Non-linear step strain of branched polymer melts

Non-linear step strain of branched polymer meltsD. M. Hoyle, O. G. Harlen, D. Auhl, and T. C. B. McLeish

Citation: Journal of Rheology (1978-present) 53, 917 (2009); doi: 10.1122/1.3143794 View online: http://dx.doi.org/10.1122/1.3143794 View Table of Contents: http://scitation.aip.org/content/sor/journal/jor2/53/4?ver=pdfcov Published by the The Society of Rheology Articles you may be interested in Elongational flow of polymer melts at constant strain rate, constant stress and constantforce AIP Conf. Proc. 1526, 168 (2013); 10.1063/1.4802612 NonLinear Rheology of Polymer Melts AIP Conf. Proc. 1152, 168 (2009); 10.1063/1.3203265 Nonlinear Step Strain of Branched Polymer Melts AIP Conf. Proc. 1027, 451 (2008); 10.1063/1.2964725 Stress Relaxation of a Branched Polybutadiene in DoubleStep Shear Deformations J. Rheol. 29, 533 (1985); 10.1122/1.549801 Converging Flow of Polymer Melts J. Rheol. 25, 605 (1981); 10.1122/1.549651

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Non-linear step strain of branched polymer melts

D. M. Hoylea) and O. G. Harlenb)

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

D. Auhl and T. C. B. McLeish

epartment of Physics and Astronomy, Interdisciplinary Research Centre inPolymer Science and Technology, University of Leeds, Leeds LS2 9JT,

United Kingdom

(Received 23 November 2008; final revision received 7 May 2009�

Synopsis

ong-chain branched polymer melts such as low density polyethylene �LDPE� and branchedetallocene polyethylenes show strong time-strain separability in step strain. Constitutive models

f the multi-mode Pom-pom form are highly successful in modeling the stress generated byeneral flow histories for these materials. However, a single Pom-pom mode is not time-straineparable and reconciling this to the step-strain phenomenon has been a challenge. We investigateulti-mode integral Pom-pom models and a differential approximation to compare time-strain

eparation, with respect to mode density. Here we show that for a wide class of branchedistributions, a family of damping functions can be derived with a response that is very close toeparable. We evaluate the family for both LDPE and branched high density polyethylene meltsnd show that a damping function derived from the multi-mode Pom-pom model gives an accuraterediction of the damping behavior in step-strain experiments. © 2009 The Society ofheology. �DOI: 10.1122/1.3143794�

. INTRODUCTION

An experimentally observed rheological property of many branched polymer melts ishat the stress relaxation following a step strain satisfies time-strain separability �TSS�. Inhese cases, the relaxation modulus G�� , t�, after a step strain of �, can be factorized to aood approximation as

G��,t� = G�t�h�� , �1.1�

here G�t� is the linear relaxation spectrum and h�� is a function of strain only, whichs known as the “damping function.”

This observation is used as the basis of the Kaye–Bernstein Kearsley Zappas �K-BKZ�amily of integral constitutive models �Bernstein et al. �1963�; Wagner and Stephenson1979��, where TSS is “built in.” However, these models do not perform well in geo-

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2009 by The Society of Rheology, Inc.917. Rheol. 53�4�, 917-942 July/August �2009� 0148-6055/2009/53�4�/917/26/$27.00

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etrical flows of long-chain branched �LCB� melts. For LCB polymer melts, a multi-ode version of the Pom-pom model introduced by McLeish and Larson �Bishko et al.

1997�; McLeish and Larson �1998�; McLeish �2002�� has been highly successful inodeling the non-linear rheology. This model, which is based on tube theory for a simple

ranched molecule, does not explicitly satisfy TSS. Larson showed for a restricted classf models �Larson �1985a, 1985b�� that apparent TSS may arise from the superposition ofodes, but whether this applies to the Pom-pom model is still open to discussion.Indeed, in a comprehensive review of an early version of the Pom-pom equations,

ubio and Wagner �2000� found no time-strain separable region in the non-linear relax-tion modulus. However, this version of the model did not incorporate the importantechanism of the branch-point withdrawal introduced by Blackwell et al. �2000�. Thisakes a considerable difference to the smoothness of response in strong flows.Chodankar et al. �2003� reviewed the differential and integral forms of the Pom-pom

odel and showed that by incorporating the branch-point withdrawal, multi-mode ver-ions of both models display apparent TSS in step strain. A numerical damping functionould be produced at various times that was in better agreement with the experimentallyetermined damping function of a branched low density polyethylene �LDPE�, Lupolen810H, than the tube theory damping function for linear melts h��= �1+ 4

15�2�−1. As wells the original differential and integral forms, they also analyzed the Öttinger differentialersion of the model �Öttinger �2001�� and found that this gave a better approximation tohe integral model at the terminal time than the original differential approximation of

cLeish and Larson �1988�.In further work, Venerus �2005� and Venerus and Nair �2006� used the Öttinger ver-

ion without the maximum stretch condition to model the stress relaxation of entangledinear polymers, where the strain is imposed over a finite time interval, and found gooduantitative agreement for a variety of different melts and solutions, even though thisodel was designed for branched polymers.However, the more sensitive technique of Fourier transform rheology �Fleury et al.

2004�; Vittorias and Wilhelm �2007�� in large amplitude oscillatory shear shows thatSS does not, after all, hold perfectly for branched polymers. These observations raise aeries of questions related to the general issue of how apparent TSS arises:

For what subsets, if any, of multi-mode Pom-pom models are the predictions in non-linear step-strain time-strain separable?When these conditions are relaxed, how close to separability are the predictions for awider class of models?To what extent do multi-mode Pom-pom models derived from fits to extensional dataon long-chain branching melts satisfy the conditions of the classes stated above?

In this paper, we will address these issues. We review the original integral and differ-ntial Pom-pom equations and assess how the range and density of modes affects theccuracy of the differential model. In Sec. III we also examine how the range of relax-tion times in the multi-mode spectrum affects the time scales over which apparent TSSs observed and the behavior at the terminal time. In Sec. IV we derive an analyticamping function approximation using certain model and material assumptions, underhich the Pom-pom model is time-strain separable. This damping function is then com-ared against “idealized fluids” that incorporate some or all of the assumptions madeuring the derivation. Finally, in Sec. V, we survey a range of branched polyethylene
elts produced by two different synthesis routes to examine how close these materials

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re to the conditions for TSS. Comparisons are then made between the derived dampingunction and the modeled stress relaxation over a wide range of times and strains.

