Stretching polymer chains - Malcolm · PDF file444 Rheol Acta (2010) 49:443–458 polyethylene (HMP) and also, the understanding by others, resulted in a deeper understanding of the

Rheol Acta (2010) 49:443–458DOI 10.1007/s00397-010-0435-z

REVIEW

Stretching polymer chains

Malcolm Mackley

Received: 16 October 2009 / Accepted: 7 January 2010 / Published online: 4 February 2010© Springer-Verlag 2010

Abstract This paper is based on a presentation givenat the de Gennes discussion meeting held at Chamonixin February 2009. The paper gives a personal review ofthe way developments relating to the stretching of poly-mer chains has taken place over the last 40 years. deGennes was very influential in relation to chain dynam-ics concepts at the University of Bristol, where manypioneering concepts relating to chain stretching weredeveloped by the late Sir Charles Frank and AndrewKeller. The paper reviews basic concepts on extensionalrheology, droplet deformation, chain extension fromextensional flow and the achievement of high chain ex-tension for high-modulus polyethylene. The paper alsoreviews recent developments concerning the influenceof chain stretching on polymer melt processing.

Keywords Polymer chains · Rheology · Stretching

Introduction

This paper gives a personal account of how think-ing behind polymer chain extension developed withthe culmination of achieving ultra-high-modulus poly-ethylene and also an appreciation of the way polymer

Paper presented at the De Gennes Discussion Conferenceheld February 2–5, 2009 in Chamonix, France.

M. Mackley (B)Department of Chemical Engineering and Biotechnology,University of Cambridge, Cambridge, CB2 3RA, UKe-mail: [email protected]

chains can be stretched by flow altering the rheology,processing and in some cases, final product. In 1970, asa PhD student at the H.H. Wills Physics Laboratory,Bristol, I was privileged to have Sir Charles Frankand Andrew Keller as supervisors in an exciting pe-riod of polymer science relating to chain stretching.It was Frank’s paper of 1970 (Frank 1970) that setthe firm scientific foundation in the quest for ultimatemechanical properties from low-cost polymers such aspolyethylene. His ideas coupled with the pioneeringwork of Keller (see, for example, Keller 1968) were toopen up complete new fields of polymer science thatwould also influence the world of rheology, morphol-ogy and polymer crystallisation. These were excitingtimes and de Gennes would often visit the laboratoryfreely exchanging ideas and showing great interest inthe work. de Gennes brought with him an appreciationof polymer dynamics and relaxation times which at thetime was not a particular specialty of anyone at Bristol.

The paper starts in the 1900s with foundation work ofTrouton (1906) on extensional viscosity. It then followsthe 1930s pioneering experiments and modelling ofTaylor (1934) on droplet deformation in extensionaland simple shear flow. Kuhn and Kuhn (1943) laida theoretical foundation in the 1940s in relation topolymer solution chain behaviour, and this was trans-lated into general process understanding by Ziabickiand Peterlin in the 1960s. In 1970, Albert Pennings(Pennings et al. 1970) made an unexpected experimen-tal observation on the way extensional flow influencespolyethylene solution growth, and from this point, SirCharles Frank directed the scientific community in thedirection of “extreme chain stretching”. Subsequently,in the 1980s, Smith and Lemstra (1980) made thecommercial breakthrough in relation to high-modulus

444 Rheol Acta (2010) 49:443–458

polyethylene (HMP) and also, the understanding byothers, resulted in a deeper understanding of theway chain stretching influenced polymer rheology andprocessing.

The paper has a main thread but includes a fewside stream indulgences of which Catastrophe Theoryis one.

Newtonian extensional viscosity

Extensional flow has played an important part in thestory of chain stretching from polymer solutions andTrouton (1906) was one of the first to recognise that“stretching” a fluid was different to “shearing a fluid”.The so-called Trouton Newtonian viscosity ratio of 3is the ratio of extensional viscosity to simple shearviscosity for a Newtonian fluid, and as a PhD student,I spent some time worrying about where the factor 3came from, although there are of course several textreferences to the Trouton viscosity ratio (see for exam-ple Bird et al. 1987). The late Sir Charles Frank was oneof my PhD supervisors, and I remember him saying,“It’s only a matter of geometry and nothing special.”It is in fact easy to derive the Trouton ratio from ageneralised description of a Newtonian fluid.

