Stress relaxation dynamics of an entangled polystyrene solution following step strain flow

Stress relaxation dynamics of an entangled polystyrene solution following step strainflowDavid C. Venerus and Ritesh Nair

Citation: Journal of Rheology (1978-present) 50, 59 (2006); doi: 10.1122/1.2135331 View online: http://dx.doi.org/10.1122/1.2135331 View Table of Contents: http://scitation.aip.org/content/sor/journal/jor2/50/1?ver=pdfcov Published by the The Society of Rheology Articles you may be interested in Monte Carlo simulations of stress relaxation of entanglement-free Fraenkel chains. II. Nonlinear polymerviscoelasticity J. Chem. Phys. 126, 074903 (2007); 10.1063/1.2431649 Monte Carlo simulations of stress relaxation of entanglement-free Fraenkel chains. I. Linear polymerviscoelasticity J. Chem. Phys. 126, 074902 (2007); 10.1063/1.2431648 Shape recovery of a dispersed droplet phase and stress relaxation after application of step shear strains in apolystyrene/polycarbonate blend melt J. Rheol. 43, 951 (1999); 10.1122/1.551035 Stress Relaxation and Differential Dynamic Modulus of Polyisobutylene in Large Shearing Deformations J. Rheol. 29, 273 (1985); 10.1122/1.549791 Relaxation of Shear and Normal Stresses in DoubleStep Shear Deformations for a Polystyrene Solution. A Testof the DoiEdwards Theory for Polymer Rheology J. Rheol. 25, 549 (1981); 10.1122/1.549650

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Stress relaxation dynamics of an entangled polystyrenesolution following step strain flow

David C. Venerusa) and Ritesh Nair

Department of Chemical Engineering and Center of Excellence in Polymer Scienceand Engineering, Illinois Institute of Technology, Chicago, Illinois 60616

(Received 8 August 2005; final revision received 16 September 2005�

Synopsis

The stress relaxation behavior of a well-entangled polystyrene following step shear deformationsis investigated. New experimental data are reported for the relaxation of shear stress andfirst normal stress difference that are not affected by imperfect strain history or transducercompliance. Anomalous �type C� shear stress relaxation behavior is observed indicating significantdiscrepancies from the tube model prediction for the damping function. First normal stressdifference data are used to evaluate the Lodge-Meissner relation and small deviations are observed.We find that slip between the test fluid and cone-and-plate fixtures is responsible for the observedtype C behavior, but only leads to small departures from the Lodge-Meissner relation. Simulationsof step strain flows with wall slip show qualitative agreement with experimental results. Inaddition, we examine the issue of polydispersity in the polystyrene standards used to preparewell-entangled solutions for this and similar studies. © 2006 The Society of Rheology.�DOI: 10.1122/1.2135331�

I. INTRODUCTION

The response of entangled polymer liquids to step strain deformations has attracted theattention of experimental and theoretical rheologists for many years. In particular, com-parisons of predictions from the celebrated tube model to step strain experiments onmonodisperse, linear polymers have been scrutinized for more than three decades. De-spite this large effort, there remains controversy over observed discrepancies betweentheory and experiment.

The tube model �Doi and Edwards �1986�� is based on a course-grained polymer chainwith Z entanglements per chain. Z=M /Me, where M is the molecular weight of thepolymer and Me is the entanglement molecular weight �Ferry �1980��. During flow, thetube and primitive chain inside it deform affinely causing the chain to orient and stretch.On a relatively short time scale �R, the stretched chain retracts within the tube to itsequilibrium length. On a relatively long time scale �d, orientation is lost by reptation untilthe tube orientation becomes isotropic. The relationship between the retraction time �R

and reptation time �d is given by �R=�d /3Z �Doi and Edwards �1986��.Step shear flow involves the strain history ��t�=�0H�t�, where H�t� is the Heaviside

step function and �0 is the applied shear strain. For t�0, the shear stress �12, first normal

a�
Author to whom all correspondence should be addressed; electronic mail: [email protected]
© 2006 by The Society of Rheology, Inc.59J. Rheol. 50�1�, 59-75 January/February �2006� 0148-6055/2006/50�1�/59/17/$27.00

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60 D. C. VENERUS AND R. NAIR


stress difference N1=�11−�22 and second normal stress difference N2=�22−�33 are ob-served as functions of �0 and time. It is common to report the shear stress in terms of ashear relaxation modulus G��0 , t�,

G��0,t� =�12��0,t�

�0, �1�

which, in the limit of small strains ��0�1�, is the linear shear relaxation modulus G�t�.The tube model predicts that for t��R the ratio G��0 , t� /G�t� is independent of time.Experimental G��0 , t� data on many entangled polymer liquids display this “time-strainfactorable” behavior. It such cases, it is common to introduce the damping function h��0�,

h��0� =G��0,t�

G�t�. �2�

Numerous studies have been published with G��0 , t� data on linear and branched poly-mers with both narrow and broad molecular weight distributions. In Osaki’s �1993�review of h��0� experiments, a classification scheme was introduced that is based onpredictions of the tube model.