I. CONSTITUTIVE EQUATIONS

Polymer rheology for highly entangled molecules can be successfully characterized byolecular topology. The concept of the tube model introduced by Edwards and deennes �cf. Doi and Edwards �1986�� can be applied to branched polymers, in particular,cLeish and Larson �McLeish and Larson �1998�; McLeish �2002�� developed a mo-

ecular theory for non-linear theory for a melt of Pom-pom molecules. A Pom-pomonsists of a backbone with an equal number of arms attached to branch points at eithernd. As with the tube model for linear polymers, the entanglements around the backbonere smoothed to a tube that moves affinely with the applied deformation and restrictsateral movement. However, unlike the linear case where polymers can diffuse freelylong its tube, the presence of two branch points impedes motion along the tube giving aifferent rheology. In particular, the stretch relaxation time �which controls the triggeringf extensional hardening� is much longer than for a comparable linear melt.

In extensional flows of sufficient high rate, the backbone is stretched until its effectiveurvilinear tension matches the cumulative equilibrium tensions in the q arms, at whichoint the branch points retract into the backbone tube. Equating the tension in the back-one with the maximum tension the arms can hold gives the supremum for the stretch toe �=q.

Blackwell et al. �2000� introduced the concept of branch-point displacement. Thisodels the reduction in average arm length associated with fluctuations of the positions

f a branch point. This happens over length scales of the tube diameter at extensions upo maximum q. Incorporating this behavior into the stretch dynamics reduces the time fortretch relaxation and smoothes the unphysical cusps in the steady state extensionaliscosity. This result also gives better steady state extensional results for LDPE whenodeled by a superposition of Pom-pom modes. This multi-mode Pom-pom model was

ntroduced by Inkson et al. �1999� to account for polydispersity and high complexities ofultilevel branching.At the heart of the model, in either single- or multi-mode forms, is the separation of

he two relaxation processes of stretch �fast� and orientation �slow�. Physically, the sepa-ation is the consequence of entangled dynamics.

There have been several suggested algorithms for calculating the orientation tensor inhe Pom-pom model. It was originally derived in integral form, with a differential ap-roximation based on the upper-convected Maxwell model. The integral form is complexnd computationally expensive, restricting its use to simple flow geometries. The differ-ntial model is more commonly used, particularly, in complex flow calculations as it isomputationally simpler, but it is not in quantitative agreement with the integral modelMcLeish and Larson �1998��.

Other differential constitutive models were subsequently developed. A thermodynami-ally motivated differential model was suggested by Öttinger �2001�. Verbeeten et al.2001� also suggested the extended Pom-pom �XPP� differential model, which was theotivation for other subsequent forms such as the double convected Pom-pom �DCPP�

Clemeur et al. �2003��. These models were developed to avoid the maximum stretchondition and improve the quantitative agreement with the integral model, in particular,
o give a non-zero second normal stress difference in shear.

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In this work, we look at the Pom-pom constitutive equations for a “step” strain, wherenearly instantaneous shear step strain is imposed at t=0. The relaxation is observed in

he form of the relaxation modulus G�� , t�=�xy /�.In the multi-mode Pom-pom, the extra stress tensor is formed as the sum contributions

rom different relaxation modes �Inkson et al. �1999��. The stress contribution from eachode is the product of the corresponding backbone stretch ��t� and orientation tensor

= �t�,

�= = N−1�i

n

gi�i2�t�S= i�t� , �2.1�

here gi is the relaxation rate and i is the mode index. The dimensionless constant N−1

154 for the integral model and N−1=3 for the differential approximation.The integral model has orientation tensor S= , for each mode i given by

S= �t� = �−�

t

dt�e−t−t�/�b

�b�E= �t�,t� · u��E= �t�,t� · u��

�E= �t�,t� · u��2 , �2.2�

here E= �t� , t� is the deformation tensor for the local flow between times t� and t, u�� is thesotropic unit vector at time t� averaged over within the brackets . . . �.

The differential approximation uses the upper convected Maxwell constitutive equa-ion for the orientational degrees of freedom only, where the auxiliary tensor A= satisfies

DA=

Dt= K= · A= + A= · K= T −

1

�b�A= − I=� . �2.3�

he orientation is given by the unit tensor

S= =A=

trA=. �2.4�

n the original Pom-pom model, the tension in the backbone is derived from a forcealance between the drag force on each branching point and the backbone of the mol-cule acting as a Hookean spring, imposing an elastic stretch relaxing toward the equi-ibrium length of the backbone. Writing the stretch as the dimensionless parameter ��t�L�t� /L0 gives

D

Dt��t� = ��t�K= :S= −

1

�s��t� − 1� . �2.5�

ach arm has a maximum thermal tension that it can hold before it becomes entropicallyore favorable to withdraw the arms into the backbone orientation tube. This gives aaximum stretch of �i=qi, where qi is the parameter specifying the effective degree of

ranching of mode i.The flow induced branch-point displacement discussed by Blackwell et al. �2000�

odifies the relaxation rate for the stretch before the critical condition �i=qi is met. Sincehe relaxation time has an exponential dependence on the length of the arms, there is aeduction in the relaxation time �s→�se

−��−1�, with ��=2 / �q−1� �McLeish �2002��. Thetretch equation is now nonlinear in ��t�,

D��t� = ��t�K= :S= −

1��t� − 1�e��i−1�. �2.6�

Dt �s


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Equation �2.1� together with either Eq. �2.2� �integral model� or Eq. �2.3�, Eq. �2.4�differential model�, and Eq. �2.6� constitute the Pom-pom models employed here.

II. POM-POM IN STEP STRAIN

We model a step-strain flow by imposing a shear-rate �̇=��t�, so that no relaxationccurs during deformation. The calculation of the initial orientation and stretch can beound by neglecting terms associated with the characteristic relaxation times �b and �s inheir corresponding dynamical equations. After this initial stage, we assume that no moreeformation occurs.

In the differential model, the initial orientation is given by

Sxy��,0+� =�

�2 + 3, �3.1�

ith stretch given by McKinley and Hassager �1999�,

��,0+�D = �1 +1

3�2 1/2

. �3.2�

or the integral Pom-pom model,

Q12�� =4�

15�1 + 415�2�1/2 , ��,0+�I = �1 +

4

15�2 1/2

. �3.3�

Using the initial orientation as a boundary condition, the relaxation of the fluid can beomputed for the integral and differential model shear relaxation moduli and is given,espectively, by

GI��,t� =�xy

�=

g0�I2��,t�e−t/�b

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, �3.4�

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GD��,t� =�xy

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1 + 13�2e−t/�b

. �3.5�

he differences in the models become apparent at late times �t��s�. The orientation ofhe integral model alone appears to show TSS; this is observed at terminal times when theackbone stretch has relaxed. The differential orientation is, however, not TSS as GD

ecomes independent of strain in the long time limit. This unphysical behavior can bebserved in Fig. 1.

