Predicting low density polyethylene melt rheology in elongational and shear flows with “pom-pom” constitutive equations

Predicting low density polyethylene melt rheologyin elongational and shear flows with

‘‘pom-pom’’ constitutive equations

N. J. Inkson and T. C. B. McLeisha)

Interdisciplinary Research Centre in Polymer Science and Technology,Department of Physics and Astronomy, The University of Leeds, Leeds LS2 9JT,

United Kingdom

O. G. Harlen

Department of Applied Mathematics, The University of Leeds, Leeds LS2 9JT,United Kingdom

D. J. Groves

Interdisciplinary Research Centre in Polymer Science and Technology,Department of Physics and Astronomy, The University of Leeds, Leeds LS2 9JT,

United Kingdom

(Received 11 November 1998; final revision received 2 February 1999)

Synopsis

A recent constitutive equation derived from molecular considerations on a model architecturecontaining two branch points a ‘‘pom-pom’’ captures the qualitative rheological behavior of lowdensity polyethylene~LDPE! in shear and extension for the first time@T. C. B. McLeish and R. C.Larson, J. Rheol.42, 82 ~1998!#. We use a hypothetical melt of pom-poms with different numbersof arms to model the behavior of LDPE. The linear relaxation spectra for various LDPE samples aremapped to the backbone relaxation times of the pom-pom modes. Data from start-up flow inuniaxial extension fixes the nonlinear parameters of each mode giving predictions for shear andplanar extension with no free parameters. This process was carried out for data in the literature andfor our own measurements. We find that multimode versions of the pom-pom equation, withphysically reasonable distributions of branching, are able to account quantitatively for LDPErheology over four decades in the deformation rate in three different geometries of flows. Themethod suggests a concise and functional method of characterizing long chain branching in polymermelts. © 1999 The Society of Rheology.@S0148-6055~99!01103-7#

I. INTRODUCTION

The branching topology of polymer molecules has a marked effect on the flow prop-erties of the melt, and in general branched polymers have very different rheologicalbehavior from that of linear polymers, especially in extensional flows. The ‘‘pom-pommodel’’ of McLeish and Larson~1998!, presents a theoretical approach to the behavior ofthe macromolecular segments that lie between the multiple branch points of large

a!Author to whom all correspondence should be addressed. Electronic mail: [email protected]

© 1999 by The Society of Rheology, Inc.J. Rheol. 43~4!, July/August 1999 8730148-6055/99/43~4!/873/24/$20.00

branched molecules. In this article this model is applied to industrial grade polymericmaterials such as low density polyethylene~LDPE!. The molecular structure of LDPE isknown to contain multiple irregularly spaced long chain branches, although detailedinformation on its molecular topology is not available. Its rheological behavior has poseda long-term challenge for modeling.

The most accurate modeling to date has employed integral Kaye-Bernstein-Kearsley-Zapas~K-BKZ ! @Kaye ~1962!, Bernstein, Kearsley, and Zapas, 1963# constitutive equa-tions @see Ahmedet al. 1995#. These contain sufficiently flexible kernel functions topermit the response in uniaxial extension and shear to be fitted closely. However, oncethis is done they fail even qualitatively to predict the rheology in planar extension.~Alternatively, they may be arranged to fit both extensional geometries@Wagneret al.~1998!# but in this case they fail in shear.! In experiments LDPE strain hardens underplanar extension, as in uniaxial extension; by contrast the integral models predict asoftening response similar to that of simple shear flow. The failure of single integralconstitutive equations such as K-BKZ or Wagner equations@Wagner and Laun 1978# topredict hardening in planar extension is discussed by Samurkaset al. ~1989!. The termhardening~or thinning! refers to the rise~or drop! in transient viscosity at strain rates inthe nonlinear regime. The reason for the failure of this class of equations must be thatthey do not take into account the molecular dynamics of entangled branched polymermelts under flow.

The molecular issues are greatly clarified by considering model architectures. Forexample, the pom-pom molecule is an idealized branched polymer consisting of a back-bone~or crossbar! that ends in two branch points consisting of a number,q, of arms@seeFig. 1~a!#. A tube model for the effect of entanglements around a monodisperse melt ofthis structure gave rise to a set of integral and differential equations for internal structuralvariables such as segment stretch and orientation@McLeish and Larson~1998!, Bishkoet al. ~1997!#. Together these provide a constitutive relation for the stress on deformationof the melt. It displays the correct qualitative behavior by simultaneously predictingstrain hardening in both planar and uniaxial extensional deformation and shear thinning.A key feature of molecular theories of entangled polymer melts, which is essential in thecase of branched polymers, is the natural separation of relaxation times for stretch and fororientation@McLeish and Larson~1998!#. This is not typically an ingredient in phenom-enological constitutive equations, such as Phan-Thien-Tanner and other Oldroyd typemodels where the same relaxation time applies to stretch and orientation.

A rigorous quantitative calculation of the stress relaxation in polydisperse LDPE mol-ecules with many layers of branching within each molecule would be too complicated toform a useful constitutive equation. However a tube model would predict, quite gener-ally, that different parts of the molecule have widely seperated timescales for orienta-tional relaxation@McLeish ~1988a!, Rubinsteinet al. ~1990!#. These parts would in turnbe expected to possess different relaxation times for stretch and orientation. This suggeststhat we can ‘‘decouple’’ the structure into an equivalent set of pom-pom molecules witha range of relaxation times and arm numbers. The orientational relaxation time distribu-tion would be set by the linear relaxation spectra, with nonlinear measurements fixing theremaining parameters.

In this article we report the results of applying this procedure for three intensivelystudied batches of LDPE taken from the literature: melt 1, IUPAC A and IUPAC X. Wealso include a commercial sample with a different degree of long chain branching, calledLDPE B, which was characterized in our own laboratory~see Fig. 14 for its complexshear modulus!. This polymer is a high Mw and broad molecular weight distribution~MWD! LDPE with Mw ; 250 000 and polydispersity of~Mw/Mn! ; 15.

874 INKSON ET AL.

II. RHEOLOGICAL DATA AND EXPERIMENTS

A. Data taken from the literature

Melt 1 is the LDPE sample on which Meissner performed early rheological measure-ments@extensional viscosity is presented in Meissner~1971! and the shear viscosity dataare in Meissner~1972!#. IUPAC A is characterized in Meissner~1975!, with uniaxialextension data given in Laun and Mu¨nstedt~1979!; and IUPAC X data are taken fromLaun and Schuch~1989!. Note that IUPAC X is a batch of the same material as IUPACA @which is discussed in Laun and Schuch~1989!#.

