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# 75 Class Note for MATSE 259 at PSU

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This 50 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 34 views.

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Date Created: 02/06/15

Topics to be Covered Polymer Melt Rheology Introduction to Viscoelastic Behavior T imeT emperature Equivalence Chapter 11 in CD Polymer Science and Engineering hue 06 65 505201 Newton s Law The mosT convenienT way To describe defor39maTion under39 shear39 is in Terms of The angle 6 Through which The maTer39ial is deformed tan0 2 zyxy Then if you shear39 a fluid aT a consTanT r39aTe 6v0 7w 7 x ny dzdt dz JusT as sTr39ess is pr39opor39Tional To sTr39ain in Hooke39s law for39 a solid The shear39 sTr39ess is pr39opor39Tional To The r39aTe of sTr39ain for39 a fluid Ey 0quot yx quot3w m xy Newtonian and Non Newtonian Fluids TXY na Y Shear Thinning Newtonian Fluid 7 Slope Shear Thickening Strain Rate When the chips are down what do you use Variation of Melt Viscosity with Strain Rate Log 17a 5 Zero Shear Rate Viscosrty Pa 4 3 2 1 I I I I I I I gt 3 2 1 0 1 2 3 4 Log jseC39I SHEAR RA T ES EN C0 UN T ERED IN PROCESSING Compression Injection Spin Moldin alenderin I Extrusionl Molding I Drawingl 100 101 102 103 104 105 Strain Rate sec I How Do Chains Move Log 11m constant Variation of Melt Viscosity with Molecular Weight Polydimethylsiloxane Polyisobntylene Polyethylene Polybntadiene Poly tetramethyl psilph enyl Si e Polymethyl meth acrylate Polyethylene glycol Polyvinyl acetate Polystyrene 39 nm KLDP10 DDOI ODDO nm 34 Log M constant Entanglements ShorT chains d on39T enTangle buT ong ones do Think of The difference beTween a nice linguini and spagheTTios The iTTe r39ound Things you can geT ouT of a Tin we have some value judgemenTs concerning The r39eIaTive mer39iTs of These Two forms of pasTa buT on The advice of our lawyers we shall refrain from commenT Entanglements Viscosity a measure of the frictional forces acting on a molecule nm 10 Small molecules the viscosity varies directly with size At a critical chain length chains start to become tangled up with one another however Km 2 Th en nm Entanglements and the Elastic Properties of Polymer Melts EMTe 39J Depending upon The raTe aT which chains disenTangle reIaTive To The raTe aT which They sTreTch ouT There is an elasTic componenT To The behavior of polymer meITs There are various consequences as a resulT of This ddo W i 01 d 160 C 20 1800C 18 16 200 C 14 12 39 I 39 I 39 I 39 I 10391 10392 100 101 102 103 Strain Rate at Wall seal Jet Swelling a h H o a h u L a E Viscoelasticity If we s rr39e rch a cr39ys ralline solid If we apply a shear39 stress To The energy is stored in The A fluidener39gy is dissipa red Chemical bonds In flow I VISCOELASTIC I Idealy easfic IdealY Viscous behaviour behaviour Viscoelasticity Homer knew fhaf le firs ling To do on ge ing your chariof ouf in le moming was 0 put le wheels back In T elemachusin The Odyssey would p his chariof againsf a wall Robin hood knew never fo leave his bow sfr ung Experimental Observations Creep Stress Relaxation Dynamic Mechanical Analysis Creep and Stress Relaxation gt TIME Creep deformaTion under a consTcmT load as a funcTion of Time STr39ess Relaxcn ion consTcmT deformaTion experimem TIME Redrawn from the data of W N Findley Modern Plastics 19 71 August 1942 50 I I I A 2695psi 