pom-pom model predictions on nonlinear stress relaxation

Pom-pom model predictions on nonlinear stress relaxationin single-step strain flow

Sheng Cheng Shie & Tzy Ming Yang & Chi Chung Hua

Received: 3 July 2007 /Accepted: 10 October 2007 /Published online: 20 November 2007# Springer Science + Business Media B.V. 2007

Abstract A phenomenological pom-pom model, whichutilizes linear stress relaxation data as an input, is proposedfor simulating the detailed retraction behavior of a fullpom-pom chain as well as the associated nonlinear stressrelaxation in single-step shear flows. Specifically, theinitial, Rouse-like arm and/or backbone retraction as wellas the long-time, renormalized (dumbbell-like) backboneretraction is simulated at one time, and a possible couplingbetween linear orientation relaxation and nonlinear stretchrelaxation due to diffusive or convective constraint releaseis self-consistently accounted for. The model’s predictionsare systematically tested against nonlinear stress relaxationdata on three nearly monodisperse pom-pom melts, includ-ing two six-arm pom-pom polybutadienes (PPBDs) and anH-shaped polyisoprene (PPI). The model can describe thepresent data reasonably well by incorporating a previouslyproposed effect of drag–strain coupling. Accounting furtherfor the effect of constraint release, due primarily to armdiffusive motion and its coupling with a known polydis-persity in arm molecular weight distribution, appreciablybroadens the predicted nonlinear stress relaxation andsubstantially improves the results of theory/data compar-isons. The significance of the current findings is discussed.

Keywords Computer simulation . Entangled branchedpolymer . Nonlinear viscoelasticity . The pom-pommodel

Introduction

Several peculiar rheological features of branched polymermelts have recently been demonstrated to be well capturedby the pom-pom molecular theory established by McLeishet al. [1–3]. So far, applications of the pom-pom model fallinto two main categories: The first was primarily concernedwith linear viscoelastic properties of entangled branched orpom-pom polymer liquids [4–9]. The second took advan-tage of multi-mode pom-pom constitutive equations foranalyzing the flow behaviors of commercial branchedpolymers [10, 11], or for a direct evaluation of theconstitutive equations themselves [12, 13]. As a scrutinyinto the full-chain stretch relaxation behavior, we haverecently constructed a stochastic pom-pom model forinvestigating the nonlinear stress relaxation of a nearlymonodisperse H-shaped polyisoprene melt in single-stepstrain flows [14]. The previous study largely confirmed thefundamental proposal in the pom-pom model as to apeculiar partial arm withdrawal (PAW) after a criticalbackbone stretching. It, nevertheless, appeared that moreextensive studies were necessary in order to fully explorethe essential relaxation mechanisms available to a pom-pomchain under nonlinear step–strain flows.

The present work is interested in making use of a self-consistent, phenomenological formulation of the pom-pommodel for simulating the full-chain stretch relaxation of anentangled pom-pom chain in single-step strain flows.Through direct theory/data comparisons using nonlinearstress relaxation data reported in the literature on the meltsof two six-arm pom-pom polybutadienes [4, 7] and anH-shaped polyisoprene [2], we show that the individualeffects of essential nonlinear relaxation mechanisms of anentangled pom-pom chain can be systematically explored,and, on the other hand, the performance of the proposedpom-pom constitutive equations can be evaluated.

J Polym Res (2008) 15:213–224DOI 10.1007/s10965-007-9161-3

Electronic supplementary material The online version of this article(doi: 10.1007/s10965-007-9161-3) contains supplementary material,which is available to authorized users.

S. C. Shie : T. M. Yang : C. C. Hua (*)Department of Chemical Engineering,National Chung Cheng University,Chia Yi 621 Taiwan, Republic of Chinae-mail: [email protected]

http://dx.doi.org/10.1007/s10965-007-9161-3

The special features of the proposed pom-pom constitu-tive equations lie in that they utilize a whole set of linearrelaxation data as an input to predict the associatednonlinear relaxation behavior, and that the effects of apossible coupling between linear orientation relaxation andnonlinear stretch relaxation are self-consistently accountedfor using an existing formulation of constraint release forbidisperse linear-chain systems [15]. Similar to a previouslyproposed stochastic pom-pom model [14], the detailedretraction behavior of a full pom-pom chain, including theinitial Rouse-like arm and/or backbone retraction as well asthe long-time renormalized (dumbbell-like) backbone re-traction, is simulated at one time. In addition, severalnonlinear effects not previously considered within the pom-pom model are systematically examined. They include apossible nonaffine deformation [16, 17] or partial strandextension (PSE) [18], as well as a known polydispersity inthe molecular weight distribution of the pom-pom samplesunder investigation.

The central observations of this study are as follows: Atstrains exceeding the predicted onset of PAW, nonlinearrelaxation features characteristic of pom-pom chain retrac-tion become evident for all three cases examined. In thisrespect, the experimental data seem to be well described bythe theoretical predictions using a previously proposedeffect of drag–strain coupling [3]. Accounting further for aneffect of constraint release due to arm diffusive motion,along with its coupling with a known polydispersity in armmolecular weight distribution, appreciably broadens thepredicted backbone stretch relaxation and substantiallyimproves the results of theory/data comparisons.