The Lodge-Meissner relation �Lodge and Meissner �1972�� relates the shear stress �12

and first normal stress difference N1 relaxation in step strain flows. For shear flows, the�extinction� angle that diagonalizes the finite strain tensor �� is given by

�� =1

2cot−1��

2� . �3�

The extinction angle for the stress tensor �� in shear flows is given by

�� =1

2cot−1� N1

2�12� . �4�

For an instantaneous, affine deformation applied to an isotropic fluid at t=0, the principalaxes of the stress and strain tensors are coaxial after the deformation: ��0+�=��0+�.Lodge and Meissner �1972� showed that for an arbitrary step strain �0, the principal axesof the stress and strain tensors remain coaxial for t�0: ��t�=��t�. Hence, from Eqs. �3�and �4�, one obtains �Lodge and Meissner �1972��:

N1��0,t��12��0,t�

= �0. �5�

It is well known that measurements of N1��0 , t� are complicated by transducer compli-ance. However, Eq. �5�, has been verified for several moderately entangled �Z�10�polymeric liquids �Vrentas and Graessley �1981�; Venerus and Kahvand �1994��. Predic-tions from the tube model �Doi and Edwards �1986�� are consistent with the result in Eq.�5�.

There are numerous experimental data for linear, entangled and monodisperse polymerliquids that are in nearly quantitative agreement with the tube model predictions for thedamping function h��0�. However, as noted by Venerus �2005�, there is also a largeamount of experimental data for the same class of polymer liquids �typically with Z�20� that are qualitatively different from tube model predictions. Most cases showingthese discrepancies are classified by Osaki �1993� as “type C” behavior where the ratioG��0 , t� /G�t� is less than the tube model prediction for some range of strain and time. Insome of these cases, “kinked” relaxation curves are observed where G��0 , t� is nearly
constant, or even increases, with time.

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61STRESS RELAXATION DYNAMICS OF AN ENTANGLED…


The most widely accepted explanation for anomalous �type C and kinked� stressrelaxation behavior in linear, entangled and monodisperse polymers is wall slip betweenthe fluid and shearing surfaces. The evidence supporting this hypothesis is, however,indirect and in some cases conflicting. For example, Larson et al. �1988� performedbirefringence measurements in Couette flow and found violations of the optical analogueof the Lodge-Meissner relation for a highly entangled �Z�65� polystyrene solution.Particle tracking measurements on essentially the same fluid showed significant motionof tracer particles near the solid-liquid interface �Archer et al. �1995��. On the other hand,Islam et al. �2001� reported birefringence measurements from planar Couette flow of ahighly entangled �Z�31� polystyrene solution that were consistent with the Lodge-Meissner relation, leading the authors to conclude that slip was not the cause for theobserved type C behavior. However, more recent experiments �Archer et al. �2002�;Sanchez-Reyes et al. �2002��, made with cone and plate fixtures coated with micron-sizedglass beads, did not show type C behavior for PS solutions having as many as 37entanglements per chain.

The experimental approaches described above are inconclusive tests for wall slip forseveral reasons. First, birefringence and particle tracking measurements are performed inCouette flow rather than cone and plate flow—the geometry used to obtain virtually allreported G��0 , t� data. Second, in these optical techniques the test fluid is in contact withglass rather than metal, as in cone and plate flow. Third, birefringence measurementsmade with the beam propagating along the vorticity direction �Larson et al. �1988�; Islamet al. �2001�� can be strongly affected by edge effects, and, because of the finite size ofthe laser beam, reflect the average orientation �stress� field of a possibly complex flow.Finally, while it is clear the surface modification techniques implemented by Archer andco-workers �Archer et al. �2002�; Sanchez-Reyes et al. �2002�� influence the solid-fluidinteractions and presumably reduce wall slip, the magnitude and dynamics of wall slip instep strain flows are not well understood.

At present, there does not exist a single case where N1��0 , t� data have been obtainedfrom mechanical measurements in cone and plate flow where anomalous G��0 , t� behav-ior is observed. Hence, direct tests of the Lodge-Meissner relation, Eq. �5�, have not beenpossible for highly entangled polymer liquids that display type C behavior. If such mea-surements did exist, however, it would be possible to examine ��t� for flows showinganomalous stress relaxation behavior.

Venerus �2005� recently examined the large body of experimental stress relaxationdata on linear, entangled and monodisperse polymer solutions and melts. In this work,three new criteria were proposed for anticipating and/or avoiding anomalous step strainexperiments. Two of these criteria are ti /�R�1 and �T /�R�1, where ti is the time re-quired to impose the step strain and �T is the transducer torsional response time. Thesetwo criteria were found to be consistent with a majority of the 60 experiments on linear,entangled, and monodisperse polymers found in the literature �Venerus �2005��. A thirdcriterion was proposed by Venerus �2005� for wall slip

S =ksGN

2 �d

h� 1, �6�

where ks is the slip velocity coefficient and h is the gap between the shearing surfaces.The criterion for wall slip, which is based on slip in steady flow, was not applied toliterature data because of the difficulty in obtaining ks. However, modeling results pre-sented by Venerus �2005� did demonstrate wall slip could explain observations of type C
behavior.

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We now consider the cases not consistent with criteria for imperfect step strain andtransducer compliance proposed by Venerus �2005�. If we focus on false positive cases�i.e., cases where ti /�R�1 and �T /�R�1, but type C or type A/C behavior was observed�,the following observations can be made: they are all for moderately entangled �10�Z�25� polystyrene �PS� solutions composed of the same PS standards �see Tables 1 and 2and Fig. 6 of Venerus �2005��. Also, of these 15 false positive cases, all but two weremade using a strain-controlled rheometer. As noted by Venerus �2005�, a possible expla-nation for these discrepancies is wall slip.