The relaxation spectra for IUPAC A is given in Laun~1986!, and Khan and Larson~1987!; IUPAC X spectra are in Samurkas, Larson and Dealy~1989! and melt 1 in Laun

FIG. 1. ~a! A pom-pom with three arms at each branch point (q 5 3). At small times the polymer chains areconfined to the Doi-Edwards tube.sc is the dimensionless length of branch point retraction into the tube.l isthe stretch ratio whereL is the curvilinear length of the crossbar andL0 is the curvilinear equilibrium length.~b! At longer timescales the pom-poms arms are relaxed and fluctuate on timescales such that the crossbarcannot entangle with them. The crossbar can now reptate out of its widened tube.

875LDPE RHEOLOGY WITH ‘‘POM-POM’’ EQUATIONS

~1986!. The spectra are given as sets of relaxation timest i and associated moduligi .Note that the spectra are taken at different temperatures: 125 °C for IUPAC X and 150 °Cfor IUPAC A.

We obtained a relaxation spectrum for LDPE B from oscillatory shear stress growth ofthe equivalent shear viscosity using a Rheometrics RDA II rotational rheometer in thesteady shear mode. Measurements for strain rates from 0.001 to 0.3 s21 were made at140 °C in nitrogen using a 25 mm diam 5° cone and plate geometry. The LDPE B wasmelt pressed into disc specimens again at 180 °C. This provides both a convenient sampleform to load into the rheometer and the same melt history as that for the elongationalmeasurements.

B. New extensional data

The transient uniaxial viscosity measurements for LDPE B were obtained from aRheometrics RME elongational rheometer. This is essentially the instrument described byMeissner and Hostettler~1994!. The sample, in the form of a rectangular strip, is grippedby horizontal metal caterpillar belt clamps and pulled symmetrically over a supportcushion of nitrogen gas, which passes though a sintered metal frit that forms the surfaceof the sample table. The force require to extend the sample is measured by a transducerunit attached to the right-hand belt grips, together with the drive motor. The transducerunit itself is, in effect, a horizontal moving beam monitored by a low voltage displace-ment transducer~LVDT ! and suspended from a fixed beam by twin vertical leaf springs.The sample table, caterpillar grips and transducer are all mounted in a small oven with anitrogen atmosphere controlled at the melt extension temperature.

The strain measure used is the Hencky strain«H 5 ln(Lt /Lo) with a maximum of 7.The stress measurement assumes that the sample remains uniform in cross section, and soit is essential to ensure that the polymer specimens are of high quality and extend uni-formly. Therefore, great care is needed in preparing the samples to avoid inhomogene-ities. The polymer was cryogenically ground to make a coarse powder feedstock in orderto remove effects of granule memory. This was then pressed at 180 °C into rectangularbars 60 mm37 mm31.5 mm and allowed to cool naturally between the press platens atfull moulding pressure.

Satisfactory uniform extension was obtained at 140 °C for a range of strain rates from0.001 to 0.1 s21. All measurements were made ‘‘undamped,’’ that is, without the oildashpot provided to reduce oscillation during the initial tensioning of the polymersample.

III. MOLECULAR RHEOLOGICAL MODEL

A. Stress relaxation in branched polymers

A linear polymer chain in a melt has the freedom to move but is constrained by itsneighbors, which act to keep it in a tube centered on the chain’s primitive path. In orderto diffuse the chain ‘‘reptates’’~slides! in a curvilinear motion along the primitive path@Doi and Edwards~1986!#. The presence of a branch point affects the ability of the chainto move in its tube. A star polymer cannot reptate like a linear polymer chain and mustrely on star arm fluctuations in order to relax its stress@de Gennes~1975!, Pearson andHelfand ~1984!, Ball and McLeish~1989!#. In addition, the presence of branch pointsaffects the stress relaxation of polymer chains after it has undergone a deforming strain.For a linear chain, the tube will deform with the bulk strain and experience a change incontour length followed by the chain retracting to its original length@Doi and Edwards1986#. For a branched polymer only the free ends can retract back into its stretched tube

876 INKSON ET AL.

after deformation. Deeper segments will only retract when it is entropically favorable towithdraw the branch points connecting them to the outer segments into the tube@Bick andMcLeish ~1996!#. This occurs only at high strains when the tension in the deeper seg-ments exceeds the total force at the branch point from the dangling arms. The entropicforce arises from the ends ability to explore more of their surroundings~by Brownianmotion! than more confined inner segments@Doi and Edwards~1986!#. Thus, in thepom-pom molecule, a stretched segment connected toq outer ~unstretched! ones by abranch point can support a tension ofq times the entropic force of the free ends justbefore arm withdrawal. Consequently the maximum stretch of this segment is equal toq~providing this is still within its Hookian regime!. These general consequences of the tubetheory of polymer melts provide the basic physics underlying the pom-pom model.

B. Pom-pom model

Here we briefly review the details of the molecular constitutive equation derived for amonodisperse melt of pom-pom molecules. A pom-pom molecule has a ‘‘crossbar’’ ofsbdimensionless entanglement ‘‘lengths’’ (sb 5 Mb /Me , whereMb is the molecular massof the crossbar andMe is the entanglement molecular mass!. The crossbar has a branchpoint at each end connecting it toq arms, each ofsa entanglement lengths~likewise,sa 5 Ma /Me ); see Fig. 1.

The dominant contribution to the stress is assumed to arise from the crossbar segment,since the arms relax on much faster timescales~since they behave like star polymerarms!. There are three variables that quantify the state of a pom-pom crossbar at anytime, t: l(t), the dimensionless stretch ratio of the crossbar;S(t), the orientation tensor,which measures the distribution of unit vectors of the crossbar tube segments(S 5 ^uu&, averaged over all orientations!; andsc(t), the dimensionless distance of armwithdrawal into the backbone’s tube~see Fig. 1!. Additionally there are five structureparameters: the relaxation time for orientation of the crossbar,tb , the relaxation time forstretch (l) of the crossbar,ts , the number of dangling arms~or priority q! and thedimensionless molecular masses of the crossbarsb and armssa . We noted above that thestretch of a segment is limited by the balance of entropic tensions tol(t) < q. Insituations that would generate larger stretches, the dangling arms are withdrawn into thecrossbar tube so thatsc(t) becomes nonzero. The crossbar stretch is then locked atq untilsc becomes zero again. In this way only one ofl or sc is changing in time@McLeish andLarson~1998!#.