This paper is organized in the following way: We firstintroduce the background theory and a phenomenologicalformulation of the pom-pom model for simulating nonlinearrelaxation properties in single-step strain flows. Afterward,we discuss the results of theory/data comparisons, andconclude the central findings and implication of this study.

Background theory

The pom-pom model

Pom-pom polymers belong to a class of branched polymersthat are composed of two symmetric arm polymers beingconnected by a backbone polymer. The simplest such apolymer is the H-shaped polymer, for which the linear andthe nonlinear stress relaxations have been thoroughlyinvestigated by neutron scattering and viscoelastic measure-ments [2] as well as by a stochastic simulation [14]. Thecentral breakthrough in modeling the rheological propertiesof pom-pom melts has been initiated by McLeish et al. [1,2], who proposed that the arm polymer essentially behaves

as a usual entangled star arm, while the long-time,renormalized dynamics of the backbone polymer resemblesthat of an entangled linear chain with, however, thefrictional drag being concentrated on the two branch points.As an important consequence, the backbone polymershould exceptionally behave dumbbell-like. Moreover, asthe chain tension on the backbone polymer becomessufficient to pull the arm polymers into the tube previouslyoccupied by the backbone, a peculiar partial arm withdrawal(PAW) is predicted to occur. In this case, it has been notedthat the initial arm and backbone retractions should beRouse-like until PAW is complete, since then a renormalized,dumbbell-like backbone retraction starts to take over [14].According to the pom-pom model, the critical backbonestretch for the onset of PAW is solely determined by thenumber of arms of each pom-pom chain [1], as we introducelater.

The peculiar features of the pom-pom polymer dynamicsoutlined above may, in principle, be directly tested in linear ornonlinear stress relaxation experiment. For instance, existinglinear viscoelastic data on pom-pom melts seem to wellconfirm a reptation-like feature for the long-time backbonedynamics [2, 4–7]. On the other hand, the characteristicbackbone retraction may, in principle, be discriminated insingle-step strain experiments, as we demonstrated in anearly stochastic simulation [14]. However, due to anpostulated effect of the so-called drag–strain coupling [3]as well as the effects of constraint release, it becomesdifficult in practice to discriminate the proposed dumbbell-like feature for the long-time, renormalized backboneretraction. These issues will be further discussed later.

As far as a monodisperse pom-pom sample is concerned(the effect of polydispersity will be considered later), theprincipal time constants underlying the linear and thenonlinear relaxations of a pom-pom chain are as follows:arm retraction time (C as), arm orientation relaxation time(C a), (renormalized) backbone retraction time (C bs, which isto be contrasted with the usual backbone Rouse time,denoted as C b,R), and backbone orientation relaxation time(C b). These four time constants, altogether, have beenproposed to underlie a hierarchy of the principal relaxationsof an entangled pom-pom chain subjected to a nonlinearstep deformation [1, 2]. If one further assumes that armretraction follows the usual Rouse behavior, the followingexpressions can be arrived at [1, 2]:

ta ¼ t0S1:5a exp

15

4Sa

1

2� 1� wbð Þ 1

3

� �� ; ð1Þ

tb ¼ qtaw2bS

2b ; ð2Þ

214 S. C. Shie, et al.

tbs ¼ 5

2qtawbSb; ð3Þ

tas ¼ S2a t0; ð4Þwhere Sa and Sb are the numbers of entanglements perarm or backbone polymer, respectively, C0 is the Rousetime of one entanglement, wb is the weight fraction ofthe backbone polymer, and q is the number of arms oneach side of a pom-pom backbone. The previous expres-sions for Cb and Cbs, respectively, have been employed inan early stochastic simulation to well capture the linearand the nonlinear stress relaxations of an H-shapedpolyisoprene (q=2) [14]. For a six-arm pom-pom polybu-tadiene (q=3), it is assumed here that the separationbetween backbone stretch and orientation relaxations,currently given by C b=C bs ¼ 2=5wbSb , is independent ofthe number of arms, q, as implied by the original theory.Moreover, since linear relaxation data will be directlyutilized to describe the primary orientation relaxation,the apparent value of C b can thus be determined fromthe fit to data, and the rest of time constants followimmediately the relations prescribed in Eqs. 1–4.

For the case of the renormalized, dumbbell-like back-bone retraction, which generally dominates the nonlinearrelaxation of a pom-pom chain on typical experimental timescales, the following formula accounting for the effect ofdrag–strain coupling due to local branch-point displacementhas been proposed in a later modification to the originalpom-pom model [3]:

1

Leq

@L tð Þ@t

¼ � L tð Þ�Leq � 1

C bsexp ν�

L tð ÞLeq

� 1

� �� ; ð5Þ

with an initial condition L t ¼ 0þð Þ�Leq ¼ h E � ujj ieq. Inthis formula, L(t) denotes the mean instantaneous chainlength, E is the displacement gradient tensor of the fingerstrain tensor, u is the unit orientation vector of a polymerentanglement, :::h i represents taking the ensemble average,and the subscript associated with the brackets means thatthe quantity within it is evaluated at equilibrium. Note thatν� :¼ 81

1401� 1�wað Þ7=3 1þ7wa=3ð Þ

w2aqk

�

� is a very important parameter in

the pom-pom model for describing a postulated effect oflocal branch-point displacement, where wa ¼ 1� wbð Þ is theweight fraction of the arm polymer, and k* is an unknownconstant (independent of molecular weight) describing themagnitude of an assumed quadratic potential for localbranch-point displacement. Later, the previous formula willbe used to find the theoretical value of ν* for each of thethree pom-pom samples investigated in this work. Moredetailed discussion about the theory of drag–strain couplingand its general impact on the model’s predictions can befound in Blackwell et al. [3].

