Fundamentals of Polymer Engineering Part 15 doc

14 FlowBehaviorofPolymericFluids 14.1INTRODUCTION Inordertousepolymers(whetheravailableintheformofpelletsorasmeltfrom apolymerizationreactor),thematerialhastobeconvertedintousefulshapessuch asfibers,films,ormoldedarticles.Thisisdoneusingunitoperationssuchasfiber spinningandinjectionmolding,whichareanalyzedindetailinChapter15.Here, wesimplymentionthatflowisanintegralpartofanyshapingoperation,and, veryfrequently,itisusefultoknowquantitiessuchasthepressuredropneededto pumpapolymericfluidataspecifiedflowratethroughachannelofagiven geometry.Theansweriseasytoobtainifweareworkingwithlow-molecular- weightliquidsthatbehaveinaNewtonianmanner;allweneed,bywayof materialproperties,isinformationaboutthetemperaturedependenceoftheshear viscosity.Iftheprocessisisothermal,theshearviscosityisaconstantanditcan bemeasuredinanyoneofseveralways.Polymermeltsandsolutions,however, haveasteadyshearviscositythatdependsontheshearrate.Therefore,itisa materialfunctionratherthanamaterialconstant.Forpolymericfluidsthetypical shapeofthesteadyshearviscositycurveasafunctionofshearrateisshownin Figure14.1.Atsteadystate,theviscosityisconstantatlowshearrates.Itusually decreases with increasing shear rate and often becomes constant again at high shear rates. There is, therefore, a lower Newtonian region characterized by the zero shear rate viscosity Z 0 and an upper Newtonian region characterized by an 573 Copyright © 2003 Marcel Dekker, Inc. infinite shear viscosity Z 1 . Between these two regions, the viscosity versus shear rate behavior can usually be represented as a straight line on logarithmic coordinates—this is the power-law region. In the foregoing, we have been careful to append the word ‘‘steady’’ to the shear viscosity. Unlike with Newtonian liquids, the shear stress takes some time to reach a steady value upon inception of shear flow at a constant shear rate. This is sketched in Figure 14.2, which shows that the shear stress can also overshoot the steady-state value. Polymeric fluids are therefore non-Newtonian in the sense that the shear viscosity depends on both shear rate and time. Obtaining the pressure drop corresponding to a given flow rate is, consequently, a slightly more complicated process. FIGURE 14.1 Qualitative behavior of the steady shear viscosity of polymeric fluids. FIGURE 14.2 Start-up and shutdown of shearing at constant shear rate. 574 Chapter 14 Copyright © 2003 Marcel Dekker, Inc. Ashear-thinningviscosityisnottheonlynon-Newtonianfeatureofthe behaviorofpolymericfluids;severalotherunusualphenomenaareobserved.If, inthesituationdepictedinFigure14.2,theshearrateissuddenlyreducedtozero aftertheattainmentofasteadystate,low-andhigh-molecular-weightliquids againbehavedifferently.ThestressintheNewtonianfluidgoestozeroinstantly, butittakessometimetodisappearinthepolymer.Thetimescaleoverwhichthis stressrelaxationoccursisknownastherelaxationtimeandisdenotedbythe symboly.Additionally,ifasmall-amplitudesinusoidalstrainisimposedonthe polymer,theresultingstressisneitherinphasewiththestrainnoroutofphase withthestrain:Thereisanout-of-phasecomponentrepresentingenergydissipa- tionandanin-phasecomponentrepresentingenergystorage(seeSect.12.4). Bothstressrelaxationandthephasedifferenceindynamicexperimentsareelastic effects;wesaythatthepolymersarebothviscousandelastic(i.e.,viscoelastic). Intime-dependentflow,therelativeextentofthesetwoeffectsdependsonthe valueofthedimensionlessgroupknownastheDeborahnumber(De)anddefined