The focus of this investigation is stress relaxation dynamics of highly entangled poly-mer solutions following step shear deformations. We report measurements of bothG��0 , t� and N1��0 , t� on a highly entangled �Z25� polystyrene solution in cone andplate flow. These data, and results from a model of step strain experiments, are used toexamine how wall slip affects the stress relaxation behavior of entangled polymer liquids.In the following section, the preparation and characterization of the test fluid and experi-mental techniques are discussed. In Sec. III, both experimental and modeling results arepresented and discussed; consideration is given to the effects of molecular weight distri-bution on step strain behavior. The conclusions of this study are summarized in the finalsection of this paper.

II. EXPERIMENT

The test fluid used in this study is a polystyrene solution in diethyl phthalate �DEP�.The polystyrene �Tosoh Corp., F-2000� has a reported weight-average molecular weightMw=20 600 kDa and ratio of weight-to number-average molecular weight Mw /Mn=1.2.This is the same PS standard used extensively by Archer and co-workers �Islam et al.�2001�; Sanchez-Reyes et al. �2002��. A 5.0% w/w PS solution was prepared by directlydissolving the PS in DEP in a sealable container. The mixture was gently stirred by handevery few weeks over a period of 9 months. Assuming no volume change on mixing, thePS concentration is 0.056 g/cm3 and the PS volume fraction is �1=0.053. For entangledpolymer solutions, the entanglement molecular weight Me=Me

0 /�11.3, where Me

0

=18 kDa is the entanglement molecular weight for undiluted PS �Ferry �1980��. Hence,for the 5.0% PS-DEP solution, which we designate as PS20M05DEP, the average numberof entanglements per chain is Z25.

PS20M05DEP samples were loaded by placing a blob of fluid on the lower plate andallowing any trapped bubbles to escape, which would require 2 or 3 days. To preventsolvent evaporation during debubbling, the sample was enclosed in a sealed chamberwith a small dead space. No special treatment of the cone and plate fixture surfaces wasused other than multiple cleanings with acetone. All reported data are the average of threeto five repeat experiments.

Measurements of the shear storage G�� and loss G�� moduli from small-amplitude oscillatory shear were conducted in parallel plate flow on a Rheometrics Me-chanical Spectrometer �RMS-800� at temperatures of 25 °C, 50 °C, and 75 °C. An en-vironmental chamber bathed the sample in N2 gas saturated with DEP to minimizesolvent evaporation. The principle of time-temperature superposition was used to shift thedata to a temperature of 25 °C to obtain the dynamic shear modulus data forPS20M05DEP shown in Fig. 1. From these data, we estimate the plateau modulus GN

220 Pa. Also shown in Fig. 1 is a fit to the dynamic modulus data using the discretespectrum of relaxation times �gi ,�i� given in Table I. From this spectrum, we obtain thezero-shear viscosity 022.5 kPa s and the average relaxation time �d415 s. These
characteristic fluid properties are comparable with those obtained by Archer and co-

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workers on a PS solution having the same composition �Islam et al. �2001�; Sanchez-Reyes et al. �2002��. An estimate of �R for this fluid is �R=�d /3Z5.5 s.

All step strain experiments were performed in cone and plate flow on the RMS-800. Aseries of cone and plate fixtures were used with radius R=25.0 mm and different coneangles =0.05, 0.10, and 0.15 rad. These fixtures were machined by Hostettler �ETH,Zürich� from stainless steel and the cone and plate surfaces have a roughness of 1–2 �m.The fixtures were fabricated with an internal channel that allowed oil from a temperature-controlled bath to be circulated through them. Using this configuration, temperature con-trol was maintained to within ±0.1 °C. All step strain experiments were conducted at atemperature of 25.0±0.1 °C.

Figure 2 shows the measured strain �m�t� for a series of applied strains �0 from theRMS-800. From this figure, we see that the time required to impose the strain for thelargest �0 is about 90 ms. Hence, ti�0.9 s and, according to the criterion ti /�R�1�Venerus �2005��, all step strain experiments should be free from the effects of a finitestep imposition time. Figure 3 shows the relaxation of shear stress for �0=2.5 measured

FIG. 1. Dynamic shear storage G� �� and loss G� �� moduli from small-amplitude oscillatory shear ofa PS-DEP solution �PS20MDEP05� at 25 °C. Solid lines are calculated moduli using relaxation spectrum inTable I.

TABLE I. Discrete relaxation spectrum for PS20MDEP05 at 25 °C.

i �i �s� gi �Pa�

1 0.010 86.22 0.032 15.33 0.10 20.04 0.32 26.05 1.0 27.96 3.2 35.77 10.0 42.28 31.6 38.09 100 25.6

10 316 41.511 1000 4.78


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using cone and plate fixtures with three different cone angles �=0.05,0.10,0.15 rad�,which translates to different step imposition times ti. The curves in Fig. 3 show that�12��0 , t� is independent of time for t 0.2 s.