According to the original theory, which assumes k*=0.36, one has ν� ’ 1 and ν� ’ 2, respectively, for thePPBD samples and the PPI sample investigated in thiswork. In practice, this parameter has sometimes beentreated as adjustable, in particular in simulating the flowbehavior of commercial branched polymers [10, 11]. Thus,the effect of the value assigned to this parameter will alsobe briefly examined in later theory/data comparisons. Notealso that ν*=0 corresponds to the case of the usualdumbbell retraction, and is currently used as an approxi-mation to describe the initial, Rouse-like arm or backboneretraction as PAW is permitted to proceed.

A phenomenological formulation of nonlinear stressrelaxation in single-step strain flows

After describing the basic features of the pom-pom model,we proceed with a phenomenological formulation of it thatis especially convenient and versatile for simulating thenonlinear relaxation of an entangled pom-pom chain innonlinear step–strain flows, such as single-step shear flows.Generally speaking, solving the original set of pom-pomconstitutive equations is a tedious work to do. As animportant consequence, it becomes difficult for people totest their own data against the predictions of the originalpom-pom model. By contrast, an important feature of thepresent reformulation is that the linear orientation relaxationor, equivalently, the so-called tube survival probability,8 tð Þ ¼ G tð Þ

.G 0ð Þ

N , is directly fitted from stress relaxationdata at a small strain, γ≈0.1, where G(t) and G 0ð Þ

N are thelinear relaxation modulus and the plateau modulus, respec-tively. This strategy not only greatly simplifies theconstruction of pom-pom constitutive equations for anonlinear step–strain flow, but it also permits a scrutinyinto the stress relaxation properties pertinent to theunderlying chain retraction behavior, as we demonstrate inthis work. The detailed formulation and its centralassumptions/justifications are discussed below.

Analogous to the case of a binary mixture of entangledlinear chains [15], 8 (t) is split into two parts accounting forthe contributions from arm (8 a(t)) and backbone (8 b(t))polymers, respectively:

G tð Þ.G 0ð Þ

N ¼ 8 tð Þ ¼ waPNa

i¼1

wiwa

exp �t=C ið Þ

þwbPNaþNb

j¼Naþ1

wj

wbexp �t

�C j

� ¼ wa8 a tð Þ

þ wb8 b tð Þ

ð6Þ

where wi(wj) denotes the weighting of the ith ( jth)relaxation mode in linear stress relaxation, Ci(Cj) is theassociated time constant, and Na and Nb are the totalnumbers of Maxwell modes used for arm and backbone

Pom-pom model predictions on nonlinear stress relaxation in single-step strain flow 215

polymers, respectively. Table 1 summarizes the basicinformation of the three pom-pom samples investigated.Note that the linear relaxation function, 8 a(t) or 8 b(t), soconstructed has a value of unity at t=0 (i.e., the onset of theplateau regime) and a vanishing value on a time scalecorresponding to arm or backbone orientation relaxation.Moreover, the present choice of relaxation modes (seedetailed specifications in Table 2) leads to results inagreement with the well-known dilution effect, G tð Þ �G 0ð Þ

N waþ1b , upon complete arm relaxation (i.e., at t � 5 �

10ta ), where α ¼ 4=3 is the experimentally observeddilution exponent accounting for an important couplingbetween the linear relaxations of the arm and backbonepolymers, respectively [7]. The previous agreement, in fact,might also justify the formulation in Eq. 6 treating the armand backbone materials independently. A similar couplingdue to constraint release during nonlinear chain relaxationwill be considered later.

Table 3 compiles relevant model and experimentalparameters. Two things should be noted here. First, sincethe value of τb fitted from experimental data must besubjected to the influences of contour-length fluctuationsand double reptation, it should be an underestimation of theactual time constant of interest [19]. As a result, the actualbackbone retraction times, Cbs, should also be somewhatunderestimated as the previously given relation for Cbs/Cb isemployed. Fortunately, the main features seen in subsequenttheory/data comparisons seem to be largely unaffected withina small factor compensating for the aforementioned effects.Second, the estimated torsional compliance time, CT, andimposition time of a step strain, CI, indicate that the dataunder consideration should be largely free from theinfluences of the experimental artifacts associated with theseinstrumental parameters [20].

Similar to typical mean-field formulations for entangledpom-pom [10, 12] or linear [21] polymers, the nonlinear

relaxation modulus, G(t, γ), may be cast into a formconsisting of the nominal contributions from both arm andbackbone materials:

G t; γð Þ ¼ 15

4G 0ð Þ

N ½ wa8 a tð Þ12a tð Þ �þ wb8 b tð Þ12b tð Þ ��Qyx γð Þ=γ;

ð7Þ

where 1a(t)(1b(t)) represents the normalized mean seg-mental stretch for the arm (backbone) polymer, as de-scribed by Eq. 5, and it has a value right after a step strain1a;b t ¼ 0þð Þ ¼ E �uj jh ieq; Q(γ) is the strain-dependent yxcomponent of the universal orientation tensor of the Doi–Edwards model without assuming independent alignment.Note that the value of E �uj jh ieq is obtained by a directnumerical integration without making usual approxima-tions. As the effect of a possible nonaffine deformation isconcerned, Eq. 7 can be consistently modified with theappropriate strain and stretch measures (see, for example,Chen et al. [22]). Several other nonlinear effects can also besimultaneously incorporated within the current formulationof the pom-pom model, as we discuss later.