asfollows: De¼ y T ð14:1:1Þ whereTisthecharacteristictimeconstantfortheprocessofinterest.Forlow valuesofDe,thepolymerresponseisessentiallyliquidlike(viscous),whereasfor highvalues,itissolidlike(elastic).Afurthermanifestationofviscoelasticityis theswellingofajetofpolymeronemergingfroma‘‘die’’orcapillary.Thisis showninFigure14.3.Dieswell,orjetswell,canbesuchthatD j =Deasily exceeds2;thecorrespondingNewtonianvalueis1.13.Thisistrueatverylow flowrates.Athighflowrates,dieswellreducesbutunstablebehaviorcalledmelt fracturecanoccur.Thejetcanbecomewavyorthesurfacecanbecomegrossly distorted,assketchedinFigure14.4;theextentofdistortionisalsoinfluencedby thegeometryofthecapillary,itssurfacecharacter,andthepropertiesofthe polymer.NotethatmeltfractureisneverobservedwithNewtonianliquids. Thephenomenajustdescribedareinterestingtoobserveandexplain.A quantitativedescriptionofthemis,however,essentialfordevelopingmodelsof FIGURE14.3Thedie-swellphenomenon. Flow Behavior of Polymeric Fluids 575 Copyright © 2003 Marcel Dekker, Inc. variouspolymerprocessingoperations.Someofthesemodelsarediscussedin Chapter15,andthesearebasedontheconservationprinciplesofmass, momentum,andenergy,togetherwithappropriateconstitutiveequationsand boundaryconditions.Theyareusefulforprocessoptimizationandfordetermin- ingtheeffectofthevariousmaterial,geometrical,andprocessingvariablesonthe propertiesofthepolymericproduct.Themodelsalsoallowustorelate performancevariablestomachinevariables.Theyarealsousefulforpredicting theonsetofflowinstabilities.Inthischapter,though,wedescribemethodsof measuringthestressresponseofpolymericfluidsinwell-characterizedflow situations,presenttheassociatedmethodsofdataanalysis,andgivetypical results.Thisnaturallyleadstoadiscussionoftheoriesavailabletoexplainthe observedbehaviorintermsofmaterialmicrostructureandtomethodsof mathematicallyrepresentingthestress–deformationrelationsorconstitutive behavior.Thisistherealmofrheology,whereinweexaminebothpolymer solutionsandpolymermelts.Notethat,atalessfundamentallevel,these measurementscanbeemployedforproductcharacterizationandquality-control purposes.AsuccincttreatmentmaybefoundinRef.[1]. 14.2VISCOMETRICFLOWS Theflowfieldthatisgeneratedinmoststandardinstrumentsusedtomeasure rheologicalorflowpropertiesisaparticularkindofshearflowcalledviscometric flow.Allofthemotioninaviscometricflow,whetherinCartesianorcurvilinear coordinates,isalongonecoordinatedirection(say,x 1 inFig.14.5),thevelocity variesalongasecondcoordinatedirection(say,x 2 ),andthethirddirectionis neutral.Anillustrationofthis,showninFigure14.5,istheshearingofaliquid betweentwoparallelplatesduetothemotionofoneplaterelativetotheother. Thevelocitygradientorshearrate, _ gg,thenisdv 1 =dx 2 ,wherev 1 istheonly nonzerocomponentofthevelocityvector.Iftheshearrateisindependentof position,theflowfieldishomogeneousandthecomponentsoftheextrastress [seeEq.(10.5.6)ofChap.10]arealsoindependentofposition.Thisisintuitively obvious(theextrastressdependsontherateofshearstrain),andbecausethe shearrateisthesameeverywhere,somustbetheextrastress.Thisfactisusedto FIGURE14.4Extrudatemeltfracture. 