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# Introduction to the Materials Science of Polymers MATSE 443

Penn State

GPA 3.83

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This 0 page Class Notes was uploaded by Kacie O'Keefe on Sunday November 1, 2015. The Class Notes belongs to MATSE 443 at Pennsylvania State University taught by Staff in Fall. Since its upload, it has received 28 views. For similar materials see /class/233032/matse-443-pennsylvania-state-university in Materials Science Engineering at Pennsylvania State University.

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Date Created: 11/01/15

IOIGT MODEL Maxwell mdel essenTiolly assumes a uniform disTribuTion Of sTress Now assume uniform disTribuTion of sTroin VOIGT MODEL PicTur39e r39epr39esenToTion gt EquoTion so Ego nd8 m E H CH I l STr39oin in boTh elemenTs of The 239 model is The same and The ToTaI I sTress is The sum of The Two conTr39ibuTions IOIGT MODEL creep and stress relaxatio Gives a r39e rar39ded elasTic response bu r does no r allow for quotidealquot s rress relaxaTionin ThaT The model cannoT be insTan raneously deformed To a given s rrain Bu r in CREEP G cons rcm rcso om so E80 n 13 ED d l39 It39 11 gm 1 exp 1r Tt r39eTar39daTion Time E SUMMARY A E E il l Tl Tl Maxwell Voigt Sprmg Dashpot element element Strain Spring Dashpot Maxwell VOigt model model V PROBLEMS WITH SIMPLE MODELS The moxwell model connoT occounT for39 o re rorded elos ric response The voigT model does noT descr39ibe s rr39ess r eloxo rion BoTh models are chor oc rer ized by single r eloxo rion Times a specTrum of r eloxo rion Times would provide a be r rer39 descripTion NEXT CONSIDER THE FIRST TWO PROBLEMS THEN THE PROBLEM OF A SPECTRUM OF RELAXATION TIMES 8 FOUR PARAMETER MODEL ELASTIC VISCOUS FLOW RETARDED ELASTIC eg CREEP EM go 6o r go EMTIM EM1 exp M EV I nv Strain Retarded elastic response Elastic Permanent response 1 deformation gt t1 5 Time t applied removed DISTRIBUTIONS OF RELAXATION AND RETARDATION TIMES The Maxwell Wiechert Model d8 C71 1 G d r 39 n1E1d E G 1d6 1 2 g3 2 2 n2 13ij 111 112 in E31d63 LF n3 Ed l 8 Consnder s rress relaxa rlon 5 1 0 G1 60 exp Tz fl 62 60 exp 1112 63 60 exp 1113 DISTRIBUTIONS OF RELAXATION AND RETARDATION TIMES STress relaxa rion modulus E G 39go J 5 61 62 53 9 Lvlnz J ng EU gio 1 exp 43 gio 2 exp 030 gio 3 exp 550 Or39 in general A E EEZ E 2 En exp 17ft where E m quot o SIMILARLY FOR CREEP COMPLIANCE COMBINE VOIGT ELEMENTS TO OBTAIN DZD1 exp 1 DISTRIBUTIONS OF RELAXATION AND RETARDATION TIMES Example The Maxwell Wiechert Model with 2 En exp lift 10 n 2 gt 8 Glassy 3 6 I region Equot E 4 10 Transmon E0 g Rubbery 2 m 8 plateau E g 0 e 6 2 1 0 1 2 3 4 0L Log time min 3 Low gh gt molecular molecular 4 weight weight 10 8 6 4 2 0 2 Log time sec TIME TEMPERATURE equivalence in behaviour SUPERPOSITION PRINCIPLE Reccll ThaT we have seen ThaT There is a Time Temperature Log E 0 71 3 we L Tempemmre This can be expressed formally in Terms of a suprposiTion principle LogE t dynes m2 0 00 0 I Glassy region Transluon Rubbery p laleau TIME TEMPERATURE SUPERPOSITION PRINCIPLE creep logtlog 211 T TIME TEMPERATURE SUPERPOSITION PRINCIPLE stress relaxation wrwz nukz smrr cm LOG smrT a muss muxmm MODULuz nvnucw VINE HOURS SIGNIFICANCE OF SHIFT FACTOR WhaT is The significance of The log scale forra and whaT does This Tell us abouT The