Kinetic Modeling of Thiol-Ene Reactions with Both Step and Chain Growth Aspects

Transcription

1 Full Paper 267 Summary: A kietic model is preseted for thiol-ee crosslikig photopolymerizatios icludig the allowace for chai growth reactio of the ee, i.e., homopolymerizatio. The kietic model is based o a descriptio of the average chai legths derived from differetial equatios of the type of Smoluchowski coagulatio equatios. The method of momets was applied to obtai average properties of thiolee reactio systems. The model predicts the molecular weight distributio of active ad iactive species i the pregel regime of thiol-ees, as well as the gel poits depedig o the sythesis parameters. It is show that, whe o homopolymerizatio is allowed, the average molecular weights ad the gel poit coversio are give by the typical equatios valid for the step-growth polymerizatio. Icreasig the extet of homopolymerizatio also icreases the average molecular weights ad shifts the gel poit toward lower coversios ad shorter reactio times. It is also show that the ratio of thiyl radical propagatio to the chai trasfer kietic parameter (k p1 /k tr ) affects the gelatio time, t cr. Gelatio occurs earlier as the k p1 /k tr ratio is icreased due to the predomiat attack of thiyl radicals o the viyl groups ad formatio of more stable carbo radicals. The gel poit i thiol-ee reactios is also foud to be very sesitive to the extet of cyclizatio, particularly, if the moomer fuctioalities are low. Number-average chai legth of carbo radicals X 1 (solid curves) ad thiyl radicals X 1 0 (dashed curves) plotted agaist the viyl group coversio, x M, durig thiol-ee polymerizatio. Calculatios were for six differet k p /k tr ratios. Kietic Modelig of Thiol-Ee Reactios with Both Step ad Chai Growth Aspects Oguz Okay,* 1 Christopher N. Bowma* 2 1 Departmet of Chemistry, Istabul Techical Uiversity, Maslak, Istabul, Turkey okayo@itu.edu.tr 2 Departmet of Chemical ad Biological Egieerig, Uiversity of Colorado, Egieerig Ceter, ECCH 111, 424 UCB, Boulder, Colorado 80309, USA bowmac@colorado.edu Received: Jauary 14, 2005; Revised: March 14, 2005; Accepted: March 15, 2005; DOI: /mats Keywords: crosslikig; gelatio; kietic modelig; thiol-ee reactios Itroductio Thiol-ee photopolymerizatios are step-growth radical polymerizatios ivolvig a reactio betwee multifuctioal thiol ad ee (viyl) moomers. [1 3] Previous work has demostrated sigificat polymerizatio advatages of thiol-ee systems, icludig a rapid reactio, [4,5] low shrikage, little or o oxyge ihibitio, [1,5,6] selfiitiatio, [5] accessibility of a large umber of thiol-ee comoomer pairs, [3] ad the formatio of highly crossliked etworks havig good physical, optical, ad mechaical properties. The step-growth ature of thiol-ee photopolymerizatios was first suggested by Kharasch i [7] The polymerizatio reactio proceeds via propagatio of a thiyl radical ( S) through the viyl fuctioal group. Rather Macromol. Theory Simul. 2005, 14, DOI: /mats ß 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

2 268 O. Okay, C. N. Bowma tha beig followed by additioal propagatio, this propagatio step is cotiually followed by chai trasfer of the carbo radical ( CH ), thus formed, to the thiol fuctioal group, regeeratig a thiyl radical, i.e., --S þch 2 CH! k p1 --S--CH 2 -- C H-- þ SH--! k tr --S--CH 2 --C H-- --S--CH 2 --CH 2 -- þ --S ð1aþ ð1bþ These successive propagatio ad chai trasfer reactios serve as the basis for the step-growth ature of traditioal thiol-ee polymerizatio. As a result, the gel formatio dyamics of thiol-ees are quite differet from the