σ + I P = 2 η ε̇ (1)

Where σ is the symmetric stress tensor, I the devi-atoric identity matrix, P hydrostatic pressure, η theNewtonian simple shear viscosity and ε̇ the symmetriccomponent of the general strain rate tensor γ̇ . Thestrain rate (velocity gradient) tensor γ̇ can be dividedinto two components, a symmetric deformation com-ponent ε̇ij and an antisymmetric rotation component ωij

given by

ε̇ij = 1

2

(γ̇ij + γ̇ ji

), ωij = 1

2

(γ̇ij − γ̇ ji

)(2)

Simple shearing flow, depicted by Fig. 1, is a com-bination of pure shear and rotation, and this rotationin fact results in simple shear flow not being simpleat all! The appropriate values for simple shear tensorcomponents are given in Eq. 3.

γ̇ij =∣∣∣∣∣∣

0 0 0γ̇ 0 00 0 0

∣∣∣∣∣∣, ε̇ij =

∣∣∣∣∣∣

0 γ̇ /2 0γ̇ /2 0 0

0 0 0

∣∣∣∣∣∣,

ωij =∣∣∣∣∣∣

0 −γ̇ /2 0γ̇ /2 0 0

0 0 0

∣∣∣∣∣∣σij =

∣∣∣∣∣∣

0 τ 0τ 0 00 0 0

∣∣∣∣∣∣

(3)

substituting (3) into (1) gives the linear Newtonianshear viscosity equation familiar to all engineers,

τ = 2ηγ̇

2= ηγ̇ (4)

Pure extensional deformations are rotation free, andthis, in many ways, makes them more straightforwardto understand; however, in terms of liquid flow, theyare in general more difficult to create in the laboratorythan simple shear.

Uniaxial extension depicted in Fig. 2 is a classicextensional deformation which is easily and universallyapplied to the mechanical testing of solids but is morechallenging to realise in relation to fluid flow.

The deformation tensor components of an incom-pressible fluid for uniaxial extension are,

Incompressibility γ̇11 + γ̇22 + γ̇33 = 0 (5)

γ̇ij =∣∣∣∣∣∣

γ̇ 0 00 −γ̇ /2 00 0 −γ̇ /2

∣∣∣∣∣∣ε̇ij =

∣∣∣∣∣∣

ε̇ 0 00 −ε̇/2 00 0 −ε̇/2

∣∣∣∣∣∣

ωij =∣∣∣∣∣∣

0 0 00 0 00 0 0

∣∣∣∣∣∣σij =

∣∣∣∣∣∣

σ11 0 00 σ22 00 0 σ33

∣∣∣∣∣∣

(6)

Fig. 1 Simple shearing flow,illustrated as a combinationof pure shear and solid bodyrotation

Pure shear deformation withno rotation

Solid bodyrotation

d x2

d uSimple shear

X

X2

21

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Fig. 2 Uniaxial extension

X

X

X

Substitution of the equations given in Eq. 6 into thegeneralised equation for Newtonian fluids given inEq. 1 results in,

σii + P = 2η ε̇ii

σ11 + P = 2η ε̇

σ22 + P = −2ηε̇

2σ11 − σ22 = 3η ε̇ = ηe ε̇ (7)

So the extensional viscosity ηe for uniaxial extensiongiven by, ηe = σ11−σ22

ε̇11is thus, ηe = 3 η, the Trouton

ratio.

Droplet deformation in extensional and shear flow

As a precursor to polymer chain stretching, the de-formation of droplets within a viscous suspension hasmany similarities and is instructive. Much of the un-derstanding of droplet deformation in flowing systemsoriginates from the work of Taylor (1934), and it was hewho invented a Four Roll mill apparatus for generatinga 2-dimensional pure shear velocity field. The FourRoll mill is shown schematically in Fig. 3a togetherwith a parallel band apparatus in Fig. 3b; he used tocreate simple shearing flow. Taylor demonstrated thatdroplet deformation depended on the capillary numberCa of the flow defined as Ca = ηc γ̇ D

�where ηc is the

Fig. 3 Droplet deformation;G.I.Taylor, a Four Roll Milland b Parallel bandapparatus. c Grace diagramfor droplet deformationregimes

a) Four roll mill b) Parallel Band

1Ca

Viscosity ratio of drop to continuous phase

1

1

pure shear

Simple shear

c) Schematic of Grace diagram

CapillaryNumber

DC c

a

Dropdeformationregion

Stable undeformed drop region

446 Rheol Acta (2010) 49:443–458

continuous phase viscosity, γ̇ the applied shear rate,and D the initial diameter of the drop and � the surfacetension of the fluid.

The deformation response of a drop in relation tothe viscosity ratio of the drop to continuous phase isdemonstrated clearly by the so called “Grace diagram”(Grace 1982) shown in Fig. 3c. This diagram indicates anumber of features, namely: in order to stretch drops, acapillary number greater than 1 is generally required inorder for the viscous forces to overcome that of surfacetension. In addition, because of the rotary componentof flow within simple shear, it is more difficult toelongate drops in simple shear than in an elongationalflow. In fact, if the viscosity ratio exceeds 3.4, it isimpossible to deform a drop significantly in a simpleshearing flow. The limitation on droplet deformationin simple shearing flow is related to the additionalrotational component of the flow; this results in thedroplet rotating at the same time as being stretched inthe 45◦ direction with the result that the drop rotationpreventing persistent stretching.