Axial and torsional compliance are known to affect mechanically measured transientstresses in viscoelastic liquids. The dynamics of axial and torsional motion of the trans-ducer must be decoupled from the dynamic response of the fluid for meaningful transientstress data to be obtained. Vrentas and Graessley �1981� suggested that adequate criteriafor this decoupling are �A /�d�1 and �T /�d�1, where �A and �T are the axial and tor-sional response times of the transducer, respectively. Venerus �2005� examined the effectof torsional compliance in more detail and proposed a modified criterion �T /�R�1.Estimates for these response times in cone and plate flow, which are based on modelsusing Newtonian fluids, are as follows: �A=6�0R /3KA and �T=20�0R3 /3KT �Vren-

FIG. 2. Applied strain versus time for step strain experiments with applied strains �0=0.25, 0.5, 1.0, 1.5, 2.5,4.0, 6.3, 10 �bottom to top curves� showing step imposition time ti�0.09 s.

FIG. 3. Shear stress relaxation of PS20MDEP05 for �0=2.5 measured in cone and plate flow with R
=25 mm and cone angles =0.05 rad �solid line�; 0.10 rad �dashed line�; 0.15 rad �dashed-dotted line�.

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tas and Graessley �1981��. For the RMS-800 used in this study, the axial and torsionaltransducer compliances are conservatively estimated to be KA=1 N/�m and KT

=100 N m/rad. For the PS20M05DEP test fluid and a cone angle =0.10 rad, we obtainthe following estimates: �A=10.6 s and �T=0.7 s. The Vrentas and Graessley �1981�criterion for torsional compliance ��T /�d�1� is clearly satisfied, and the more strictcriterion ��T /�R�1� proposed by Venerus �2005� is also satisfied. The Vrentas andGraessley �1981� criterion for axial compliance ��A /�d�1� is also satisfied. A direct testof the effect of axial compliance is given in Fig. 4 where N1��0 , t� data for �0=2.5obtained with three different cone and plate fixtures are shown. Only a small range of �A

values are covered �3.1, 5.9, and 10.6 s�, but it is clear from Fig. 4 that N1��0 , t� areinsensitive to the ratio �A /�d. The deviation at long times for one of the N1��0 , t� curvesshown in Fig. 4 �=0.10 rad, R=14 mm� is most likely due to a slight misalignment ofthe sample, which partially fills the gap.

Based on the results presented in this section, we conclude that both �12��0 , t� andN1��0 , t� data for the PS20M05DEP test fluid obtained in cone and plate flow �R=25.0 mm, =0.10 rad� are free from the effects of finite step imposition time andtransducer compliance.

III. RESULTS AND DISCUSSION

A. Stress relaxation experiments

The shear relaxation modulus G��0 , t� of PS20M05DEP for applied strains �0 in therange 0.1–10.0 are shown in Fig. 5�a�. The solid line in Fig. 5�a� is the linear shearrelaxation modulus G�t� obtained from the spectrum of relaxation times shown in TableI, which is in good agreement with experiment for �0=0.1. The data shown in Fig. 5�a�are typical of an entangled, linear polymer liquid, with G��0 , t� curves having a roughly�0-independent shape for t 200 s. In order to examine more carefully G��0 , t� dynam-ics, Fig. 5�b� shows the ratio G��0 , t� /G�t� versus time. From this figure, we see thatwithin experimental uncertainty, the ratio G��0 , t� /G�t� is independent of time for t

FIG. 4. First normal stress difference relaxation of PS20MDEP05 for �0=2.5 measured in cone and plate flowwith R=25 mm, =0.10 rad �solid line�; R=25 mm, =0.15 rad �dashed line�; R=14 mm, =0.10 rad�dashed-dotted-line�.

200 s, which is consistent with the estimate �d415 s. Figure 5�b� also shows clear


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evidence of chain retraction for t�10 s, which is expected based on the estimate for�R5.5 s. The G��0 , t� /G�t� data in Fig. 5�b� also show signs of an additional relaxationmechanism for 10 s� t�100 s.

Isochronal values of G��0 , t� /G�t� versus applied �0 strain are shown in Fig. 6. Thesolid line in Fig. 6 is the damping function h��0� prediction from the tube model with theindependent alignment approximation �Doi and Edwards �1986��. As expected, for t�3.0 s, the ratio G��0 , t� /G�t� is larger than the tube model prediction because theretraction of stretched polymer chains is not yet complete. For 10 s� t�30 s at strains inthe range 0.5��0�2.5, the ratio G��0 , t� /G�t� starts near the predicted value and thenmonotonically decreases with increasing time. This behavior, which has been observedpreviously for a number of highly entangled polymer solutions, is what Osaki �1993�

FIG. 5. Time dependence of �a� shear modulus G��0 , t� and �b� reduced modulus G��0 , t� /G�t� ofPS20MDEP05 for �0=0.10 ��; 0.25 ��; 0.5 ��; 1.0 ��; 1.5 ��; 2.5 ��; 4.0 ��; 6.3 ��; 10.0 ��. Solidline in �a� is calculated modulus using relaxation spectrum in Table I.

designates as type C behavior. Relative to the results in Fig. 6, Islam et al. �2001�


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observed more pronounced type C and kinked behavior for a slightly more entangled�Z=31� PS-DEP solution. For the same PS-DEP solution, Sanchez-Reyes et al. �2002�reported G��0 , t� /G�t� versus �0 data obtained with bead-coated surfaces displaying verysubtle type C behavior.