TemperaTure dependence of relaxaTion behaviour in amorphous polymers 9 Consider sTress relaxaTion ET 2 En exp Tft LeT a parTicuIar mode of relaxaTion have a characTerisTic Time Ito aT I and a characTerisTic T39rrqe aT If Th JEFINE G 3 T 39 T Term can be meTen 1 GT Ito In 50 ThaT The exponenTiaI I39 I39 C t t0 Hence Taking logs 3909 TIu 3909 1 3909 f1 SIGNIFICANCE OF SHIFT FACTOR log mu log we gt log in ie relaxaTion behaviour aT one TemperaTure can be superimposed on ThaT aT anoTher by shifTing an amounT or along a log scale BUT real behaviour is characTerized by a disTribuTion of relaxaTion Times and relaxaTion mechanisms vary and have differenT lengTh scales as a funcTion of TemperaTure This implies ThaT all The relaxaTion processes involved have more or less The same TemperaTure dependence RELAXATION PROCESSES ABOVE Tg THE WLF EQUATION From empirical observa rion q 5T N o 62TSTForTggtTlt Tg100C Log aT Originally Though r rha r andZC were universal cons ran rs 1744 and 516 respec rivelywhesn T 2 T9 Now known rha r rhese vary from polymer ro polymer Homework problem show how The WLF equa rion can be ob rained from The rela rionship of viscosi ry ro free volume as expressed in The Dooli r rle equa rion DYNAMICS OF POLYMER CHAINS An advanced Topic That we will noT discuss in detail Rouse Bouche model A chain as a sfr ing of may Beads linked by springs 12 Repfafionscaing concepfs And ofher advanced Theories LIQUIDS i39e 9 a ltag1ss scatyerers Polarizability per unIt volume Geometry of Characteristics of observation the incident beam TO A FIRST APPROXIMATION a 317 HENCE PURE LIQUIDS SHOULD NOT SCATTER LIGHT FLUCTUATIONS Pure liquids density Solutions concentration K 1 c0529 0 R Miw12r2c But a polymer is too large to Be considered a single oscillato ie a point source of radiation Must consider interference effects ZIMM PLOT K 1 cosze c 2 R Ml21 2c lSsm 9 1 W 4 1 Experimentally measured parameters 5 4 Weight Average 4 39 a 3quot ll 39Ill 39 MolecularWeight 9 011 7 39 5 I 39 Fquot 2 quot39 I II I Klcos 9c K 4quot J I I 7 l Virial e 39 39 Expansion Dependence upon angle of observation THE VISCOSITY OF POLYMER SOLUTIONS Measure the time taken to flow between fixed marks in a capillary tube under the draining effect of gravity The volume rate of flow 0 is then related to the viscosity by Poiseuille39s equation 4 TEPI D 8111 where P is the pressure difference maintaining the flow r and l are the radius and length of the Capillary and n is the viscosity of the liquid Relative Viscosity Defined as the viscosity of a polymer solution divided by that of the pure solvent and for dilute solutions where t is the time taken for a volume V of solution no subscript or solvent subscript o to flow between the marks Relative Viscosity as a 16 Function of Concentration 39 39 14 39 A 39 A power series similar to that used in the treatment of HM 39 39 osmotic pressure and light scattering data is commonly used to fit relative Viscosity data 1392 l 2 nm no 1 nckc 10 Both 1 and k are constants 39 I I I 0 l 2 3 4 5 n is called the intrinstc viscostty Concentration gIOO cc Plot of 11m versus c for PMMA in chloroform Plotted from the data of G V Schultz and F Blaschke If Viscosity measurements are confined to dilute solution so that we can neglect terms in a and higher 11 1 nn n0 n1kc The Specific Viscosity isde nedas nsp nrell Note also that as c