Polymer chain deformation in extensionaland shear flow

The concept of polymer chains acting like random coilswas developed in the 1940s, and mathematicians soondeveloped equations that are now called Fokker Planckdiffusion type equations (Kuhn and Kuhn 1943) wherechain end probability distributions were obtained fordilute chains. The chain was modelled as random coilwith a characteristic Rouse relaxation time τ . Resultsof the modelling are shown in Fig. 4 where the steady-state chain extension is plotted as function of β, whereβ = γ̇ τ and γ̇ is the applied strain rate. If β (oftenreferred in Rheology as a Weissenburg number) is lessthan 1, very little chain stretching occurs for either asimple shear, (transverse) shear rate or an extensional(longitudinal) shear rate. For the case of β < 1 in bothsimple shear and elongational flow, the viscous forcesare too weak compete with entropic elasticity which isdriving the chain towards a random configuration and,in both cases of shear and extension chain stretching, isonly a very weak function of β. The situation is, how-ever, very different for β > 1. Here, chain extension insimple shear is modest and again limited by the factthat the presence of rotation in the flow prevents persis-tent chain stretching in any one orientation. However,according to early models for longitudinal extensionalflow, at β = 1, chain extension became infinite becauseat this point, the fluid viscous forces are greater than theentropic elastic retraction forces and the chain is free

Fig. 4 Graph of dilute solution steady state polymer chain ex-tension as a function of β = ε̇ τ where ε̇ is the applied transverseor longitudinal strain rate and τ the relaxation time of the chain.(Mackley and Keller 1975)

to stretch. Here, the viscous forces exceed the entropicrelaxation process and fluid rotation does not impedestretching. Further refinement of modelling introduc-ing limiting chain extensibility (the FENE model, Birdet al. 1980) later showed that the chain extension atβ > 1 reached a limiting high value. The absence ofrotation within extensional flows enabled full stretchingof the chain to be achieved.

Chain stretching in extensional flows was appreci-ated at an early stage by fibre spinning companieswho used uniaxial flow from spinneret dies followed bymechanical drawing to induce orientation, and scientistsuch as Ziabicki (1959) and Peterlin (1966) made thelink between extensional flow and processing behav-iour in relation to achieving orientation and stretchingby liquid processing.

At a similar time to the work of Kuhn, statistical the-ories of polymer rubber elasticity were developing (seefor example Treloar 1975). The concept of a polymerbeing an entropy spring became understood, and thesuccess of statistical mechanics in predicting the solid-state stress strain behaviour was a remarkable achieve-ment and up to this time remains one of the greattriumphs of relating molecular structure to macroscopicmechanical properties.

Albert Pennings and chain stretching of polyethylenein solution

My own arrival at Bristol closely coincided with SirCharles Frank and Andrew Keller, receiving a pre-

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Fig. 5 Photographs of SirCharles Frank (left) andAndrew Keller (right)

view of a paper written by Albert Pennings and hisresearches from Groningen, Holland (Pennings et al.1970). Photographs of both Sir Charles Frank andAndrew Keller are shown in Fig. 5, and two figuresfrom the Pennings paper are shown in Fig. 6. Kellerhad a thriving Polymer Physics group whose main inter-est centered on his brilliant discovery of chain foldingcrystallisation (Keller 1968). He was also interested inpolymer orientation, and at the time, Frank Willmouth(Willmouth et al. 1968) and Mary Machin (Hill) (Keller

and Machin 1967) were working on aspects associ-ated with oriented crystallisation. Willmouth had madediscoveries of so-called shish kebab crystallisation instirred vessels, but it was the Pennings paper thathighlighted the importance of extensional flow. ThePennings team discovered that fibrous crystallisationonly occurred within their Couette type apparatuswhen “Taylor vortices” (Taylor 1923) of the typeshown in Fig. 6a were present. Figure 6b shows thefinal fibrous crystals that remained after crystallisation.

Fig. 6 a Schematic of TaylorVortices. b Fibrous flowinduced crystallisationinduced by Taylor Vortices(Pennings et al. 1970)

a b

448 Rheol Acta (2010) 49:443–458

Pennings conjectured that it was the extensional chainstretching flow component of the Taylor vortices thatinduced the onset of fibrous crystallisation and that thenormal simple shearing, Couette flow between the in-ner rotor and outer wall was unable to induce adequateorientation to nucleate fibrous crystals. These key ex-perimental observations marked the genesis of ideaswith Sir Charles Frank concerning chain extension andultimate polymer properties.

