-screw extruder, a fine dispersion can be generated

within a residence time which is in the order of magnitude

of one or two minutes. At first sight, it is surprising that

the enormous increase in the number of particles can be

realized within this short time scale. Consicter for instanee

the example of a 30/70 PS/HDPE blend, starting with one

large granule PS in every gram of HOPE. An average partiele

diameter of 1 ~ in the final blend implies that the number

of particles per cm3 has been increased to roughly 1011 .

These particles originate from the desintegration of highly

extended polymer threads.

The aim of this Section is to show some of the mechanisms

that produce this large reduction in length scale in such a

short time. Calculations are presented concerning (i) the

combined action of affine deformation plus reorientation,

(ii) the highly simplified dispersion process of an isolated

-182-

droplet in a screw extruder, and (ili) the total shear in the

kneading sectien of a corotating twin-screw extruder.

9.1.1. COMBINED AFFINE DEFORMATION AND REORIENTATION

Consicter a Couette flow between two concentric cylinders, in

which a droplet with an initia! diameter of 1 mm is affinely

deformed into a thread with diameter 1 ~· The concentric

cylinders can be part of a screw extruder, with typical

dimensions: diameter D = 25 mm, gap width H = 2.5 mm, and

screw speed N = 200 rpm.

The available shear rate is:

V 'II'DN/60

y z H H

With Eq. 6.2, the required amount of shear can be

calculated:

10-6 10-3 B/a

10-3

(1 + 2 -1/4

y ) 10-3 + y z 106

(9 .1)

(9 .2)

The required time to establish this total shear amounts to:

t = y/y = 104 s, which is almest three hours.

If the same droplet is perfectly reoriented once per

revolution (once per tn seconds), then the decreasein

droplet width (after n-1 reorientations) is given by:

B/a

where n

2 -1/4 n ( (1 + Yn l l

the nuffiber of reorientations

(9 .3)

the amount of shear between two reorientations

the time between two reorientations

-183-

60/N 0.3 s (9.4)

100•0.3 30 (9.5)

The required number of reorientations can be calculated by

combining Eqs. 9.1 - 9.5:

(9.6)

yielding n 5.

This means: the required reduction in droplet size can be

achieved within t = n.tn 1.5 seconds, provided

Ca >> Cacrit' and provided a mixing apparatus which

combines shearing and reorientation.

For equation 9.3 to be valid it is assumed that after each

reorientation, the extended droplet is cut in pieces which

have the size 'B' of the original droplet. In the flow

inside extruders, stretching and folding is the only

mechanism that produces an increase of the interfacial

area, see Section 2.2.1.

It is therefore more likely to expect the droplet to be

folded at each reorientation, followed by stretching, as is

shown in Figure 2.5. In this case, the value of Yn (which is

30 in the present example) only applies during the time

preceding the first reorientation. For long times, Yn = 2,

as in some static mixers.

Substituting Yn = 2 in Eq . 9.6 yields a number of

reorientations n = 18, and hence a mixing time of

t = n.tn z 6 seconds for the required reduction in droplet

size.

9.1.2. DI SPERSION OF AN ISOLATED DROPLET INA SCREW

EXTRUDER

Consicter an extruder, with dimensions:

-184-

diameter D 25 mm

flight clearance ó 0.25 mm

tip width b 2 mm

length L/D 5

channel depth H 2.5 mm

pitch angle Cj' 20°

relative flight width e = b/ (11'DsinC!') :::: 0.1

die diameter d 3mm

and further:

throughput Q 50 g/min (single screw extruder,

Section 9.1.2)

Q 100 g/min (corotating twin-screw extruder,

Section 9.1.3)

screw speed N 200 rpm

density p 800 kg/m3

It is assumed that the material makes one pass through the

nip between screw flight (or barrier flight, as present in

Maddock-type of mixers) and barrel wal!. The material is

then transported towards the die by the drag flow in the

screw channel.

The shear rate over the flight is:

y :::: 'II'DN/60

ó :::: 1000 s - 1 ( 9. 7)

Polystyrene (PS), which has a viscosity nd = 100 Pa.s at

this shear rate, is dispersed in high-density polyethylene

(HOPE) with corresponding viscosity nc = 150 Pa.s. Hence,

the viscosity ratio p = 0.7. The interfacial tension is

about 5 mN.m-1 (1), so for droplets having a radius

R = 1 ~ the local capillary number Ca can be calculated:

Ca n y R ~

(1 30 (9.8)

-185-

This capillary number exceeds Cacrit' required for breakup,

by a factor 40 (Cacrit z 0.7, campare Figure 1.8 in Sectien

1.4). However, the residence time in the clearance is very

short:

B t z 0.015 s.