To examine this effect further, it is convenient to look at the ratio of the non-linear andinear stress relaxation modulus by defining the damping ratio H�� , t� as

H��,t� =G��,t�G�t�

, �3.6�

here G�t�=�igie−t/�bi is the linear relaxation modulus. For a time-strain separable fluid,

�� , t� is independent of t and is equal to the damping function.The results for the one-mode model can be seen in Fig. 2. Note for both integral

dashed� and differential �solid� models that H�� , t� displays two phases of relaxation, forimes t�b and t�b. No relaxation occurs until a time on the order �s when the stretch
egins to relax. During this phase, both models show a similar decay of H�� , t�. Differ-

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nces between the models appear at times beyond �b where the orientation relaxes. In thentegral model, the damping ratio tends to a plateau, which is a Doi–Edwards-type damp-ng regime. However, the differential model shows an increase in the damping ratio andltimately tends to the limit of H�� , t�=1. This occurs because the non-linear denomina-or in the differential relaxation modulus �Eq. �3.5�� decays with time, unlike in thentegral case �Eq. �3.4��.

IG. 1. The relaxation modulus for the integral �dashed� and differential �solid� Pom-pom model depicting theifferences in terminal time behavior between the models. The strains used were �=0.1,5 ,10 with parameters

0=10, �b=10, �s=5, and q=6.

IG. 2. The damping ratio H�� against time for a one-mode integral �dashed� and differential �solid� Pom-
om. The strains used were �=0.1,5 ,10 with parameters g0=10, �b=10, �s=5, and q=6.

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Clearly neither model displays TSS as found experimentally for LDPE materials. Thiss, however, to be expected since real “Pom-pom” polymer melts constructed from mono-isperse controlled-architecture polymers display exactly this failure of TSS arising fromhe separation of stretch and orientation �McLeish et al. �1999��. More importantly, theifferential model does not properly approximate the behavior of the integral model atong times. We need to assess the seriousness of this failure in the context of multi-mode

odels relevant to polydisperse branched melts.

ulti-mode model

The multi-mode Pom-pom model �Inkson et al. �1999�� was introduced to account forolydispersity and multi-level branching in LDPE. Rubio and Wagner �2000� used thisodel to describe an LDPE material labelled IUPAC-A in step strain. This model, how-

ver, did not incorporate the modified stretch equation �Blackwell et al. �2000�� whichas instrumental in achieving quantitative fits to start up flows of LDPEs.

ABLE I. Parameters used for IUPAC-A. Both the cases of ��=0 and ��=2 /q are listed. Linear data producedrom Laun �1986�, ��=0 parameters from Inkson et al. �1999�, and ��=2 / �q−1� parameters are from Blackwellt al. �2000�.

Mode No., iModulus, gi

�Pa��b,i

�s�

��=0 �i�=2 /qi−1

qi �b,i /�s,i qi �b,i /�s,i

1 1.520�105 1.0�10−3 1.0 – 1.0 –2 4.005�104 5.0�10−3 1.0 – 1.0 –3 3.326�104 2.8�10−2 1.0 – 2.0 2.04 1.659�104 1.4�10−1 1.0 – 2.0 2.55 8.690�103 7.0�10−1 2.0 2.0 4.0 2.06 3.151�103 3.8�100 6.0 1.7 7.0 2.07 8.596�102 2.0�101 6.0 2.15 8.0 1.58 1.283�102 1.0�102 9.0 1.25 12.0 1.09 1.8495�100 5.0�102 22.0 1.1 30.0 1.0

IG. 3. The relaxation modulus and damping ratio parameters for IUPAC-A. The modifications to the equationor backbone stretch relaxation by Blackwell improve the plateau modulus showing TSS over three orders ofagnitude in time from 10−1 to 102 s �from left to right ��=0 and ��=2 / �q−1��. Strains of 0.1, 10, and 20 were

sed.


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Figure 3 compares the shear relaxation modulus and damping ratio for the two fits toUPAC-A. This shows the differences between the unmodified ��=0� and modified��=2 / �q−1�� backbone stretch �Eq. �2.6�� given by Blackwell et al. �2000� and in Table. Note that the parameters for IUPAC-A in Blackwell et al. �2000� used a strain param-ter ��=2 /q, which was later corrected to ��=2 / �q−1� in McLeish �2002�. In Fig. 3�a�,or ��=0, the damping ratio shows no plateau with respect to time, showing no TSS. Inig. 3�b�, the damping ratio plateaus, after initial early relaxation of faster modes, for up

o three decades of time. The damping ratio H�� , t� shows clearly improved TSS given byhe improved modeling of branch-point withdrawal.

When the faster modes have relaxed, they contribute little to the total relaxationodulus. However, as the number of modes left to relax reduces, the terminal time

ehavior, in which H�� , t� tends to one, becomes more dominant until the final modeisplays the behavior as seen in single-mode relaxation. This causes spurious oscillationsn the damping ratio for the differential Pom-pom model near the terminal time, as can beeen in Fig. 3 after the plateau of TSS.

To examine the effect of different discretization of the material spectrum on the pre-ictions of TSS, in Table II we give details of two different spectra for an LDPE materialhat we have labelled LDPE1. LDPE1 is a similar material to IUPAC-A and its material

ABLE II. Parameters used for LDPE1–9 and 12 mode models. The 9 mode parameters were used in Sec. IIInd compared with the 12 mode model to show that an increase in the number of modes gives an increase in theeriod of TSS predictions.

Mode i

LDPE1 at 150 °C 9 modes LDPE1 at 150 °C 12 modes

gi

�Pa��b,i

�s� �b,i /�s,i qi

gi

�Pa��b,i


1 92497 0.00316 – 1 64373 0.00316 – 12 27781 0.0154 1.0 2 37846 0.01 – 13 19747 0.0750 1.7 3 13408 0.0316 5.0 44 9610 0.365 4.4 3 14122 0.1 4.0 55 4326 1.778 5 5 7155 0.316 4.0 56 1583 8.660 2.3 5 4417 1.00 4.0 57 405.2 42.17 2.3 8 2191 3.162 3.0 58 30.44 205.4 1.6 11 1034 10.0 2.0 69 0.407 1000.0 1.6 14 404.7 31.62 1.7 810 88.19 100 2.0 1011 7.911 316.2 1.2 1212 0.3402 1000 1.0 14

ABLE III. Material properties of polyethylenes investigated �Das et al. �2006�; Hassell et al. �2008��. Theverage relaxation time �̄b is a viscosity averaged quantity, where �̄b=�iGi�bi

2 /�iGi�bi.