goes to zero infinite dilution then the intercept on the yaXis of a plot of aspC against c is the intrinsic Viscosity n MEASURING THE INTRINSIC VISCOSITY In practice we use two semiempirical equations suggested by Huggins and Kraemer Tlsp v 2 7c 11 k 11 C E c 111 nkun2c 2 o E E s V V 3 to V lnnrel I ll c 0 0 n 11ml I I ll c 0 Concentration c gdl 025 Schematic diagram illustrating the effect of strong intermolecular interactions Concentration c g d1 025 Schematic diagram illustrating the graphical determination of the intrinsic viscosity THE MARKHOUWINKSAKURADA EQUATION The RelaTionship BeTween InTrinsic ViscosiTy and Molecular WeighT If The log of The inTrinsic viscosiTies of a range of sample SChEma39l39C kdlagr am f Thke DeJermInaTIon is ploTTed againsT The log of Their molecular weighTs of l 8 Mar quotHouw39quot quot5 um a consmnls Then linear ploTs are obTained ThaT obey equaTion K and a V Monodisperse standards Ternp 10g Solvmt 0 PS 3 PMMA Slope a 239 2 A Intercept log K Log 11 1 M 1 1 U c 4 5 e 7 4 5 e NoTe ThaT K and IIaII are noT universal consTanTs Log Molecular Weight buT vary wiTh The naTure of The polymer PloTs of The log h versus log M for PS and PMMA The solvenT and The TemperaTure ReploTTed from The dam of Z Grubisic P Rempp and H THE VISCOSITY AVERAGE MOLECULAR WEIGHT For Osmotic Pressure and Light Scattering we saw that there is a clear relationship between experimental measurement and the number and weight molecular weight average respectively Viscosity measurements are related to molecular weight by a semiempirical relationship and a new average the Viscosity Average for polydisperse polymer samples is defined In very dilute solutions nsp nsp nSPi n a a N W 0 KM i Hence nspK ZNIici And n esp If By substitution and rearranging we obtain Note that the Viscosity Molecular Weight is Not an Absolute Measure as it is a function of the solvent through the MarkHouwink parameter quotaquot FRACTIONATION PDLMER SDLUIIDN l COLUMN Collect lractions Measurehow much P I o ymer is in each lractio SIZE EXCLUSION OR GEL PERMEATION CHROMATOGRAPHY Schematic diagram depicting the separation of Schematic diagram of an SEC instrument molecules of different size by SEC Solvent Reservoir Mixing Valve Injection Port SEC Columns Small Permeating Molecules Lar e Exc uded Molecules Volume Void 8 O O Pores Bulk Movemeni of Solvent For a given volume of solvent flow molecules of different size travel different pathlengths within the columnThe smaller ones travel greater distances than the larger molecules due to permeation into the molecular mazeHence the large molecules are eluted first from the column followed by smaller and smaller molecules THE CALCULATION OF MOLECULAR WEIGHT BY SEC The Simplest Case where Monodispersed Standards of the Polymer are Availab Area normalized Polydisperse Sample Intensity Monodisperse XV Standards J L Elution Volum e V h i i 2 hi Selective Permeation L i M E W M bi ecular w Z l 1 Weight Exclusion u Totai Permeatlon 7 4 ri Mn Z Wi V l Elution Volume a Schematic diagram depicting the calibration of an SEC instrument k n Z Wi ni K 2 W i M1 HOW DOES SEC SEPARATE MOLECULES Benoit and his coworkers recognized that SEC separates not on the basis of molecular weight but rather on the basis of hydrodynamic volume of the Calibration Curves for LinearP01yB polymer molecule In solution Starshaped Same solvent