The mathematical constitutive equations of the pom-pom model are comprised ofthree dynamical integro-differential equations~previously constitutive equations havealways taken the form of one integral or differential equation! for each of the dynamicalstructure variablesS(t), l(t) andsc(t) together with an expression for bulk stress. Thesedynamical equations are supplemented by expressions for the relaxation times for orien-tation,tb , and stretch,ts . Earlier work@McLeish and Larson~1998!# has shown that therigorous integral equation forS(t) may be approximated by a differential equation whichshares the same asymptotic structure and greatly reduces computational time. In thisarticle we use these simplified pom-pom equations and ignore the small contribution tothe stress arising from oriented arm segments~the dimensionless arm withdrawal,sc 5 0). In order to preventl from increasing beyondq, dl/dt is set to zero whenl 5 q and the deformation rate would tend to stretch it further. In the differentialapproximation the orientation tensorS(t) is defined by


S~ t ! 5A~ t !

trace~A~ t !!, ~1!

whereA(t) evolves as

]

]tA~ t !1u–¹A 5 K–A1A–KT2

1

tb~A2I !, ~2!

whereK is the deformation rate tensor~velocity gradient,K 5 ¹u).This differential approximation ensures the correct asymptotic form of the orientation

tensor in shear flow, and gives the same qualitative behavior in shear and extension as thefull integral expression.

In flow, the dissipative drag on the molecule, which is entirely due to the effectivefriction at the branch point, is in balance with the elastic recovery of the backbone. Theelastic recovery of a Gaussian chain ofsb steps of sizea on relaxation of the arm materialhas a spring constant equal to 3kT/sba2. Equating frictional drag forces on the branchpoints from the drag of their surroundings to the elastic Brownian force leads to

zb

2 SK :S2]L

]t D 53kT

sba2~L2sba!, ~3!

whereL 5 sbal is the curvilinear distance of separation of the branch points along thetube, andzb is the friction coefficientzb @see McLeish and Larson~1998!#, giving

]

]tl 5 l~K:S!2

1

ts~l21!, for l , q. ~4!

In this simplified model, the stress of the melt is entirely due to the crossbar segments andis given by

s 515

4G0fb

2l2~ t !S~ t !, ~5!

whereG0 is the plateau modulus, andfb is the fraction of molecular weight contained inthe crossbar. The stress is quadratic in the backbone extensionl. One factor ofl arisesbecause the stretched backbone occupies a greater length of the tube equal tolsba. Theother comes from the tension within the backbone, which is proportional tol. Theprefactor of 15/4 arises from the tube model calculation of the plateau modulus in termsof the molecular parametera @Doi and Edwards~1986!#.

McLeish and Larson~1998! discuss the behavior of this model in start up of shear andextension. The viscosities in different flow geometries are given by

h 5sxy

g, in shear flow, ~6!

and

hu 5 hp 5sxx2syy

2«, in uniaxial and planar extension. ~7!

The behavior in uniaxial and planar extension is found to be very similar and quitedifferent from that in shear. The viscosity in the start up of shear is shown in Fig. 2~a!. In

878 INKSON ET AL.

all these single mode figures showing calculations for a single pom-pom modetb /ts 5 3, ts 5 1, G0fb

2 5 1 andq 5 5, with varying dimensionless strain rates of

tsg 5 1, 3, 10, and 30. In shear, increasing the shear rate lowers the maximum viscosity.This is due to the behavior ofl shown in Fig. 2~b!. At high strain ratesl has a largetransient overshoot before decaying to a steady state value of

l ;1

12ts

2tb

, for g @ ts21.

Thus whents ! 2tb , the crossbars are not permanently stretched as one might have

expected at shear ratesg > ts21 because the orientational alignment reduces their com-

ponent along the shear gradient. As a consequence the pom-pom model shares the strongshear-thinning behavior of linear polymer melts.

FIG. 2. ~a! Transient shear viscosity of a pom-pom melt (q 5 5, tb /ts 5 30) in start-up shear flow at fourdifferent shear rates.~b! Crossbar stretch ratiol~t! in start-up of shear flow against time at four different shearrates.


The behavior in transient extension is very different from that of shear; see Fig. 3~a!.The transient viscosity increases with time~extension hardens! until it reaches its steady

state value. For strain rates greater than the critical strain rate,«crit 5 (121/q)/ts thesteady value ofl will achieve its maximum value ofq; see Fig. 3~b!. At these high strainrates the transient viscosity separates from the low strain rate viscosity curves and in-creases rapidly, only to level out suddenly oncel reachesq. This rapid increase inviscosity is seen in the extensional viscosity data of LDPE.

We now consider the effect of changing the value ofq in this model. In shear flowthere is no difference in the behavior of the viscosity except at very high shear rates whenl may briefly attain a value ofq. However, in extensional flows the value ofq limits the

maximum value ofl and hence the plateau viscosity. Figure 4 shows that for« 5 1.0s21 there is a clear increase in the steady state viscosity withq (q 5 1, 2, 5, 10! due to

the pom-pom backbones becoming fully stretched. In fact, sincehplateau} q2/ « high qpom-poms contribute much more stress and hence have much higher viscosities than a

FIG. 3. ~a! Transient uniaxial extensional viscosity of a pom-pom melt (q 5 5,tb /ts 5 3) in start-up at fourdifferent extension rates.~b! Crossbar stretch ratiol(t) in start-up of uniaxial extension flow against time, atfour different extension rates.

880 INKSON ET AL.

melt of low q pom-poms at« > ts21 . Another consequence of this relation is that the

value of the plateau in the viscosity decreases with the extension rate. Thus beyond the

critical strain rate,«crit , the viscosity decreases with the extension rate as the stresssaturates. Consequently the steady state extensional viscosity has a sharply peaked maxi-

mum at«crit ~see Fig. 5!.

C. Multiple levels of branching

In a large molecule, with multiple branch points, the segmental tension can be tracedfrom the free ends inwards by summing the tensions at each branch point@Bick andMcLeish ~1996!#. This value of summed multiples of the entropic force on a givensegment is named itspriority. It is defined as the ratio of the maximum tension of asegment to its equilibrium tension. Thus in the pom-pom model the cross bar has a

FIG. 4. Transient uniaxial extensional viscosity of a pom-pom melt (tb /ts 5 3) in startup at a constant

extension rate of« 5 1.0 as a function of arm number (q 5 1, 2, 5, 10!.

FIG. 5. Steady state uniaxial extensional viscosity,h( «), plotted against rate of extension,«, for a pom-pommelt (q 5 5, tb /ts 5 3).


priority q. The maximum tension of a segment depends on the number of free ends thatcan be traced back to it. The immediate response of a branched molecule to a large straindepends on thepriority distribution of the segments within it.