CodeMW

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�-�T

�°C� 0

�kPa/s��̄b

�s�

LDPE1 240 9 150 51 50LDPE2 242 11 160 368 428LDPE3 146 7 150 2 1.0HDPE1 68 2.68 155 50 28HDPE2 84 2.2 155 35 18


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roperties are listed in Table III. The two different spectra were prepared by fitting 9 and2 modes Pom-pom models to linear and extensional rheology measurements for relax-tion times in the range 0.003–1000 s. Figure 4 shows a comparison between both modelsnd experimental measurements at 150 °C, in extensional and shear stress. Extensionates of 0.001, 0.003, 0.01, 0.03, 0.1, 0.3 and shear rates of 0.003, 0.01, 0.03, 0.1, 0.3, 1,

IG. 4. A comparison of the 9 �dashed lines� and 12 �solid lines� mode Pom-pom fits to extensional and shearata taken at 150 °C for LDPE1. Extension rates of 0.003, 0.01, 0.03, 0.1, 0.3, 1, 3, 10, and 30 and shear ratesf 0.03, 0.1, 0.3, 1, 3, and 10 were used. There appears little difference between the two fits; however, thencreased density of modes in the 12 mode fit gives a longer region of TSS.

IG. 5. �a� Relaxation modulus and damping ratio for a 12 mode model of LDPE1. The comparison ofifferential �solid� and integral �dashed� models shows that the differential model approximates the TSS of thentegral model correctly until terminal time behavior becomes dominant. �b� Relaxation modulus and dampingatio for a 9 and 12 mode model of LDPE1. The comparison of 9 modes �dashed lines� and 12 �solid lines�odels is shown. The increase from 9 to 12 modes of relaxation improves the plateau of TSS. Comparing the

amping ratio near terminal time in �a� and �b� shows a reduction in oscillations for the denser 12 mode
pectrum. Strains of �=0.1 and 7 are used.

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were used. Although there is little difference in the quality of the fits to the rheologicalata, in Fig. 5�b� it can be seen that the denser 12 mode spectrum has a smoother plateaun the damping ratio and later transition to the terminal damping. These results areompared to the experimental damping behavior of LDPE1 in Fig. 11.

Figure 5�a� compares the differential and integral model for the 12 mode spectrum forDPE1. The damping ratio shows that in a multi-mode model, the differential modelpproximates the plateau of the integral model correctly. This happens because the oscil-ations in the differential model occur at a similar time scale to the integral modelransitioning to a Doi–Edwards damping regime and occur once most of the stress in theode has relaxed. Consequently neither affects the overall stress significantly, except in

he terminal zone.We conclude that, empirically, a sufficiently dense mode spectrum of either the dif-

erential or integral multi-mode Pom-pom model does exhibit TSS, over four decades ofime, when parameters are extracted from representative LDPE materials. This takes us tohe question of what properties of both material and model cause this behavior to arise,hen the individual modes are not TSS?

V. DAMPING FUNCTION

. Derivation

In the previous section, it was observed that increasing the number of modes in theifferential Pom-pom equations improves the modeling of the step-strain experiment.his in turn gives an apparent TSS over several decades of time. We now derive anpproximate analytical expression for the damping function h�� in the region of TSS. Toroceed, we need to make a number of simplifying assumptions about both the model andhe material properties. The effect of deviations from these approximations will then behecked.

In order to provide an analytic solution of the backbone stretch Eq. �2.6�, we initiallyet the parameter ��=0, so that we do not incorporate the improvement made by Black-ell et al. �2000� Although, in general, not incorporating the branch-point withdrawalill not give TSS, the other approximations used in the derivation of the damping func-

ion together with ��=0 give a special case of TSS. However, in modeling a general fluid,e restore the important smoothing behavior of ��0. This is incorporated when we

pproximate the piecewise continuous initial stretch with a fully continuous �relaxationime�-strain separable equation.

If the material is assumed to be time-strain separable then

G��,t� = �i

Gi�i2��,t�e−t/�bi

1 + N�2e−t/�bi

= h��i

Gie−t/�bi = h��G�t� . �4.1�

he use of a sufficiently dense spectrum of modes implies that the differential andntegral models are equivalent, up to the factor in the denominator N, equal to 4

15 for thentegral model and 1

3 for the differential model.We will now look for constraints on the family of parameters Gi, �bi

, �si, and qi that

llow Eq. �4.1� to hold. We take the continuous limit of the sum and for convenience
rite �=1 /�b so that

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s

s

sf


Redistribution

�0

�

d�G��2��,�,t�e−t�

1 + N�2e−t� = h��0

�

d�G��e−t�. �4.2�

his effectively increases the number of modes to infinity and so suppresses the oscilla-ions associated with the final modes of relaxation �that is, reminiscent of a Gibbs phe-omenon�. The exponential term in the denominator of Eq. �4.2� is responsible for thenphysical terminal behavior of the differential approximation. So we remove this term inrder to focus on the conditions for TSS during the many decades of relaxation prior tohe terminal zone.

In order to obtain an analytic solution for �� , t�, we revert to the unmodified stretchquation with ��=0� so that

��,t� = 1 + ��0 − 1�e−t�s. �4.3�

e will restore the important smoothing behavior provided by ��0 below. With thesepproximations, Eq. �4.2� becomes

h��0

�

d�G��e−t� =1

1 + N�2��0

�

d�G��e−t� + 2�0

�

d�G��0 − 1�e−t�−t�s

+ �0

�

d�G��0 − 1�2e−t�−2t�s . �4.4�

otice that the dependence on the q-spectrum enters Eq. �4.4� via the piece-wise-ontinuous equation for the initial stretch

�0 = ��1 + N�2�1/2 for �0 q

q otherwise.� �4.5�

In the Appendix we show that this function may be approximated by a continuouseparable expression of the form

�0 − 1 =��1+a1/a2�q − 1�

�qma2 + ��1+a1/a2�a2�1/a2

= ��q − 1� . �4.6�

By smoothing out the initial stretch as a function of � in this way, we recover themoothing behavior of ��0 that was lost by the approximation of Eq. �4.3�.

This approximation produces separability at the level of stretch only �which is not theame as TSS for the full constitutive equation�. The expression containing the dampingunction is now fully separable in terms of strain and time,

h��0

�

d�G��e−t� =1

1 + N�2��0

�

d�G��e−t�

+2��1+a1/a2

�qma2 + ��1+a1/a2�a2�1/a2

�0

�

d�G��q − 1�e−t�−t�s

+��1+a1/a2�2

�qma2 + ��1+a1/a2�a2�2/a2

�0

�

d�G��q − 1�2e−t�−2t�s .

