3 Step-Growth Polymerization 3.1 INTRODUCTION As described in the previous chapters, the properties of polymeric materials depend considerably on their molecular-weight distribution (MWD). This in turn is completely determined by the mechanism of polymerization. There are various mechanisms by which polymer chains grow; this chapter focuses on one of them: step-growth polymerization. Monomer molecules consisting of at least two functional groups can undergo step-growth polymerization. In order to keep mathematics tractable, this chapter will focus on polymerization of bifunctional monomers. The two reacting functional groups can either be on the same monomer molecule, as in amino caproic acid, NH 2 ðCH 2 Þ 5 COOH, or on two separate molecules, as in the reaction between ethylene glycol, OHðCH 2 Þ 2 OH, and adipic acid, COOHðCH 2 Þ 4 COOH. If they are located on the same monomer molecule, represented schematically as ARB, the concentrations of the functional groups remain equimolar throughout the course of the reaction, which can be schema- tically represented as follows [1–7]: nðARBÞ!ARB½ARB n2 ARB ð3:1:1Þ Here, R represents an alkyl or aryl group to which the two functional groups A and B are attached. In case the functional groups are located on two different monomers, ARA and BR 0 B, an analysis similar to the one for ARB 103 Copyright © 2003 Marcel Dekker, Inc. polymerizationcanbeconducted.AspointedoutinChapter1,theoverall reactionrepresentedbyEq.(3.1.1)consistsofseveralelementaryreactions, whichcanberepresentedasfollows: P m þP n ! k m;n k 0 mþn P mþn þW;m;n¼1;2; ð3:1:2Þ whereP n representsARBðARBÞ n2 ARB,P m representsthe ARBðARBÞ m2 ARBmolecule,andWrepresentsthecondensation product.Theforwardandreverserateconstantsk m;n andk 0 mþn are,ingeneral, chain-lengthdependent,asdiscussedinthefollowingparagraphs. Iftwosmallmolecularspecies,AandB,reactas AþB ! Productsð3:1:3Þ itisevidentthatthereactionwillproceedonlyafteramoleculeofAdiffuses closetoamoleculeofBfromthebulk.Thus,theoverallreactionbetweenAand Bconsistsoftwoconsecutivesteps:(1)thediffusionofmoleculesfromthebulk ofthemixturetowithincloseproximityofeachotherand(2)thechemical interactionleadingtoproductformation.Thisisrepresentedschematicallyas AþB ! Diffusion ½AB ! Chemical reaction ABðProductÞð3:1:4Þ However,polymermoleculesareverylongandgenerallyexistinahighlycoiled stateinthereactionmasswiththefunctionalgroupssituatedatthechainends. Therefore,inadditiontothe‘‘bulk’’moleculardiffusionofP m andP n ,thechain endsmustdiffuseclosetoeachother(calledsegmentalmotion)beforethe chemicalreactioncanoccur.Thiscanberepresentedschematicallyas P m þP n ! Bulk diffusion ½P m P n ! Segmental diffusion ½P m :P n ! Chemical reaction P mþn ð3:1:5Þ Becausethebulkandsegmentaldiffusionstepsdependonthechainlengthsof thetwopolymermoleculesinvolved,theoverallrateconstantsinEq.(3.1.2)are, ingeneral,afunctionofmandn.Theexactnatureofthisdependencecanbe deducedfromthefollowingexperiments. 104Chapter3 Copyright © 2003 Marcel Dekker, Inc. 3.2ESTERIFICATIONOFHOMOLOGOUSSERIES ANDTHEEQUALREACTIVITYHYPOTHESIS [1,4,5] Thefollowingesterificationreactionsofmonobasicanddibasicacidsofhomo- logousseriesillustratetheeffectofmolecularsizeontherateconstants: HðCH 2 Þ n COOHþC 2 H 5 OH ! HCl HðCH 2 Þ n COOC 2 H 5 þH 2 O ð3:2:1aÞ COOHðCH 2 Þ n COOHþC 2 H 5 OH ! HCl COOC 2 H 5 ðCH 2 Þ n COOHþH 2 O þCOOC 2 H 5 ðCH 2 Þ n COOC 2 H 5 þH 2 OþC 2 H 5 OHþHCl ð3:2:1bÞ ThesereactionshavebeencarriedoutinexcessofethanolwithHClcatalyst,and theratesofreactionhavebeenmeasuredforvariousvaluesofthechainlengthn. Thereactionrateconstantsareevaluatedusingthefollowingrateexpression: r e ¼ d½COOH dt ¼k A ½COOH½H þ ð3:2:2Þ wherer e istherateofesterificationand[]representsmolarconcentrations.The concentrationofethanoldoesnotenterintoEq.