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A New Proposal for Polymer Dynamics in SteadyShearing Flows
VIJAY R. MHETAR, L. A. ARCHER
Department of Chemical Engineering, Texas A & M University, College Station, Texas 77843
Received 22 March 1999; revised 20 September 1999; accepted 4 October 1999
ABSTRACT: Beginning with a recently proposed expression for the drag force on a singlemacromolecule pulled with constant velocity through a fluid of long-entangled mole-cules (V. R. Mhetar and L. A. Archer, Macromolecules 1998, 31, 6639), we investigatethe effect of entanglement loss on polymer dynamics in steady shearing flows. Atsteady-state, a balance between the elastic restoring force and viscous drag acting onentangled polymer segments reveals a critical molecular strain gm,c beyond which thedrag force exerted on polymer molecules by their neighbors is insufficient to supportarbitrarily small orientation angles. Specifically, we find that in fast steady shear flowstd
21 , g , tRouse21 , polymer orientation in the shear plane approaches a limiting angle
xc ' atau(1/(1 1 gm,c)) beyond which flow becomes incapable of producing furthermolecular alignment. Shear flow experiments using a series of concentrated polysty-rene/diethyl phthalate solutions with fixed entanglement spacing, but variable polymermolecular weight 0.94 3 106 # Mw # 5.48 3 10
6, reveal a limiting steady-stateorientation angle between 6 and 9 over a range of shear rates; confirming thetheoretical result. Orientation angle undershoots observed during start-up of faststeady shearing flows are also explained in terms of a transient imbalance of elasticrestoring force and viscous drag on oriented polymer molecules. Our findings suggestthat the DoiEdwards affine orientation tensor (Q) is not universal, but rather dependson deformation type and deformation history through a balance of elastic force andviscous drag on polymer molecules. 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys38: 222233, 2000Keywords: polymer dynamics; steady shear flow; orientation angle; DoiEdwardstheory; entanglement loss; partial retraction; convected constraint release; viscous drag
INTRODUCTION
The DoiEdwards (D-E) theory contends that inmelts and solutions of entangled macromoleculestranslational motion of individual molecules isconfined to tube-like regions surrounding the mo-lecular contour. When an entangled polymer issubjected to sudden straining, the tube and chaindeform affinely. Stresses induced by the deforma-tion relax on two time-scales. At short times of
order the longest Rouse relaxation time of the freepolymer, tRouse, stretched chain segments are as-sumed to retract along their respective tubes torestore the equilibrium contour length. Followingequilibration of contour length, affine orientationof chain segments relax by reptation. For t. tRouse the instantaneous stress in the materialis then,
s 5 ~15/4!GNQ~E!c~t!, (1)
where GN is the elastic modulus of the entangledpolymer network; E the deformation gradienttensor; Q(E) the second moment tensor of seg-
Correspondence to: L. A. Archer (E-mail: [email protected])Journal of Polymer Science: Part B: Polymer Physics, Vol. 38, 222233 (2000) 2000 John Wiley & Sons, Inc.
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ment orientation vectors u; Q 5 (1/^uE z uu&)^(Ez u)(E z u)/uE z uu&; and c(t) a reptation relaxa-tion function, c(t) 5 p;odd 8/p
2p2exp(2p2t/td),where td0 is the terminal orientation relaxationor reptation time.1
Equation 1 predicts that for t . tRouse stress isfactorizable into time-dependent, GNc(t), andstrain-dependent, Q(E), parts. For a shear strainof magnitude g, the strain-dependent shear stresscontribution may be approximated as Qxy(g) '
4/15g~1 1 g2/5!.
2 Experimental results using weak to
moderately entangled solutions of narrow molec-ular weight distribution poly(styrene) providesupport for the latter result.3,4 This functionalform of Qxy(g) has nonetheless been the source ofa long-standing paradox in the D-E theory. Spe-cifically, the prediction for Qxy(g) implies that forshear strains g . =5 a larger imposed strainyields a smaller shear stress, sxy(g) ; 1/g at highstrains. This unusual result follows directly fromthe assumption that polymer chain segments ori-ent affinely in response to an imposed strain,regardless of the size of that strain. Monotonicdecrease in the second moment tensor Qxy(g) athigh shear strains reflects enhanced segmentalignment parallel to the direction of shear, x,compared with alignment parallel to the directionof the shear gradient, y.
The multivalued shear stress result observedin step shear flows carries over to D-E stresspredictions in all time-dependent shearing flows.In these flows, the shear stress at any instant intime is taken to be the affine stress accumulatedover the history of the deformation, weighted bythe materials memory (c(t 2 t9)) of that stress,sxy(g) 5 15/4 GN
(0) *2`t Qxy(E(t, t9))c(t 2 t9)/t9
dt9.1 Thus, a continuous shearing deformationimposed at a rate g , tRouse
21 is assumed to orientpolymer chain segments in proportion to the ef-fective strain accumulated by molecular segmentsat steady-state. The orientation of polymer seg-ment vectors is therefore anticipated to monoton-ically approach the direction of shear as shearingbecomes progressively faster. The consequent de-crease in shear stress with increasing rate yieldsthe unusual prediction that at high rates appar-ent steady-state viscosity h 5 (sxy(g)/g) scaleswith shear rate as h ; g23/2. Thus, the very sameDoiEdwards orientation tensor Q that providesfavorable predictions in step-shear deformationsyields unrealistic levels of shear thinning insteady shear flow. Any attempt to fix Q to im-prove predictions in steady shearing has the un-
desirable effect of reducing the quality of the D-Estep-strain predictions.