Polymer chain deformation in uniaxial extension andthe opposed jet

One of Sir Charles Franks’ many research interestsconcerned diamonds, and he realised that in certain di-rections the unit cell of diamond, shown in Fig. 7a, wassimilar to that of polyethylene shown in Fig. 7b. The

dotted lines in the figures indicate the all trans-carbonback for each unit cell. After carrying out a simplecalculation, Frank (1970) concluded that the Youngsmodulus for a fully extended polyethylene chain shouldbe of order 285 GPa. Semicrystalline unoriented poly-ethylene has a Youngs modulus of 1 GPa and me-chanically drawing polyethylene might increase this to10 Gpa, still well below the Frank calculated 285 GPa.The Frank conjecture for high-modulus polyethyleneset a scientific objective that was to excite much re-search in the following decade. He had set down a goalto produce fully extended oriented polyethylene chainsrather than the usual chain folded lamellae of the typeidentified by Keller and others.

Following from the Pennings paper, Charles Frankconcluded that extensional flow was an effective wayof stretching chains within a polymer solution; how-ever, because polymer relaxation times τ in solution

a

b

Fig. 7 Unit cell of a polyethylene and b diamond. Dashed arrow in each figure indicates direction of carbon backbone

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Fig. 8 a Streamline flow foruniaxial extension createdby flow into opposed jets;b localised flow birefringenceassociated with polyethylenesolution flowing within regionof uniaxial extension.(Mackley and Keller 1975)

a b

were typically of order 10−3 s, in order to satisfy theβ = γ̇ τ > 1 condition, a strain rate γ̇ of order 103 s−1

would be necessary to stretch chains. The Four Roll millapparatus is generally not able to reach these levels ofstrain rates and so Frank (Frank et al. 1971; Mackleyand Keller 1975) proposed an opposed jet arrangementshown in Fig. 8a, where with 1-mm diameter nozzlesand flow velocities of order m/s, extensional stretchrates of order 103 s−1 could be achieved. Frank’s origi-nal suggestion was to create biaxial extension by firingtwo jets into the faces of each other; however, it wasquickly experimentally established that the uniaxial ex-tension “suction” mode was more stable and amenableto closer examination. The symmetry and nature of theflow pattern is shown in Fig. 8a, and the associatedlocalised flow birefringence for a polyethylene solutionis shown in Fig. 8b. Whilst at the time, it was difficultto know exactly the relaxation time of the solution, the

flow birefringence, seen as a localised strand along theexit symmetry axis of the flow, appeared to developwhen the β > 1 condition was satisfied. The experi-ment demonstrated that localised birefringence couldbe achieved, and this birefringence was associated withhigh levels of chain stretching. Further, experimentscarried out at lower temperatures where crystallisationcould occur demonstrated that fibrous crystallisationoccurred within the regions of localised birefringence(Mackley and Keller 1975).

The reason for the observed localisation was notimmediately apparent; however, a later series of exper-iments using a Four Roll Mill apparatus (Crowley et al.1976) resulted in the discovery of a simple explanationfor localisation in both the double jet and Four Rollmill. The geometry of the Four Roll mill apparatus isgiven in Fig. 9a, and the associated localised birefrin-gence for a polyethylene oxide solution is shown in

Fig. 9 a Schematic of fourroll mill and localisedbirefringence. b Photographof localised birefringence(Crowley et al. 1976)

a b

450 Rheol Acta (2010) 49:443–458

Fig. 9b. In order to stretch chains, not only must theβ > 1 condition be satisfied but the chain must also bein the flow field for a sufficient length of time in order tobecome fully stretched. Only chains that approach theexit symmetry axis in the case of the double jet and thesymmetry plane in the case of the Four Roll mill satisfythis condition.

In order to achieve high chain extension within anextension flow, two conditions must, therefore, be sat-isfied, namely:

1. β > 1 , where β = γ̇ τ . This ensures that the ap-plied extensional strain rate γ̇ = ε̇ is sufficientlystrong to overcome the entropic elasticity of thechain that normally keeps the chain in a randomcoil configuration.

2. γ̇ t >> 1. This ensures that the applied extensionalstrain rate acts for a sufficient length of time inorder for the chain to have sufficient time tobecome fully stretched from its original randomconfiguration. This is equivalent to saying that anadequate strain must be applied in order to stretchthe chain.