polymerizatio of multifuctioal acrylate moomers. I a typical free-radical polymerizatio of multiacrylate moomers, high molecular weight polymers form at earzero moomer coversio because of the chai-growth ature of the polymerizatio, leadig to excessive cyclizatio, multiple cross-likig, microgelatio, low gel poit coversios, diffusio- ad reactio diffusio-cotrolled reactios, ad ultimately, the formatio of ihomogeeous etworks. [8 11] However, i thiol-ee systems, very low molecular weight species domiate the pre-gel regime, leadig to higher gel poit coversios ad the formatio of homogeeous etworks. Although the traditioal free-radical reactio of a thiol ad a ee proceeds via a step-growth process (Equatio (1a) (1b)), for some ees the propagatio of a carbo radical via homopolymerizatio of the ee accompaies the traditioal thiol-ee photopolymerizatio, i.e., --S--CH 2 -- C H-- þ CH 2 --S--CH 2 --CH 2 --CH 2 -- C H CH! k p ð1cþ This propagatio step, referred to i the remaiig of this mauscript as homopolymerizatio, is particularly importat i thiol-acrylate polymerizatios ad leads to the formatio of homopolymer groups i the copolymer. [4 6,12 14] Oe may expect that, at a high ratio of homopolymerizatio to chai trasfer kietic parameter k p /k tr, the reactio mechaism of thiol-ees chages from predomiatly stepgrowth to predomiatly chai-growth ature. While thiol-ee polymerizatios have bee examied extesively i recet years, may fudametal aspects of these reactios, such as the molecular weight developmet durig the pre ad post-gel regimes, as well as the effect of the homopolymerizatio o the gelatio process remai relatively uexplored. The aim of the preset work is primarily to develop a kietic model for predictio of the molecular weight averages of thiyl ad carbo radicals as well as of polymer molecules durig thiol-ee reactios with homopolymerizatio prior to the oset of gelatio. Here, we report a kietic model for thiol-ee photopolymerizatios utilizig multifuctioal thiol ad ee moomers. The kietic model is based o a descriptio of the average chai legths derived from differetial equatios of the type of Smoluchowski coagulatio equatios. [15] The method of momets was applied to obtai average properties i the pre-gel period ad to predict the gel poit. I the kietic treatmet that follows, the mai assumptios made are: (i) the steady-state approximatio for each of the radical species i the system; (ii) the reactios are chemically cotrolled rather tha diffusio cotrolled; ad (iii) active (radical) species cotai oly oe radical ceter. The secod assumptio is reasoable for thiol-ee systems due to the very low molecular weight of polymers over a large rage of the pre-gel regime, while the validity of the third assumptio (mooradical assumptio) was previously demostrated for free-radical cross-likig copolymerizatio systems. [16] Kietic Mechaism Notatio ad Rate Equatios The depedet variables S r., R r., ad P r represet thiyl radicals, carbo radicals, ad dead polymer or moomer molecules, respectively. The subscript r describes the total umber of moomer uits i the molecule. Thus, P 1 represets the ureacted moomers i the reactio system. The fuctioalities of the viyl ad thiol moomers are represeted by f 1 ad f 2, respectively, which deote the iitial umber of the fuctioal groups of molecules. Furthermore, the symbols M r ad SH r are used to represet the viyl ad thiol groups o molecules P r, respectively. The total cocetratios of carbo radicals [R.], thiyl radicals [S.], viyl groups [M] ad thiol groups [SH] i the reactio system are related to the variables defied above by the followig equatios: ½RŠ ¼ X1 ½SŠ ¼ X1 ½MŠ ¼ X1 ½SHŠ ¼ X1 ½R r Š ½S r Š ½M r Š ½SH r Š ð2aþ ð2bþ ð2cþ ð2dþ A set of kietic mechaisms is preseted for thiol-ee photopolymerizatios of multifuctioal thiol ad ee moomers havig symmetric fuctioal groups. The mechaism cosists of four steps: iitiatio, propagatio, chai