Several proposals have been advanced in theliterature to correct the D-E steady shear predic-tions without changing Q. All but one,5,6 achievethis by including more realistic descriptions ofshear stress contributions due to flow-inducedchanges in polymer contour length. Reviews out-lining the successes and shortcomings of each ofthese proposals are available,5,7,8 and will not bediscussed here. Instead, a different approach forevaluating polymer orientation dynamics insteady flows is addressed. In this approach theorientation of entangled polymer chain segmentsin a deformation field is contended to be deter-mined by a balance between the elastic restoringforce due to molecular alignment parallel to thedirection of shear and viscous drag exerted onchain segments by their neighbors, which retardsrestoration of the equilibrium random distribu-tion of molecular segment vectors. Specifically, wepropose that arbitrarily small orientation anglesx between entangled polymer segments and amacroscopic flow are possible only if the viscousdrag force on segments is sufficiently large tobalance the entropic, elastic restoring force thatcontinually attempts to return segments to theirequilibrium orientation distribution.
This approach is different from that taken inthe DoiEdwards theory, where orientation ofpolymer segments in any flow field is completelydescribed by the affine orientation tensor, Q, forthat flow field. A consequence of the new proposalis that at moderate shear rates, in a steady shearflow, the orientation of polymer chain segmentsapproaches a limiting angle beyond which theflow becomes incapable of inducing further molec-ular alignment. Our result appears to resolve thelong-standing discrepancy between good agree-ment between DoiEdwards step-strain predic-tions and experimental results, and the unusuallyhigh levels of shear thinning predicted in steadyshearing flows.1,5 Experimental support for theproposal is discussed in the Results and Discus-sion section of the article.
Theory
Consider a homogeneous melt of flexible, entan-gled macromolecules each with degree of polymer-ization N and entanglement spacing Ne0. At equi-librium, each of the N-mer molecules will on av-erage entangle with (N/Ne0 2 1) surrounding
POLYMER DYNAMICS IN STEADY SHEARING FLOWS 223
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molecules. If a uniform shear flow is applied tothe system (N-mer chain and surrounding mole-cules at equilibrium), such that g ! td0
21, thefrictional drag force experienced by each N-mer isidentical to the force that would arise if the mol-ecule is pulled by its center of mass with a di-rected velocity Vc 5 Rg through a sea of chemi-cally identical, but stationary molecules (P-mers),(see Fig. 1).
Recently, Mhetar and Archer,9 showed that apulled macromolecule (N-mer) will not instinc-tively follow a curvilinear path around entangle-ment points. These authors contended, as didBueche almost half a century ago,10 that the N-mer will instead continually attempt to drag sur-rounding molecules entangled with it (P-mers) asit translates through the entangled polymer net-work. Because P-mers are themselves entangledand the N-mer is here free at both its ends thepulled N-mer cannot drag P-mers indefinitely asargued by Bueche,10 rather the N-mer will onaverage only induce P-mer translation over ashort distance, of order the mean entanglementspacing a 5 =Ne0b. Beyond that point, the largefrictional resistance offered by polymer moleculesentangled with the P-mer arrests its motion and,thereby, compels the pulled N-mer to slidethrough its entanglement with the P-mer to main-tain the applied velocity. This sliding motiontraces out a curvilinear (tube-like) path aroundthe N-mer contour with diameter equal to themesh size of the entanglement network =Ne0b.The frictional resistance the N-mer experiencesper entanglement site is therefore Pzm, the dragcoefficient per N-mer molecule caused by the com-bined dragging and sliding motions is then,
zN 5N
Ne0Nzm, (2)
where zm is the monomeric friction coefficient ofthe N-mer, and we have assumed P 5 N.
The predicted halting curvilinear motion of theN-mer molecule progresses at an average speedVT that is related to the induced center of massvelocity Vc by Lc(N)/VT 5 R/Vc, where Lc(N) andR are the N-mers mean contour length and ra-dius of gyration, respectively. The total drag forceoffered by the surrounding molecules to the N-mers motion is therefore not the usual Rousedrag assumed for polymer chains sliding intu-itively through monomer-filled tubes: fdrag5 VTzRouse 5 VTNzm, rather fdrag 5 VTzN.
Before proceeding, it is useful to derive thelimiting Newtonian viscosity h0 and terminal mo-lecular relaxation time td0 predicted by eq 2 foran entangled melt of equal-sized molecules. In aslow steady shear flow, the rate of viscous dissi-pation per unit volume is h0g
2. The dissipationdue to a single N-mer is zCVC
2 5 zCR2g2, where zC
is the center of mass friction coefficient. The rateof dissipation per unit volume is thereforezCR
2g2/Nb3, yielding h0 5 zC/b where b is themonomer size. When the N-mer moves by Lc(N)along its curvilinear path, its center of masstranslates a distance R. Equating dissipationrates yields,
zC 5 zNSVTVCD2
5 zNSLc~N!R D2