The requirement for high levels of chain stretch isillustrated schematically in Figs. 10 and 11. Figure 10illustrates a random coil chain where R0, the routemean square end to end distance of a random coil,is given by R0 = an1/2, where a is the length of eachrandom link and n the number of links. If the chain wasfully extended, its extended length Lm would be Lm =an and so the extensional strain required to stretch thechain is λm = a n

a n1/2 = n1/2.For a typical polymer chain of, say, 3,000 repeat

units, an extensional strain order 55 of is required tostretch the chain. In the case of the Four Roll mill andthe opposed jet geometries, this kind of large strain can

x0 ,y0

x1 , y1

Fig. 11 Schematic diagram illustrating chain stretching indifferent regions of the Roll Mill

only be achieved for fluid lines that are close to the exitsymmetry plane or axis, and this is shown schematicallyfor the Four Roll mill in Fig. 11. The central flow fieldof the Four Roll mill approximates to a pure sheardeformation where the stream function � is given by� = ε̇ x y. The velocity components are given by

vx = d�

dy= ε̇ x , vy = −d�

dy= −ε̇ y

resulting in the link between coordinates 1 and 2, givenby

x1 = x0 eε̇ t, y1 = y0 e−ε̇ t

In travelling from 1 to 2, a horizontal fluid elementwould strained by an amount

γ = eε̇ t

Fig. 10 Schematic of randomcoil and fully extended chain R0

a

Lm a n

1/2na0R n links

1/2n1/2na

n am

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Fig. 12 a Streamlinephotograph of Two Roll millflow pattern. b Photograph oflocalised flow birefringencewithin a two roll mill. (Frankand Mackley 1976)

a b

Thus, in order to achieve a strain of, say, 55, ε̇ tneeds to be off order 4. And this in turn means thaty1/y0 needs to be y1

y0≈ 1

55 ≈ 0.02 which is why the bire-fringence observed in Fig. 11 is so localised along theexit symmetry plane. Streamlines that do not approachthe symmetry have low strains and consequently lowstretch.

The need for high strains to stretch chains was alsodemonstrated in a Two Roll mill configuration (Frankand Mackley 1976) shown as a flow pattern in Fig. 12aand as flow birefringence in Fig. 12b. Again, high local-isation of birefringence was seen along the exit stream-line of the flow. The consistent conclusion that can bedrawn from these experiments is that adequate exten-sional strain rates and strain are required to stretchchains.

In the 1970s, the scientific interest in chain stretchingprompted de Gennes to realise that the relaxation timeof the stretched chain could be substantially different tothat of the random coil, and he wrote a seminal paper(de Gennes 1974) explaining how polymer relaxationtimes can vary with chain stretch and how this canin turn lead to hysteresis effects for flowing polymersolutions. This concept has subsequently been appliedby others to explain a broad range of polymer flowsituations.

The Six Roll mill: a digression into catastrophetheory

The exploration of stagnation point flow studies atBristol culminated in building a flow within a Six Rollmill (Berry and Mackley 1977). This came about be-cause Sir Michael Berry realised that the potential func-tion of the so-called elliptic umbilic catastrophe surface

matched the stream function �(x, y) for the Six Rollmill given below.

� (x, y)=γ

(1

3x3−x y2

)− 1

2ω

(x2+y2

)−Vyx+Vx y

where γ is a constant and ω, Vx, andVy are controlvariables.

During the early 1970s, Christopher Zeeman waspioneering what he called Catastrophe Theory (see,for example, Zeeman 1997). Zeeman gave popularlectures, and one example of a cusp catastrophe hesometimes chose was of dogs meeting which is shownschematically in Fig. 13. At a long distance apart, twodogs are generally neutral to each other. As they ap-proach, they may become friendly; however, if theyget too close, they may suddenly turn aggressive. Thepoint he was making was that although there may be asmooth change in a so-called control parameter space,

Friendly

Aggresive

1/ distance apart

Dogs meeting

Fig. 13 A Zeeman “cusp” catastrophe illustrating how a smoothvariation can in some cases leads to a “catastrophic” change

452 Rheol Acta (2010) 49:443–458

this may lead to a sudden change in “the topology” ofbehaviour within the control space.

On a more pragmatic and scientific note, the Six Rollmill provided a quantitative platform to test catastro-phe theory predictions. The Six Roll mill apparatusis shown in Fig. 14a, and the symmetric flow pattern,generated by the flow when ω, Vx, and Vy all arezero, is shown in Fig. 14b. The “catastrophe surface”for the elliptic umbilic catastrophe is shown schemati-cally in Fig. 14c, and this represents regions in controlspace where the Six Roll mill flow pattern will changetopology and when critical points will exist in the flow.Figure 14d is an example where by adjusting variousroller speeds on the Six Roll mill, it was possible tochange ω, Vx, and Vy and detect a location in controlspace on the catastrophe surface where a critical pointexisted. At the centre of the photograph, there is astreamline that goes to an unusual cusp. This representsa position on the catastrophe surface, and in eitherside of this point in control space, the topology ofthe flow changes. The Six Roll mill did not provide

much greater insight into chain stretching, but it wasinstructive in relation to providing a quantitative ratherthan qualitative example of catastrophe theory.