V

This yields a dimensionless timet*:

t* 0

t Rz0.5, 11c

(9. 6)

(9.7)

which is too short for the droplet to break up. Campare

Figure 6.8, where a dimensionless time t* ~ 9 is required

(at p = 1) for droplet breakup at Ca/Cacrit z 40. Because of

the large capillary number, the droplet will deform

affinely into an extended thread. Its dimensions fellow

from the total shear over the flight: y = y.t z 15, hence

the elangation ratio L/2R becomes (with Eq. 6.1):

L/2R

After passing through the nip, the droplet will be

subjected to much lower shear stresses in the e xtruder

channel. The shear rate becomes (Eq. 9.1): y = 100 -1 s ,

( 9. 8)

and

the capillary number (with a matrix viscosity 11c = 700 Pa.s)

Ca z 14, which is high enough to keep the droplet extended.

In this case, the value Ca/ Cacrit z 14/0.7 = 20. The

dimensionless time for breakup is about 100, see Figure 6.8.

At a throughput Q = 10-6 m3/s, the average residence time

in the extruder channel amounts to

'II'DLH

t Q

z 24 s, hence t* z 170 (9. 9)

The available time is in the same order of magnitude as

the required time, allowing for interfacial tension-driven

disturbances to cause breakup.

-186-

9.1.3. TOTAL SHEAR IN A COROTATING TWIN-SCREW EXTRUDER

The total shear (with and without reorientations) in a

neutral kneading section of a corotating twin-screw

extruder can be calculated. Consicter a kneading section as

given in Figure 9.1.

The shear rate Yf between a kneading flight and the barrel

wall can be calculated to be: Yf z 1000 s-1 (Eq. 9.7).

Figure 9.1. Cross-sectien of a neutral kneading sectien of a

corotating twin-screw extruder.

The shear rate Yr in the region between two flight passages

(in tangential direction) amounts to: Yr = 100 s- 1

(Eq. 9.8) and the residence time tt is estimated to be

'II'D/2 tt = l,21!'DN/60

Assuming that all

reg ion, the total

60 = 0.3 s N

of the material passes

shear Yt in tangential

(9.11)

through the nip

direction becomes:

(9.11)

The average residence time ta in axial direction over the

kneading section* is

2 'II'DLH (1-2e)

Q z 15 s

* the relative flight width e 0.2 here.

(9.12)

-187-

Tagether with the number of (tangential) passes over the tip

of the kneading flight, v,

N v :::: 2 60 tv = 100 (9.13)

this fina1ly yields for the total shear y Yt·v:::: 4500

Therefore, if no reorientation is assumed, a droplet with

initia! diameter 'a' = 3 mm is deformed into a thread with

diameter B:

2 -1/4 -1/2 B = a. ( 1 + y ) :::: a. y :::: 45 J.1ffi (9.14)

Suppose that all of the material is perfectly reoriented

once per revolution of the screw, then the total number of

reorientations over the kneading section is:

N

n = 60 .tv 50 (9.15)

Hence, the amount of shear between two reorientations, Yn'

is given by

1 1 Yn = n ·Y = 50 . 4500 :::: 90 (9.16)

This leads to a drastic, although irrealistic reduction in

droplet diameter:

B a. 2 -n/4 (1 + y )

-25 a. (90) :::: 10-52 m. (9.17)

In reality, the mechanism of reorientation is far less

efficient. A material line passing through the intermeshing

region may be stretched and folded as indicated in Figure

9.2. Suppose that every line is subjected to Yn = 2 (see

Figure 9.2 b) then the calculation of the last paragraph of

Sectien 9.1.1 yields a droplet diameter B:

-188-

B 2 -n/4 a. (1 + 2 ) z 1o-11 m (9.18)

Even this is an overestimation, since the major part of the

material 'escapes' in axial direction by flowing between the

kneading flights. It can be expected that only a minor part

is reoriented. Moreover, from the experiments with the

Couette device there proves to be a limiting value for the

ratio B/a, especially for viscoelastic threads, and at smal!

local radii R (~ B), Rayleigh disturbances wil! start to

occur. Yet these calculations once more underline the

necessity of reorientation in reducing the length scale of a

blend. 2

[SJ[Z] CSJEJ CS]~ [SJ[!]