576Chapter14 Copyright © 2003 Marcel Dekker, Inc. great advantage in several viscometers or rheometers, as such instruments are called. As discussed in Section 10.5, there are, in general, only six independent stress components. For a Newtonian liquid, these can be evaluated using Newton’s law of viscosity, T ij ¼ t ij ¼ Z @v i @x j þ @v j @x i ! ð14:2:1Þ provided that i 6¼ j and T ii ¼Àp þt ii ¼Àp þ 2Z @v i @x i ð14:2:2Þ in which both i and j can be 1, 2, or 3. For the situation depicted in Figure 14.5, therefore, only t 12 is nonzero for a Newtonian liquid. For polymeric fluids, we can use symmetry arguments and show that t 13 and t 23 are still zero [2], but the normal stresses t ii can be nonzero. Furthermore, if the t ii exist, they are even functions of the shear rate. The shear stress, however, is an odd function. Now, note that due to incompressibility, p is indeterminate and we cannot obtain t ii from a measurement of T ii . However, p can be eliminated if we take stress differences. We can, therefore, define the first and second normal stress differ- ences as follows: N 1 ð _ ggÞ¼T 11 À T 22 ¼ t 11 À t 22 ð14:2:3Þ N 2 ð _ ggÞ¼T 22 À T 33 ¼ t 22 À t 33 ð14:2:4Þ and these must depend uniquely on the shear rate _ gg, because each of the extra- stress components is a unique function of _ gg. The existence of a positive first normal stress difference during shear flow can be used to explain die swell. If the fluid being sheared between parallel plates in Figure 14.5 were to emerge into the atmosphere, T 11 would obviously equal FIGURE 14.5 Viscometric flow in rectangular Cartesian coordinates. Flow Behavior of Polymeric Fluids 577 Copyright © 2003 Marcel Dekker, Inc. Àp a ,wherep a isatmosphericpressureandisacompressivestress.Apositivefirst normalstressdifferencewouldthenimplythatT 22 isnegative(compressive)and greaterthanp a inmagnitude.Inotherwords,theupperplatepushesdownonthe liquidbeingshearedandtheliquidpushesupontheplatewithastressthat exceedsp a inmagnitude.Becauseonlyatmosphericpressureactsontheoutside oftheupperplate,ithastobehelddownbyanexternallyappliedforcetoprevent thefluidfrompushingthetwoplatesapart.Whenthefluidemergesintothe atmosphere,thereisnoplatepresenttopushdownonit,andit,therefore, expandsandweobservedieswell. Becausetheshearstressisanoddfunctionoftheshearrateandthenormal stressdifferencesareevenfunctions,itiscustomarytodefinetheviscosity functionandthefirstandsecondnormalstresscoefficientsasfollows: Zð _ ggÞ¼ t 12 _ gg ð14:2:5Þ C 1 ð _ ggÞ¼ N 1 _ gg 2 ð14:2:6Þ C 2 ð _ ggÞ¼ N 2 _ gg 2 ð14:2:7Þ whichtendtoattainconstantvaluesZ 0 ,C 10 ,andC 20 astheshearratetendsto zero.Next,weexaminetwoofthemostpopularmethodsofexperimentally determiningsomeorallofthesequantities. 14.3CONE-AND-PLATEVISCOMETER Inthisinstrument,theliquidsampleisplacedinthegapbetweenatruncatedcone andacoaxialdisk,asshowninFigure14.6a.Theconeistruncatedsothatthereis no physical contact between the two members. The disk radius R is typically a couple of centimeters, whereas the cone angle a is usually a few degrees. Either of the two members can be rotated or oscillated, and we measure the torque M needed to keep the other member stationary. We also measure the downward force F needed to hold the apex of the truncated cone at the center of the disk. From these measurements, we can determine the three material functions defined by Eqs. (14.2.5)–(14.2.7). Note that F equals zero for a Newtonian liquid. This flow is a viscometric flow when viewed in a spherical coordinate system, and there is only one nonzero component of the velocity. This component is v f , which varies with both r and y; the streamlines are closed circles. If we rotate the plate at an angular velocity O, the linear velocity on the plate surface at any radial position is Or. On the cone surface at the same radial position, however, the velocity is