A related picture shows how a large molecule containing multiple branch points re-laxes after deformation. The free chain ends will relax rapidly, behaving in the same wayas the arms of the pom-pom up to the branch point connecting them to the rest of themolecule. This branch point is able to move one diffusive step after a deep retraction ofthe chain ends connected to it. This allows the molecular segment up to the next branchpoint to relax as a star arm, but on the much longer timescale set by diffusion of the outerbranch point. In turn, deep retractions of this segment allow the inner branch point tomove, and so on. Such a hierarchical process continues until the deepest~innermost!segments of the molecule relax at the longest timescales~see Fig. 6!. Therefore, therelaxation time of a segment is determined by the path distance to the nearest free endthat is able to release it from its tube constraint by retraction. This statistic, calledseniority@Rubinsteinet al. ~1990!#, also increases, like its priority, towards the middle ofa complex branched molecule. Both statistics are required to calculate the full molecularrheology of a branched polymer. Although real LDPE molecules are highly asymmetric,every segment within them does possess a well-defined seniority and priority. This isbecause both the relaxation time~seniority! and the critical stress for branch-point with-drawal ~priority! are both determined by relaxations from one of the two tree structuresconnected to it. In either process the relaxation arrives first from one of the segment’sends; it is this that determines its priority or seniority. Since relaxation times increaseexponentially with seniority of segments, an appropriate physical picture of a molecule at

FIG. 6. Relaxation process of a long chain branched molecule such as LDPE. At a given flow rate« the

molecule contains an unrelaxed core of relaxation timest . «21 connected to an outer ‘‘fuzz’’ of relaxed

material of relaxation timest , «21 behaving as solvent.

882 INKSON ET AL.

a given flow rate« contains an unrelaxed core of relaxation timest . «21 connected to

an outer ‘‘fuzz’’ of relaxed material of relaxation timest , «21 behaving as solvent~Fig. 6!.

IV. A MULTIMODE POM-POM MODEL

LDPE has a random, polydisperse, branched structure as a result of the high-pressurefree-radical polymerization process in which it is produced. This also results in LDPE’shigh proportion of long chain branches~LCBs!. The precise structure and degree ofbranching are still unknown and there do not exist accurate experimental techniques tomeasure them@see Axelsonet al. ~1979! for a discussion of nuclear magnetic resonance~NMR! techniques#. It is therefore tempting to ask whether the structure of LDPE can beinferred from its rheological behavior.

Even if the exact molecular structure were known, a rigorous calculation of the fullpriority and seniority distributions for a randomly branched polymer would be prohibi-tively expensive computationally. Instead, we chose to approximate it by taking a theo-retical blend of the relatively simple pom-pom molecules with differing number of arms,assigning orientational relaxation times from the linear relaxation spectra. The high pri-ority segments of LDPE are thus represented by pom-poms with larger numbers of armsand longer relaxation times than the outer segments. We expect high priority~maximumstretch,q! segments to share a high seniority~relaxation time for orientationtb). The

TABLE I. IUPAC A.

t i ~s! gi ~Pa! q tb /ts

0.001 1.523105 1 2.00.005 4.0053104 1 2.00.028 3.3263104 1 2.00.14 1.6593104 1 2.00.7 8.693103 2 2.03.8 3.1513103 6 1.7

20 8.5963102 6 2.15100 1.2833102 9 1.25500 1.8495 22 1.1

TABLE II. IUPAC X.


0.001 9.23104 1 2.00.003 6.553104 1 2.00.01 3.83104 1 2.00.03 2.573104 1 2.00.1 1.773104 1 2.00.3 1.103104 2 2.01.0 7.313103 2 2.03.0 3.763103 3 1.9

10.0 2.13103 4 1.830.0 9.43102 5 1.03

100.0 3.53102 5 1.0


central approximation is that of ‘‘decoupling’’ the different levels of branching on thesame molecules — interactions between segments of different priorities on the samemolecule are neglected. Our justification for this approximation is that for a given flow

rate «, the flow behavior is dominated by those segments whose relaxation timestb and

ts are of order«21. As noted earlier, modes with relaxation times much smaller than

«21 remain relaxed, whereas longer relaxation time modes become ‘‘saturated’’ in strain.Since the relaxation times increase exponentially with seniority, and both seniority andpriority increase in a similar fashion from the outside of a molecule, segments of differentpriorities have widely separated relaxation times. Consequently, the effects of couplingsof stretch and orientation between them are expected to be small.

In the multimode model, each~ith! mode is characterized by four parameters: thestretch and orientation relaxation timestsi andtbi , respectively; the number of armsqi ;and the fraction of the stress relaxing,gi . There are now two evolution equations permode equivalent to Eqs.~2! and ~4! for the tube orientationSi (t)and stretchl i (t).

The stress contribution of all the~n! modes is additive:

s 5 (i 5 1

n

si 5 (i 5 1

n

gil i2~ t !Si~ t !. ~8!

The plateau moduligi and the orientation relaxation time may be found from the linearrelaxation spectrum. Published relaxation spectra are available for both IUPAC A andIUPAC X and are shown in Tables I and II. To test the robustness of our approach weobtained three relaxation spectra for the LDPE B sample from its complex shear modu-lus, and they are shown in Tables III–V.

A. Relaxation spectra and pom-pom model parameters

In Tables I–VI we present the linear spectra of three materials decorated with nonlin-ear pom-pom parameters for each mode that best fit the data in uniaxial extension.

The values oftbi in these spectra are essentially arbitrary. We have chosen to usepom-pom modes that directly correspond to these linear spectra. Thus each pom-pommode corresponds to a mode in the linear spectrum that determines both the value ofgiandtbi of this mode. The two remaining parameters,tsi andqi , must be found from thenonlinear response. However, to be consistent with the physical model they must satisfythe following constraints. First, the ratiotbi /tsi is 4/p2 times the average number of

TABLE III. LDPE B ~12 mode relaxation spectrum!.


0.01 7.20623104 1 5.00.030 305 3.30983104 1 5.00.091 841 2.79423104 1 4.00.278 33 1.73163104 1 3.50.843 48 1.18383104 2 3.32.5562 6.6293103 3 3.27.7466 3.90583103 4 3.0

23.467 1.58353103 6 2.971.146 7.13693102 8 2.0

215.61 2.26483102 10 1.92653.42 87.078 21 1.52

1980.2 15.915 75 1.01

884 INKSON ET AL.

entanglements of backbone section. Although the number of entanglements is unknown,it should lie within a range of roughly 2–10. The physical limit for a completely unen-tangled segment istbi 5 tsi . Moreover, sincetb( i 21) sets the fundamental diffusiontime for the branch point controlling the relaxation of segmenti, we must also satisfytb( i 21) , tsi . Thus thetsi are physically constrained to lie in the intervaltb( i 21), tsi , tbi . The number of arms,qi , is determined by the priority. Again this is

unknown, but as the priority and seniority increase towards the inner segments,qi shouldincrease withtbi .