The ratio N1��0 , t� /�12��0 , t� versus time for PS20M05DEP for applied strains �0 inthe range 0.5–10.0 are shown in Fig. 7. The dashed lines in Fig. 5�a� indicate the expectedresult based on the Lodge-Meissner �1972� relation, Eq. �5�. To our knowledge, these arethe only mechanical N1��0 , t� data for a highly entangled, linear and monodisperse poly-mer liquid. At first glance, the data in Fig. 7 appear within experimental uncertainty to beconsistent with Eq. �5�. These results are consistent with the birefringence data obtained

FIG. 6. Isochronal plots of G��0 , t� /G�t� versus strain �0 for PS20MDEP05 at t�s�: 0.3 ��; 1.0 ��; 3.0 ��;10.0 ��; 30.0 ��; 100 ��; 300 ��; 500 ��; 800 ��. Thin solid lines connecting symbols are drawn forclarity; heavy solid line is the tube model prediction �with independent alignment� for the damping functionh��0�=1/ �1+4�0

2 /15�.

FIG. 7. Time dependence of ratio of first normal stress difference to shear stress for PS20MDEP05 for �0

=0.5 ��; 1.0 ��; 1.5 ��; 2.5 ��; 4.0 ��; 6.3 ��; 10.0 ��. Dashed lines are predictions of Lodge-
Meissner relation Eq. �5�.

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by Islam et al. �2001�. If the Lodge-Meissner �1972� relation is satisfied, this would seemto suggest that wall-slip was not significant in step strain flows of entangled polymerliquids. In the next section we present results, however, suggesting wall slip does indeedoccur.

B. Stress relaxation simulations

Using a model designed to simulate step strain experiments of entangled linear poly-mers, Venerus �2005� examined the effects of finite step imposition time, torsional trans-ducer compliance and wall slip. The model fluid rheology is given by a modified versionof the differential pompon �DPN� model proposed by Öttinger �2001�. Although devel-oped to describe branched polymers, the DPN model captures the essential features ofstep strain flow behavior of entangled linear polymer liquids �Venerus �2005��. Since wehave ruled out the effects of finite step imposition time �ti /�R�1� and torsional trans-ducer compliance ��T /�R�1�, our focus here is on wall slip. We use Z=25, which means�d /�R=75, to match the simulation results with experiments.

The imposed strain rate is given by �̇0=�0�H�t�−H�t− ti�� / ti, where H�t� is the Heavi-side step function. In this study, we fix the parameters ti /�d=10−3 and �T /�d=0. In theabsence of compliance, the actual strain rate in the fluid �̇ differs from the imposed strainrate �̇0 according to �̇= �̇0−2d�s /dt, where �s is the strain lost by slip between the fluidand plate. As in Venerus �2005�, �s is obtained from an expression suggested by recentsteady flow experiments �Sanchez-Reyes and Archer �2002��

d�s

d�t/�d�= S� �12

GN�2

, �7�

where S is the dimensionless wall slip parameter defined in Eq. �6�. To estimate S we usethe properties of the PS20M05DEP fluid �GN=220 Pa, �d=415 s� and cone and plategeometry �hR=2 mm�. From steady flow experiments on a similar PS solution,Sanchez-Reyes and Archer �2002� obtain a value for the slip velocity coefficient ks

10−3 �m/Pa2 s, which gives S10. As will be seen shortly, this value for S is too largeto be consistent with our experiments. Hence, for the simulations presented in this sec-tion, we use S=1.0.

Simulated G��0 , t� curves for applied strains �0 in the range 0.1–10.0 are shown assolid lines in Fig. 8. For comparison, dashed lines for the no slip case �S=0� are shownfor three strains. Similar results were presented by Venerus �2005� using a different valuefor S�=0.5�. The solid curves in Fig. 8 are qualitatively consistent with the experimentalG��0 , t� data in Fig. 5�a�. These curves are typical of entangled polymer liquids; bythemselves �i.e., if the dashed lines were not present� they do not show any strongevidence that wall slip was occurring. In fact, it is only for the pair of curves �dashed andsolid� for �0=1.0 that significant differences are observed. The curves for S=1.0 shownin Fig. 8, and other simulations done with larger values of S�=10.0�, do not show any signof a kinked shape that has been observed in several published G��0 , t� data.

Isochronal values of G��0 , t� /G�t� from Fig. 8 versus applied �0 strain are shown inFig. 9 along with the damping function h��0� prediction from the tube model. Thesimulation results in Fig. 9, which are nearly in quantitative agreement with the experi-mental data in Fig. 7, clearly show type C behavior. Hence, as noted by Venerus �2005�,wall slip can lead to type C behavior in step strain flows. These results are also consistent
with the studies of Archer and co-workers �Archer et al. �2002�; Sanchez-Reyes et al.

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�2002�� who demonstrated type C behavior is significantly reduced in highly entangledPS solutions if experiments are performed with bead-coated shearing surfaces, whichpresumably reduces the effective slip velocity coefficient ks.