Figure 9.2. Stretching and folding of a material line upon passing

through the intermeshing region of a corotating twin-screw

extruder.

9.2. BREAKUP OF THREADS

Once a certain length scale is reached, the relatively

slower breakup processes govern the course of the mixing

process. This means: except for threads having a very smal!

diameter, the time scale at which breakup occurs is still

rather large, compared with the overall residence time. This

is shown in Table 9.1, in which initia! distortions of

relative amplitude a 0 /Ro = 10-3 are assumed to cause thread

-189-

breakup in the fastest process, i.e. in a stationary matrix.

With the typical data ~c = 1400 Pa.s and c = 5 mN.m- 1 , the

time for breakup in the absence of flow can be calculated

from Eq. 6.5.

Table 9.1. Time for breakup (tb) for PS filaments in a HDPE matrix

which is at rest. Parameters: filament diameter Do and

viscosity ratio p.

p Do (~) tb (s)

1 10 300

5 150

1 30

0.5 15

0.1 3

0.2 10 100

5 50

1 10

0.5 5

0.1 1

Apart from coalescence phenomena, the resulting draplets

wil! have a diameter which is approximately twice the

diameter of the thread (2).

This simple calculation is valid for polymers exhibiting

Newtonian behaviour as the shear rate approaches zero. E.g.

on a two rol! mil!, this is the case for the time interval

spent by the material on the rolls between two passages

through the nip. In a corotating twin-screw extruder, low

shear rates may be expected in partly filled channels

between two kneading sections (see Chapter 3). In this

respect, also the contraction flow into the die during

stranding and the subsequent cooling conditions may

introduce morphological transitions. See Chapter 7, where

long ultrathin fibers are formed in the strand at a

viscosity ratio p = 1 for a broad range of compositions.

Upon renewed heating, these fibers may either desintegrate

into draplets or coalesce to form a co-continuous

-190-

structure (3) . In the next Section, experiments will be

discussed which illustrate some of these time-dependent

transitions in the model system PS/HDPE.

9.3. EXPERIMENTAL

The polymers used in the experiments are HOPE Stamylan 7058

and 7359 (DSM), and PS 638 (DOW Chemical). The properties of

these materials are given in Sectiens 7.2.1 and 8.2.1.

Blends are made on a Berstorff ZE25 laboratory corotating

twin-screw extruder at a throughput Q = 100 g/min and screw

speed N = 200 rpm. The extruder geometry is shown in Figure

9.1. While eperating the machine, samples can betaken from

the vent port. After exiting the die, the strand is quenched

in water. It is possible to vary the position of the water

bath, while keeping the diameter of the strand constant. The

time between the die and the water level can thus be varied

from 1 second to approximately 50 seconds.

9.4. RESULTS

0 :..: 96g/min

90% HDPE 7058 63!3

200'C

... --~ ~'" ~ q, "'~y~

----.-- ..... ~ ~ -~ -

Figure 9 . 3. Extruder geometry, processing conditions and morphology

while processing HDPE/PS blends, with different cooling

conditions. The average residence times are indicated

where necessary.

-191-

Figu~e 9.3 shows the resulting morphology of a 90/10 HOPE

7058/PS 638 blend (viscosity ratio p = 1), depending on

cooling conditions . When the strand is immediately quenched

in water, an oriented morphology is formed. However, a

droplet-in-matrix structure is found if the strand touches

the water after 6 seconds.

In the kneading section, the material is molten and

subsequently deformed. The total shear generated in this

kneading sectien is calculated to be y = 4500 (see Sectien

9.1.3). This means (Eq. 6.2): a droplet with initia!

diameter 'a' = 3 mm is deformed into a thread with diameter

B = 45 ~, neglecting the effect of reorientations. The size

of the spherical particles that are found in the sample

taken from the vent port is in the order of 1.5 ~· This

clearly indicates the presence of reorientations in the

kneading section according to the mechanism proposed in

Sectien 9.1.3.

The draplets are extended again in the contraction flow

through the conical die. A theoretica! value for the maximum

elangation ratio À follows from the surface ratio of the

rectangular screw channel and the die. With typica1 data as

in Sectien 9.1.2, À becomes:

3 Y.~Dsin~(1-2e)

~/4 d2 z 25 (9.19)

For the blend morphology in the experiments, this means:

Droplets with a diameter 'a' = 1.5 ~are present in the

extruder channels preceding the die. Upon passing the die,

these draplets are extended into threads having a diameter B

2 -1/4 = a. (1 + À ) = 0.3 ~· Cooling conditions and coalescence

wil! determine the final features of the morphology.