2505pst 5 23 05 1 2008 psi A A A MA A u e A r 1690psi II I I n 39 39 nunquot l I IF F 1320 psi 181 o 000090000000000 000 o F lll IIIIIIIIIIIII I IIll 2000 4000 6000 Iinne htzuis 8000 Creep and Recovery 0 VISCOELASTIC RESPONSE Strain Recovery fe manent e armation Time Stress Stre s removed app ied Strain vs a PURELY ELASTIC RESPONSE Strain Strain Time Plots Creep b PURELY VISCO US RESPONSE I Permanent p Deformation Stre s Stress I Time appfigea removed 5 gt i hear gear Time tress tress applied removea c VISCOELASTIC RESPONSE Strain Creep Permanent a rmation Stress Time I cigeliyesa removed Stress Relaxation The daTa are noT usually reported as a sTressTime plo i but as a modulusTime plo i This Time dependen i modulus called The relaxa on modulus is simply The Time dependen i s iress divided by The cons ian i strain 0 Redrawn from the data ofJR McLaughlin and AV Log Et dynesom 2 Tobolsky J Cuttmd Sn 7 555 1952 0 C 10 60 C 92 C 110 C 115 C I I 0 100 C 80 9 Stress relaxation ofPMMA 0001 Time hours Amorphous Polymers Range of Viscoelastic Behaviour Liquid A Vs Semi Like Solid Rubber39y LeaTher39 i SofT Like Hard Glass L y gt Viscoelastic Properties 0f Amorphous Polymers LogE A Glassy Pa 10 Region 9 39 Mea3ur39ed over39 8 Some ar39bi rr39ar39y 7 Time period say Crosslinked 10 secs 6 quotElaslomers Rubbery 5 Plateau Low 4 39 Molecular Melt Weight 3 gt Temperature Viscoelastic Properties 0f Amorphous Polymers Glassy A Region E 10 Glass I S rr39e rch sample an g TmnS I O g 9 ar39bi rr39ar39y amoun r EA Rummy me05ur39e The s rr39ess E 8 Plateau required To main rain 0 7 This s rr39ain 4 6 LOW Then E r 6Te M01 WW gt High 5 Weight Molecular Weight 4 3 I I I I I I 10 8 6 4 2 0 2 Log Time Sec I L0 E A Glass TI m e 010 Region 9 Temperature 8 I 7 Equ Iva39ence CrossImked 6 39 Elastomers 5 Rubber Plateau Low N 4 39 Molecular Melt A 525 G 3 Weight gt g 10 39 Trgnsssiuon Temperature 3 9 K R bb S 8 Pllategtlzy g0 7 6 Ml levgulara Hi I 5 Weight M0 ecular Weight 4 3 I I I I I I 10 8 6 4 2 Relaxation in Polymers FirsT consider a hypoTheTical isolaTed chain in spaceThen imagine sTreTching This chain insTanTaneously so ThaT There is a new end To end disTanceThe disTribuTion of bond angles Trans gauche eTc changes To accommodaTe The conformaTions ThaT are allowed by The new consTrainTs on The ends Because iT Takes Time for bond roTaTions To occur parTicularly when we also add in The viscous forces due To neighbors we say The chain RELAXES To The new sTaTe and The relaxaTion is described by a characTerisTic Time quotC How quickly can I do these things Amorphous Polymers the Four Reions of Viseoelastic Behavior L E 533 A GLASSY STA TE conformaTional 10 39 changes severely inhibiTed 9 8 Tg REGION cooperaTive moTions of segmenTs now occurbuT The 7 moTions are sluggish a maximum in 6 Tan 8 curves are observed in DMA experimenTs 5 Low 4 Molecular P 01W RUBBERY PLA TEAU quotcf becomes Weight Melt 3 gt shorTerbuT sTIll longer Than The empemfure Time scale for disenTanglemenT TERMINAL FLOW The Time scale for disenTanglemenT becomes shorTer and The melT becomes more fluid like in iTs behavior Semicrystalline Polymers s00 030 p 025 a 0 o o o O 020 000 3 o 00 015 0 39 oLDPE 3 o oLPE 0 o o 010 9 O 7 o o 00 005 O O 0 000888300 0 000 200 150 100 50 0 50 100 150 Redrawn from the data of H A F locke Kolloidiz Z Polym 180 188 1962 NoTion in The amorphous domains consTrained by crysTalliTes NoTions above T9 