Figure 10 shows the ratio N1��0 , t� /�12��0 , t� versus time from the simulations withS=1.0 for applied strains �0 in the range 0.1–10.0. The curves in this figure are remark-ably consistent with the experimental data shown in Fig. 7. In particular, the curves inFig. 10 and data in Fig. 7 both show good agreement with the Lodge-Meissner relation�dashed lines� initially, followed by a subtle transient where N1��0 , t� /�12��0 , t� fallsbelow the applied strain. For t /�d 1, the ratio N1��0 , t� /�12��0 , t� increases, which seemssomewhat unphysical. Given the difficulty in obtaining N1��0 , t�, and the increased un-certainty in taking the ratio of decaying functions, it was concluded earlier that the small

FIG. 8. Simulated stress relaxation modulus for �0=0.1, 0.5, 1.0, 1.5, 2.5, 4.0, 6.3, 10 �top to bottom curves�for the DPN model �Z=25� with slip parameter S=1.0. Dashed lines are for �0=0.1, 1.0, 10 �top to bottom� forno slip case �S=0�.

FIG. 9. Isochronal plots of G��0 , t� /G�t� versus strain �0 for the DPN model �Z=25� with slip parameter S=1.0 at t /�d: 0.01 ��; 0.03 ��; 0.1 ��; 0.3 ��; 1.0 ��; �3.0 ��. Thin solid lines connecting symbols aredrawn for clarity; heavy solid line is the tube model prediction �with independent alignment� for the damping

2
function h��0�=1/ �1+4�0 /15�.

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deviations from the Lodge-Meissner relation were within experimental error. The consis-tency between the experiments and simulations suggests that the small deviations fromthe Lodge-Meissner relation are significant.

The dynamic behavior of the strain lost due to wall slip �s for S=1.0 are shown in Fig.11�a�. The simulation results in this figure show that for small to moderate applied strains�0.5��0�1.5� most of the slip occurs well after chain retraction �t /�d�10−1��R /�d�,while for larger strains �4��0�10� slip occurs before chain retraction �t /�d�10−2

��R /�d�. The “delayed slip” shown in Fig. 11�a� for moderate applied strains is consis-tent with tracer particle measurements reported by Archer et al. �1995�. Also, from Fig.11�a�, we see that the equilibrium slip strain �s�� is a nonmonotonic function of appliedstrain �0. This behavior can be understood as follows. The strain lost during the imposi-tion of the strain is �s��0

2ti2 and after the step is �s��0

2�d2h��0�2. For small strains, since

�0 is not large and h��0��1, most of the strain is lost after the step. For large �0 whereh��0��1, most of the strain is lost during the step. A plot of 2�s�� /�0 versus appliedstrain from the simulation is shown in Fig. 11�b�. These results show the fraction ofapplied strain lost due to slip reaches a maximum of 75% for �01.2 and saturates to avalue of roughly 15% for large �0. Also shown in this figure are the measurements ofArcher et al. �1995� for a very highly entangled �Z�65� PS solution, which are inqualitative agreement with the simulation results. The dynamics of �s and its dependenceon �0 provide an explanation for the type C damping behavior shown in Figs. 6 and 9.

We now address the question of why step strain flows with significant levels of wallslip still exhibit approximate agreement with the Lodge-Meissner relation. Indeed, evenin cases when as much as 75% of the applied strain is lost due to wall slip, relativelysmall deviations from the Lodge-Meissner relation are observed. In Fig. 12, the extinc-tion angles for strain �� and stress �� defined in Eqs. �3� and �4�, respectively, are shownfor �0=1 and 10. Figure 12 also shows the experimental �� for the PS20M05DEP fluidand for the very highly entangled �Z�65� PS solution studied by Larson et al. �1988�.Immediately following the step, ��; then both increase towards the equilibrium value�� /4� as a result of wall slip with �� increasing at a slightly faster rate than ��. Hence,even though slip occurs after the step, the extinction angles for strain �� and stress �� are

FIG. 10. Simulated time dependence of ratio of first normal stress difference to shear stress for �0=0.5, 1.0,1.5, 2.5, 4.0, 6.3, 10 �top to bottom curves� for the DPN model �Z=25� with slip parameter S=1.0. Dashed linesare predictions of Lodge-Meissner relation Eq. �5�.

nearly coaxial the Lodge-Meissner relation, Eq. �5�, is approximately satisfied.


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C. Molecular weight distribution effects

In an attempt to study highly entangled polymer liquids while avoiding the transducercompliance difficulties associated with high-modulus fluids, many researchers have usedultrahigh molecular weight �Mw�4000 kDa� polystyrene standards �Mw /Mn�1.2� dis-solved in organic solvents. A large percentage of published step strain experiments on PSsolutions �including the present study� have used standards with Mw=4480, 8420, and20600 kDa from Tosoh Corp. with Mw /Mn1.2. In this section, we consider the effect ofpoly-dispersity on the behavior of entangled, linear and “nearly” monodisperse polymersin step strain flows.

First, we re-examine the G�� and loss G�� measurements for PS20M05DEPshown in Fig. 1. Close inspection of these data reveals a broadened transition between theterminal and rubbery regions for 10−2 rad/s��10−1 rad/s. Published dynamic modu-lus measurements from similar polystyrene solutions made with the same PS standardsalso show this feature �Islam et al. �2001�; Sanchez-Reyes et al. �2002�; Sanchez-Reyesand Archer �2002��.