The difference in resulting morphology can be explained by

using the breakup times of Table 9.1. When the strand is

immediately quenched upon leaving the die, threads are

-192-

visib1e (diameter ~ 0.3 ~) having a breakup time of about 9

seconds, see Table 9.1. When a time of about 6 seconds is

allowed between the die and the water bath, the same strand

exhibits a morphology which consists of finely dispersed

droplets (diameter ~ 0.6 ~). This is in fair agreement with

the predicted time for breakup.

If the volume fraction is increased, the morphology of the

blend contains threads with a larger diameter due to

coalescence (4). Consequently, larger breakup times {up to

30 seconds for threads of 1 ~) are required for the

transition of these threads into droplets. Figures 9.4 and

9.5 show the morpbologies of HDPE/PS blends with different

cooling conditions. The breakup times, as predicted in Table

9.1, are confirmed. The 50/50 HDPE/PS blend, however, still

forms a highly oriented morphology, which transforms into a

droplet-in-matrix structure after annealing for 10 minutes.

Apparently, the breakup of the threads is decelerated by the

presence of neighboring threads.

10/90 20/80 30/70 40/60 50/50

%PS/% HOPE

Figure 9.3. Scanning electron micrographs of the microtomed

surfaces of PS/HDPE blends, parallel to the direction

of extrusion. The time between the die and the water

bath is 7 seconds.

-193-

20/80 30/70 40/60 50/50 %PS/% HOPE

Figure 9.4. As Figure 9.3, but now with a time of 50 seconds

between die and water bath.

9.5. CONCLUSIONS

From the calculations and experiments, the following

conclusions concerning the coupling of detailed and overall

rnadelling can be stated.

The complicated flow field inside extruders generate

deformation processes having an inherently transient

character. (Results of studies concerning deformation under

equilibrium and steady conditions have only limited

practical value) . Upon passing through the high shear zone

between flight tip and barrel wall, the draplets are

stepwise subjected to a supercritical flow , i.e.

Ca>>Cacrit' and are deformed into threads with a finite

elangation ratio L/almax' limited by the interfacial tension

and the total deformation y. Upon leaving the nip, the

extended dropiets may either return to their original shape

or desintegrate into droplets.

Even in kneading elements in corotating twin-screw

extruders, only a minor part of the material passes through

the nip, which repesents the high-shear zone, and

-194-

reorientation is not perfect. This can be abducted from the

ca1culations, but can also be seen in flow visualization

experiments in Plexiglas-walled extruders. It is therefore

worthwhile to study the flow inside these geometries in yet

more detail (5) . The emphasis should be on the process of

reorientation upon passing the material from one screw to

the other, and on the the material flowing sideways in axial

direction, thus avoiding the high-shear zones. Perhaps this

can give an optimum kneading flight width and clearance &, as in the Tactmor model for internal mixers (6).

Experiments on deformation and breakup of droplets in

various combinations of shear flow and elongational flow (as

present in most extruder geometries) have been reported by

Bentley et al. (7,8). Systems with viscosity ratios p > 1,

often encountered in the rubber toughening of glassy

polymers, can thus be investigated.

Systems with viscoelastic fluids have been investigated

elsewhere (9-13). The modelling of coalescence in polymer

systems is still in its initial stage, although a few

successful attempts have been made (12,15,16).

9.6. REFERENCES

1. S. Wu, Polymer Interface and Adhesion, Marcel Dekker, New

York and Basel (1979)

2. s. Tomotika, Proc. Roy. Soc. (London), ~. 302 (1936)

3. J.J. Elmendorp, in Mixing in Polymer Processing, Ed. C.

Rauwendaal, Marcel Dekker, New York and Basel, to appear

4. W.M. Barentsen, Ph.D. Thesis, Eindhoven University of

Technology (1972)

5. W. Szydlowski and J.L. White, J. Non-Newtonian Fluid

~eh., ~. 29 (1988)

6. I. Manas-Zloczower, A. Nir and z. Tadmor, Rubber Chem.

Techn., ~. 1250 (1982)

-195-

7. B.J. Bentley and L.G. Leal, J. Fluid Mech., lQl, 241

(1986)

8. H.A. Stone, B.J. Bentley and L.G. Leal, ibid., ~, 131

(1986)

9. R.W. Flumerfelt, Ind. Eng. Chem. Fund., ~, 312 (1972)

10.W.K. Lee, K.L. Yu and R.W. Flumerfelt, Int. J. Mult.