The validity of a phenomenological formulation like Eq. 7has, in fact, rested upon a decoupling approximation for theorientation and stretch relaxations of an entangled chain.Such a treatment, typical of existing pom-pom constitutiveequations, could nevertheless be further refined usingexisting theories of constraint release, as described below.On the basis of a straightforward extension of the popularMead–Larson–Doi model [23] for monodisperse systems,Pattamaprom and Larson [15] have later proposed ageneralization for a binary mixture of entangled linear chainsto account for a possible coupling in chain relaxations thatarises from the effect of constraint release. Specifically, asthe arm and backbone polymers are treated in an analogous

Table 1 Material properties of the investigated pom-pom polymer samples

Polymera T (°C) Mw,b×10−4 (g/mol) Mw,a×10

−4 (g/mol) PDIb Me×10−3 (g/mol) Sb Sa q wb

gcc gdc

PPBD-1 24.5 4.7 1.95 1.15 2.1 22 9 3 0.29 5.5 7.7PPBD-2 23.0 8.9 2.1 1.07 (1.04) 2.1 40 10 3 0.4 5.5 7.7PPI 25.0 11.1 2.0 1.13 (1.01) 5.0 22 4 2 0.58 3.4 4.6

a The material properties of PPBD-1, PPBD-2, and PPI were taken from Archer and Varshney [4], Archer and Juliani [7], and McLeish et al. [2],respectively.b The polydispersity index (PDI) of the PPBD-1 sample was given for the entire polymer, whereas the individual values for backbone and arm(those given within the parentheses) polymers, respectively, were given for PPBD-2 and PPI.c The critical strain for partial arm withdrawal,γc, was estimated using the following relation: 1 t ¼ 0þð Þ ¼ E �uj jh ieq� 1þ 0:271+2ð Þ0:492¼ q[12]d The corresponding relation for nonaffine deformation was given by 1 t ¼ 0þð Þ ¼ tr C�1=2

��3 ¼ 1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4þ γ2ð Þp� .3 ¼ q [16].


way as for a binary linear-chain system, the governingequations for the backbone polymer may be rewritten as

1

C0b

¼ 1

12bC bþ 1

1b�wa

1:

a

1a� wb

1:

b

1b

!; ð8Þ

1:

b ¼ � 1b � 1

C bsexp ν� 1b � 1ð Þ½ � � 1

21b � 1ð Þ

� wa � 1:

a

1aþ 1

12aC a

!þ wb � 1

:

b

1bþ 1

12bC b

!" #:

ð9Þ

Note that, for the case ν*=0, Eqs. 8 and 9 are basically thesame as the constitutive equations utilized by Pattamapromand Larson for bidisperse linear polymer melts [15], giventhat the arm and backbone polymers are treated as twodifferent components. In Eq. 8, the modified backboneorientation relaxation time, t

0b, now reflects the change in the

primitive path length (the first term on the r.h.s) due to chainstretch as well as the impact of convective constraint release(CCR; the second term on the r.h.s) due to chain retraction[23, 24]. Similarly, Eq. 9 accounts for the enhanced stretchrelaxation due to diffusive constraint release (DCR or doublereptation) and CCR, respectively. The reason for the absence

Table 2 Properties of relaxation modes used to fit the experimental linear relaxation modulus of Eq. 6

Mode numbering PPBD-1a PPBD-2b PPIc

wi, j Ci, j/τb wi, j Ci, j/τb wi, j Ci, j/τb

1 0.563436 0.0005 0.44944 5.08×10−6 0.2755 2.41×10−5

2 0.039774 0.0008 0.033595 2.03×10−5 0.017488 9.63×10−5

3 0.063639 0.002 0.063417 8.13×10−5 0.097839 0.0003854 0.043751 0.005 0.047451 0.000325 0.029172 0.001545 0.078228 0.008 0.006098 0.0013 0.051504 0.0039066 0.052773 0.02 0.068827 0.003906 0.094626 0.015637 0.022545 0.05 0.050868 0.015625 0.078058 0.06258 0.058341 0.08 0.069915 0.0625 0.19618 0.259 0.046409 0.2 0.12732 0.25 0.12909 110 0.021876 0.5 0.068856 1 0.030544 411 0.005171 0.8 0.014216 412 0.001989 213 0.002068 5

aNa=4 and Nb=9bNa=5 and Nb=6cNa=4 and Nb=6

Table 3 Fundamental time constants and essential model or experimental parameters

Polymer taa;s ta;bb;R taa tabs tab tcb (s) G 0ð ÞN � 10�6d (Pa) tfT (s) τI (s)

PPBD-1 3.0×10−6 2.0×10−5 0.008 0.39 1.0 163.7 0.305e 0.02g (1.2×10−4) 0.05–0.20.27g (1.6×10−3)

PPBD-2 5.4×10−8 8.6×10−7 0.0013 0.156 1.0 1300 1.2 0.01h (7.7×10−7) 0.1–0.30.08h (6.2×10−6)