3 Kietic Modelig of Thiol-Ee Reactios with Both Step ad Chai Growth Aspects 269 trasfer, ad termiatio. The reactio equatios describig the steps of the polymerizatio ca be writte as follows: Iitiatio: I! h A ð3aþ A þsh r! k i A þm r! k i Propagatio: S j þm r j! k p1 R j þm r j! k p Chai Trasfer: R r þsh j! k tr Termiatio: S r R r R r R r P r þ S j S j þs r j k tc ad=or k td! P r ad=or P j þ P r j S j þr r j k tc ad=or k td! P r ad=or P j þ P r j R j þr r j k tc ad=or k td! P r ad=or P j þ P r j ðr; j; r j ¼ 1; 2; 3;...Þ ð3bþ ð3cþ ð4aþ ð4bþ ð5þ ð6aþ ð6bþ ð6cþ Decompositio of the iitiator I accordig to Equatio (3a) produces primary radicals A., which may react either with a thiol or viyl group o rmers, deoted by SH r ad M r, respectively (Equatio (3b) (3c)). Note that r ¼ 1, 2, 3,..., where r ¼ 1 correspods to the fuctioal groups o moomers. The thiyl (S r.) ad carbo radicals (R r.) thus formed may propagate accordig to Equatio (4a) ad (4b), respectively. Equatio (4a) accouts for the propagatio of a thiyl radical through the viyl fuctioal group, while Equatio (4b) accouts for the homopolymerizatio reactio of carbo radicals. The chai trasfer reactios represeted by Equatio (5) produce iactive polymer molecules of chai legth r(p r ) durig which carbo radicals R r. become thiyl radicals S j. of a differet chai legth. Fially, the termiatio reactios betwee thiyl ad carbo radicals occur by couplig ad/or by disproportioatio mechaisms (Equatio (6a) (6c)). The rate costats k i, k p1, k p, k tr, k tc, ad k td are for iitiatio, for the formatio of carbo radicals from thiyl radicals, for propagatio by homopolymerizatio, for chai trasfer to thiol, ad for termiatio by couplig ad by disproportioatio, respectively. For the sake of clarity ad simplicity, we assume that the termiatio rate costat does ot deped o the type of the radical. I additio to the itermolecular reactios metioed above, itramolecular reactios may also occur durig the thiol-ee polymerizatio. These reactios are distiguished as cyclizatio ad itramolecular chai trasfer reactios, as schematically illustrated i Figure 1. Cyclizatio, that is itramolecular propagatio, may occur by the attack of a thiyl or carbo radical ceter o oe of the pedat viyl groups o the same molecule ad leads to the formatio of Figure 1. Schematic represetatio of cyclizatio ad itramolecular chai trasfer reactios i thiol-ee copolymerizatio of viyl ad thiol moomers (f 1 ¼ 2, f 2 ¼ 3). The filled ad ope circles represet thiol ad viyl groups, respectively. The dashed lies show the homopolymer blocks. The arrows show the possible routes for itramolecular reactios. Radical ceters are idicated by the dots.

4 270 O. Okay, C. N. Bowma cycles. I typical thiol-ee reactios, the cosumptio rate of viyl fuctioal groups via propagatio domiates over the cosumptio rate via homopolymerizatio. [4,5] Therefore, i the followig aalysis, cycles are assumed to form oly by the attack of thiyl radicals o the pedat viyl groups located o the same oligomer molecule. I cotrast to cyclizatio, itramolecular chai-trasfer reactios produce o cycles; they oly covert carbo radicals ito thiyl radicals (Figure 1). Sice the itramolecular reactios occurrig i similar molecules proceed i a similar microeviromet, it is reasoable to assume equal rates