are ofTen more complexofTen involving coupled processes in The crysTalline and amorphous domains Less easy To generalize polymers ofTen have To be considered individually see DMA daTa opposiTe Topics to be Covered Simple models of Viscoelastic Behavior T ime Temperature Superposition Principle Chapfer39 11 in CD Polymer Science and Engineering Mechanical and Theoretical Models of Viscoelastic Behavior GOAL r elaTe GT and JT To relaxa on behavior We will only consider LINEAR MODELS ie if we double GT or G l39 Then 71 or eT also increases by a facTor of 2 small loads and sTr39ains T ENSILE SHEAR EXPERIMENT EXPERIMENT Stress Relaxation t T t Ea 00 Ga 30 70 Creep Dt M Linear Time Independent Behavior 1 1 G D J Time Dependent Behavior 1 1 Et D t Gt 70 Simple Models of the Viscoelastic Behavior of Amorphous Polymers Keep in mind Tho r simple creep and recovery do ro for39 viscoelos ric mo rer39iols looks some rhing like This c VISCOELASTIC RESPONSE Strain Creep Permanent drman0n Stre s Stress Time app zed removed Simple Models of the Viscoelastic Behavior of Amorphous Polymers While stress relaxation da ra look some rhing like This Log Et a ynescm2 10 100 kUIQ Glassy A Region Glass 39 Transition Rubbery Plateau Low 39 Molecular gt High Weight Molecular Weight I I I I I I 8 6 4 2 0 2 Log Time Sec Simple Models of the Viscoelastic Behavior of Amorphous Polymers Extension 7 t Impment linear cumin behavinr UE Simple Models of the Viscoelastic Behavior of Amorphous Polymers mscous flow 7 l VD Newtonian uid gt gt Iquot Txy 1W d5 a 17 Strain vs Time for Simple Models PURELYELASTIC RESPONSE PUREL Y VISCO US RESPONSE a E 5 18 7 a 11 Strum Straw nme St St Permanent y s 1655 app ted remaved Deformman She She 309 Srre Tm applted remaved Maxwell Model Maxwell was inTer39esTed in creep and sTr39ess r39elaxaTion and developed a differenTial equaTion To describe These pr39oper39Ties Maxwell sTar39Ted wiTh Hooke39s law Then allowed 6 To var39y wiTh Time Wr39iTing for39 a NewTonian fluid Then assuming ThaT The r39aTe of sTr39ain is simply a sum of These Two conTr39ibuTions GE8 d5 d8 dT quotEOE d8 6 on d8 ELd6 dT39 Tl Ed l39 MAXWELL MODEL Creep and Recovery 7 Strain Creep and gt recovery d8 9 df T1 Strain RecaH that rea vwscoe asnc bzhawour ooks somzmmg Mk2 mm A pmmr e represemmmn of MaxweH s equanon Maxwell Model Stress Relaxation d8 E Ld6 d f n E T LogEE0A In a stress relaxationyexper imen r O 1 de o d r 39 3 Hence d5 gdI 4 5 n 5 c5 coeporf 396 239 Where Og T 11 TT E Relaxa rion Time Maxwell Model Stress Relaxation Log Et P6910 Real data looks like this 8 39 The Maxwell 7 model gives curves like this 6 0R like this Time MAXWELL MODEL 1 t Et0 ex 80 80 It Voigt Model Maxwell model essentially assumes a uniform distribution Of stressNow assume uniform distribution of strain lOIGT MODEL Picture representation gt Equation d T 6T E80 n dT Strain in both elements of the model is the same and the total stress is the sum of the two contributions Voigt Model Creep and Stress Relaxation em a retarded 2mm response but does not aHow 7 for dzaV mess r2axanonm that m mode cannot be msmmanzous y deformed m 0 gm strum But in REEF 1 constant 10 0391 0392 6T so Es r n jg ff d 90 d r T n L RETARDED ELASTICRESPONSE a a 62 gm 1 exp TTL TL refardafion Time TlE What do the straintime plots look like Problems with can t do creep And can t do men relaxation Simple Models Tiie muxwell model cannot aooount for a