(3.2.2)becauseitispresentinthe reactionmassinlargeexcess. InEq.(3.2.2)[COOH]representsthetotalconcentrationofthecarboxylic acidgroupsinthereactionmassatanytime,whetherpresentintheformofa monobasicofdibasicacid;thisisusuallydeterminedbytitration.½H þ isthe concentrationofprotonsliberatedbythehydrochloricacid.Useoftherate equationintheformshowninEq.(3.2.2),togetherwithexperimentson monobasicanddibasicacidshavingdifferentn,makesitpossibletoisolatethe effectofthesizeofthemoleculeonk A . TherateconstantsforvariousvaluesofnaretabulatedinTable3.1.Two important conclusions can be drawn from the experimental results: 1. The reactivity of larger molecules does not depend on the size of the molecule for n > 8. [2,7] 2. For larger molecules, the rate constant is independent of whether there are one, two, or more carboxylic acid groups per molecule. Similar conclusions have also been obtained on the saponification of esters and etherification reactions [4,5]. If, in the chemical reaction step of Eq. (3.1.5), Step-Growth Polymerization 105 Copyright © 2003 Marcel Dekker, Inc. thereactivityofaCOOHgroupwithanOHgroupisassumedtobe independentofn,theseobservationsimplythattherateofdiffusionoflarge moleculesisnotaffectedbythevalueofn.However,weknowintuitivelythatthe largerthemolecule,theslowerisitsrateofdiffusion.Consequently,itisexpected that,asnincreases,thediffusionalrateshoulddecrease,implyingthatk A must decreasewithincreasingn,aconclusioninapparentcontradictionwiththe observedbehavior. AsshowninEq.(3.1.5),therearetwotypesofdiffusionalmechanisms associatedwiththereactionofpolymermolecules.Althoughtherateofbulk diffusionoftwomoleculesdecreaseswithn,therateoftheotherstep,called segmentaldiffusion,isindependentofn.Theindependenceofnisduetothefact thatthereissomeflexibilityofrotationaroundanycovalentbondinapolymer molecule(seeChapter1),andthereisrestrictedmotionofasmallsequenceof bondsneartheends,whichconstitutessegmentaldiffusion.Thisbringsthe functionalgroupsoftwoneighboringmoleculesneareachother,regardlessofthe chainlengthoftheentiremolecule.Thus,withincreasingn,twopolymer moleculesdiffuseslowlytowardeachotherbybulkdiffusionbutstaytogether foralongertime(thetwoeffectscancelingout),duringwhich,segmental diffusionmaybringthefunctionalgroupstogetherforpossiblereaction. BasedontheexperimentalresultsofTable3.1,wecanpostulateasimple kineticmodelforthestudyofstep-growthpolymerizationinwhichalloftherate constantsareassumedtobeindependentofchainlength.Thisisreferredtoasthe equalreactivityhypothesis.Thefollowingsectionshowsthatthisassumption leadstoaconsiderablesimplificationofthemathematicalanalysis.However, thereareseveralsystemsinwhichthishypothesisdoesnotholdaccurately,and theanalysispresentedheremustbeaccordinglymodified[2,8–14]. TABLE3.1RateConstantsfortheEsterificationofMonobasic Chainlengthk A 10 4 ð250 CÞ a k A 10 4 ð250 CÞ a (n)(monobasicacid)(dibasicacid) 122.1— 215.36.0 37.58.7 47.458.5 57.427.8 6—7.3 87.5— 97.47— Higher7.6— a Inliterspermole(offunctionalgroup)second. 