Flow, ]_, 385 (1981)

11.S.L. Goren andM. Gottlieb, J. FluidMech., 12..Q., 245

(1989)

12. J.J. Elmendorp, Ph.D. Thesis Delft University of

Technology (1986)

13.C. van der Reijden-Stolk, Ph.D . Thesis Delft University

of Technology (1989)

14.R, de Bruin, Ph.D. Thesis Eindhoven University of

Technology (1989)

15.B.K. Chi and L.G. Leal, J. Fluid Mech., 2Ql, 123 (1989)

16.A.K. Chesters, Proc. Conf. Turbulent Two Phase Flow

Systems, Toulouse, France (1988)

-196-

SUMMARY

Mechanica! blending of two (or more) incompatible polymers

offers interesting routes towards new materials with

specific properties. Since desired properties of a blend

strongly depend on the resulting morphology, it is necessary

to understand the parameters governing the blending process.

Two main areas are of concern, namely the modelling of

compounding equipment and the studying of the dispersion

processes. This thesis addresses topics in both these areas.

A simple computational model is proposed for the two

important continuous mixers: The corotating twin-screw

extruder and the Co-kneader. By distinguishing between

extruder channels that have a degree of fill f < 1 and these

that are completely filled, it is possible to calculate

locally filled lengths, power, specific energy and

temperature rise. With regard to mixing, the shear rate,

shear stress, residence time and the number of

reorientations can be determined. Because of the

comparatively simple geometry in single screw extruders with

segmented screw design, including mixing sections, the model

can be used to analyse these extruders as well.

Detailed studies of deformation and breakup processes of

droplets in well-defined flow fields can be incorporated in

the practical model derived above if the time-effects during

the different stages are taken into account. The time

dependent deformation of Newtonian droplets into extended

threads is studied by means of a Plexiglas-walled

Couette-apparatus . When the shear rate is slowly increased,

allowing for a nearly equilibrium deformation, the

dependenee of the critica! capillary number Ca (ratio of

viscous and interfacial forces) on the viscosity ratio, as

reported in the literature, is reproduced.

-197-

However,in transient flows at Ca>>Cacrit' draplets are

deformed into long slender bocties which remain extended

until the shear has stopped.

Extended polymer threads suspended in another polymer matrix

desintegrate into arrays of draplets driven by Rayleigh

disturbances due to the interfacial tension. This process

can be studied using an optical microscope. From the growth

rate of the disturbances, the interfacial tension can be

determined. The time for breakup is greatly increased if a

suitable compatibilizer, acting as an interfacial agent, is

introduced in the system. Connected to this is the stability

of polymer layers in a matrix. Although seemingly stable,

these layers also break up, in a way similar to that of

polymer threads in a matrix.

Several non-equilibrium morphologies as obtained with blends

of the model system polyethylene/polystyrene are discussed.

These blends are made on a corotating twin-screw extruder.

By varying the viscosity ratio and composition, an

interesting spectrum of non-equilibrium morphologies is

obtained. In general, these morphologies are present in

blends of incompatible polymers (e.g polystyrene/polyamide-6

or polypropylene/ethylene-vinylalcohol copolymer) . Due to

coalescence, no stable dispersions are found when the volume

fraction is increased. Co-continuous morphologies are formed

upon blending or after compression moulding.

The rate-determining step of the coalescence process can be

investigated by observing the thinning behaviour of the

layer between two colliding spheres, or between a sphere

approaching a plane of the same polymer. A relatively simple

model can give insight into the dynamics of the thinning

process.

When block copolymers are introduced in the model system

polyethylene/polystyrene, more finely scaled morphologies

are obtained. The non-equilibrium character, however,

remains present.

To study the coupling of overall and detailed modelling,

-198-

the development of the morphologies of blends is followed

during the mixing process. It can be concluded that in the

nip of a kneading element in corotating twin-screw

extruders, which represents the high-shear zone, the

critical capillary number is exceeded by at least one order

of magnitude, resulting consequently in affine deformation.

However, the local residence time is far toa short for

breakup to occur. A second conclusion, drawn from the

diameter of the droplets, is that only a minor part of the

material passes through the nip, and reorientation is nat

complete.