zero. If the cone angle is small, we can assume that v f 578 Chapter 14 Copyright © 2003 Marcel Dekker, Inc. varies linearly across the gap between the cone and the plate. The shear rate _ gg at any value of r is then given as follows: _ gg ¼ Or À 0 ra ¼ O a ð14:3:1Þ in which a is measured in radians and O in radians per second. Thus, the shear rate is independent of position within the gap. As a consequence, the stress components resulting from fluid deformation do not depend on position either. Also, because this is a viscometric flow, the only nonzero stress components are the shear stress t fy and the normal stresses t ff , t yy , and t rr . Note that, due to symmetry, all derivatives with respect to f are zero. Also note that, by a similar line of reasoning, the stresses will be independent of position even when the plate is oscillated or given a step strain. If we integrate the shear stress over the cone surface, we can get an expression for the torque M as follows: M ¼ ð R 0 rt fy 2prdr¼ 2pR 3 t fy 3 ð14:3:2Þ so that t fy ¼ 3M 2pR 3 ð14:3:3Þ and the viscosity is Z ¼ 3Ma 2pR 3 O ð14:3:4Þ FIGURE 14.6 (a) A cone and plate viscometer; (b) force balance on the plate. Flow Behavior of Polymeric Fluids 579 Copyright © 2003 Marcel Dekker, Inc. Inordertoobtainthenormalstressfunctions,weneedtosolvethe equationsofmotioninsphericalcoordinates[3].Anexaminationofthey componentofthisequationshows@p=@y¼0.Thus,pdependsonralone, becausederivativeswithrespecttofarezero.Further,becausemostpolymer fluidsarefairlyviscous,wecanneglectinertiaand,asaresult,thercomponentof theequationofmotionyieldsthefollowing[3]: dp dr ¼ 2 r t rr À t yy þt ff r ð14:3:5Þ IntegratingwithrespecttorfromRtorgives pðrÞ¼pðRÞþð2t rr Àt yy þt ff Þln r R  ð14:3:6Þ Atr¼R,T rr equalsÀp a ,wherep a isatmosphericpressure.Thus,fromthe definitionoftheextrastress,wehave ÀpðRÞþt rr ¼Àp a ð14:3:7Þ Tomakefurtherprogress,weexaminetheequilibriumoftheplateandbalance forcesintheydirection(Fig.14.6b).Theresultis F þ pR 2 p a þ ð R 0 ½ÀpðrÞþt yy 2prdr¼ 0 or F þ pR 2 p a þ pR 2 t yy ¼ ð R 0 pðrÞ2prdr ð14:3:8Þ Introducing Eqs. (14.3.6) and (14.3.7) into Eq. (14.3.8), integrating by parts, and simplifying the result gives the first normal stress difference: N 1 ¼ t ff À t yy ¼ 2F pR 2 ð14:3:9Þ Clearly, the first normal stress difference in shear is a positive quantity. Finally, if we were to use a pressure transducer to measure T yy on the plate surface, we would have T yy ¼ÀpðrÞþt yy ¼Àp a À t rr þ t yy Àð2t rr À t yy À t ff Þln r R  ¼Àp a þ N 2 þðN 1 þ 2N 2 Þln r R  ð14:3:10Þ Knowing N 1 , we can get N 2 from Eq. (14.3.10). This measurement is not easy to make, but the general consensus is that N 2 is negative and about 0.1–0.25 times the magnitude of N 1 . 580 Chapter 14 Copyright © 2003 Marcel Dekker, Inc. Shown in Figure 14.7 are data for the shear viscosity of a low-density polyethylene sample (called IUPAC A) as a function of the shear rate over a range of temperatures [4]. These data were collected as part of an international study that involved several investigators and numerous instruments in different labora- tories. For this polymer sample,  MM n is 2 Â10 4 and  MM w exceeds 10 6 . It is seen that the shear rates attained are low enough that, at each temperature, the zero shear rate viscosity can be identified easily; these Z 0 values are noted in Figure 14.7 itself. Clearly, the viscosity values increase with decreasing temperature. If we plot Z 0 as a function of the reciprocal of the