PPI 2.8×10−5 8.5×10−4 0.003 0.196 1.0 285.7 0.24 0.015i (5.3×10−5) <0.1

a Time constants normalized by the corresponding value of τbb The Rouse-like backbone retraction time, τb,R, is set to be S2bt0.c For PPBD-1 and PPBD-2, experimentally reported terminal relaxation times were directly used; for PPI, value was taken from the horizontalshift for a best fit of the linear relaxation data.d For PPBD-1 and PPI, values were taken from the vertical shift for a best fit of the linear relaxation data; for PPBD-2, the experimentally reportedvalue was directly used.e This value of plateau modulus is considerably lower than the typical one for 1,4-polybutadiene melts due to the presence of ca. 45% 1–2 additionin the sample.f The formula tT ¼ 20ph0R

3

3a kT has been used, where η0 is the zero-shear viscosity of the polymer fluid, α is cone angle, R is cone radius, and kT isthe torsional transducer compliance; the values given within the parentheses represent the dimensionless ones, τT/τb.g h0 ¼ 1:2� 106 Pa � s, R=4 mm (or 7.5 mm), α=2° (or 1°) and kT=0.00045 rad/Nm (provided by Paar Physica).h h0 ¼ 6:2� 104 Pa � s, R=4 mm (or 7.5 mm), α=1.2° (or 1°) and kT=0.0025 rad/Nm. The zero-shear viscosity was estimated from the linearrelaxation data, and kT took the value given by Venerus [20].i h0 ¼ 5:0� 104 Pa � s, R=5 mm, α=1° and kT=0.002 rad/Nm. The zero-shear viscosity was estimated from the linear relaxation data, and kTtook the value given by Venerus [20].


of a similar term of DCR in Eq. 8 is that such a contributionshould have already been included in the fitted linearrelaxation function, 8 b(t), in Eq. 6. Note also that only thelongest relaxation mode (i.e., τb) is considered to bemodified by the effect of chain stretch or constraint release.On the other hand, since the nonlinear stress relaxationassociated with the arm polymer typically falls out of thewindow of experimentally accessible data (see, for example,the time constants listed in Table 3), we have omitted similarmodifications for the arm polymer for the sake of simplicity.The more general cases may be formulated, of course, as thecontributions of the arm polymer turn out to be essential in aspecific application.

Several essential points should be further remarked.First, we assume that the aforementioned formulas forconstraint release apply equally well to entangled pom-pomchains as to entangled linear chains. This assumption is anessential one and, of course, deserves further corroboration.It is also very important to note that although the originalpom-pom molecular theory has accounted for certain mean-field effect of constraint release in arriving at the expres-sions of the fundamental constants given in Eqs. 1 to 3, theeffects now added in Eq. 9 are believed to capture somemore detailed consequences of constraint release. Second,the importance of the modifications made in Eq. 8 can tellus whether it is, in general, appropriate to utilize anexperimentally fitted linear relaxation function alone indescribing the primary orientation relaxation in a nonlinearrelaxation experiment. More precisely, if chain stretch andCCR turn out to have negligible impact on the linearrelaxation, one can, in principle, take the easy way todirectly utilize the experimentally fitted linear relaxationmodulus in formulating the orientation relaxation, ascurrently done. Third, since the CCR arising from armretraction typically occurs at extremely short times (see, forexample, Table 3), its effect generally becomes irrelevant inpractice. In fact, it came to our notice later that the onlyimportant modification to the original decoupled formula-tion, Eq. 7, arises from the enhanced backbone retractiondue to DCR of the arm polymer, as now accounted for inEq. 9. This might be expected, considering that theseparation between (renormalized) backbone retractiontime, Cbs, and arm orientation relaxation time, Ca, is givenby a factor about Sb (see Eq. 3). That is, because theconstraint release due to arm diffusive motion typicallytakes place on a time scale close to that of the principalbackbone retraction, its effect might be very important inthe general cases of nonlinear relaxation of a pom-pomchain. A similar situation can, in principle, be encounteredin a polydisperse entangled linear-chain systems.

In summary, a later scrutiny into the model’s predictionssuggested that the coupling effect introduced in Eq. 8 ispractically unimportant, thus justifying a factorized formu-

lation as Eq. 7. On the other hand, according to Eq. 9, itturned out that the constraint release due to arm diffusivemotion has some appreciable impact on the predictednonlinear stress relaxation. In particular, through a furthercoupling with a known polydispersity in arm molecularweight distribution, the overall effect of the aforementionedDCR mechanism becomes very prominent on the result ofthe current theory/data comparisons, as we show in the nextsection.

Results and discussion

We are now in a position to proceed with theory/datacomparisons using the pom-pom constitutive equationsconstructed in the preceding section, Eqs. 1–4 and 6–9.Since the two pom-pom polybutadiene samples showsimilar features in such comparisons, we discuss in detailonly the results for PPBD-1. Afterward, results for PPBD-2and PPI will be summarized for comparison. In order tosave computational efforts, which is essential for simulatingbranched polymer melts that generally bear a broadspectrum of relaxation times, adaptive timestep sizes asdescribed elsewhere [14] have been employed to substan-tially reduce the simulation time of each run to a couple ofminutes on a personal desktop.