Stretching polyethylene chains to their limit

The 1970 Frank paper had a big effect on a number ofresearch groups in that there was now a high modulustarget to aim for if fully stretched polyethylene couldbe captured within the solid state. Extensional flow so-lution processing of the type described in the previoussection was not successful in producing a continuouscommercial flow process to manufacture high moduluspolyethylene. Fibrous “shish kebab” of the type shownin Fig. 15 were produced, but not in a continuous fibreform where mechanical testing could be carried out.Professor Ian Ward who was initially at Bristol in thelate 1960s and then subsequently went to Leeds Univer-sity was very active in applying mechanical solid-statedeformation to semi-crystalline polyethylene, and by

Fig. 14 The Six Roll mill.a Photograph of apparatus.b Photograph of symmetricalSix Roll mill flow pattern, thegerm of the elliptic umbiliccatastrophe surface.c Schematic of the ellipticumbilic catastrophe surface.d Photograph of flow withcontrol settings set on thecatastrophe surface andshowing a flow singularitywithin the flow. (Berry andMackley 1977)

a b

c d

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Fig. 15 Shish kebab polyethylene fibrous crystals

controlling draw temperature, drawing rate and mole-cular weight distribution, he had success in achievingsignificantly enhanced Youngs modulus fibres (see, forexample, Andrews and Ward 1970; Capaccio and Ward1975). It was the Pennings Group (Zwijnenburg andPennings 1975), however, who first achieved a 100 GPahigh-modulus fibre, and this was done by the processingof Ultra High Molecular Weight (UHMWPE) solu-tions. They developed a “surface growth” techniquewhere a seed crystal was introduced into a weakly en-tangled polymer solution, and the seed crystal was thenpositioned close to the inner rotor of a Couette con-centric cylinder apparatus. Chain stretching appearedto occur between the seed and moving surface, andthe continuous growth of a 100-GPa fibre could beachieved. Unfortunately, the growth rate of the fibrewas of order mms/min which made the process notviable at a commercial level.

It was Smith and Lemstra (1980) who made thebreakthrough in inventing a process that could be com-mercialised. Paul Smith had studied at Groningen, andPiet Lemstra had spent time with Andrew Keller atBristol. Smith and Lemstra were given great freedomat DSM Research to develop their own ideas, and theycame up with an ingenious way of stretching chainsinvolving weakly entangled UHMWPE solutions. Theirinitial batch process is shown in Fig. 16a. A low entan-glement gel solution of polyethylene was prepared at

about 1% concentration of polymer. This gel solutionwas then extruded as a weak fibre and quenched intoa low temperature solvent bath. The UHMWPE withinthe fibre crystallised in an essentially unoriented statebut retained elements of the low level of entanglementwithin the polymer and some mechanical integrity. Thisfibre was then hot drawn to high draw ratios of theorder 50 or greater. Normally melt crystallised poly-ethylene cannot be drawn in the solid state to drawratios of more than 5–10; however, because of the lowentanglement solution processing of the UHMWPE,Smith and Lemstra obtained sufficiently high drawratios to produce HMP with a modulus greater than100 GPa. Figure 16b shows a continuous flow variationon the process which now forms the basis for the pro-duction of the HMP Dyneema fibre which is now a ma-jor commercial success. (http://www.dsm.com/en_US/html/hpf/home_dyneema.htm). The use of a screw ex-truder to produce a gel solution with the direct feedingof solvent and polymer into the upstream end of theextruder was at the time a new and inventive step. Thelow polymer concentration means that a very high levelof solvent recovery was necessary; however, both theprocess and product are now fully established.

The Smith and Lemstra process avoided issues ofchain relaxation in the solution state because all thechain stretching was carried out in the semi-solidstate. Because of the initial low entanglement condi-tion, higher draw ratios were achieved than is possiblefrom drawing melt crystallised highly entangled poly-ethylene. de Gennes acted as a consultant to DSM dur-ing the exciting period of the commercial developmentof Dyneema, whilst I, the author, worked on competingprocesses to the DSM technology (see, for example,Mackley and Solbai 1987). Sadly for the author, DSMspatenting was consistently more advanced and broadenough for our own efforts to be thwarted by the DSMpatent claims. I was, however, a beneficiary of thetechnology, as by the year 2000 I was using severalDyneema HMP ropes in my racing dinghy.