LOg PolyA gt Same temperature Molecular Weight Linear PolyA gt M 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 Elution Volume a The Universal Calibration Curve If we model the properties of the polymer con in terms of an equivalent hydrodynamic sphere then the intrinsic viscosity 1 is related to the hydrodynamic volumth via the equation 25 A V h Tl M Benoit and his coworkers recognized that the product of intrinsic viscosity and molecular weight was directly proportional to hydrodynamic volume A is Avogadro39s number and M is the molecular weight I El linear polystyrene 10 linear PMIVIA I I I linear PVC polystyrene combs I polystyrene stars 9 393 PSPMlA graft copolymers comb PSPMMA heterograft copolymers A Polyphenyl siloxane ladder polymers 8 I PSPMMA statistical copolymers Log 11 M 7 6 5 I I I I I 18 20 22 24 26 28 30 Elution Volume A universal calibration plot of log 11M vs elution volume for various polymers Redrawn from the data of Z Grubisic P Rempp and H Benoit POLYMERIZATION KINETICS THERMODYNAMICS KINETICS Tells us where the system Tells us how fast the would like to go eventually system takes various ie defines relationships reaction paths between macroscopic variables at equilibrium EXAMPLES SUGAR OXYGEN gt PRODUCTS ENERGY CRYSTALLIZATION IS ALSO A PROCESS CONTROLLED BY KINETICS AS WE WILL SEE LATER POLYMERIZATION KINETICS STEP GROWTH CHAIN Polymerization SLOW Can use statistical methods as well as kinetics to describe mol wt distributions more on this later FAST Can apply statistical methods to an analysis of the microstructure of the products but not the polymerization process and things like mol wt KINETICS OF STEP GROWTH POLYMERIZATION WHY BOTHER How long does it take to make polymer Can we speed up the reaction What is the relationshi between kinetics and the Mal t 01 the product REVISION RATE OF CONSTANT x CONCENTRATIONquotTERMs REACTION RATE OF DISAPPEARANCE dM OF MONOMER a k x CONCENTRATION quot TERMs KINETICS OF POLYCONDENSATION KEY ASSUMPTION FLORY The reactivity of a functional group is independent of the len th 0 the chain to which it is attac ed EXAMPLE DIBASIC ACID GLYCOL gt POLYESTER o o This group HO ICIW NNNNNNNNNNNNNNNNNN m C OH reacts as readily as O O HO NNNN OH This group l l l l gt HOCCOH HONNMNNNNNNNNNNNOH 41 sec 41 g equiv 1 k x104 A WAS FLORY RIGHT 25 2039 10 Chain Length N UI 15 20 Redrawn from The da ra of FIoryPJ39 Principles of Polymer Chemis rryCorne Universi ry Press 1953 p71 KINETICS OF POLYCONDENSATION AABB AABB KineTic equaTion for This Ty of reacTion Is usually of The arm d A Reaction Rate 7 kZAB NB A AND B ARE THE CONCENTRATIONS OF FUNCTIONAL GROUPS However es l39er39ificaTions are acid caTalyzed and in The absence of added sTrong acid dA 2 T k3A B MORE KINETICS dA 2 7 kgw B If A B C A 13 Hence dc 2 k3 03 dt EXTENT OF REACTION Define p EXTENT OF REACTION In fhis example OF COOH GROUPS REACT ED L T E p OF COOH GROUPS ORIGINALLY PRESENT Then C Co And 2 1 2cok3f 1 HO2 Redrawn from The da ra of Flor39yPJ39 JACS61 3334 1939 E u 100 l 1000 Time mins 2000 ACID CATALYZED REACTION d CdofH k39 COOH 1 OH 1 80 39 dc k39c2 dt 39 60 1 1 1 p cokT 1 P constant 40 20 39 Note the concentration of the acid catalyst a constant is included in k39 0 o 260 460 660 Time mins Redrawn from the data of Flor39yPJ39 JACS6133341939 NUMBER AVERAGE DEGREE OF POLYMERIZATION N xn N 0 0 c c01p 1 Ie xn 