The values forqi and tsi were found by fitting the transient uniaxial extensionalviscosity. As noted earlier the shear viscosity is insensitive toqi . Although transient fullextension can occur in the single mode model at sufficiently high shear rates, this occursrarely in the multimode model as increasing the shear rate merely excites faster relaxingmodes. For each mode, the evolution ofSi (t) may be calculated analytically, while thestretchl(t) was found by integrating Eq.~4! using a fourth-order Runge-Kutta scheme.The values ofqi andtsi were then adjusted by ‘‘trial and error.’’ Although with 10 ormore modes this optimization might appear to be a complex task, the nonlinear behavior

is dominated by modes whose stretch Weissenberg number is of order one (ts« ' 1).

Therefore at each particular strain rate only those modes for whichts« is close to unitybehave nonlinearly. Recall that the single mode extensional viscosity has a sharp maxi-

mum atts« ' 1. Thus at each specific strain rate we can isolate the one or two modesthat are active. For each strain rate, the level of the plateau in the extensional viscosity is

TABLE IV. LDPE B ~10 mode relaxation spectrum!.


0.0136 1.23105 1 4.00.145 4.533104 1 4.01.03 1.5033104 1 3.54.86 5.1963103 1 3.36.72 3.03103 2 3.2

17.53 1.7443103 3 3.1100.0 9.553102 5 2.9500.0 80.0 10 2.5

1000.0 50.0 23 1.95000.0 4.0 150 1.4

TABLE V. LDPE B ~8 mode relaxation spectrum!.


0.01 8.88423104 1 3.50.0571 4.81093104 1 3.30.3261 2.64833104 2 3.21.8622 1.26663104 3 3.0

10.634 4.74533103 4 2.960.724 1.1923103 9 2.5

346.77 251.45 15 2.31980.2 24.445 60 1.15


determined by the value ofqi , whereas the form of the growth of the viscosity towardsthe plateau determines the ratiotbi /tsi .

The results of matching the uniaxial extension viscosity with measurements of IUPACA, by Laun and Mu¨nstedt~1979!, are shown in Fig. 7 and the shear viscosity fit is shownin Fig. 8. The corresponding values ofqi and tbi /tsi are shown in Table I. Note thesuccessful fitting of turn ups and thinning~end points!. The level of agreement of theshear viscosity is very pleasing as the parameters in the model were chosen only to fit theextensional data and the shear fit follows with no further adjustment of the parameters. Infact the shear response is rather weakly dependent on the nonlinear parameters: adjustingqi makes little difference since maximum extension is often not achieved in shear, andadjustingts only varies the position of the stress overshoot peak slightly. The shearbehavior of the multimodal model matches the start-up viscosity well and the presence ofmultiple modes smoothes out the large overshoot that is apparent in the single modepompom model at high shear rates@Fig. 2~a!#. The fit of the extensional viscosity data formelt 1 @Meissner~1972!# is shown in Fig. 9. The extensional viscosity peaks do notmatch all the curves precisely, since the data set covers many strain rates where only afew modes are dominating the response.

The IUPAC X data of Laun and Schuch~1989! is significant in that it provides theonly measurements of both uniaxial extension and planar extension@which is comprised

FIG. 7. Transient uniaxial extensional viscosity of a 9 mode pom-pom melt in start up plotted against data forIUPAC A LDPE. The extension rates range from 0.01 to 1.0 s21.

TABLE VI. Melt 1.


1024 1.293105 1 4.5

1023 9.483104 1 3.9

1022 5.863104 2 3.7

1021 2.673104 2 3.6

100 9.803103 3 2.9

101 1.893103 7 2.0

102 1.803102 8 2.1

103 1.03100 9 1.3

886 INKSON ET AL.

of ‘‘stressing viscosities’’ in two directions,m11(t), in the direction of greatest extension

andm21(t) in the direction which has a constant length; see Meissneret al.1982!# as well

as simple shear. Now both of the fits to the latter two data sets are entirely free ofparameter choices. Data are only available for two different strain rates of 0.01 s21 ~Fig.10! and 0.05 s21 ~Figure 11!. By choosing suitable values ofqi andtbi /tsi we are ableto fit all three experiments simultaneously. Note that there is quite a lot of scatter in thedata form2

1(t). The values for the parameters are shown in Table II.Both the IUPAC A and IUPAC X parameter sets are in accord with the constraints

imposed on the values ofqi andtbi /tsi . The value ofq increases with mode number andhencetb . The values oftb /ts correspond to a range of between two and six entangle-ments. These are physically reasonable values, and correspond to long chain branchfrequencies from 1 per 4000 carbon atoms to 1 per 10 000 carbon atoms.

FIG. 8. Transient shear viscosity of a 9 mode multimodal pom-pom melt in start up plotted against data forIUPAC A LDPE. The shear rates range from 0.001 to 20.0 s21.

FIG. 9. Transient uniaxial extensional viscosity of an 8 mode pom-pom melt in startup plotted against data formelt 1 LDPE. Extension rates range from 0.01 to 1.0 s21.


The values ofq andts were obtained through isolation of the most active modes at agiven strain rate. Since extensional data are not available at strain rates which correspondto the action of the shortest time modes, varying the values ofq for these modes makesvirtually no difference to the quality of the fit. We have assigned the value ofq 5 1 tothese modes on the grounds that they represent the outermost segments of the moleculeand are expected to retract as linear polymers. The values ofq for the long time modesare important only at very low rates of strain.

The parameter values for the LDPE B sample are shown in Tables III–V. We usedthree different relaxation spectra in order to establish a connection between the choice of

FIG. 10. Transient uniaxial extensional, planar extensional and shear viscosity of an 11 mode pom-pom meltin start up plotted against data for IUPAC X LDPE. The shear/elongation rate is 0.01 s21.

FIG. 11. Transient uniaxial extensional, planar extensional and shear viscosity of an 11 mode pom-pom meltin start up plotted against data for IUPAC X LDPE. The shear/elongation rate is 0.05 s21.