106Chapter3 Copyright © 2003 Marcel Dekker, Inc. 3.3KINETICSOFARBPOLYMERIZATION USINGEQUALREACTIVITYHYPOTHESIS[2] Achemicalreactioncanoccuronlywhenthereactingfunctionalgroupscollide withsufficientforcethattheactivationenergyforthereactionisavailable.The rateofreaction,r,canthusbewrittenasproportionaltotheproductofthe collisionfrequency,o mn ,betweenP m andP n andtheprobabilityofreaction,Z mn (whichaccountsforthefractionofsuccessfulcollisions),asfollows: R¼ao mn Z mn ð3:3:1Þ whereaisaconstantofproportionality.Accordingtotheequalreactivity hypothesis,Z mn isindependentofmandnandis,say,equaltoZ.Ifthe functionalgroupsofthetwomoleculesP m andP n canreactinsdistinctways,the probabilityofareactionbetweenP m andP n isgivenbysZ.Thecollision frequencyo mn betweentwodissimilarmoleculesP m andP n intheforwardstepis proportionalto½P m ½P n ,whereasthatforP m andP m isproportionalto½P m 2 =2 (thefactorofone-halfhasbeenusedtoavoidcountingcollisionstwice).Thus,if k p istherateconstantassociatedwiththereactionbetweenfunctionalgroups, thenundertheequalreactivityhypothesis,k m;n ,therateconstantassociatedwith moleculesP m andP n intheforwardstep,isgivenby k m;n ¼ r ½P m ½P n ¼sk p ;m6¼n;m;n¼1;2; r ½P m 2 ¼ sk p 2 ;m¼n;n¼1;2; 8 > > < > > : ð3:3:2Þ ForlinearchainswithfunctionalgroupsAandBlocatedatthechainends,there aretwodistinctwaysinwhichpolymerchainscanreact,asshowninFigure3.1. This fact implies the following for such cases: k m;n ¼ 2k p m 6¼ n; m; n ¼ 1; 2 k p m ¼ n; n ¼ 1; 2; ( ð3:3:3aÞ ð3:3:3bÞ The various (distinct) elementary reactions in the forward step can now be written as follows: P m þ P n ! 2k p P mþn þ W ; m 6¼ n; n ¼ 1; 2; 3; P m þ P n ! k p P 2m þ W ; m ¼ n; n ¼ 1; 2; ð3:3:4aÞ ð3:3:4bÞ The reverse step in Eq. (3.1.2) involves a reaction between polymer molecule P n and condensation product W; there is a bond scission in this process. It may be observed that P n has n 1 equivalent chemical bonds where the reaction can occur with equal likelihood. It is thus seen that if k 0 p is the reactivity of a bond Step-Growth Polymerization 107 Copyright © 2003 Marcel Dekker, Inc. with W, the reactivity of an oligomer P n is ðn 1Þk 0 p . The mole balance equations for various molecular species in a constant-density batch reactor can now be easily written. Species P 1 is depleted in the forward step when it reacts with any other molecule in the reaction mass. However, P n ðn 2Þ is formed in the forward step when a molecule P r ðr < nÞ reacts with P nr and is depleted by reaction with any other molecule. In the reverse step P n is depleted when any of its chemical bonds are reacted and it is formed whenever a P q ðq > nÞ reacts at a specified bond position. For example, if we are focusing our attention on the formation of P 4 , a molecule having chain length greater than 4, say, P 6 , would lead to the formation of P 4 if W reacts at the second or fourth position of P 6 . The mole balance relations are therefore given by the following: d½P 1 dt ¼2k p ½P 1 f½P 1 þ½P 2 þgþ2k 0 p ½Wf½P 2 þ½P 3 þg ð3:3:5aÞ d½P n dt ¼ k p P n1 r¼1 ½P r ½P nr 2k p ½P n f½P 1 þ½P 2 þg k 0 p ½Wðn 1Þ½P n þ2k 0 p ½Wf½P nþ1 þ½P nþ2 þg; n ¼ 2; 3; 4; ð3:3:5bÞ There is no factor of two in the first term of Eq. (3.3.5b) because of the symmetry, as shown through an example of the formation of P 6 . This occurs at a rate given by ð2k p ½P 1 ½P 5 þ2k p ½P 2 ½P 4 þ2k p ½P 3 ½P 3 Þ. The factor of the first two terms arises because k m;n is 2k p , whereas the factor of the last term, 2k p ½P 3 =2, arises because of the fact that two molecules of P 3 are consumed simulataneously when P 3 reacts with P 3 . The first term in Eq. (3.3.5b) for this is k p P 5 r¼1 ½P r ½P nr ,as shown. If the concentration of all the reactive molecules in the batch reactor is defined as l 0 ¼ P 1 n¼1 ½P n ð3:3:6Þ FIGURE 3.1 The two distinct ways in which two linear bifunctional chains can react. 