It is instructive to begin with a general theory/datacomparison for the case in which the simulation makes onlyuse of the essential time constants given in Eqs. 1–4,together with Eq. 5 for the case ν*=0. In this way, thesimulation would mimic the linear-chain analogy of theinvestigated pom-pom, so that the individual effects ofthe nonlinear relaxation mechanisms subsequently intro-duced may be clearly perceived. The comparison is shownin Fig. 1, where the case with a small strain, γ=0.1, serves toillustrate the fitting of linear stress relaxation. From the timeconstants listed in Table 3, it is evident that the experimen-tally accessible data for PPBD-1 fall primarily in the regimewhere the renormalized backbone retraction dominates thenonlinear stress relaxation. This situation is similar to thecase with PPBD-2, but is to be contrasted with that of PPIfor which arm orientation relaxation and, in particular, theinitial Rouse-like backbone retraction play important rolesat accessibly short times. For clarity, the simulation resultand experimental data at each strain value have beenvertically shifted by the same amount in all the figuresshown in this work, so that one can better see thecomparisons at individual strain values.

As implied by the reasoning noted above, any deviationseen in the comparison made in Fig. 1 might be ascribed tothe special features of the renormalized backbone retrac-tion. Indeed, a considerably broader relaxation can be seenat strains exceeding γ≈5, which is close to the predicted


onset for PAW, i.e., γ≈5.5. Similar features have, in fact,been observed for the other two pom-pom samples (see theElectronic supplementary material ). Thus, the aforemen-tioned phenomenon might be regarded as rheologicalevidence of PAW, in addition to that manifested by aprevious neutron scattering experiment [2]. Below, weexamine first the effect of drag–strain coupling, which hasbeen suggested to be an essential mechanism in describingthe backbone stretch of a pom-pom chain at highdeformation rates [3].

The physical origin of drag–strain coupling has beensuggested to arise from the fact that local branch-pointdisplacement will reduce the primitive path associated witharm orientation relaxation, thus reducing the effective dragon the branch point and expediting subsequent backbonestress relaxation. Moreover, a general coupling betweensuch a drag reduction and the imposed strain might berationalized by the argument that increasing the stretch pullsmore arm material into the backbone tube [2]. Figure 2shows that drag–strain coupling may considerably broadenthe predicted backbone stretch relaxation and, in general,lead to a better agreement with the data. In addition, thisfigure shows the effect of various parameter values assignedto ν*. In order not to cast doubt on the physical significanceof the drag–strain coupling mechanism, only the theoreticalparameter values (i.e., ν*=0.9, 1, and 2 for PPBD-1, PPBD-2, and PPI, respectively) will be utilized in the subsequenttheory/data comparisons.

As has been mentioned earlier, when PAW is permitted toproceed, Rouse-like arm and backbone retractions would firsttake place before the long-time, renormalized (dumbbell-like)backbone retraction sets in at a later stage. Although it is notpossible in practice to directly discern the effect of PAW for

the two PPBD samples studied here, the data at accessiblyshort times for PPI indeed have some overlap with this regime.Therefore, below we briefly describe how the process of PAWcan be approximately simulated within the current constitutiveequations. Before a critical backbone retraction has beenreached during PAW, i.e., 1b=3 for the PPBD samples and1b=2 for the PPI sample, Eq. 5 utilizes a parameter valueν*=0 along with the theoretical Rouse time of the arm orbackbone polymer (see Table 3). Afterward, Eq. 5 utilizesthe renormalized backbone retraction time, τbs, along withthe effect of drag–strain coupling to describe the long-timebackbone retraction. For the reason mentioned above, theeffect of PAW will be examined only in a later theory/datacomparison for PPI.

Among all additional effects later incorporated, it hasbeen found that only an effect of constraint release seems tohave some appreciable influence on the current theory/datacomparisons. Figure 3 shows a typical comparison as theeffect of constraint release prescribed in Eq. 8 or 9 wasselectively incorporated. Firstly, we note that the predic-tions using both Eqs. 8 and 9 collapse with those using onlyEq. 9, suggesting that the effect of constraint release(denoted as CR in this figure) on backbone orientationrelaxation is negligible. The significance of this finding hasbeen discussed earlier. On the other hand, it can also beseen that the DCR due to arm diffusive motion has adominant effect, since Eq. 9 without this particular effectleads to results not much different from the case in whichconstraint release is fully absent. Specifically, it appearsthat DCR arising from arm diffusive motion somewhatbroadens the predicted nonlinear stress relaxation and, ingeneral, leads to a better agreement in theory/data compar-isons. Momentarily, the effect of this particular DCR

t / τb

10-3 10-2 10-1 100 101G

(t,γ

) /

GN

(0)

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

γ = 0.1

2

3

5

6

18

simulation

Fig. 1 An overall comparisonof nonlinear stress relaxationbetween experimental data andthe simulation, Eqs. 1–7 withν*=0, for the PPBD-1 sample.Data have all been shifted forclarity


mechanism will be considered along with a knownpolydispersity in arm molecular weight distribution.