19 Mo 0X 19 113 10039 KINETICS OF FREE RADICAL POLYMERIZATION We need To consider The following steps INITIATION PROPAGATION CHAIN TRANSFER TERMINATION INITIATION ooE Benzoyl peroxxde I Z EO Decomposition H H ki R39 CHz gt R CHz 39 R39 M M1 X X Addition INITIATION kd ASSUME DECOMPOSITION IS THE I 2R39 RATE LIMITING STEP ki ie gtgt R M gt M1 kl kd THEN WE SHOULD ONLY HAVE TO CONSIDER kd d I 1 am 1391 leW d t BUT ONLY A FRACTION 1 OF RADICALS INITIATE CHAIN GROWTH O a cs O ridr 2fkdI PROPAGATION k M1 M quotgt M k M2 M pgt M3 dM rp Tr klellM In general k Mx M quotgt M21 Assumption reactiviw is independent of chain eng rh TERMINATION COMBINATION H H mmeHZ cl39 39Cl Cszwwvw k X X Tc km Ill PII Mquot I Mxy gt mmCHZ C C CHZMMWM gt39lt gt39lt DISPROPORTIONATION H H mvaHz Cl39 39 CHZMMMIN k X X fd km H H MkM9 gt MxMy wwvvaHC H C CHZvawwt X X RATE OF TERMINATION 1 II I 392kM39M39 OBTAINED FROM 39 Both reactions are second order 39 Rate of removal of chain radicals sum of the rates of the two termination reactions SUMMARY ri daft 391 2fk dI d pp kptMnMJ d rt 2kT M39M39 PROBLEM We don39fknow M SOLUTION Assume a sfead state concepfra on q TraHSIenT speaes STEADY STATE ASSUMPTION M CONSTANT This means 1 ha1L radicals are consumed a1L The same raTe as They are generated Pi PT kadI2kTM2 fk I 2 W kd H 139 RATE OF PROPAGATION RATE OF PROPAGATION RATE OF POLYMERIZATION R P p p substituting rp kp f 12 M kt BUT I Is NOT CONSTANT fromfiEi fkdl obtain IIoe kd r HENCE RP kp fkd12 M 10112 ede2 WHAT DOES THIS TELL US 1 IF WE WANT TO INCREASE RP INCREASE M OR I 12 2 RP kpkf 3 TROMSDORFF EFFECT But changing I also changes mol wt more on this later For ethylene at 130 06 and 1 bar pressure kpk2 005 For ethylene at 2000 C and 2500 bar pressure kpk2 30 0 500 1000 1500 2000 Time mins CONVERSION MolM Mo DEFINITION in initial stages of reaction we can assume I Io constant dM T pr Rd lu t Integrating A kpfk dl Io 112 T M01 kT Ln Amount of monomer used up Amount of monomer at start A kConversion quotv MAXIMUM CONVERSION USUALLY THERE IS A FIRST ORDER DECAY IN INITIATOR CONCENTRATION ie d IkdI IIoekdf if and M 2 k fk 16 2 k Ln Pd 11ed1 Mo kd kt hence AConversion CONVERSION 10 1 2 kp 1 21 MAX CONVERSION 139 gt 0quot 1 exp 2kpf10 112 kT kd AVERAGE CHAIN LENGTH DEFINE KINETIC CHAIN LENGTH RATE OF MONOMER ADDITION TO GROWING CHAINS RATE AT WHICH CHAINS ARE STARTED This is The average number of monomers polymerized per chain radical aT a parTIcular insTanT of Time during The polymerizaTion KINETIC CHAIN LENGTH CONSIDER A TIME PERIOD t let us say that 1 100 chains are started 2 1 000 000 monomers are reacted in this time period Then the average degree of PolYmerization of these chains 39 1 000 IS 10000 KINETIC CHAIN LENGTH THERE WILL BE SOME OBVIOUS ERRORS eg What about chains that were initiated but did not terminate just before the start of the chosen period 1 BUT THESE DECREASE AS t gt SMALL IN THE LIMIT OF A TIME PERIOD dt Pp kp v Pi 2 kd k112 I 112 0 12 lev Eli2 Cf THE DEGREE OF POLYMERIZATION THEN DEPENDS UPON THE MECHANISM OF TERMINATION v disproportionation 2 v combination XI XI 3 INSTANTANEOUS NUMBER AVERAGE CHAIN LENGTH What if Termination occurs by both mechanisms define an average number of dead chains per Termination reaction rate of dead chain formation rate of termination l2kd knllM39lz my kn kfd knn M12 kt HENCE kptMIIMI xn I2ktd ktcHMIz kle 39 