888 INKSON ET AL.

spectra and the values of the parameters used in the model to fit the extensional rheology.This is discussed in Sec. IV B. The LDPE B extension and shear viscosity data and theorycurves for the 12 mode spectrum are given in Figs. 12 and 13, respectively~the fits usingthe other relaxation spectra look much the same!. Note that in extension, the theorycurves at the higher strain rates of 0.03 and 0.1 s21 rise significantly above the experi-mental data. It is not possible to obtain a better fit here because only the slowest~12th!mode contributes significant stress at all the strain rates. It is possible, however, that thesamples broke at these two strain rates before reaching their maximum strain value~which is set by an extension-thinning instability to be discussed below!.

FIG. 12. Diagram of the transient uniaxial extensional viscosity of a 12 mode pompom melt in start-up plottedagainst data for LDPE B. The extension rates range from 0.001 to 0.1 s21.

FIG. 13. Transient shear viscosity of a 12 mode pom-pom melt in start up plotted against data for LDPE B.Shear rates range from 0.001 to 0.3 s21.


B. Discussion

The most remarkable feature of our model is the consistent coincidence of the rapid

changes in gradient ofh1(t,«) with the filament failure that ends the extensional ex-periments.

The model continues to make predictions for the stress growth after the severe‘‘knees’’ in the curves, the data cease at those points because the filament breaks via anecking instability. This may also be the cause of the slight overshoots in some of theexperiments. This constitutes another remarkable feature of this approach to long chainbranched polymers — a suitable decomposition into pom-pom modes seems able topredict both the upturn in extensional viscosity and the point at which the material willfail due to a necking instability. We conjecture that the strain and strain rates at which along chain branched melt fails by necking is an inherent property of the material. Thisfailure can be predicted by analogy with the Conside`re criterion@Nadai ~1950!, Vincent~1960!, and Cogswell and Moore~1974!# for elastic solids. This predicts that necking willoccur at strain when the tangent to the stress-strain curve passes through the origin. Thisprocedure would be expected to hold, approximately, for viscoelastic materials where the

stress is primarily elastic, and would predict failure precisely at point whereh1(t,«) hasa rapid change in gradient.

In the preceding models, the number of modes was determined by the number ofmodes in the linear relaxation spectra of each sample. Therefore the frequency range ofthe complex viscosity data sets the range of relaxation times. This can lead to problemsif, as is the case with the IUPAC X data, there is insufficient low frequency data to probethe longest relaxation time modes. These modes correspond to the innermost sections ofthe molecules that are the most highly branched. Although these modes have compara-tively little effect on the linear rheology~and hence are often neglected!, they dominatethe nonlinear rheology at low extension rates. A lack of long-time modes may be com-pensated for by allowing the value ofts of the longest mode to approachtb . Thisproduces a greater stress contribution for that mode~see the choice of parameters forIUPAC X in Table II! at low extension rates.

Although, in this casets has to be set equal totb to compensate for the lack of lowfrequency modes, it is expected on physical grounds thattb /ts should decrease for suchhigher modes. At the longer timescales associated with the higher modes, entanglementsare only produced by other unrelaxed molecular sections. Thus the effective entangle-ment concentration decreases with mode number because segments with faster relaxationtimes do not act as constraints. The effect of thisdynamic dilutionis to decrease thenumber of entanglementssb and hencetb /ts @sincetb /ts 5 (4/p2) sb , McLeish andLarson~1998!#. In fact the innermost segments may not be entangled at all at the longestrelaxation times.

In addition to the range of relaxation times, the density of relaxation times is alsocritical. Too high a density results in the solution for the moduli being highly ill posed@see, e.g., Friedrich and Hofmann~1983!, Winter et al. ~1993!#, whereas too low a den-sity can lead to a single mode being active over a large range of strain rates. Thisproduces problems in fitting the nonlinear response~an example is the overprediction ofthe stress maximum of melt 1 at the particular rate of 0.01 s21, see Fig. 9!. However, ifour model is truly to reflect the underlying molecular physics, it must be possible toexpress the model parameters in a form that is independent of the choice of thetbi . Thisleads to the idea of a ‘‘molecular fingerprint’’ of a long chain branched melt based uponthe parameters of the multimode pom-pom model.

We obtained three different relaxation spectra from LDPE B data with 8, 10, and 12

890 INKSON ET AL.

modes, respectively. Thetbi andgi of the 8 and 12 mode spectra were chosen using anumerical error-minimizing routine to fit of theG8(v) curve of the LDPE B sample~seeFig. 14!. The 10 mode spectrum was ‘‘hand picked’’ to ensure a very different selectionof time parameterstb . In order to visualize how the individual modes in each of thespectra contribute to the viscosity, we plotgt2/Dt againstt in Fig. 15. Provided thatmodes are chosen so that the spectra remain stable,gt2/Dt should follow a smoothcurve, characteristic of the material, without much fluctuation of individualgi . Includedin Fig. 15 is a 15 mode spectrum of the same LDPE B sample to illustrate how theill-posed nature of the discretization leads to an instability if the number of modes is toolarge for the available data.

FIG. 14. Storage and loss modulii,G8(v),G9(v) of the LDPE B sample obtained from oscillatory shear stressgrowth of the equivalent shear viscosity using a Rheometrics RDA II rotational rheometer in the steady shearmode.

FIG. 15. Plot of different choices of linear relaxation spectra for LDPE B in the form ofgitbi2 /Dtb plotted

againsttb .


Since the parameter values required to fit the multimode pom-pom model should bedirectly related to the branching structure of the material, we wish to identify a materialcharacteristic or ‘‘fingerprint’’ from these parameter values. However, it is clearly essen-tial that this should be independent of the discretization of the linear spectrum. In Fig. 16we plot the values ofgi , qi andr i (r i 5 tbi/tsi) for each fit of the extensional viscositydata for the three different relaxation spectra. It can be seen that theqi distributions areindeed very similar, irrespective of the number of modes. Although the values ofqi forthe lower modes can vary slightly without affecting the fit to the extensional viscosity itis the higher modes which reflect the degree of branching deep within the LDPE mol-

FIG. 16. 8, 10 and 12 mode pom-pom parametersgi , qi , r i of LDPE B plotted againsttb . The parameterswere obtained from fitting of the extensional viscosity data.

FIG. 17. Multi-modal pom-pom parametersgi , qi , r i of LDPE IUPAC A and the time-temperature shiftedparameters of IUPAC X plotted againsttb .