retarded elastic response Tiie voigi model does not describe stress relaxation Botli models are oliaraoterized by single relaxation times aspectrutn of relaxation times would provide u better description NEXT 7 ONSIDER THE FIRST TWO PROBLEMS THENrTHE PROBLEM OFA SREcTRLM OF RELAXA HON TIMES Four Parameter 7 Elasfic viscous flow refa ded elasfic 90259 039 a f a g 0 0 5 EMWM EMU emum Retardedar AnelasttcReSpame Strum Permanent 2quotde r55 d Tm Distributions of Relaxation and Retardation Times Stress Relaxation We have menTioned ThaT alThough The Maxwell and VoigT models are seriously 151 E0 exp tTt flawed The equaTions have The righT 7 form El ETexp tTd139 0 WhaT we mean by ThaT is shown Creep opposiTe where equaTions describing The Maxwell model for sTress relaxaTion and Dt D01 exp tT The VoigT model for creep are compared co To equaTions ThaT accounT for a D fDTt91 exptTt9drt 0 conTinuous range of relaxaTion Times These equaTions can be obTained by assuming ThaT relaxaTion occurs aT a raTe ThaT is linearly proporTional To The disTance from equilibrium and use of The Boszmann superposiTion principle We will show how The same equaTions are obTained from models The Maxwell Wiechert Model CL 91 l 5103 dT n1 E1d7 92 l 10quot 712er Consider stress relaxation d5 Er 39 0 61 agexprr a aoexprrr2 a3 aoexprr3 Distributions of Relaxation and Retardation Times S rr ess r e oncrl39ion modulus E39l39 G80 61 51 52 53 W 11 E gt1exp Jr101 2 exp TO2 3 exp 1O3 O O 0 Or in general an 2 En exp TquotCm where En 3011 Similarly for creep compliance combine voig39l39 elements 1390 obfain Dltgt an 1 exp 4111 Log E I a ynescm2 Distributions of Relaxation and Retardation Times Example the Maxwell Wiechert Model With n 2 10 39S Glassy 91 8 A Region 3 Glass La 10 39 Transition go 6 9 N Rubbery 4 8 Plateau 2 7 0 6 5 Melt gt 4 3 I I I I I I 39 10 8 6 4 2 0 2 Log Time Sec 2 1 0 1 2 3 4 Log time min Em 2 En exp Jrrm n 2 Log Et dynescmZ Time Temperature Superposition Principle Recall Thcn we have seen Thcn g A Glassy There is a Time TemperaTure 10 39 Region equivalence in behavior 9 39 8 7 C 139 k d l FOSSm e A 19693711 6 Elastomers Glass R bb 10 Transmon 5 u er Plateau Low 9 T Rubbew 4 quot Molecular Melt 8 Plateau 3 Weight 7 Temperature 6 Afolevgular gt High 5 39 Weight M This can be expressed 4 formally in Terms of a 3 I I I I I I superposiTion principle 10 8 6 4 2L0g lgme 33923 Time Temperature Superposition Principle Creep Stress Relaxation Modulus Time Temperature Superposition Principle Stress Relaxation a yrzescm2 a 2 2 o H O o SHIFT FACTOR vs TEMPERATURE Stress Relaxation Data x Lag Shift Factor 60 39l39 f 10392 100 102 10121010108 10396 10394 10392 10390 102 Time hours 103914 Relaxation Processes above T9 the WLF Equation From empirical observaTion quot61T39 0 CZTI ForTg gtTlt Tg 100 C Log GT Originally ThoughT ThaT C1 and C2 were universal consTanTs 1744 and 516 respecTivelywhen TS 2 T9 Now known ThaT These vary from polymer To polymer Homework problem show how The WLF equaTion can be obTained from The relaTionship of viscosiTy To free volume as expressed in The DooliTTle equaTion Semi Crystalline Polymers NON LINEAR RESPONSE TO STRESSSIMPLE MODELS AND THE TIME TEMPERATURE SUPERPOSITION PRINCIPLE DO NOT APPLY Temperature A Amorphous meh Tm Rigid crysfaline domains Rubbery amorphous domains Tg Rigid crysfaline domains Glassy amorphous domains

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