A possible nonaffine deformation on the tube, proposedby Marrucci et al. [16, 17], or a possible partial strandextension (PSE) due to flow-induced tube shrinkage,proposed by Mhetar and Archer [18], has also beensystematically examined for their effects on the currenttheory/data comparisons. The overall finding is that theyplay only minor roles. The reader is referred to an earlierwork [22] for a detailed description of the basic ideasbehind these proposals, as well as of how their effects may,in principle, be simulated within a set of constitutiveequations similar to the current ones. It is, nevertheless,

noteworthy that the theory of nonaffine deformationpredicts an onset of PAW at γ≈7.7 for the PPBD samplesand γ≈4.6 for the PPI sample, in contrast with the previous(affine-deformation) predictions, γ≈5.5 and 3.4, respective-ly (see also Table 1). Thus, the two assumptions might, inprinciple, be discriminated if the characteristic features ofPAW could be identified more precisely. Tentatively, itappears that the predictions based on affine deformationbetter correlate with the peculiar nonlinear features noted inthe current data, such as what shown in Fig. 1 (see also thesupporting information).

Figure 4 illustrates the exact impact of the two additionalmechanisms mentioned above. It is clear that the effect of

t / τb

10-4 10-3 10-2 10-1 100 101

G(t

, γ)

/ G

N

(0)

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

γ = 5

6

8

18

ν* = 0

0.9

3

7

Fig. 2 This plot illustrates theeffect of drag–strain coupling,Eq. 5, with various parametervalues of ν* for the PPBD-1sample. Data have all beenshifted for clarity

t / τb

10-4 10-3 10-2 10-1 100 101

G(t

,γ)

/ G

N

(0)

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

γ = 5

6

8

w/o CR

w/ CR(Eq. 9, w/o DCR)

w/ CR(Eq. 9)

w/ CR(Eqs 8 & 9)

Fig. 3 This plot illustrates theeffects of CR, Eqs. 8 and/or 9,for the PPBD-1 sample, wheredrag–strain coupling (ν*=0.9)has been included. Data have allbeen shifted for clarity


PSE is slim here, and adoption of the nonaffine deformationassumption modifies the predictions mainly through thepredicted damping function. For the data considered here,these two mechanisms seem to lead to a slightly bettershape of the damping function at long times. At interme-diate times, since the nonlinear stress relaxation is domi-nated by PAW and drag–strain coupling, the merit of addingPSE or nonaffine deformation becomes difficult to per-ceive. Still, these two mechanisms might play importantroles at much shorter times before the renormalizedbackbone retraction commences, since then only moderatechain stretch remains.

In general, the presence of a slight polydispersity in themolecular weight distribution of a pom-pom sample mightconsiderably modify its rheological properties. In particular,since polydispersity may considerably modify the armorientation relaxation time, Ca, due to its exponentialdependence on the arm molecular weight, the frictionaldrag on the subsequent backbone relaxations can thus beconsiderably modified, too [2]. Moreover, its indirectimpact on the DCR mechanism due to arm diffusive motionis also of interest. To mimic the effect of polydispersity, wehave used forty discrete modes to approximate a log-normaldistribution for the arm or the backbone length, with thereportedMw and polydispersity index (PDI) shown in Table 1for each sample. A new estimate in the mean backbonestretch can then be obtained by re-solving Eqs. 1–3 and 9.

Figure 5 illustrates the effects of polydispersity. Ingeneral, it was observed that the direct effect of a slightpolydispersity (see Table 1) on the predicted nonlinearrelaxation properties is unimportant in view of the result oftheory/data comparisons. However, as this particular effectis further coupled with that of constraint release, its impact

on the overall theory/data comparison becomes quiteprominent, as shown in Fig. 5. Note that, besides apostulated weight distribution, there are essentially noadditional adjustable parameters introduced here in simu-lating the effect of polydispersity. To the best of ourknowledge, the currently explored impact of constraintrelease has not been reported earlier for a nonlinear flowexperiment on pom-pom melts.

Figure 6 shows the overall theory/data comparison forthe PPBD-1 sample. In general, the predictions consideringdrag–strain coupling are capable of describing the datareasonably well up to a strain as large as γ≈18. Adding theeffect of constraint release, along with its coupling with aknown polydispersity, appreciably broadens the predictedspectrum of nonlinear stress relaxation and substantiallyimproves the result of theory/data comparisons. Similarfeatures have, in fact, been observed for the case withPPBD-2, as shown in Fig. 7.

Finally, Fig. 8 shows the overall theory/data comparisonfor the PPI sample. As has been mentioned earlier, theimpact of PAW becomes discernible at accessibly shorttimes for this particular sample. Using an approximatetreatment as described earlier, the initial stress relaxationcan, in general, be captured quite well by accounting for theRouse-like arm and backbone retractions. As might beexpected, the initial stress relaxation at large strains is bettercaptured by an early stochastic simulation, which treatedmore rigorously the initial Rouse-like retraction of the fullpom-pom chain [14]. Note, however, that the previousstochastic simulation did not enforce a dumbbell-likerenormalized backbone retraction, and neither drag–straincoupling nor constraint release was considered. Similar tothe case with the two PPBD samples, at strains above the

t / τb

10-4 10-3 10-2 10-1 100 101

G(t

,γ)

/ G

N

(0)

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

γ = 5

6

8

w/ CR(Eq. 9)_DE w/o IA

w/ CR(Eq. 9)_DE w/o IA_PSE

w/ CR(Eq. 9)_Nonaffine_PSE

Fig. 4 This plot illustrates theeffects of PSE and nonaffinedeformation (to be contrastedwith the Doi–Edwards formulawithout assuming independentalignment, denoted as DE w/oIA), respectively, for the PPBD-1 sample, where drag–straincoupling (ν*=0.9) and CR havebeen included. Data have allbeen shifted for clarity


predicted onset of PAW, γ≈3.4, nonlinear features charac-teristic of pom-pom chain retraction become quite evident,and the effect of drag–strain coupling becomes essential(see the supporting information). Also, inclusion of theeffect of constraint release along with its coupling with aknown polydispersity in arm molecular weight distributionfurther improves the result of theory/data comparisons.Overall, the common features of nonlinear stress relaxationas currently revealed for three pom-pom melts seem tosuggest that the explored properties could be universal forentangled pom-pom liquids.