fkdkf12I12 xn CHAIN TRANSFER Chain Transfer can occur To solvent added agentse1 c d M39 R39 H M gt M H x R39 rquot d1 39 kquot RIWl M39 dM39 2 2 T 2kd M 391 k39rcl M 39 kn TJI39M CAN THEN OBTAIN 2quot M 160 Ethylbenzene xquot 12 12 L5121 OR Xquot 80 1 1 C kn 4o xn xn0 M o SJM Redrawn from The dafa of RA6regg FRMayo Faraday Soc Discussions23281947 COPOLYMERIZATION POLYMERIZATION OF TWO OR MORE MONOMERS Examples Polyethylenecopropylene EP Polystyrenecobutadiene SBR Polyvinylidine chloridecovinyl chloride Saran Microstructure depends on R d I method of polymerization 0quot 0m C P Ymers ABBBAABABAAABABB Alternating Copolymers ABABABABABABABAB Block Copolymers AAAAAAAABBBBBBBBBAAAAAA POST POLYMERIZATION OR HOW TO MAKE A GRAFT COPOLYMER Examples High Impact Polystyrene HIPS Acrylonitrile Butadiene Styrene ABS Dissolve polymer A in a solvent Add monomer B and initiator Polymerize m B R B AAAAAAAAAA E AAAAAAA A AA cu B 1 c39u AAAABAAAAAAB Rs B B gt AAAAAAAA A A Eu w B AAAAAAAAAA EU 3 B B B B B BBBBBB up EU to AAAAAAAA AA EU B 3333 B B R 63 FREE B lll lll P lYMEIIIIATIIIN ARE quotRANDOMquot COPOLYMERS REALLY RANDOM A A Alt B A 3 V B A B DO A39s ADD TO A39s AS EASILY AS B39s ADD TO A39s and vice verso POSSIBlE PIIIIIIIIBTS COPOLYMERIZATION OF MONOMERS M1 AND M2 Homo po lymer39s ATer39naTing copolymer39s Ideaquot or39 Truly random copolymer39s Non ideal copolymer39s Tendency To quotblockinessquot or39 alTer39naTing KINETIGS 0F P lYIVIEIIIIATIIIN TENDENCIES 1 BLOCKS AND OR HOMOPOLYMER IF k11 gt k12 AND k22 gt k21 2 ALTERNATING IF k12 gt k11 AND 3 RANDOM COPOLYMERS IF k11 k12 AND k22 k21 k21 gt k22 BEAGTIIIITV RATIOS DEFINE 22 1 k12 2 k21 WHAT IF r1quot2 gtgt1 r1quot2 ltlt 1 r1 1 KINETIGS 0F BOPIIlYIVIEIIIIATI N 1 k11M1M1k21M2M1 321 k22 M2M2 k12M1M2 Divide and eliminaTe M Ter39ms using STEADY STATE ASSUMPTION M 1 Mgener aTed and consumed aT equal r39aTes we only need focus on one Type of radical eg M 1 k12M139M2 k21M239M1 P lYIVIEII EQUATION dM1 M1 r1M1 M29 d39V2 39 W21 V1 9 th or dV1 r1V1V21 dIle 39 r2M2139M11139 remember P lYIVIEII EQUATION IT is ofTen more convenienT To work in Terms of mole fracTions dM1 define dquotquot11 0 le F1 moe fracTion of monomer 1 in The polymer aT some insTanT of M1 1quot 39 lme M1 quot39 M2 f1 moe fracTion of monomer 1 in The feed aT The same insTanT of Time P lYIVIEII EQUATION F 39139 1 1 1 f12 1 39 2 P1P22f1 21quot2f1quot2 1 NoTe Thcn This equaTion describes The insTonToneous F copolymer composiTion 1 immune r17r21 0 l 0 f1 In a baTch copolymerizaTion composiTion will quotdrifT quot wiTh conversion To Trecn This properly we need To firsT do some sToTisTicsbuT here we will jusT give a couple of illusTraTions DETERMINATION OF IIEIIIITIIIITV IIIITIIIS There are some older39 meThods based on rearrangemenTs of The copolymer39 equaTion Mll M LET Mg X and dMz y 1rX THEN y 1 X XI 1X1 r2TX MAYO LEWIS PLOT FINEMAN ROSS PLOT DETERMINATION OF IIEIIIITIIIITV IIIITIIIS ALSO KELEN woos PLOT Q e SCHEME APPLICATION OF PROBABILITY THEORY AND nmr39 SPECTROSCOPY POLYMER SYNTHESIS COPOLYMERIZATION THE COPOLYMER EQUATION 1 lAX X Composition of the Feed y r B P A 1 B y Composition of the Instaneously Formed Polymer 1 X P1B Copolymer Composition as a Function of Conversion VDC Polymer Composition Degree of Conversion Compositional variation for vinylidine chloridevinyl chloride copolymers MIXING WILL A POLYMER DISSOLVE IN