FIG. 11. Simulated slip strain for the DPN model �Z=25� with slip parameter S=1.0. �a� time dependence ofslip strain for �0=0.5, 1.0, 1.5, 2.5, 4.0, 6.3, 10 �bottom to top curves�; �b� equilibrium fractional slip strainversus applied strain. Symbols are tracer particle measurements on a PS/TCP solution from Archer et al. �1995�.

It is sometimes more convenient to examine subtle features of dynamic modulus data


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by plotting the dynamic viscosity ��=G� /� ,�=G� /�� in the form of a Cole-Cole plot.A dimensionless Cole-Cole plot � /0 versus � /0 for a single mode Maxwell fluid isa semicircle with origin at �0,1 /2�. This case is effectively that of the original tube model�i.e., an infinitely long chain undergoing with reptative motion only�. Other relaxationmechanisms such as constraint release and polydispersity and branching tend to withflatten the semicircular shape of � /0 versus � /0 �Garcia-Franco and Mead �1999��.The dimensionless Cole-Cole plot for a well-entangled, linear and nearly monodispersepolymer liquid should be a symmetric, slightly flattened semicircle.

Figure 13 shows a plot of � /0 versus � /0 for PS20M05DEP �using the data in

FIG. 12. Simulated dynamics of extinction angles for strain �� and stress �� tensors defined in Eqs. �3� and �4�for �0=1.0 and 10 �top and bottom sets of curves� for the DPN model �Z=25� with slip parameter S=1.0.Symbols are experimental �� results for PS20MDEP05 �0=1.0 ��, �0=10.0 ��; PS/TCP solution fromLarson et al. �1988� �0=1.0 ��.

FIG. 13. Cole-Cole plot comparison of single element Maxwell fluid behavior �solid line� and experimentaldata �symbols� for different PS systems with Z25: �� PS20MDEP05 from this study; �� PS-DEP solutionfrom Sanchez-Reyes and Archer �2002�; �� PS465K melt from Luap et al., �2005�. Inset shows GPC trace
from Tosoh Corp. for PS standard �F-2000� used to make PS-DEP solutions.

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Fig. 1�. Also shown in this figure are data for a PS-DEP solution with the same compo-sition taken from Fig. 1 of Sanchez-Reyes and Archer �2002�, and data for a PS465Kmelt �Mw /Mn=1.08� with Z25 taken from Fig. 1 of Luap et al. �2005�. As expected,the small differences in polymer concentration and temperature are absorbed into 0, andthe data for the two PS-DEP solutions shown in Fig. 13 are in good agreement. Moreimportantly, the shapes of the Cole-Cole plots in Fig. 13 for the narrow molecular weightdistribution melt and broader molecular weight distribution solutions are qualitativelydifferent: the former is nearly symmetric about the point �0,1 /2� while the latter areskewed to the right. This difference is an indication of the higher polydispersity in thePS-DEP solution. The inset to Fig. 13 shows GPC traces from Tosoh Corp. for several PSincluding the Mw=20600 kDa �F-2000�, which show a significant high-elution time tailindicating the presence of a small molecular weight tail.

Additional insight on the effect of polydispersity can be gained by examination of therelaxation spectra obtained from dynamic modulus data. Figure 14 shows the discrete�gi ,�i� spectrum of relaxation times from in Table I and continuous spectrum H�� ob-tained using the nonlinear regularization method of Honnerkamp and Weese �1993� forPS20M05DEP. This figure clearly shows good agreement between the two spectra andthe presence of a second peak in the relaxation spectrum 10–20 times smaller than thelongest relaxation time � /�d=1. This peak is consistent with the GPC trace shown in Fig.13 and the suggestion that there is a significant amount of shorter chains in the20600 kDa PS standard. Signs of a relaxation process with a time of roughly �d /10 is alsoevident in the stress relaxation data shown Fig. 5. The continuous spectra H�� for thePS-DEP solution of Sanchez-Reyes and Archer �2002� shown in Fig. 14 has a shapesimilar to that of the PS20M05DEP fluid from this study. Figure 14 also shows H�� forthe PS465K melt of Luap et al., �2005�, which does not show clear evidence of a secondpeak.

IV. CONCLUSIONS

The stress relaxation behavior of a well-entangled polystyrene following step shear

FIG. 14. Relaxation spectra for different PS systems with Z25: �� and heavy solid line� PS20MDEP05 fromthis study; �thin solid line� PS-DEP solution from Sanchez-Reyes and Archer �2002�; �dashed-dotted line�PS465K melt from Luap et al., �2005�.

deformations has been investigated. The polystyrene-diethyl phthalate solution used in


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this work has the same composition as that used in several previous investigations and, inthe context of the tube model, has roughly 25 entanglements per chain. New experimentaldata for the relaxation of shear stress and first normal stress difference have been re-ported. These data have been obtained from cone-and-plate flow experiments designedsuch that they satisfy two of the three criteria recently proposed by Venerus �2005� foravoiding artifacts due to an imperfect strain history and transducer compliance.