-199-

SAMENVATTING

Het mechanisch mengen van twee (of meer) polymeren die in

thermodynamische zin niet mengbaar zijn biedt interessante

routes om te komen tot nieuwe materialen met specifieke

eigenschappen. Omdat de gewenste eigenschappen van een

mengsel sterk afhankelijk zijn van de resulterende

morfologie is het nodig de parameters die het mengproces

beheersen te begrijpen. Dit proefschrift beschrijft

onderzoek aan onderwerpen uit de twee belangrijkste

aandachtsgebieden: modelvorming van kompoundeerapparatuur en

bestudering van het dispergeerproces.

Een eenvoudig rekenmodel wordt gepresenteerd voor twee

belangrijke kontinue mengers: de meedraaiende

dubbelschroefsextruder en de Kc-kneder. Door onderscheid te

maken tussen extruderkanalen die een vulgraad 'f' hebben ten

opzichte van geheel gevulde, is het mogelijk om lokaal de

gevulde lengte, vermogen, specifieke energie en

temperatuurstijging te berekenen. Ten behoeve van de

mengwerking kunnen worden bepaald: de afschuifsnelheid,

afschuifspanning, verblijftijd en het aantal herorientaties.

Vanwege de minder ingewikkelde geometrie in

enkelschroefsextruders, die voorzien zijn van een

opbouwschroef (inklusief mengsekties), kan het model ook

gebruikt worden om deze machines te analyseren.

Gedetailleerde studies naar de deformatie- en

opbreekprocessen van druppels in goed gedefinieerde

stromingsvelden kunnen in het hierboven beschreven model

worden ingebracht mits de tijdseffekten gedurende de

verschillende stadia bekend zijn. De tijdsafhankelijke

deformatie van Newtonse druppels tot lange draden is

bestudeerd in een Couette-apparaat, voorzien van een

plexiglazen wand. Bij een zodanig langzaam opvoeren van de

afschuifsnelheid, dat de druppel steeds een

evenwichtsdeformatie kan aannemen, kan het uit de literatuur

-200-

bekende verloop van het kritische kapillairgetal Cacrit (de

verhouding van viskeuze krachten en grensvlakkrachten bij

breuk) als funktie van de viskositeitsverhouding worden

gereproduceerd. In tijdsafhankelijke stromingen bij Ca >>

Cacrit worden de druppels gedeformeerd tot lange draden die

uitgerekt blijven totdat de afschuiving gestopt is.

Gesmolten polymere draden, gesuspendeerd in een polymere

matrix veranderen in rijen druppels ten gevolge van Rayleigh

verstoringen onder invloed van de grensvlakspanning. Dit

proces kan worden bestudeerd onder een lichtmikroskoop. De

grensvlakspanning kan worden bepaald uit de aangroeisnelheid

van de verstoringen. De opbreektijd (tijd tot breuk) wordt

sterk vergroot door het introduceren van een geschikte

compatibilizer, die als grensvlakaktieve stof fungeert in

het systeem. Gerelateerd hieraan is de stabiliteit van

gesmolten polymere lagen in een matrix. Alhoewel

ogenschijnlijk stabiel, breken deze lagen op eenzelfde wijze

op als polymere draden in een matrix.

Verschillende niet-evenwichtsmorfologieën, verkregen met

mengsels van het modelsysteem polyetheen/polystyreen worden

besproken. Deze mengsels zijn gemaakt op een meedraaiende

dubbelschroefsextruder. Door de viskositeitsverhouding en

samenstelling van de mengsels te variëren ontstaat een

interessant spektrum van niet-evenwichtsmorfologieën. In het

algemeen zijn deze morfologieën aanwezig in mengsels van

niet mengbare polymeren (bijvoorbeeld polystyreen/nylon-6 of

etheen-vinylalkohol kopolymeer) . Bij grotere volumefrakties

zijn de gevormde dispersies niet stabiel als gevolg van

koalescentie. De snelheidsbepalende stap van het

koalescentieproces kan worden onderzocht door de afname van

de filmdikte tussen twee elkaar naderende grensvlakken te

volgen. Een relatief eenvoudig model kan inzicht geven in

de tijdschaal van dit proces.

Door de introduktie van blokkopolymeren die fungeren als

compatibilizer in het modelsysteem polyetheen/polystyreen

ontstaan veel fijnere morfologieën. Het

niet-evenwichtskarakter blijft echter gehandhaafd.