absolute temperature, we get a straight line, showing that the Arrhenius relation is obeyed; that is, Z 0 / exp E RT  ð14:3:11Þ where the activation energy E equals 13.6 kcal=mol in the present case. The activation energy typically increases as the polymer chain stiffness increases. FIGURE 14.7 Determination of zero shear viscosity Z 0 ; sample A. I: u WRG (25-mm diameter, cone angle a ¼ 4  ); j sample A stabilized, temperature shift of 130  C data using E ¼ 11:7 kcal=mol [4]. II:  Kepes (26.15-mm diameter. a ¼ 21  4 0 ). IV: s WRG modified (72-mm diameter, a ¼ 4  ). (From Ref. 4.) (Reprinted with permission from Meissner, J.: ‘‘Basic Parameters, Melt Rheology, Processing and End-Use Properties of Three Similar Low Density Polyethylene Samples,’’ Pure Appl. Chem., vol. 42, pp. 553– 612, 1975.) Flow Behavior of Polymeric Fluids 581 Copyright © 2003 Marcel Dekker, Inc. Note,though,thatclosetotheglasstransitiontemperature,theWLFequation [seealsoEqs.(12.5.4)and(13.6.13)ofChapters12and13,respectively] log Z 0 ðTÞ Z 0 ðT g Þ ! ¼À 17:44ðTÀT g Þ 51:6þðTÀT g Þ ð14:3:12Þ maybeamoreappropriateequationtouse.Thisisbecauseattemperatures betweenT g andT g þ100  C,theviscosityisstronglyinfluencedbyincreasesin freevolume;athighertemperatures,weessentiallyhaveanactivatedjump process. Inadditiontotemperature,thezero-shearviscosityisalsoinfluencedbythe pressure,especiallyathighpressuresandespeciallyclosetotheglasstransition temperature.ThepressuredependenceisagainoftheArrheniustype, Z 0 /expðBpÞð14:3:13Þ andistheresultofthetendencyofthefreevolumetodecreaseontheapplication ofalargehydrostaticpressure. Thefirstnormalstressdifferenceinshear,N 1 ,ontheIUPACAsampleand measuredat130  CisdisplayedonlogarithmiccoordinatesinFigure14.8.Ifwe comparethisfiguretoFigure14.7,wedeterminethatN 1 iscomparabletothe shearstressatlowshearratesbutsignificantlyexceedsthisquantityathigher shearrates.ThishappensbecauseofthestrongerdependenceofN 1 on _ gg comparedwiththedependenceoft fy on _ gg.Indeed,straightlinestypically FIGURE14.8FirstnormalstressdifferenceinshearforIUPACALDPEat130  C. (FromRef. 4.) 582 Chapter 14 Copyright © 2003 Marcel Dekker, Inc. [...]...Flow Behavior of Polymeric Fluids 583 result when these two functions are plotted in terms of the shear rate on logarithmic coordinates Although the slope of the N1 plot usually lies between 1 and 2, the maximum value of the slope of the shear stress plot is unity Although all of the data discussed so far have been melt data, the shear behavior of polymer solutions is similar to that of polymer melts,... decreases to a value even below that of the zeroshear value The behavior of polymer solutions is qualitatively similar to that of polymer melts, except that the extensional viscosity of polymer solutions can exceed the corresponding shear viscosity by a far wider margin than does the extensional viscosity of polymer melts [15] Example 14.3: How is the extensional viscosity of a Newtonian fluid related to... behavior of polymer solutions becomes identical to that of polymer melts if data are plotted against the product of molecular weight with polymer concentration With increasing molecular weight, polymer polymer entanglements occur, and the zero-shear viscosity of melts increases at the 3.4–3.6 power of the molecular weight It has been shown experimentally that this relationship holds for polydisperse polymers... decade of frequency Figure 14.14 shows a master curve of G0 and G00 values in a temperature range of 130–250 