The experimental shear modulus shows monotonic �non-kinked� relaxation and time-strain factorability for times comparable to the longest relaxation time ��d� estimatedfrom linear viscoelastic measurements. However, the experimental damping function fallswell below the tube model prediction �type C behavior� at modest applied strains ��0

�1�. First normal stress difference data are used to evaluate the Lodge-Meissner relationand small deviations are observed. It is suggested that slip between the test fluid andcone-and-plate fixtures is responsible for the observed type C behavior. Interestingly, wallslip does not appear to have a significant effect on the validity of the Lodge-Meissnerrelation.

Simulations of step strain flows with wall slip show qualitative agreement with ex-perimental results. This model involves an unknown slip parameter that is related to thethird criterion proposed by Venerus �2005� for wall slip. In particular, the simulationsshow that while wall slip leads to rather pronounced type C behavior in the dampingfunction, it has only minor effects on the Lodge-Meissner relation. Simulation and ex-perimental results also indicate the strain lost due to slip, relative to the applied strain,goes through a pronounced maximum at applied strains �0�1.

The issue of polydispersity in the polystyrene standards used to prepare well-entangledsolutions for this and similar studies is also considered. It appears that some commonlyused, high molecular weight polystyrene standards �Mw�4000 kDa�, have high levels ofsmaller molecular weight chains that result in intermediate time scale relaxation pro-cesses.

The results found in this study are confirmation of the importance of wall slip in stepstrain flows of well-entangled polymer liquids. It should be noted that this result in noway contradicts the results found by Venerus �2005�—one must pay attention to thepossibility that both an imperfect strain history and transducer compliance can lead toanomalous experiments. Wall slip, a complex phenomenon that is not well understood,appears to be equally important. What remains unanswered is if wall-slip can be elimi-nated by careful test fluid design �polymer chemistry, molecular weight concentration andsolvent� and experimental configuration, or if surface modification techniques such asthose developed by Archer and co-workers are necessary.

ACKNOWLEDGMENTS

The authors acknowledge helpful discussions with Professor H.C. Öttinger �ETH,Zürich� and Professor J.D. Schieber �IIT, Chicago�, and thank Dr. T. Schweizer �ETH,Zürich� and Professor C. Friedrich �Albert-Ludwigs-Universität, Freiburg� for the relax-ation spectrum calculations and Mr. Jürg Hosttetler �ETH, Zürich� for machining the coneand plate fixtures.

References

Archer, L. A., Y.-L. Chen, and R. G. Larson, “Delayed lip after step strains in highly entangled polystyrene

solutions,” J. Rheol. 39, 519–525 �1995�.


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Archer, L. A., J. Sanchez-Reyes, and M. T. Islam, “Relaxation dynamics of polymer liquids in nonlinear step

strain,” Macromolecules 35, 10216–10224 �2002�.Doi, M., and S. F. Edwards, The Theory of Polymer Dynamics �Oxford University Press, Oxford, 1986�.Ferry, J. D., Viscoelastic Properties of Polymers, 3rd ed. �Wiley, New York, 1980�.Garcia-Franco, C. A., and D. W. Mead, “Rheological and molecular characterization of linear backbone flexible

polymers with the Cole-Cole model relaxation spectrum,” Rheol. Acta 38, 34–47 �1999�.Honnerkamp, J. and J. Weese, “A nonlinear regularization method for the calculation of relaxation spectra,”

Rheol. Acta 39, 163–173 �1993�.Islam, M. T., J. Sanchez-Reyes, and L. A. Archer, “Nonlinear rheology of highly entangled polymer liquids:

Step shear damping function,” J. Rheol. 45, 61–82 �2001�.Luap, C., C. Müller, T. Schweizer, and D. C. Venerus, “Simultaneous stress and birefringence measurements

during uniaxial elongation of polystyrene melts with narrow molecular weight distribution,” J. Rheol. 45,

83–91 �2005�.Larson, R. G., S. A. Khan, and V. R. Raju, “Relaxation of stress and birefringence in polymers of high

molecular,” J. Rheol. 32, 145–151 �1988�.Lodge, A. S. and J. Meissner, “On the use of instantaneous strains, superposed on shear and elongational flows

of polymeric liquids, to test the Gaussian network hypothesis and to estimate segment concentration and its

variation during flow,” Rheol. Acta 11, 351–352 �1972�.Osaki, K., “On the damping function of shear relaxation modulus for entangled polymers,” Rheol. Acta 32,

429–437 �1993�.Öttinger, H. C., “Thermodynamic admissibility of the pompon model for branched polymers,” Rheol. Acta 40,

317–321 �2001�.Sanchez-Reyes, J., M. T. Islam, and L. A. Archer, “Step shear dynamics of highly entangled polymer liquids,”

Macromolecules 35, 5194–5202 �2002�.Sanchez-Reyes, J. and L. A. Archer, “Steady shear rheology of entangled polymer liquids: Implications of

interfacial slip,” J. Rheol. 46, 1239–1262 �2002�.Venerus, D. C. and H. Kahvand, “Normal stress relaxation in reversing double-step strain flows,” J. Rheol. 38,

1297–1315 �1994�.Vrentas, C. M. and W. W. Graessley, “Relaxation of shear and normal stress components in step strain experi-

ments,” J. Non-Newtonian Fluid Mech. 9, 339–355 �1981�.Venerus, D. C., “A critical evaluation of step strain flows of entangled linear polymer liquids,” J. Rheol. 49,

277–295 �2005�.


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Stress relaxation dynamics of an entangled polystyrene solution following step strain flow

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