A GIVEN SOLVENT WILL A PARTICULAR POLYMER MIX WITH ANOTHER POLYMER POLYMER SOLUTIONS AND BLENDS MISCIBLE IMMISCIBLE OO o O o o O o 0 Single phase Phase separated Phase separated solution gt lt Blends T LP Solvent or polymer molecules Solvent rich phase Solvent poor phase CAN WE PREDICT PHASE BEHAVIOUR THE SECOND LAW OF THERMODYNAMICS AND FREE ENERGY For a process To occur spontaneously must increase eg mixing AStot 2 AS sys ASsurr Bu l39if heat is released by The system A AH Assn 3T5 Tm Define free energy AG TAStot AH TAS WHY BOTHER TO CALCULATE THE FREE ENERGY 1 BECAUSE IF WE DON39T THE GHOSTS OF BOLTZMANN AND EHRENFEST WILL COME BACK TO HAUNT US 2 ONE CONDITION FOR FORMING A SINGLE PHASE IS THAT THE FREE ENERGY MUST BE NEGATIVE 3 THE OTHER CONDITION IS THAT THE SECOND DERIVATIVE OF THE FREE ENERGY MUST BE POSITIVE THEORIES OF MIXING AGm AHm TASm Assume The in rer39ac rions beTween The molecules Can be approxima red by a mean fieldThis allows The en rhalpy and enTropy To be Tr ea red seperaTely BOLTZMANN39S TOMB EnTropy is associaTed wiTh The disTribuTion of energy and maTTer in a sysTemThis can be expressed formally in Terms of The equaTion carved on Boszmann39s Tomb S k an WhichTodayis normally wr39iTTen S kan Where Q is The number39 of arrangemenTs Available To The sysTemAT a given V E N LATTICE MODELS Summary 1 NBm1 NBm1 NUMBER OF CONFIGURATIONS Equalsize small molecules 11 l Flory model for polymer solutions V V V V IV V V V V V AL AL AL AL AL AL AL AL L LAL AL AL AL AL AL ALA r r r r r r wr A wr r r lr r lr lr AL L A AL AL LALALALALALALALAL 4 4L 4 VV IV IV IV IV IVWV IV W I V V V V V lr r r O O A LALALALALAL LAL AL AL AL AL AL AL AL Vquot r 7 7 17 l V Ir lr v v lr v lr lr r l AAAAA V V V V WYVV V VV IV I LALAL AL AL AL ALAL AL V V VVV IV V IV le IV AL V 7 V A r LALALA 7 71 1717 7 7 WV 1 OSolvent AL V V V 7 V V 7 l39r f A 0 AL AL LALALALALALALAL L A r 1r w V V V V 1r V IV 1 L AL AL AL AL AL AL AL AL V V VVV IV V IV le IV ALALALALALALALALA VVVWVWVVVWVVVWVWVVVW DA A A AOA A A AOA A 1 A 1 LALALALALALALALALALA lt A 1 A V 7 AL How are These expresions obtained Polymer chain segment HENCE R ASm nA n manglan 71717171717171717171 71717171717171717171 LALALALALALALALALALA 71717171717171717171 LALALALALALALALALA 71717 1717 171717 171 1 ALALALALALALALALA A quot0 ALA 7 1717171717 L AL ALAL AL AL AL AL AL 7 17 AL AL 17 717171717171717171 L 1L JLQLQLJLQL JLJLQ 7 17 1717 171717 17 17 1 A AOL LALALALALALALALALA 7 17 17 17 17 17 17 17 17 L JLQL AL AL JLQL ALQL 717171717171717171 L 4L JLQL 4L ALOAL ALOALQ 717171717171717171 A A 1 A 71 ALA 01 LAL AL ALAL AL AL AL AL QNANB NA NB NUMBER OF CONFIGURATIONS REGULAR SOLUTION THEORY SOME NOTES CONCERNING THE ENTROPY OF MIXING THE QUANTITIES N A OR N 3 CAN EITHER BE THE NUMBER OF MOLECULES OF A AND B RESPECTIVELY OR THE NUMBER OF MOLES THE CONSTANT CHANGES DEPENDING ON CHOICE MOLES gt R or RT MOLECULES gt k or kT IN CALCULATING ASm WE USED STIRLING39S APPROXIMATION nN N In N N THE FINAL ANSWER IS WRITTEN IN TERMS OF MOLE FRACTIONS x A x B N N X 2 A X 2 B A N N B N N POLYMER SOLUTIONS AND BLENDS L AL WV V V AL AL COUNTING THE CONFIGURATIONS IS MORE DIFFICULT BUT FLORY AND HUGGINS OBTAINED L AL AL AL AL AL AL AL K K KOX X K K K A L OSolvent Polymer chain segment A Tm nAn AnBn B DEFINITION OF VOLUME FRACTIONS AS m R nsln snpln p 75100 25100

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