-201-

De koppeling tussen de globale en de gedetailleerde

modellering is bestudeerd door het ontstaan te volgen van

morfologieën van blends tijdens het mengproces. Een eerste

konklusie is dat in de spleet van een kneeddeel in

meedraaiende dubbelschroefsextruders (een gebied van hoge

afschuiving) het kritische kapillairgetal met tenminste een

grootte-orde wordt overschreden. Dit resulteert in affiene

deformatie. De lokale verblijftijd is echter veel te kort om

opbreken te laten plaatsvinden. Een tweede konklusie, die

volgt uit de uiteindelijke diameter van de druppels is dat

slechts een klein gedeelte van het materiaal de zone van

hoge afschuiving passeert. Volledige herorientatie vindt

niet plaats.

NOMENCLATURE

A constant in Eq. 7.2

A

B

(interfacial) surface

drop width

B coefficient in Eq. 5.22

Br Brinkman number

-202-

~ Cauchy-Green deformation tensor

ca capillary number

D deformation

D

r. F

diameter

deformation gradient

force

F (cumulative) residence time distribution

Fx correction factors (F0 , F0 c, Fp, Fpc, FN)

Gz Graetz number

H

L

Lc M

N

p

p

p

Q

(local) channel depth

(channel) length

centerline distance

molecular weight

screw speed

power

pressure

coefficient in Eq. 5.22

throughput

R radius of curvature

R

s T

T

V

w a

b

b

c

radius

area

torque

temperature

circumferential speed

channel width

droplet diameter

radius of curvature

temperature coefficient of viscosity

specific heat

c constant in Eq. 6.7

m

m

N

m

m

m

Pa

m

m

m

-203-

d diameter

e relative flight width

f degree of fill

constant of gravity

film thickness

gap height

Boltzmann constant

(flow) resistance factor

displacement factor

relative length

(axial) length

m orientation vector

n number of flights

n power law index

n number of reorientations

nr number of reorientations

p

p

p

q

r

t

t

u

V

x

z

pressure

viscosity ratio

number of pins in Co-kneader

growth rate of a distortien

relative remaining flight length

time

(average) residence time

velocity

velocity

position vector

coordinate in transverse direction

coordinate in down-channel direction

Greek symbols

ll = difference

Q(À,p) tabulated growth function

dimensionless parameter in Eq. 6.9

distortien amplitude

ex aspect ratio

<ex> = average heat transfer coefficient

m

m

m

m

Eq. 1.2

Pa

s

m

m

m

m

y y

s s

shear rate

total shear

(flight) clearance

loss angle

c efficiency parameter

-204-

c relative void space between aggregates

n viscosity

e orientation angle

e circumferential angle

heat conduction coefficient

À elongation ratio

wavelength

length stretch

viscosity

v constant in Eq. 6. 7

p

0

0

density

interfacial tension

Liapunov exponent

shear stress

~ orientation angle

~ pitch angle

x wavenumber

x shape factor

Subscripts

D drag

DC drag, (due to) curvature

L leakage (over flights)

LD leakage (through flights)

N number of flights p pressure

PC pressure, (due to) curvature

a average

a axial

a,b,c =part a, b, c

b = breakup

m

Pa.s

m

Eq. 2. 6

Pa.s

kg.m-3

N.m- 1

Eq . 2. 7

N.m- 2

Fo

Foc

QL

QLD

FN

Fp

Fpc

ta

Qa, ka

La,Q.a

tb

c continuous phase

c cohesive

c channel

d drag

d droplet

d disperse phase

f flight

f flight

h hydrodynamic

h hole

i intermeshing region

2. local

m dominant

m metered

m ma ss

m melt

n between two reorientations

0 reference

0 initial

p pressure

r residence

r region between two flights

s substrate

sp specific

t tot al

t tangential

t tot al

x,z= x, z direction

w wall

Superscripts

e energy

es specific energy

g shear ra te

gs tot al shear

h channel depth

-205-

'lcrlf'c Fe (Eq. 2.9)

Ec,J.lc

Qd

Pct

llctrlf'ct

Ef,Tf,Ff,J.lf,Lf

'Yt Fh (Eq. 2.8)

Lh

Fi

f2.