Exploring polymer melts in extensional flow

The issue of whether chains can be significantlystretched by polymer melt processing remains an es-sentially unanswered question. Over the last 50 years,much energy has gone into describing the rheologyof polymers and significant molecular understandingoriginated from the reptation concept of de Gennes(1971) and its incorporation into a molecular theory byDoi and Edwards (1986). The whole idea of polymerentanglement captured the imagination of the poly-

454 Rheol Acta (2010) 49:443–458

Fig. 16 Schematic diagramsof the HMP gel drawingprocess. a batch process;b continuous process

1. Low entanglement UHMWPE polymer gel

2. Unoriented Gel fibre

Quench bath

3. Unoriented semi crystalline fibre

4. Hot draw

5. Oriented High Modulus Polyethylene

Solvent recovery

Piston

Screw extruder

UHMWPE Polymer powder SolventLow entanglement polymer gel

Spinneret

Gel fibres

Quench bath

Low entanglement semi crystalline fibre

Hot draw

Solvent recovery

High ModulusPolyethylene

b

mer community and when later, I was at Cambridge,de Gennes would also be a frequent visitor to theCavendish Physics laboratory where there was veryfertile scientific exchanges on polymer dynamics.

The original Doi Edwards theory was a theory basedsolely on reptation and orientation. The entropic elas-ticity generated by flow was associated with orientationchanges and not chain stretching. Ball and Mcleish(1986) subsequently extended the theory to incorporatechain stretching and contemporary molecular models ofthe Pom Pom (McLeish and Larson 1998), MolecularStress Function, Marruci (Pearson et al. 1991), Wagneret al. (2004) and Rolie Poly (Likhtman and Graham2003) type now contain many additional chain orienta-tion and relaxation mechanisms when compared to theoriginal Doi Edwards model.

Experimentally, the reality of polymer melt behav-iour is clouded by a broad spectrum of relaxation timesdue to polydispersity. Even monodisperse polymersshow a range of relaxation modes, and it is this polydis-persity of relaxation modes that make definitive state-ments difficult in relation to chain orientation within amelt.

Experiments of different kinds by many differentresearchers have shown that the rheology of polymersmelts in simple shear and in extension can be different.In general, polymer melts shear thin in simple shearflow whilst in extension, they can shear thicken or thinin a different way. Branched polymers in particularshow strong differences.

A way of exploring the extensional flow behaviourof a polymer melt is to follow the flow birefringence be-haviour of a melt in Cross Slot geometry. This geometrywas first used for polymer solutions (Scrivener et al.1979; Odell and Carrington 2006) and then appliedto melts by the Eindhoven Group led by Han Meijer(Verbeeten et al. 2002). Figure 17a is a schematic of aCross Slot apparatus with superimposed “pure shear”flow streamlines. The presence of the outer walls of theCross Slot means that the flow does not perfectly matchpure shear; however, the flow near the central stagna-tion point is a good approximation to pure shear. Crossslot flow can be realised using a Cambridge Multipassrheometer (Mackley et al. 1995; Coventry and Mackley2008) as indicated schematically in Fig. 17b, and a flowbirefringence photograph for a polymer melt is shown

Rheol Acta (2010) 49:443–458 455

Fig. 17 The Cross Slotgeometry. a schematic of pureshear streamlinessuperimposed on Cross Slotgeometry. b Schematic ofCross Slot MPRconfiguration. c Flowbirefringence pattern forDow PS680 polystyrene.Piston velocity of 0.5 mm/s(maximum extensionrate = 4.3 s−1). Inlet slitwidth = 1.5 mm. Sectiondepth = 10 mm. T = 180◦C

a

c

b

in Fig. 17c. In this case, the birefringence shows anumber of retardation bands corresponding to differentprincipal stress difference levels within the flow. Highstress levels are encountered in the central region ofthe extensional flow and at the walls. If the flow wasNewtonian, the principal stress contours would becircular emanating from the central stagnation point.Viscoelasticity causes an elliptical asymmetry to thefringes, and the development of cusping along the exitsymmetry plane means that there is higher stress andhigher chain stretching in this region.

Using modern computational models and numericalschemes, it is now possible to numerically simulate thetype of birefringence (stress) field that is experimen-tally observed. Figure 18 shows a recent example of thiswhich demonstrates the validity of current constitutiveequations in successfully predicting stresses in simple

shear regions, near the wall, extensional flow area nearthe stagnation point and mixed flows in between thetwo.