�4.7�


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In order to obtain TSS, it remains to choose the functions G��, q��, and �s�� suchhat each of the integrals has the same time dependence. We begin by assuming that theatio of orientation to stretch time for all modes is a constant �, so that

�s = �� . �4.8�

This ratio is proportional to the number of entanglements between branch points andight be expected to be constant if the probability of branching is independent of a

osition in the molecule. However, the physics of dynamic dilution predicts that � shouldecrease slightly with increasing relaxation time. In practice, when fitting a Pom-pompectrum, the ratio is adjusted to best fit the transient buildup of stress. Furthermore,hanges in the value of � have a negligible effect on the damping function as can be seenn Fig. 8.

Next we choose G�� and q�� to satisfy power laws in �, namely,

G�� = B�b, �q�� − 1� = C�c. �4.9�

he equation for the damping function now looks like

h��0

�

d��be−t� =1

1 + N�2��0

�

d��be−t� + C��0

�

d��b+ce−t��1+�� + ¯

+ �C��2�0

�

d��b+2ce−t��1+2�� , �4.10�

here the integrals may now all be written as gamma functions. Dividing by ��b+1�ives the damping function as

h�� =1

1 + N�2�1 +C��b + c + 1�

tc�1 + ��b+c+1��b + 1�+

�C��2��b + 2c + 1�t2c�1 + 2��b+2c+1��b + 1� . �4.11�

ote that this is only independent of time for the exponent c=0, with �q−1�=C=qm

eing constant. We will see below that this is a reasonable approximation for a wideamily of LDPEs, where the best fit power law c is typically �0.1. So that the dampingunction takes the form

h�� =1

1 + N�2�1 +2qm��1 + ��b+1 +

�qm��2

�1 + 2��b+1 . �4.12�

or the differential Pom-pom model �with N= 13 �, making the substitution for the best

roposed choice of parameters �Appendix� for �� =1 /4, a1=1 /2, and a2=1� in Eq.4.6� gives a damping function with three material characteristics,

hB��;qm,�,b� =1

1 + 13�2�1 +

2qm�3/2

�4qm + �3/2��1 + ��b+1 +�qm

2 �3��4qm + �3/2�2�1 + 2��b+1 .

�4.13�

s noted in the Appendix, this damping function does become greater than 1, but by lesshan 1% for low strain results. This provides the best smoothing behavior at the transitiono maximum stretch.

The family of damping functions represented by Eq. �4.13� constitutes a universal setf responses for complex branched melts in the same way that the single Doi–Edwardsamping function does for linear melts. We refer to hB as the branched damping function
BDF�.

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m

•••

Fe

Fa


Redistribution

. Ideal model

In deriving the BDF in the previous section, we assumed that parameters in theulti-mode Pom-pom model satisfied the following conditions:

G��B�−b�B�b,q��−1�C�−c�C�c,�s=��.

IG. 6. Damping ratios are shown for the ideal fluid while varying the parameter c=0,0.1,0.2 for the differ-ntial model. Strains of �=0.1,3 ,10 are used.

IG. 7. Derived BDF �solid line� plotted against strain compared with damping ratios taken from an ideal fluid�
t various times, 0.1, 1, and 10 �dashed lines�. The picture is the case for � =2 / �q−1�.

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bgc

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Fvp

930 HOYLE et al.

Redistribution

In this section, we consider the properties of an “ideal damping fluid” whose param-ters follow these relations but are described by a finite set of discrete modes.

Unless otherwise stated, the parameters used throughout this section will be B=100,=−0.5, C=qm=6, c=0, and �=5. This choice is made for illustrative purposes and alleneral features of the model appear with this parameter choice. We also restore thehoice of ��=2 / �q−1�.

During the derivation of the BDF �Eq. �4.13��, we found that the power-law coefficientor the q-distribution was required to be zero. Since the q-spectra for LDPEs generallyhow a slow increase in q with �b, we check the effect of departures from c=0 to weakower laws. In Fig. 6 the damping ratios are shown for the ideal fluid for c=0,0.1,0.2 forhe differential model at strains of �=0.1,3 ,10. For low strains, there is little differencever this range and approximate TSS is observed over four decades of time. For large �,here is a more dramatic difference due to non-linear effects. As the magnitude of c is

IG. 8. �a� Variations in the BDF �hB�� ;qm ,� ,b�� with power laws in b. �b� Variations in hB�� ;qm ,� ,b� withalues of qm. �c� Variations in hB�� ;qm ,� ,b� with values of �. Strains of 0.1–100 were used. The default
arameters of the plots are b=−0.5, qm=6, and �=5.

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Redistribution

ncreased, the time-strain separable plateau becomes less valid and a more exaggeratedrough and peak occur. However, even at �=10, provided �c��0.1, H�� , t� is constant toithin 5% over three orders of magnitude in t.In Fig. 7 we compare damping estimations H�� , t� produced for ideal fluids in the case

f c=0 and ��=2 / �q−1� at various times 0.1, 1, and 10. For all times, the coincidence ofhe damping ratios H�� , t� shows almost exact TSS that is well approximated by the BDFEq. �4.13��.

Finally, in Fig. 8, we examine how the BDF �hB� varies with the choice of parameters, qm, and �. Decreasing the power law of the elastic modulus from b=−0.2 to b−0.8 has the effect of increasing the contribution from the fast relaxing modes and

ncreases the BDF for strains in the range of 1–10. The BDF is also increased by theranching number qm= q̄i−1. Choosing the average branch number qm=0 which corre-ponds to qi=1 for all i restores the Doi–Edwards damping function because there is nohain stretching in this limit. The BDF is only weakly dependent on the ratio � ofrientation and stretch relaxation times. As � increases, the BDF decreases �Fig. 8�c��.

To test the performance of the BDF, hB�� ,qm ,� ,b�, for a commercial material weompare it to the behavior of the damping ratio, H�� , t�, for the material LDPE1. Theamping ratios are calculated from the differential Pom-pom model using the 12 modepectrum given in Table II. The parameters of the BDF are taken from the material’spectrum with the non-linear parameters for the BDF chosen as the average arm number

m= q̄i−1=5.92 and the average ratio �= r̄i=3.32 and a power law of b=−0.4. In Fig. 9he 12 mode LDPE1 damping ratios are plotted, along with the BDF �Eq. �4.13��, againsttrain for times 0.1, 1, and 10. The coincidence of the damping ratio shows that TSS isalid over these times and the BDF �heavy solid line� is found to be in very goodgreement with the predictions of H�� , t�.

The time range over which the differential Pom-pom model shows TSS begins mucharlier than the longest existing stretch time, where G�� , t� transitions into terminal time

IG. 9. A comparison of BDF �Eq. �4.13�� heavy solid line�, and the damping predictions against strain ofDPE1 at times 0.1, 1, and 10 �dashed lines�. Parameters used are qm=5.92, �=3.32, and b=−0.4. The lightolid line is the Doi–Edwards damping function.

nitary behavior. Although a finite sum of differential Pom-pom modes will not provide


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932 HOYLE et al.