Àmr Xm

Qm

Qm ( Eq. 3 . 3 8)

Tm

Yn,tn

J.lo

ao, 90, Ao

Qp

tr

Yr Ps

Esp

kt

tt

Ft

-206-

~ screw length

m torque

n screw speed

q throughput

s specific surface

t mean residence time

T transponed f.T

* dimensionless u*

average T

-207-

CURRICULUM VITAE

De schrijver van dit proefschrift werd op 8 maart 1960 in

Oss geboren. Hij behaalde het diploma Gymnasium-~ in 1978

aan het Titus Brandsma Lyceum te Oss. In datzelfde jaar

begon hij met de studie Werktuigbouwkunde aan de Technische

Hogeschool Twente (die inmiddels Universiteit Twente heet)

in Enschede. Het kandidaatsexamen werd afgelegd in september

1982. Het doktoraalexamen werd behaald op 29 mei 1985 met

als afstudeerrichting Kunststoftechnologie (prof.ir. J.F.

Ingen Housz) .

Vanaf september 1985 tot september 1989 was hij werkzaam als

wetenschappelijk medewerker aan de Technische Universiteit

Eindhoven binnen de vakgroep Polymeerchemie en

Kunststoftechnologie van de fakulteit Scheikundige

Technologie.

Stellingen.

1. Het feit dat uit het model van Manas et al. blijkt dat de

fraktie ongedispergeerde roetagglomeraten afhankelijk is

van de diameter van de gebruikte kneders, duidt erop dat

deze slecht ontworpen zijn.

I. Manas-Zloczower, A. Nir and z. Tadrnor, Rubber Chemistry and

Technology, ~. 1250 (1982)

2. Het associëren van een negatieve spoed met een negatieve

spoedhoek in dubbelschroefsextruders is onjuist, daar dit

leidt tot negatieve waarden voor kanaalsnelheid en

-breedte.

Dit proefschrift, hoofdstuk 3

3. De nauwkeurigheid van de meeste statische methoden ter

bepaling van de grensvlakspanning tussen gesmolten

polymeren staat of valt met betrouwbare

dichtheidsgegevens. Hoewel principieel beter, is het

rechtstreeks meten van het dichtheidsverschil tussen twee

gesmolten polymeren praktisch moeilijk te realiseren.

S. Wu, Polymer Interface and Adbesion, Marcel Dekker Inc.,New York

and Basel (1979)

4. De bepaling van de grensvlakspanning uit de relatie van

Taylor tussen deformatie D en afschuifsnelheid y biedt

twee voordelen: (i) de dichtheden hoeven niet bekend te

zijn en (ii) het grensvlak wordt voortdurend ververst.

G.I. Taylor, Proc. Roy. Soc. (Londen), ~. 501 (1934)

H.P. Grace, Chem. Eng. Comm., 14, 225 (1983)

5. De semi-empirische relatie van Yinxi et al. om de ligging

van het ko-kontinue gebied in een tweefasen polymeer

mengsel te beschrijven heeft geen praktische waarde.

L. Yinxi, K. Kubota, S. Amano en Y. Oyanagi, paper 13-22 op de Se PPS

Meeting, Kyoto, 1989.

6. De door Ng en Erwin beschreven exponentiële afname van de

laagdikte in gekombineerde afschuiving en heroriëntatie

betreft het optimale geval, waarbij de lengte van een

deeltje na heroriëntatie gelijk is aan de beginlengte.

Zelfs in zeer goed ontworpen dynamische mengers is dit

moeilijk te realiseren en verloopt het proces van

laagdikteverkleining veel langzamer.

K.Y. Ng en L. Erwin, Polymer Engineering and Science, 21, 4 (1981)

Dit proefschrift, hoofdstuk 9

7. Gezien het feit dat de meeste Nederlanders pas om 'het

milieu' gaan geven wanneer de vervuiling ervan hun

persoonlijke levenssfeer raakt, verdient het aanbeveling

het drinkwater op ongeregelde tijdstippen bruin te

kleuren.

8. Een onderbelicht aspekt van de beoogde verschuiving in

het privé-vervoer van auto naar fiets is het feit dat

ruim 40% van alle Nederlandse autoritten korter is dan

vijf kilometer.

Tweede Struktuurschema Verkeer en Vervoer, deel a: Beleidsvoornemen,

(Handelingen der Staten Generaal: bijlagen Tweede Kamer nr. 20992),

p. 78, Staatsuitgeverij (1988)

9.Een betrouwbare grootheid voor de stand van de ekonomie

is de dagelijkse hoeveelheid ongevraagde reklame via

brievenbus of telefax.

lD.Wanneer het jaarlijkse aantal vliegtuigongelukken wordt

uitgezet tegen het totale aantal reizigervlieguren in

plaats van reizigerkilometers, blijkt dat vliegen niet

veel veiliger is dan autorijden.

H. Antonisen, Intermediair, ~, 25 (1989)