108 Chapter 3 Copyright © 2003 Marcel Dekker, Inc. one can sum up the equations in Eq. (3.3.5) for all n to give the following: dl 0 dt ¼ d½P 1 dt þ d½P 2 dt þ d½P 3 dt þ ¼2k p l 2 0 þ k p l 2 0 k 0 p ½W P 1 n¼2 ðn 1Þ½P n þ2k 0 p ½W P 1 n¼1 P 1 i¼nþ1 ½P i ð3:3:7Þ It is recognized that P 1 n¼1 P 1 i¼nþ1 ½P i ¼½P 2 þ½P 3 þ½P 4 þþ½P 3 þ½P 4 þþ½P 4 þ ¼½P 2 þ2½P 3 þ3½P 4 þ ¼ P 1 n¼1 ðn 1Þ½P n ð3:3:8Þ Therefore, Eq. (3.3.7) can be written as dl 0 dt ¼k p l 2 0 þ k 0 p ½W P 1 n¼1 ðn 1Þ½P n ð3:3:9Þ It may be observed that P 1 n¼1 ðn 1Þ½P n represents the total number of reacted bonds in the reaction mass. It is thus seen that the infinite set of elementary reactions in step-growth polymerization in Eq. (3.1.2) can be represented kinetically by the following equivalent and simplified equation: A þB ! k p ABþW ð3:3:10Þ where AB represents a reacted bond. The representation of an infinite series of elementary reactions by only one elementary reaction [Eq. (3.3.10)] involving functional groups is a direct consequence of the equal reactivity hypothesis. This leads to a consi derable simplification of the mathematical analysis of polymer- ization reactors. Example 3.1: Consider the ARB step-growth polymerization in which monomer P 1 reacts with P n (for any n) with a different rate constant, as follows: P 1 þ P n ! k 1 P nþ1 þ W; n ¼ 1; 2; 3 ðaÞ P m þ P n ! k p P nþm þ W; m; n ¼ 2; 3; ðbÞ Derive the mole balance relations for the MWD of the polymer in a batch reactor. Step-Growth Polymerization 109 Copyright © 2003 Marcel Dekker, Inc. Solution: s in Eq. (3.3.2) is 2 because the polymer chains are linear. Let us first consider the reaction of P 1 . In the reactions of P 1 with P 1 , similar molecules are involved, and the reactivity would be 2k 1 =2. However, for the reaction of P 1 with any other molecule, the reactivity would be 2k 1 . Therefore, k 1n ¼ k 1 for n ¼ 1 2k 1 for n ¼ 1; 2; 3; & The other reactivities remain the same as in Eq. (3.3.3): k mn ¼ k p for m ¼ n 2k p for m 6¼ n; m; n ¼ 2; 3; & The mole balance of species P 1 is made by observing that two molecules of P 1 are depleted whenever there is a reaction of P 1 with itself in the forward step, whereas only one molecule of P 1 disappears in a reaction with any other molecule. Similarly, in the reverse step, whenever W reacts at the chain ends, P 1 is formed: d½P 1 dt ¼ðForward reaction of P 1 with P 1 Þ ðForward reaction of P 1 with P 2 ; P 3 etc:Þ þðreverse reaction of W at chain ends to give P 1 Þ ¼2k 1 ½P 1 ½P 1 2k 1 ½P 1 f½P 2 þ½P 3 þg þ 2k 0 p ½Wf½P 2 þ½P 3 þg ¼2k 1 ½P 1 l 0 þ 2k 0 p ½W P 1 n¼2 ½P n The rate of formation of P 2 is k 1 ½P 1 =2, and P 2 is depleted whenever it reacts with any molecule in the forward step or its bond reacts with W in the reverse step: d½P 2 dt ¼ðForward reaction forming P 2 ÞðForward reaction of P 2 with P 1 Þ ðForward reaction of P 2 with P 2 with P 2 ; P 3 ; etc: Þ ðReverse reaction of bonds of P 2 with WÞ þðreverse reaction of W with P 3 ; P 4 ; etc: to give P 2 Þ ¼ k 1 ½P 1 2 2k 1 ½P 1 ½P 2 2k p ½P 2 f½P 2 þ½P 3 þg k 0 p ½W½P 2 þ2k 0 p ½Wf½P 3 þ½P 4 þg ¼ k 1 ½P 1 2 2ðk 1 k p Þ½P 2 ½P 1 2k p ½P 2 l 0 k 0 p ½W½P 2 þ 2k 0 p ½W P 1 i¼3 ½P i 110 Chapter 3 Copyright © 2003 Marcel Dekker, Inc. Similarly, the mole balance relaxation for species P n is given by the following: d½P n dt ¼ðForward reaction of P 1 ; P 2 ; etc: with P n1 ; P n2 ; etc:Þ ðForward reaction of P n with P 1 Þ ðForward reaction of P n with P 2 ; P 3 ; etc: Þ ðReverse reaction of W with n 1 bonds of P n Þ þðReverse reaction of W with P nþ1 ; P nþ2 ; etc: to give P n Þ ¼þ2k 1 ½P 1 ½P n1 þk p P n2 r¼2 ½P r ½P nr 2k 1 ½P n ½P 1 2k p ½P n f½P 2 þ½P 3 þgk 0 p ½Wðn 1Þ½P n þ 2k 0 p ½Wf½P nþ1 þ½P nþ2 þg ¼ 2ðk p k 1 Þ½P n ½P 1 2k p ½P n l 0 þ k p P n1 r¼1 ½P r ½P nr þ 2ðk 1 k p Þ½P n1 ½P 1 k 0 p ðn 1Þ½W½P n þ2k 0 p ½W P 1 r¼nþ1 ½P r The zeroth moment of the MWD can be easily found as follows: dl 0 dt ¼ d½P 1 dt þ d½P 2 dt þ ¼ðk 1 k p Þ½P 1 2 k p l 2 0 2ðk 1 k p Þ½P 1 l 0 þ k 0 p ½W P 1 n¼2 ðn 1Þ½P n 3.4 AVERAGE MOLECULAR WEIGHT IN STEP-GROWTH POLYMERIZATION OF ARB MONOMERS Having modeled the rate of step-growth polymerization of ARB monomers, we can easily derive an expression for the average molecular weight of the polymer so formed. It is assumed that one starts with pure ARB monomer and that there are N 0 molecules present initially. After polymerization for time t, there would be fewer, say, N molecules, left in the reaction mass. This numbe r N includes both unreacted monomer molecules, P 1 , as well as dimers, trimers, tetramers, and so forth. In the computation of the average molecular weight for the system at time t, we could either consider only the dimers, trimers, and all other homologs to constitute molecules of the polymer, or, alternatively, include monomer molecules as well. Naturally, the results using the second approach would be lower than that obtained from the first one. In the following analysis, the monomer is included in Step-Growth Polymerization 111 Copyright © 2003 Marcel Dekker, Inc. the computation of the average molecular weight. This is not a drawback because, for practically important situations, the concentration of P 1 is usually negligible. It may be observed that during polymerization the total number of repeat units at any time remains unchanged and is equal to the initial number of monomer molecules, N 0 . These repeat units, however, are now disturbed over N polymer molecules at time t, so the average number of repeat units per molecule is equal to N 0 =N. This is defined as the number-average chain length , m n (sometimes called the degree of polymerization), and is given by m n ¼ N 0 N ¼ ½A 0 ½A ¼ ½B 0 ½B ð3:4:1Þ where ½A 0 , ½B 0 , and ½A and ½B are the concentrations of the functional groups A and B at times t ¼ 0 and t ¼ t, respectively. It is convenient to work in terms of the (fractional) conversion of functional group A (or B), defined as p ½A 0 ½A ½A 0 ¼ N 0 N N 0 ð3:4:2Þ which gives m n ¼ 1 1 p ð3:4:3Þ Integration of Eq. (3.3.9) can be carried out by observing that for every chemical bond formed, one molecule of condensation product, W, is formed. If ½P 1 0 and ½W 0 moles of monomer and condensation product are initially present in a batch reactor and W does not leave the reaction mass, then stoichiometry of polymer- ization gives ½Wþl 0 ¼½W 0 þ½P 1 0 ð3:4:4Þ where [W] and l 0 are the concentrations of condensation product and polymer at any instant of time. We substitute [W] from this equation into Eq. (3.3.9) to obtain dl 0 dt ¼k p l 2 0 þ k 0 p f½W 0 þ½P 0 l o g P 1 n¼1 n½P n P 1 n¼1 ½P n &' ð3:4:5Þ We further observe that P 1 n¼1 n ½P n is the first moment of the MWD and is equal to the total number of repeat units, which means that the first moment, l 10 , is time invariant. Therefore, Eq. (3.4.5) becomes dl 0 dt ¼k p l 2 0 þ k 0 p f½W 0 þ½P 1 0 l 0 gðl 10 l 0 Þð3:4:6Þ 112 Chapter 3 Copyright © 2003 Marcel Dekker, Inc. [...]