Redistribution

ny case of exact TSS, it closely estimates the existence of a region of TSS. Finally, theDF �Eq. �4.13�� still gives a good prediction to the behavior of a material even with theondition of ��=0 relaxed.

We now look for regions of TSS arising in Pom-pom spectra for a range of branchedolymer melts. We then compare experimental measurements of the damping functionith the predictions of BDF �Eq. �4.13�� where the parameters b, �, and qm are derived

rom the Pom-pom model fits of these materials obtained from the extensional rheology.

. SURVEY OF BRANCHED POLYMERS

In this section, we compare the non-linear stress relaxation of a variety of branchedolyethylenes produced by different synthesis routes. The degree of LCB dramaticallyffects the rheology of a material and in addition to analyzing the performance of theDF �Eq. �4.13�� in capturing TSS, where it exists in low and high density PEs �HDPEs�;e also examine how branching structure affects the existence of TSS in step strain and

he extent to which the relaxation modulus is damped.We investigate materials produced from two different synthesis routes; LDPEs pro-

uced by high pressure free radical polymerization and branched HDPEs produced byetallocene catalyzed polymerization �Peacock �2000��. The LDPE polymers are highly

olydisperse �Mw /Mn�O�10�� with relatively dense branching structures, whereas theetallocene HDPE polymers have a more controlled polydispersity �Mw /Mn�2� and

parse but longer branches. See Table III for material properties.Previous existence of TSS in branched materials has been well documented for LDPEs

Osaki �1993�; Chodankar et al. �2002��; we could therefore expect that the existence ofense branching structures will produce TSS. For branched HDPE, the fluid will containproportion of linear molecules as well as LCB molecules. We might therefore expect to

ee a transient transition from a linear to a branched regime and not see any region ofSS with respect to time. However, experimental evidence shows that both metalloceneaterials display TSS for at least one decade in time.Each material has linear rheological parameters �Gi ,�bi

� and Pom-pom branching pa-ameters ��si

,qi� fitted to experimental oscillatory and extensional data, respectively, us-

TABLE IV. Parameters used for HDPE1–12 mode model.

Mode i

HDPE1 at 155 °C 12 modes

gi

�Pa��b,i


1 219226 0.001 12 179387 0.003 13 37873 0.009 14 32981 0.031 15 18896 0.101 16 11820 0.333 17 6053 1.101 4.0 28 2767 3.636 4.0 29 840.6 12.01 2.0 3

10 224.0 39.67 5.0 511 26.78 131.0 5.0 512 1.946 432.6 5.0 5


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Redistribution

ng “REPTATE” software �see http://www.replate.com� with 12 relaxation modes. The fitsor LDPE1 and HDPE1 are given in Figs. 4 and 12 with parameter values detailed inables II and IV. When fitting the extensional parameters to the data, the steady statelateau is never reached experimentally. This leaves a question over the accuracy of the

values in the spectra as these primarily control the steady state behavior of LCBaterials. However, when carried out carefully, the failure of the sample at the end of

xtensional experiments tends to be reproducible and is associated with the onset of atress plateau. This was demonstrated in controlled-stress extension by Münstedt anduhl �2005�. A plot of the moduli gi and the branch parameter qi against �b for theaterials we survey is shown in Fig. 10.From Fig. 10 we can see that none of the materials perfectly satisfy the power-law

roperty used in the previous section to derive the BDF. This leads to a question in thehoice of power-law parameter b0, in the BDF �Eq. �4.13��. We choose to focus on thearly relaxation time region and take b=−0.5 for LDPE3 and b=−0.4 for the otherDPEs and the two HDPEs. The plot of the qi spectra shows that for each material, a

easonable power-law approximation could be used, with a weak power c�0.2. This is ingreement with the regime found for TSS in our earlier approximations in the BDF �Eq.4.13��.

. Experimental

For the rheological testing of the materials in both shear and uniaxial extensional flow,strain-controlled advanced rheometric expansion system rheometer �ARES� �Rheomet-

ic Scientific� with a force-rebalanced transducer �FRT� �2 K-FRT� was employed. Thepecimens were compression molded at 150 °C and the dimensions at test temperatureere corrected for the thermal expansion. Further rheological tests with respect to the

hermal stability showed that no detectable molecular structure changes took place duringhe experiments.

The non-linear elongational flow behavior was characterized using the uniaxial

IG. 10. Plot of gi �left� and qi �right� against �b for various materials. On the left, none of the materials satisfyhe power-law property used to derive the BDF. On the right, we see that the q-spectra show reasonableower-law agreement, with powers �0.2.

tretching device sentmanat elongational rheometer �SER� �Xpansion Instruments� at-


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934 HOYLE et al.

Redistribution

ached to the ARES rheometer. Different strain rates between 0.001 and 10 s−1 werepplied to compression molded specimens with a width from 3 to 10 mm and a thicknessf about 1 mm.

The step-strain and linear oscillatory shear tests were carried out using various conend plate geometries with different cone angles between 2° and 10° and a diameter of 10m. The step imposition time was about 20 ms and the maximum strain � obtainable was

5. Similar to the procedure described in Stadler et al. �2008�, a series of stress relaxationests with increasing deformation beginning with small and going to high strains waspplied to each specimen. The stress was required to vanish below the noise level at thend of each test before the next one was started. Thus, it was ensured that the remainingtress of the previous strain step was negligible compared to the stress measured in theollowing step.

For reliable stress relaxation measurements, it is necessary that the specimen does notuffer from any structural damage such as edge fracture. It is also necessary to ensure thato wall-slip �Archer et al. �1995�� or inhomogeneous flow �Wang et al. �2006�� occurithin the sample gap. The latter is particularly difficult to rule out without optical

nvestigation. However, a series of repeated relaxation tests with different cone anglesave almost identical stress relaxation curves indicating that the damping functions areime-independent and the time-strain separation principle is valid.

. Results

For each of the three LDPEs and two lightly branched metallocene HDPEs, we haveompared the predictions of the multi-mode Pom-pom model �and the BDF derived fromt� to experimental relaxation data. For all materials, the experimental damping values areetermined by averaging at least one decade of the experimental damping ratio whereSS exits. For the sake of brevity, we will only present the detailed comparison forDPE1 and HDPE1 and show only the comparison between the BDF and the experimen-

ally measured damping function for the other materials.