It has recently emerged that it is possible to obtainlarge strain “limiting” extensional viscosity measure-ments using the Cross Slot apparatus (Auhl D 2009,Private communication). Figure 19 shows both time-dependant simple shear and extensional viscosity mea-surements for a low-density polyethylene (LDPE). Thesimple shear data were obtained using TA instrumentsAres rheometer in a conventional way. The SER ex-tensional data were obtained by using a Sentmanat,stretching attachment (Sentmanat et al. 2005) that uni-axially stretched the polymer. The extensional data aredifficult to obtain particularly at the circled end pointsof the curves. Ideally, a time-independent steady-statevalue of the extensional viscosity is required; however,

456 Rheol Acta (2010) 49:443–458

Fig. 18 Comparison of experiment (LHS) and simulation (RHS)for Cross Slot geometry. Eight mode 2D, Pom Pom simulation

sample thinning followed by breakage usually preventsobtaining a definative large strain steady-state value.

From analysis carried out using the Double Jet andFour Roll Mill, it is clear that at the stagnation point theextensional strain will be by geometry infinite. There-fore, this point represents a steady-state extensionalflow, and provided that both strain rate and stress canbe measured at this point, a steady-state extensionalviscosity can be determined.

The steady-state extensional viscosity ηe,ss is givenby,

ηe,ss =(σxx − σyy

)

ε̇

10-1 100 101 102 103103

104

105

106

LDPET = 150 C 10

5

2

10.5

0.10.01

0.001

shea

r vi

scos

ity (

t), P

asel

onga

tiona

l vis

cosi

ty (

t), P

as

time t, s

10

3

10.3

0.001

0.003

0.010.030.1.

.

Fig. 19 Experimental LDPE viscoelastic data indicating limitingextensional circled viscosity value. (Courtesy of D. Auhl, Univer-sity of Leeds)

The principal stress difference can be obtained fromthe experimental birefringence data and the use of thestress optical equation below

�n = SOC(σXX − σyy

)

where �n is the measured birefringence; SOC is aknown stress optical coefficient.

The strain rate can be estimated using a simplerelation

ε̇ = AVp

where A is a calibration constant, and Vp is the pistonvelocity of the Multipass Rheometer barrels. Alterna-tively, ε̇ can be determined from matching numericalsimulation or direct experimental laser velocimetry.Figure 20 shows a pleasing comparison between theSentomet and Cross Slot rheometers. There are un-certainties in both measurements; however, both offerpotential to measure a limiting extensional viscosity fora melt which is something that in the past has been verydifficult to do.

Based on these results and other findings, the au-thor would like to make a final speculation concerningmelt processing and chain stretching within the melt.Most polymer melts have broad relaxation spectra,and this appears to be an essential ingredient towardstheir processibility. The β = γ̇ t parameter introducedfor polymer solutions plays an important role. If β isless than 1, the fluid behaves in a Newtonian manner.At β ≈ 1, viscoelasticity is present but the material istractable. For β >> 1, then a highly entangled melt has

dotted lines from 3

10-3 10-2 10-1 100 101 1025x103

104

105

106

full lines from 12-mode Pom-Pom spectrum

S

T = 155 C

E,s

td.[

Pa

s]

std.[s-1]

SER X-SlotHDB-1 HDB-2 HDB-4 HDB-6

Fig. 20 Comparison of SER rheological limiting Extensionalviscosities and limiting extensional viscosity obtained using CrossSlot centre birefringence values (Courtesy of D. Auhl, Universityof Leeds)

Rheol Acta (2010) 49:443–458 457

difficulty to be processed, unless it coexists with β <<

1 polymer that acts a low molecular mass lubricantfor the higher molecular mass material. It, therefore,appears to the author that polymer processing is abouta delicate balance in providing adequate low molecularmass material to be able to process high molecular massmaterial. Almost certainly, some of the high molecularmass material will be stretched, particularly if it sat-isfies the β >> 1 and ε̇ t >> 1 conditions in extensionalflow.

Conclusions

This brief review has omitted important contributionsof chain stretching and rheology that have been madeby other people and research groups. It has, how-ever, selectively covered a lifetime of personal researchwhere I was very fortunate to have had mentors such asSir Charles Frank, Andrew Keller and Sir Sam Edwardsand to have lived in a period when Pierre de Genneswas at the height of his powers. Since then, the worldhas changed; however, it is a personal hope that in thefuture, others will match the shear brilliance, enthusi-asm and energy of these men in the field of polymerand related science.

Acknowledgements I am most grateful to the organisers of thede Gennes discussion meeting for preparing and executing avery significant meeting held at a beautiful snowy Chamonix. Inmy view, the meeting was outstanding for its warmth, intensity,breadth, excellence and charm. Many people have been involvedin the collaborative work that has been discussed in this paperdescribing work carried out at Bristol, Sussex and Cambridge. Iam grateful to them all for their tolerance with me. Sometimes, itwas hard work, but in general, it has been a lot of fun.

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