Conclusion

A phenomenological pom-pom model has been proposedfor predicting, on the basis of linear relaxation data, thenonlinear relaxation properties of entangled pom-pomliquids in single-step strain flows. In contrast with most ofthe existing versions of pom-pom model, the proposedconstitutive equations simulate the detailed retractionbehavior of a full pom-pom chain. In addition, whileutilizing a factorized formulation analogous to that forbidisperse linear-chain systems, a possible coupling be-

t / τb

10-4 10-3 10-2 10-1 100 101

G(t

,γ)

/ G

N(0

)

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

γ = 5

6

8

w/ CR, w/o PDI

w/CR, w/ PDIa

w/CR, w/ PDIb

Fig. 5 This plot illustrates theeffects of polydispersity for thePPBD-1 sample (PDIa and PDIbdenote the respective effects ofthe arm or backbone weightdistribution), where drag–straincoupling (ν*=0.9) and CR havebeen included. Data have allbeen shifted for clarity

t / τb

10-4 10-3 10-2 10-1 100 101

G(t

, γ)

/ G

N

(0)

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

γ = 3

5

6

8

18

w/o CR

w/ CR

w/ CR_PDIaτ

bs

Fig. 6 An overall comparisonof nonlinear stress relaxationbetween experimental data andthe simulation for the PPBD-1sample, where drag–straincoupling (ν*=0.9) has beenincluded, and the effect of CR orpolydispersity has been selec-tively incorporated in the simu-lation. Data have all been shiftedfor clarity


tween linear orientation and nonlinear stretch relaxationswas self-consistently accounted for using an existingformula for constraint release. The predictions weresystematically tested against three existing sets of experi-mental data on nearly monodisperse pom-pom melts, withthe following observations that seem to be rather universalfor the data examined: In general, theory/data comparisonsclearly revealed nonlinear features characteristic of pom-pom chain retraction at strains exceeding the predictedvalue for partial arm withdrawal (i.e., γ≈5.5 for PPBDs andγ≈3.4 for PPI). In this respect, the model can describereasonably well the present data by incorporating a

previously proposed effect of drag–strain coupling. Withoutintroducing additional adjustable parameters, the agreementof theory/data comparisons seemed to be further improvedby accounting for an effect of constraint release due to armdiffusive motion, along with its coupling with a knownpolydisperity in arm molecular weight distribution. The lastfinding suggested that ‘double reptation’ of the armpolymer might as well play an important role in thenonlinear relaxation of the backbone polymer – an essentialproperty not considered in existing pom-pom theories.Overall, the generality of the currently explored featuresof nonlinear stress relaxation for three pom-pom melt

t / τb

10-5 10-4 10-3 10-2 10-1 100 101

G(t

, γ)

/ G

N

(0)

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

γ = 3

5

7

9

w/o CR

w/ CR

w/ CR_PDIa τbs

Fig. 7 An overall comparisonof nonlinear stress relaxationbetween experimental data andthe simulation for the PPBD-2sample, where drag–strain cou-pling (ν*=1) has been included,and the effect of CR or polydis-persity has been selectively in-corporated in the simulation.Data have all been shifted forclarity

t / τb

10-5 10-4 10-3 10-2 10-1 100 101

G(t

, γ)

/ G

N(0

)

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

γ = 3

4

5

10

20

w/o CR

w/ CR

w/ CR_PDIa

Stochastic simulation

τb,R

τbs

Fig. 8 An overall comparisonof nonlinear stress relaxationbetween experimental data andthe simulation for the PPI sam-ple, where drag–strain coupling(ν*=2) has been included, andthe effect of CR or polydisper-sity has been selectively incor-porated in the simulation;predictions from an early sto-chastic simulation [14] are alsoincluded for comparison. Alldata have been shifted for clarity


samples remains to be corroborated using various simula-tion schemes and for a wider variety of entangled pom-pomliquids. On the other hand, it appears thus far that theproposed pom-pom constitutive equations, Eqs. 1–4 and 6–9,provide a convenient alternative by which the detailednonlinear relaxation of an entangled pom-pom chain innonlinear step–strain flows can be modeled or systematicallystudied.

Acknowledgement The authors thank the constructive comments ofall reviewers. Financial support from MOE Program for PromotingAcademic Excellence of Universities under the Grant no. 96-2752-E-007-006-PAE as well as from the National Science Council of theROC under Grant no. 95-2221-E-194-049 is gratefully acknowledged.The resource provided by the National Center for High-PerformanceComputing of the ROC for performing part of the preliminaryinvestigation is also acknowledged.

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65:241


pom-pom model predictions on nonlinear stress relaxation

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