IG. 11. Left: the 12 mode LDPE1 relaxation modulus G�� , t� and damping ratio H�� , t� for strains 0.1, 5, and. The solid lines represent the experimental data and the dashed lines represent the differential Pom-pomredictions. Right: a comparison of BDF �Eq. �4.13�� heavy solid line�, and the damping ratio predictionsgainst strain of LDPE1 for various strains. Parameters used are qm=5.92, �=3.32, and b=−0.4 The dashed lines the Doi–Edwards damping function.

For LDPE1 we use the 12 mode spectrum presented earlier in Table II. Figure 11


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Redistribution

hows the 12 mode model LDPE1 relaxation modulus and damping ratios for strains of.1, 5, and 7. The solid and dashed lines show the experimental and differential Pom-pomodel, respectively. The differential Pom-pom model is in good agreement with experi-ents in the region of TSS. The discrepancies at early times are due to the effects of an

mperfect strain history in the experiments, which affect the results for times up to around.1 s.

On the right-hand side of Fig. 11, we compare the BDF �Eq. �4.13�� hB�� withxperimental damping values. The parameters used in hB�� are taken from the 12 modepectrum; with average q̄=6.92 so that qm=5.92, average ratio �=3.32, and a power lawf b=−0.4.

We have repeated the comparison for LDPE2 and LDPE3. The 12 mode Pom-pompectra used for LDPE2 and LDPE3 are plotted in Fig. 10. LDPE2 has a high viscosityaking experimental measurements at high strains more difficult for this material.DPE3 material has a lower viscosity than LDPE1, which allows strains of up to 15 to beeasured. The stress relaxation damping ratios predicted by the Pom-pom model show

imilar levels of agreement with those shown for LDPE1 and with TSS for times in theange 0.01–10 s. The BDFs for all three LDPEs are shown in Fig. 14 and are in goodgreement with the experimental results. The parameters used were obtained from theom-pom fit, with qm=4.83, �=3.9, b=−0.4 and qm=5.67, �=2.7, b=−0.5, for LDPE2nd LDPE3, respectively.

In contrast with the two LDPE materials, the metallocene catalyzed HDPE materialshow less extension hardening and are not fitted as accurately by the Pom-pom model.he material HDPE2 shows strain hardening for strain rates in the range 0.01–10 s−1.oth the experimental measurements and the Pom-pom model for this material give TSS

or times in the range of 0.1–30 s and the BDF, with qm=2.5, �=3.5, and b=−0.4, givesxcellent agreement with experimental results seen in Fig. 14.

Finally, we consider HDPE1, which displays the least satisfactory agreement withxperimental data. The extensional data for HDPE1 are shown in Fig. 12 for extension

−1

IG. 12. The comparison of extensional data at 155 °C is made to HDPE1 parameters in Table IV, wheretrain/shear rates of 0.03, 0.1, 0.3, 1, 3, 10, and 30 were used.

nd shear rates from 0.03 to 30 s and only show extension hardening at lower strain


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936 HOYLE et al.

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ates �although it is possible that the experiments underestimated the extensional stress atigher strain rates due to necking�. Consequently, the Pom-pom fit to this model given inable IV has only 6 modes for which qi is greater than unity. Thus, as can be seen in Fig.0, it is furthest from the ideal spectrum in which the q value is approximately constantor all modes. Also, the values of each qi are the lowest of all the spectra modeled.

The experimental and predicted values of the relaxation modulus and damping ratiosre shown in Fig. 13. The experimental data are shown as solid lines and reveal that TSS

IG. 13. Left: the 12 mode HDPE1 relaxation modulus G�� , t� and damping ratio H�� , t� for strains 0.1, 5, and. The solid lines represents the experimental data and the dashed lines represent the differential Pom-pomodel. Right: a comparison of BDF �Eq. �4.13�� heavy solid line�, and the damping predictions data for various

trains. Parameters used are qm=1.3, �=3.75, and b=−0.4 The dashed line is the Doi–Edwards dampingunction.

IG. 14. A comparison of the BDFs produced from the five materials surveyed; a summary of parameters used
an be seen in Table V.

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xists for times in the range 0.1–100 s. However, the predictions of the Pom-pom modelhown as dashed lines show no strict region of TSS. For the smallest strain, largemounts of noise were measured beyond 100 s.

At high strains, the Pom-pom fit correctly predicts the stress at early times �but noturing the step imposition� but overpredicts the stress at later times, suggesting that the qarameter distribution is more uniform than the fitted model. Nevertheless the fitted BDFbtained from the Pom-pom fit with qm=1.3, �=3.75, and b=−0.4 is in good agreementith results �although it does slightly underpredict the experimental damping values atigh strains and a slightly higher value of qm would produce a better fit�.

Figure 14 shows a comparison of the damping functions and BDFs for the five mate-ials we have analyzed. The details of the parameters used are summarized in Table V.he BDFs from top to bottom correspond to the value of qm from highest to lowest. Thisuggests that information on the extent of long-chain branching can be obtained from theamping function via the BDF. In particular, the experimental data are seen to divide intowo groups with the LDPEs showing a slower relaxation with increasing strain than the

ore lightly branched HDPEs.We have surveyed a range of materials to see which set of Pom-pom parameters

extracted from extensional data� showed a region of TSS in step strain. Although none ofhe materials exactly satisfied the conditions for the “ideal model” detailed in Sec. IV, theifferential model predicts approximate TSS for four of the five materials we surveyed;he exception being the lightly branched material HDPE1. In the case of HDPE1, thisiscrepancy could arise from errors in the extensional data to which the spectrum wastted. Furthermore, the BDF �Eq. �4.13�� provides a good approximation of the observedamping behavior for these materials, despite the relaxation in the conditions used toerive it.

I. CONCLUSIONS

We have analyzed the stress relaxation following step strain in multi-mode Pom-pomodels for branched polymers. A single-mode Pom-pom model does not show TSS and

he differential approximation shows qualitatively different terminal time behavior fromhe integral model. However, in a multi-mode Pom-pom model this discrepancy betweenntegral and differential versions does not appear until near the terminal relaxation time.

e also find that increasing the density of modes improves the accuracy of the differen-ial model in a manner that is analogous to the restriction of the “Gibbs phenomena” ofourier series with the addition of terms to the series.

TABLE V. A summary of parameters used in producing BDFs �Fig. 14�for the various materials we survey.

Material b qm �

LDPE1 �0.4 5.92 3.32LDPE2 �0.4 4.83 3.90LDPE3 �0.5 5.67 2.70HDPE2 �0.4 2.50 3.5HDPE1 �0.4 1.30 3.75

The damping ratio H�� , t� was defined to show how well parametrized models are


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ime-strain separable. This tool clearly shows how well the differential model approxi-ates the integral model and the range of time scales over which the stress relaxation is

ime-strain separable.We now turn to the questions raised in Sec. I. The first question was whether there

xist any subsets of parameters for the multi-mode Pom-pom model that give TSS in steptrain. We have shown that a material with a power-law spectrum where the ratio ofrientation to stretch relaxation time and the maximum stretch q are constant for allodes does indeed give TSS.The second question concerned how close to separability do you remain when these

onditions are relaxed. By modeling an “ideal” fluid based on the criteria above, wehowed how deviations from the constant non-linear stretch parameter q to a weak poweraw in the orientation relaxation time affect TSS. We find that provided this dependences weak, there is still a range of times over which the material shows approximate TSS.