... 1Þl0 =½P1 0 þ ð1 À 1 :41 Þ 2 À 1 :41 ¼ l0 =½P1 0 À 0 :41 0:59 After 10 min, l0 =½P1 0 À 0 :41 ¼ 0:172rÀ2:91 0:59 l ; 0 ¼ 0 :41 þ 0:172ð0:59Þð0:0 545 Þ ¼ 0 :41 6 ½P1 0 Conversion ¼ 1 À mn ¼ 2 :40 ; l0 ¼ 0:5 84 ½P1 0 Q ¼ 1 þ p ¼ 1:5 84 Copyright © 2003 Marcel Dekker, Inc 116 Chapter 3 At 200 C, kp ¼ 4: 43  10À3 ; 0 kp ¼ 2:22  10À3 4: 58 ¼ 0:0287 d ¼ 2ð2:22  10À3 Þ 0:71 l0 =½P1 0 À 0 :41 ¼ 0:172  eÀ0:287... Step-Growth Polymerization 135 TABLE 3 .4 Rate Constants for Various Reactions of 2 ,4 and 2,6 Toluene Diisocyanates with Polyols 1 04 kt L equivalentÀ1 sÀ1 Reactions Nature of OH Location of NCO 25 C 1 2 3 4 Primary hydroxyl Monomeric para Monomeric ortho Polymeric para Polymeric ortho 0.613 0.230 0.161 0.0605 4. 17 1.67 1.10 0 .43 9 5 6 7 8 Secondary hydroxyl Monomeric para Monomeric ortho Polymeric para Polymeric... forward reactions can be easily written in terms of A to D (the formation of tetrasubstituted urea does not occur) as follows: 8k 1 ð3:7:24aÞ 4k 1 ð3:7:24bÞ 2k 2 ð3:7:24cÞ k2 ð3:7:24dÞ 4k 1 ð3:7:24eÞ 2k 1 ð3:7:24f Þ 4k 2 ð3:7:24gÞ 4k 2 ð3:7:24hÞ 2k 2 ð3:7:24iÞ 2k 1 ð3:7:24jÞ U þ F À A þ CH2 OH þ H2 O ! ! U þ CH2 OH À A þ H2 O þ Z A þ F À B þ CH2 OH þ H2 O ! A þ CH2 OH À B þ H2 O þ Z ! A þ F À C þ CH2 OH... and (B B) types are present, and all of these are equally likely to occur If we know the total number of moles of unreacted A and B functional groups, the total number of moles of polymer is simply half of this In other words, the total number of moles of polymer, N , at time t is equal to 1 f2NA0 ð1 À pA Þ þ 2NB0 À 2NA0 pA g Similarly, the total number 2 of moles of polymer initially, N0 , is equal to... sequence of batch reactors, the molecular-weight distribution would be different and the polydispersity index (PDI) of the polymer formed would not necessarily be restricted to the limiting value of 2, as shown in Appendix 3.1 As a matter of fact, one of the practical methods of achieving a PDI of more than 2 is to partially recycle a portion of the product stream, as shown in Figure 3 .4 [19–22] Polymerization... independent of the chain length of the polymer molecule on which it is situated As a result [e.g., in Eq (3) of Table 3.2], the rate of formation of Z would be k3 ½COOH2 À k3 ½Z½G=K The analysis of the reactor can be performed only numerically in view of the set of nonlinear differential equations for the balance of functional groups Polyamides are formed by the polymerization of a diamine and a dicarboxylic... diisocyanates used are either 2 ,4- toluene diisocyanate (TDI) or 4, 40 -diphenyl methane diisocyanate (MDI) The diols are polyester diols formed by the polymerization of adipic acid (or phthalic anhydride) in the presence of an excess of ethylene glycol The formation of polyurethanes cannot be represented by the simple scheme of ARB polymerization because (1) the reaction of a given isocyanate group (i.e.,... formaldehyde, polymers of various lengths and structures are formed One plausible description of the progress of