892 INKSON ET AL.

ecule that are most important. It can be seen that these are very similar for the threespectra in these plots. The trend of ther i parameters is less clear and seems to dependmore on the choice of the number of very slow modes in the spectrum and their influenceon the behavior of the extensional viscosity at very low strain rates. However the shapeof the r i envelope is still consistent among the models for LDPE B, and so is also acandidate as a distinguishing feature, although not as strong asq. There is an effectivepower law for thegi distribution before the cut off. With a power of about 0.3. Rubin-stein, Zurek, McLeish, and Ball~1987! pointed out that the stress-relaxation function forentangled dynamics at or near the percolation threshold for branched polymers is not infact a power law. However, the predicted function can mimic an apparent power law thatdepends on the average number of entanglements between branch points (Mx /Me). Theapparent power index is of the order of (Mx /Me)21 so it qualitatively ties in with thematerial parameters.

We can also compare the choice of parameters for the IUPAC A and IUPAC X sincethey are batches of the same material. However, we must first perform an approximatetime-temperature shift of the IUPAC X data because its spectrum was taken at 125 °C,whereas that of IUPAC A was measured at 150 °C. This shift is not exact due to thethermorheological complexity of LDPE, but it does gives us a rough comparison of theparameter choice~see Fig. 17!. We used a value of the flow activation energy,Ea of 13.6kcal mol21 @Meissner~1975!#. This gave us a calculated shift in frequency of the IUPACX data ofaT 5 0.363. Again theqi distributions are very similar supporting a claim thatthey are a property of the material. Comparing theseqi distributions to the LDPE Bqidistributions it is apparent that they are lower, which is to be expected for a sample withless long chain branching. The parameters for melt 1 are shown in Fig. 18.

We plotted the multimodal flow curve of extensional viscosity against extensionalstrain rate for the IUPAC A model to compare to them experimental data~Fig. 19!. Thelarge peaks look rather unphysical but demonstrate the action of individual modes atdiffering rates of strain~the single mode flow curve is shown in Fig. 5!. The prominentpeaks result from the limited number of modes in the relaxation spectrum, each of whichcorresponds to single species of pom-pom molecules. The discrete nature of the model istherefore much more critical in extension than in shear. In practice, however, there is a

FIG. 18. Multimodal pom-pom parametersgi , qi , r i of melt 1 plotted againsttb . The parameters wereobtained from fitting of the extensional viscosity data.


continuous distribution of molecular weights that would smooth out the peaks. Neverthe-less with only nine modes of the IUPAC A model we do obtain a curve that follows theoverall shape of the LDPE data.

V. CONCLUSION

The remarkable result is that a small set of pom-pom modes with physically reason-able parameters is able to account simultaneously for the three functions~each of two

variables! h(t,g), h1(t,«) and hplanar1 (t,«) over four decades in time and rate. This is

quite unprecedented in modeling nonlinear LDPE rheology. This suggests strongly thatthe physics of the simple pom-pom molecule captures a generic aspect of branchedpolymer dynamics.

It seems possible to approximate very complex branched structures~which are illun-derstood themselves! using the relatively simple behavior of a dual-branched moleculewhose properties can be understood via the polymer physics derived from the tube modeland branched polymer theory. It has shown that the key aspect in the simultaneousextension hardening and shear thinning properties of LDPE can be put down to thesegments connected by branch points and their ability to sustain a large amount oftension. The model’s success in achieving hardening in planar extension while retainingstrong shear thinning demonstrates its superiority over current integral constitutive equa-tions and demonstrates the importance of including the relevant physics in the rheologicalmodels. Several further applications suggest themselves.

The use of such multimode pom-pom equations in non-Newtonian fluid dynamicssimulations would enable quantitative prediction of complex flows of LDPE. The rela-tively simple form of the approximate pom-pom model constitutive equations@involvingthe differential approximation forS(t)] could be used in such numerical simulations.Such calculations have already been performed for the single mode pom-pom model@Bishko, McLeish, Harlen, and Larson~1998!#.

Furthermore, the nonlinear spectral parameters themselves represent a method forcharacterizing branched polymer melts that could provide a discriminator for differentbranched molecular structures. They relate extensional data on branched melts directly toaveraged molecular structural quantities, and suggest that strain hardening curves and

FIG. 19. Steady state uniaxial extensional viscosity,h( «), plotted against rate of extension,«, for a 9 modepom-pom melt plotted against data for the LDPE IUPAC A.

894 INKSON ET AL.

even the strains at break in careful experiments may be characteristics of the materials.Carefully chosen sets of$tbi , tsi , qi , gi% account for all these features as well as for theshear data, and are relatively robust to the choice of mode density.

ACKNOWLEDGMENTS

This work was supported by a CASE sponsorship from BICC Cables. The authorswish to thank A. R. Blythe and M. Cassidy for useful discussions.

References

Ahmed, R., R. F. Liang, and M. R. Mackley, ‘‘The Experimental Observation and Numerical Prediction ofPlanar Entry Flow and Die Swell for Molten Polyethylenes,’’ J. Non-Newtonian Fluid Mech.59, 129–153~1995!.

Axelson, D. E., G. C. Levy, L. Mandelkern, ‘‘A Qualitative Analysis of Low Density~Branched! Polyethyleneby C13 Fourier Transform NMR at 67.9 MHz,’’ Macromolecules12, 41–52~1979!.

Ball, R. C. and T. C. B. McLeish, ‘‘Dynamic Dilution and the Viscosity of Star Polymer Melts,’’ Macromol-ecules22, 1911–1913~1989!.

Baumgaertel, M. and H. H. Winter, ‘‘Interrelation between Continuous and Discrete Relaxation Time Spectra,’’J. Non-Newtonian Fluid Mech.44, 15–36~1992!.

Bernstein, B., E. A. Kearsley, and L. J. Zapas, ‘‘A Theory of Stress Relaxation with Finite Strain,’’ Trans. Soc.Rheol.7, 391–410~1963!.

Bick, D. K. and T. C. B. McLeish, ‘‘Topological Contributions to Non-linear Elasticity in Branched Polymers,’’Phys. Rev. Lett.76, 2587–1590~1996!.

Bishko, G., T. C. B. McLeish, O. G. Harlen, and R. G. Larson, ‘‘Theoretical Molecular Rheology of BranchedPolymers in Simple and Complex Flows: The Pom-pom Model,’’ Phys. Rev. Lett.79, 2352–2355~1997!.

Bishko, G., O. G. Harlen, T. C. B. McLeish, and T. M. Nicholson, ‘‘Numerical simulation of the transient flowof branched polymer melts through a planar contraction using the pom–pom model,’’ J. Non-NewtonianFluid Mech.82, 255–273~1999!.