The final question concerned the extent to which real materials satisfy these condi-ions. We have surveyed a range of branched polyethylenes produced through two dif-erent synthesis routes to look for differences between the materials. The high pressureolymerization produces more random LCBs than the smoother metallocene catalyzedoute. For all five materials the experimental stress relaxation data showed a range ofSS, and with the exception of one material �HDPE1� the Pom-pom models showed goodgreement with experiments, approximating a region of TSS. Extracting parameters fromhe Pom-pom spectra, a BDF for each material was compared with the data. Despite nonef the materials falling in the subspace of Pom-pom models that predict exact TSS, aegion of TSS was predicted for four of our materials—the exception being HDPE1. Thiseviation arose from the step gradient in the distribution of the non-linear parameter q.ll BDFs predicted the attenuation in the stress relaxation well, including the BDF forDPE1 which captured the damping behavior of the material, despite the reduced TSSredicted by the Pom-pom model. Thus although the parameter values for these materialso not fall into the class of exact TSS, they are close enough to this parameter space sohat TSS is predicted to a good approximation.

Furthermore, both the experimental data and the corresponding BDFs split into tworoups according to the synthesis route of the polyethylene concerned, with the LDPEshowing less attenuation at high strains compared with the HDPEs. This suggests that bytting the branching parameter qm in the BDF to step-strain experiments, we can infer theverage value of q for this material. The branching parameter is an averaged value and sourther techniques, such as Fourier transform rheology, would be needed to obtain preciseetail on the branching structure.

In conclusion, while the materials surveyed did not fall strictly into the subspaceithin which the ideal BDF was derived, the BDF still predicts the experimental damping

unction to within experimental error. This suggests that the BDF is able to capture theamping behavior of a wider class of multi-mode Pom-pom models that show approxi-ate TSS. Furthermore, the two different classes of branched polyethylenes show differ-

nt damping behaviors demonstrating that the step-strain relaxation may be used to char-cterize the degree of LCB in polymer melts.

CKNOWLEDGMENT

The authors would like to acknowledge the funding of the EPSRC-Microscale Poly-
er Processing under Grant No. GR/T11807/01.

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PPENDIX: APPROXIMATING THE INITIAL STRETCH

An essential part of deriving the BDF �Eq. �4.13�� is incorporating the piecewise initialtretch Eq. �4.5�,

�0 = ��1 +1

3�2 1/2

for �0 q

q otherwise,� �A1�

s a fully continuous function dependent on the strain and the q-spectrum.An approximation that interpolates between the small and high strain limits is given by

�0 − 1 =��1+a1/a2�q − 1�

�qma2 + ��1+a1/a2�a2�1/a2

, �A2�

here a2 determines the rate of approach to the high strain limit, and a1 determines thetrain dependence at low and moderate �.

This particular function was chosen because it makes �0−1 a separable function of−1 and �, where �qm+1� is the average number of arms in the spectrum. This gives aood fit of initial stretch for �q−1� near qm �i.e., for modes whose q value is close to theverage�, but under- or overpredicts the gradient of Eq. �4.5� for low and high �q−1�,espectively. Provided that q−1 does not deviate far from qm then this approximationolds.

Figure 15 shows three choices for the parameters a1 and a2. Choosing a1=0, a2=2dotted line� gives a simpler form of the initial stretch equation but looses the � power-aw behavior at low strains. A more accurate but also more complex form of the initialtretch is also pictured �dashed line�, where a1=1 and a2=2. The dash-dotted solid linehows a1= 1

2 and a2=1, the ratio of a1 and a2 is the same as for a1=1 and a2=2 but theransition to the maximum stretch is smoother, similar to that of the simple case a1=0.

1

IG. 15. Various approximations for the initial stretch. The solid lines show the piecewise bounded stretch,espectively. The dotted and dashed lines show predictions for �a1=0 , a2=2� and �a1=1 /2, a2=1�, respec-ively. The dash-dotted line shows the prediction for a1=1 and a2=2.

or a1=0 then �=0.5, a1=1 then �=0.17, and a1= 2 then �=0.25.


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940 HOYLE et al.

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We can now insert this approximation into Eq. �4.4�, so that provided q is constant, theamping ratio is strain separable with the damping function given by

h�� =1

1 + 13�2�1 +

2qm��1 + ��b+1 +

�qm��2

�1 + 2��b+1 , �A3�

here �� is given by

� =��1+a1/a2�q − 1�

�qma2 + ��1+a1/a2�a2�1/a2

. �A4�

A plot of the derived BDF for the various parameter choices used in Fig. 15 is shownn Fig. 16. We use parameters of b=−0.5, �=5, and qm=6. The solid line shows thectual stretch �Eq. �4.5��, which displays a kink at the point of maximum stretch. Theink is captured by the case for a1=1 �dashed line�, but the behavior of the other twoases displays a smoother transition to high strain results. Note that this cusp is a con-equence of choosing ��=0, and with the inclusion of branch-point withdrawal, thisnphysical behavior is removed. Therefore, for cases with ��0 it is desirable to choosearameter values that do not produce this cusp. For the case of a1=0 �dotted line�, theDF becomes greater than one for low strains. This is because the � power-law behaviorf the initial stretch Eq. �4.5� for low strains is lost by the approximation with a1=0.ince we require the damping function to be monotonic, this choice of initial stretcharameters is not suitable for the BDF. The dashed-double dotted line shows the Doi–dwards damping function.

Therefore, in this paper, we choose to use a1= 12 , a2=1 �dash-dotted line in Fig. 16�,

nd �= 14 . This choice provides the required smoothing behavior occurring at maximum

IG. 16. A plot of the various parameter choices used in Fig. 15 substituted into the derived BDF against strain.he solid line shows the actual stretch �Eq. �4.5��. The dotted, dashed, and dash-dotted lines show damping

unctions using approximate stretch parameters �a1=0 , a2=2�, �a1=1 , a2=2�, and �a1=1 /2, a2=1�, respec-ively. The dashed-double dotted line shows the Doi–Edwards damping function. We use parameters of b−0.5, �=5, and qm=6.

tretch and is monotonic to an accuracy of less than 1%.


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Non-linear step strain of branched polymer melts