reaction might be to follow the concentration of species A to D in the reaction mass The overall polymerization represented by the reaction of functional groups can be written in terms of the following rate constants: k1 ¼ Rate constant for the reaction of primary hydrogen of urea with the... 4m0 m2 Þ1=2 1 ð3 :4: 9cÞ ð3 :4: 9dÞ q¼ 2m2 l0 þ m1 À d 2m2 ½P1 0 þ m1 þ d ð3 :4: 9eÞ q0 ¼ 2m2 ½P1 0 þ m1 À d 2m2 ½P1 0 þ m1 þ d ð3 :4: 9f Þ The number-average molecular weight can be easily obtained by multiplying mn by the molecular weight of ARB (because the molecular weight of W is usually small) Example 3.2: Suppose NA0 moles of AR1 A monomer are reacted with NB0 moles of BR2 B monomer to form the polymer. .. 10 min of polymerization at 280 C and 200 C Solution: At 280 C 15; 000 ¼ 4: 49  10À2 L=mol min kp ¼ 4: 0  10 exp 1:98ð273 þ 280Þ 4 0 kp ¼ 2:25  10À2 L=mol min Copyright © 2003 Marcel Dekker, Inc Step-Growth Polymerization 115 Because t ¼ 0, BHET does not have any W, ½W0 ¼ 0; m0 ¼ ½P0 ¼ ½P1 0 ¼ 4: 58 g mol=L 0 kp ½P2 ; 0 0 m2 ¼ kp 1 À1 Kp 0 m1 ¼ 2kp ½P1 0 ! " d¼ q0 ¼ ¼ q¼ ¼ 02 ½4kp ½P1 . Inc. thereactivityofaCOOHgroupwithanOHgroupisassumedtobe independentofn,theseobservationsimplythattherateofdiffusionoflarge moleculesisnotaffectedbythevalueofn.However,weknowintuitivelythatthe largerthemolecule,theslowerisitsrateofdiffusion.Consequently,itisexpected that,asnincreases,thediffusionalrateshoulddecrease,implyingthatk A must decreasewithincreasingn,aconclusioninapparentcontradictionwiththe observedbehavior. AsshowninEq.(3.1.5),therearetwotypesofdiffusionalmechanisms associatedwiththereactionofpolymermolecules.Althoughtherateofbulk diffusionoftwomoleculesdecreaseswithn,therateoftheotherstep,called segmentaldiffusion,isindependentofn.Theindependenceofnisduetothefact thatthereissomeflexibilityofrotationaroundanycovalentbondinapolymer molecule(seeChapter1),andthereisrestrictedmotionofasmallsequenceof bondsneartheends,whichconstitutessegmentaldiffusion.Thisbringsthe functionalgroupsoftwoneighboringmoleculesneareachother,regardlessofthe chainlengthoftheentiremolecule.Thus,withincreasingn,twopolymer moleculesdiffuseslowlytowardeachotherbybulkdiffusionbutstaytogether foralongertime(thetwoeffectscancelingout),duringwhich,segmental diffusionmaybringthefunctionalgroupstogetherforpossiblereaction. BasedontheexperimentalresultsofTable3.1,wecanpostulateasimple kineticmodelforthestudyofstep-growthpolymerizationinwhichalloftherate constantsareassumedtobeindependentofchainlength.Thisisreferredtoasthe equalreactivityhypothesis.Thefollowingsectionshowsthatthisassumption leadstoaconsiderablesimplificationofthemathematicalanalysis.However, thereareseveralsystemsinwhichthishypothesisdoesnotholdaccurately,and theanalysispresentedheremustbeaccordinglymodified[2,8– 14] . TABLE3.1RateConstantsfortheEsterificationofMonobasic Chainlengthk A 10 4 ð250 CÞ a k A 10 4 ð250 CÞ a (n)(monobasicacid)(dibasicacid) 122.1— 215.36.0 37.58.7 47 .45 8.5 57 .42 7.8 6—7.3 87.5— 97 .47 — Higher7.6— a Inliterspermole(offunctionalgroup)second. 106Chapter3 Copyright. 1 :41 Þ 2 1 :41 ¼ l 0 =½P 1 0 0 :41 0:59 After 10 min, l 0 =½P 1 0 0 :41 0:59 ¼ 0:172r 2:91 ; l 0 ½P 1 0 ¼ 0 :41 þ 0:172ð0:59Þð0:0 545 Þ¼0 :41 6 Conversion ¼ 1 l 0 ½P 1 0 ¼ 0:5 84 m n ¼. 3.1. As a matter of fact, one of the practical methods of achieving a PDI of more than 2 is to partially recycle a portion of the product stream, as shown in Figure 3 .4 [19–22]. Polymerization