Cogswell, F. and D. Moore, ‘‘A Comparison Between Simple Shear, Elongation and Equal Biaxial ExtensionDeformations,’’ Polym. Eng. Sci.14, 573–576~1974!.

de Gennes, P. G., ‘‘Reptation of Stars,’’ J. Phys.~France! 36, 1199–1203~1975!.Doi, M. and S. F. Edwards,The Theory of Polymer Dynamics~Oxford Science, Oxford, 1986!.Friedrich, C. and B. Hofmann, ‘‘Ill-posed Problems in Rheometry,’’ Rheol. Acta22, 425–434~1983!.Graessley, W. W., ‘‘Effect of Long Branches on the Temperature Dependence of Viscoelastic Properties of

Polymer Melts,’’ Macromolecules15, 1164–1167~1982!.Kaye, A., College of Aeronautics, Cranford, UK Note No. 134, 1962.Khan, S. A. and R. G. Larson, ‘‘Comparison of Simple Constitutive Equations for Polymer Melts in Shear and

Biaxial and Uniaxial Extensions,’’ J. Rheol.31, 207–234~1987!.Laun, H. M., ‘‘Predictions of Elastic Strains of Polymer Melts in Shear and Elongation,’’ J. Rheol.30, 459–501

~1986!.Laun, H. M. and H. M. Mu¨nstedt, ‘‘Comparison of the Elongational Behavior of a Polyethylene Melt at

Constant Stress and Constant Strain Rate,’’ Rheol. Acta15, 517–524~1976!.Laun, H. M. and H. M. Mu¨nstedt, ‘‘Elongational Behavior of a Low Density Polyethylene Melt 1,’’ Rheol. Acta

17, 415–425~1978!.Laun, H. M. and H. M. Mu¨nstedt, ‘‘Elongational Behavior of a Low Density Polyethylene Melt II,’’ Rheol.

Acta 18, 492–504~1979!.Laun, H. M. and H. Schuch, ‘‘Transient Elongational Viscosities and Drawability of Polymer Melts,’’ J. Rheol.

33, 119–175~1989!.Marrucci, G., ‘‘Relaxation by Reptation and Tube Enlargement: A Model for Polydisperse Polymers,’’ J.

Polym. Sci., Polym. Lett. Ed.23, 159–177~1985!.McLeish, T. C. B., ‘‘Molecular Rheology of H-polymers,’’ Macromolecules21, 1062–1069~1988a!.McLeish, T. C. B., ‘‘Hierarchical Relaxation in Tube Models of Branched Polymers,’’ Europhys. Lett.6,

511–516~1988b!.McLeish, T. C. B., ‘‘Dynamic Dilution Effects in Branched Polymers and Gels,’’ Polym. Commun.30, 4–6

~1989!.McLeish, T. C. B. and R. C. Larson, ‘‘Molecular Constitutive Equations for a Class of Branched Polymers: The

Pom-pom Polymer,’’ J. Rheol.42, 82–112~1998!.


McLeish, T. C. B. and K. P. O’Connor, ‘‘Rheology of Star-Linear Polymer Blends: Molecular Tube Models,’’Polymer34, 2998–3003~1993!.

McLeish, T. C. B.,~Ed.!, Theoretical Challenges in the Dynamics of Complex Fluids~Kluwer, Dordrecht,1997!.

Meissner, J., ‘‘Modifications of the Weissenberg Rheogoniometer for Measurement of Transient RheologicalProperties of Molten Polyethylene Under Shear. Comparison with Tensile Data,’’ J. Appl. Polym. Sci.16,2877–2899~1972!.

Meissner, J., ‘‘Basic Parameters, Melt Rheology, Processing and End Use Properties of Three Similar LowDensity Polyethylene Samples,’’ Pure Appl. Chem.42, 553–612~1975!.

Meissner, J. and J. Hostettler, ‘‘A New Elongational Rheometer for Polymer Melts and Other Highly ViscousLiquids,’’ Rheol. Acta33, 1–21~1994!.

Meissner, J., S. E. Stephenson, A. Demarmels, and P. Portmann, ‘‘Multiaxial Elongational Flows of PolymerMelts—Classification and Experimental Realization,’’ J. Non-Newtonian Fluid Mech.11, 221–237~1982!.

Milner, S. T. and T. C. B. McLeish, ‘‘Parameter-Free Theory for Stress Relaxation in Star Polymer Melts,’’Macromolecules30, 2159–2166~1997!.

Nadai, A.Theory of Flow and Fracture of Solids~McGraw-Hill, New York, 1950!.Pearson, D. S. and E. Helfand, ‘‘Viscoelastic Properties of Star-shaped Polymers,’’ Macromolecules17, 888–

895 ~1984!.Rubinstein, M., S. Zurek, T. C. B. McLeish, and R. C. Ball, ‘‘Relaxation of Entangled Polymers at the Classical

Gel Point,’’ J. Phys.~France! 51, 757–775~1990!.Rubinstein, M., E. Helfand, and D. S. Pearson, ‘‘Theory of Polydispersity Effects on Polymer Rheology—

Binary Distribution of Molecular Weights,’’ Macromolecules20, 822–829~1987!.Samurkas, T., R. G. Larson, and J. M. Dealy, ‘‘Strong Extensional and Shearing Flows of a Branched Poly-

ethylene,’’ J. Rheol.33, 559–578~1989!.Viovy, J. L., ‘‘Constraint Release in the Slip-link Model and the Viscoelastic Properties of Polymers,’’ J. Phys.

~France! 46, 847–853~1985!.Wagner, M. H. and H. M. Laun, ‘‘Nonlinear Shear Creep and Constrained Elastic Recovery of a Low Density

Polyethylene Melt,’’ Rheol. Acta17, 138–148~1978!.Wagner, M. H., ‘‘A Constitutive Analysis of Uniaxial Elongational Flow Data of a Low Density Polyethylene

Melt,’’ J. Non-Newtonian Fluid Mech.4, 39–55~1978!.Wagner, M. H., P. Ehrecke, P. Hachmann, and J. Meissner, ‘‘A Constitutive Analysis of Uniaxial, Equibiaxial

and Planar Extension of a Commercial Linear High-Density Polyethylene Melt,’’ J. Rheol.42, 621–638~1998!.

Winter, H. H., M. Baumgaertel, and P. Soskey, ‘‘A Parsimonious Model for Viscoelastic Liquids and Solids’’in Techniques in Rheological Measurement, edited by A. A. Collyer~Chapman and Hall, London, 1993!.

Vincent, P., ‘‘The Necking and Cold-Drawing of Rigid Plastics,’’ Polymer1, 7–19~1960!.

896 INKSON ET AL.

Documents

Predicting low density polyethylene melt rheology in elongational and shear flows with “pom-pom” constitutive equations