C on an injection-molding grade sample of polystyrene; all of the data have been combined by means of a horizontal shift using time–temperature superposition with a 150  C reference temperature according to the procedure of Section 12.5 The stress–relaxation modulus (calculated in the manner of. .. branching 14.9 CONSTITUTIVE BEHAVIOR OF DILUTE POLYMER SOLUTIONS When the concentration of polymer molecules in solution is sufficiently low (as shown in Sect 8.6 of Chap 8, ½ZŠc < 1 is the usual criterion), the molecules are isolated from each other and solution behavior can be predicted from a knowledge of the behavior of a single polymer molecule Because linear polymer molecules act like springs and... individual polymer molecule as a series of N þ 1 spheres, each of mass m, connected by N massless springs The polymer solution then is a noninteracting suspension of these stringy entities in a Newtonian liquid In the absence of flow, the equilibrium probability that one end of a given polymer molecule is located at a specified distance from the other end is given by Eq (10.2.12) Under the influence of flow,... only be done with the help of molecular theories These theories are examined in some detail in the remainder of this chapter 14.8 THEORIES OF SHEAR VISCOSITY The shear viscosity of polymer melts depends primarily on the molecular weight, the temperature, and the imposed shear rate; for polymer solutions, the concentration and nature of solvent are additional variables In one of the earliest theories,... viscosities at Copyright © 2003 Marcel Dekker, Inc Flow Behavior of Polymeric Fluids 601 a constant value of z, the behavior of polymer solutions also agrees with Eq (14.8.9); this is shown in Figure 14.17 for both polymer melts and polymer solutions [25] More extensive data are available in earlier reviews, which also discuss the influence of temperature, chain branching, polydispersity, and solvent viscosity... here has been modified by a large number of authors [33,34] The resulting articles, however, are too numerous to be summarized here Copyright © 2003 Marcel Dekker, Inc Flow Behavior of Polymeric Fluids 605 In closing this section, we summarize the observed viscosity behavior of polymer melts and polymer solutions The zero-shear-rate viscosity of lowmolecular-weight polymers increases linearly with molecular... on polymer melts, this is not a routine measurement The stretch-rate range of these extensional viscometers is such that the maximum stretch rate that can be achieved is of the order of 1 secÀ1 ; in polymer processing operations, a stretch rate of 100 secÀ1 is commonplace Also, not every polymer stretches uniformly, and, even when it does, steady-state stress levels are not always attained For all of . Inc. Ashear-thinningviscosityisnottheonlynon-Newtonianfeatureofthe behaviorofpolymericfluids;severalotherunusualphenomenaareobserved.If, inthesituationdepictedinFigure14.2,theshearrateissuddenlyreducedtozero aftertheattainmentofasteadystate,low-andhigh-molecular-weightliquids againbehavedifferently.ThestressintheNewtonianfluidgoestozeroinstantly, butittakessometimetodisappearinthepolymer.Thetimescaleoverwhichthis stressrelaxationoccursisknownastherelaxationtimeandisdenotedbythe symboly.Additionally,ifasmall-amplitudesinusoidalstrainisimposedonthe polymer, theresultingstressisneitherinphasewiththestrainnoroutofphase withthestrain:Thereisanout -of- phasecomponentrepresentingenergydissipa- tionandanin-phasecomponentrepresentingenergystorage(seeSect.12.4). Bothstressrelaxationandthephasedifferenceindynamicexperimentsareelastic effects;wesaythatthepolymersarebothviscousandelastic(i.e.,viscoelastic). Intime-dependentflow,therelativeextentofthesetwoeffectsdependsonthe valueofthedimensionlessgroupknownastheDeborahnumber(De)anddefined asfollows: De¼ y T ð14:1:1Þ whereTisthecharacteristictimeconstantfortheprocessofinterest.Forlow valuesofDe,thepolymerresponseisessentiallyliquidlike(viscous),whereasfor highvalues,itissolidlike(elastic).Afurthermanifestationofviscoelasticityis theswellingofajetofpolymeronemergingfroma‘‘die’’orcapillary.Thisis showninFigure14.3.Dieswell,orjetswell,canbesuchthatD j =Deasily exceeds2;thecorrespondingNewtonianvalueis1.13.Thisistrueatverylow flowrates.Athighflowrates,dieswellreducesbutunstablebehaviorcalledmelt fracturecanoccur.Thejetcanbecomewavyorthesurfacecanbecomegrossly distorted,assketchedinFigure14.4;theextentofdistortionisalsoinfluencedby thegeometryofthecapillary,itssurfacecharacter,andthepropertiesofthe polymer. NotethatmeltfractureisneverobservedwithNewtonianliquids. Thephenomenajustdescribedareinterestingtoobserveandexplain.A quantitativedescriptionofthemis,however,essentialfordevelopingmodelsof FIGURE14.3Thedie-swellphenomenon. Flow. value even below that of the zero- shear value. The behavior of polymer solutions is qualitatively similar to that of polymer melts, except that the extensional viscosity of polymer solutions can exceed. Inc. relativeslidingbetweenneighboringmaterialplanes.Inextensionalflow,as opposedtoshearflow,polymermoleculestendtouncoilandultimatelytherecan evenbestretchingofchemicalbonds,whichresultsinchainscission.Therefore, stressesintheflowdirectioncanreachfairlylargevalues. Althoughitissomewhatdifficulttovisualizehowarodofpolymermight bestretchedinthemannerofFigure14.11a,thiscanbedoneforbothpolymer melts[11]andpolymersolutions[12].Forpolymermelts,acylindricalsampleis immersedinanoilbathwithoneendattachedtoaforcetransducerandtheother endmovedoutwardsothatthestretchrateismaintainedconstant.Similarly,for polymersolutions,wemerelyplacetheliquidsamplebetweentwocoaxialdisks, oneofwhichisstationaryandconnectedtoamicrobalance,andtheotherdisk movesoutward,generatingthestretching.Thefilamentdiameterthinsprogres- sivelybutremainsindependentofposition.Simultaneously,thefilamentlengthl increasesexponentiallyasfollows: ln l l 0  ¼ _ eetð14:5:4Þ whichfollowsdirectlyfromEq.(14.5.1). Inthisflowfield,thereisnosheardeformation,andthetotalstresstensoras wellastheextrastresstensorarediagonal.Asaconsequence,thereareonlythree nonzerostresscomponents,but,duetofluidincompressibility,wecanmeasure onlytwostressdifferences.Further,inuniaxialextension,thetwodirectionsthat areperpendiculartothestretchingdirectionareidentical,sothereisonlyone measurablematerialfunction:thenettensilestresss E ,whichisthedifference T 11 ÀT 22 ort 11 Àt 22 . Forconstant-stretch-ratehomogeneousdeformation,whichbeginsfrom rest,atensilestressgrowthcoefficientisdefinedas Z þ E ðt; _ eeÞ¼ s E _ ee ð14:5:5Þ whichhasthedimensionsofviscosity.ThelimitingvalueofZ þ E astimetendsto infinityistermedthetensile,elongational,orextensionalviscosity,Z E .Ingeneral, Z E isafunctionofthestretchrate,althoughinthelimitofvanishinglylowstretch rates,wehavethefollowing[5]: lim _ ee!0 Z E Z 0  ¼3ð14:5:6Þ whereZ 0 iszero-shearrateviscosity. LaunandMunstedthaveobtainedtensilestressgrowthdataontheIUPAC ALDPEsampleat150  C[13,14],andtheseareshowninFigure14.12.Ata given