for both cyclizatio ad itramolecular chai trasfer reactios. Neglectig fuctioal group cosumptio by iitiatio relative to propagatio or chai trasfer, the rate equatios for the cocetratios of the fuctioal groups ad the moomer uits are writte as follows: d½m 1 Š ¼ f 1 ðk p1 ½SŠ þ k p ½RŠÞ½M 1 Š ½M 1 Šð0Þ ¼½MŠ 0 ð7þ d½sh 1 Š d½m 1 Š d½m 2 Š ¼ f 2 k tr ½RŠ½SH 1 Š ½SH 1 Šð0Þ ¼½SHŠ 0 ð8þ ¼ f1 1 d½m 1 Š ¼ f2 1 d½sh 1 Š ½m 1 Šð0Þ ¼0 ½m 2 Šð0Þ ¼0 ð9þ ð10þ where t is the reactio time, M 1 ad SH 1 represet viyl ad thiol groups o ureacted moomers, ad m 1 ad m 2 represet the viyl ad thiol moomer uits i the polymer. The iitial coditios of the differetial equatios are idicated with the subscript zero. Equatio (7) ad (8) accout for the cosumptio of the fuctioal groups located o ureacted moomers. The prefactors f 1 ad f 2 i these equatios accout for the fact that, if oe of the fuctioal groups o a moomer molecule has reacted, the others also disappear ad become pedat fuctioal groups o polymer molecules. Equatio (9) ad (10) describe the formatio of viyl ad thiol moomer uits i the polymer molecules, respectively. The accumulated mole fractio of viyl moomer i the polymer, F 1 is calculated from Equatio (9) (10) as: ½m 1 Š F 1 ¼ ð11þ ½m 1 Šþ½m 2 Š Usig Equatio (3a) (6c), the differetial equatios characterizig the populatio desity distributios of the radicals S r. ad R r., ad the polymers P r are give as follows: d½s r Š ¼ k i ½AŠ½SH r Šþk tr ½RŠ½SH r Š k p1 ½S r Š½MŠ k t ð½rš þ ½SŠÞ½S r Š ð12þ d½r r Š d½p r Š X r 1 X r 1 ¼ k i ½AŠ½M r Šþk p1 ½S j Š½M r j Šþk p ½R j Š½M r j Š j¼1 j¼1 ðk p ½MŠþk tr ½SHŠÞ½R r Š k t ð½rš þ ½SŠÞ½R r Š ð13þ ¼ k tr ½R r Š½SHŠ ðk p1 ½SŠ þ k p ½RŠÞ½M r Š k tr ½RŠ½SH r Šþk td ð½rš þ ½SŠÞð½R r Šþ½S r ŠÞ X r 1 þ 0:5k tc ð½r j Š½R r j Š þ 2½R j Š½S r j Š þ ½S j Š½S r j ŠÞ j¼1 ð14þ where k t ¼ k tc þ k td. The first ad last rate expressios i Equatio (12) ad (13) accout for the formatio ad cosumptio reactios of the radicals i the iitiatio ad termiatio steps, respectively. Thiyl radicals S r. also form by chai trasfer, while they disappear by propagatio, represeted by the secod ad third rate expressios i Equatio (12), respectively. Coversely, carbo radicals R r. form by propagatio ad disappear by chai trasfer, as show by Equatio (13). I additio, the homopolymerizatio reactio cotributes to the formatio ad cosumptio of the carbo radicals R r. i the reactio system. It should be oted that, sice the rate of carbo radical formatio by cyclizatio is assumed to be equal to its cosumptio rate by itramolecular chai trasfer, the itramolecular rate expressios are excluded from Equatio (12) (13). Moreover, dead polymer molecules, P r, form by the chai trasfer reactios of R r. radicals as well as by the termiatio reactios, while they disappear by the attack of thiyl ad carbo radicals o the pedat fuctioal groups o P r (Equatio (14)). Momets of Polymer Distributios The method of momets is the applied to the kietic model of the reactios represeted by Equatio (12) (14) to calculate the th momet of the active polymer, iactive (dead) polymer, ad fuctioal group distributios, defied by: Y X1 r ½R r Š ð15aþ Y 0 X1 Q X1 W X1 r ½S r Š r ½P r Š r ½M r Š ð15bþ ð15cþ ð15dþ

5 Kietic Modelig of Thiol-Ee Reactios with Both Step ad Chai Growth Aspects 271 W 0 X1 r ½SH r Š ð ¼ 0; 1; 2;...Þ ð15eþ From defiitios, zeroth momets correspod to the total cocetratio of the species, i.e., Y 0 : [R.], Y0 0 ½SŠ, W 0 : [M], ad W0 0 ½SHŠ. From the momets of the polymer distributios, the th average chai legth of the carbo radical (X ), the thiyl radical (X 0 ), ad the polymer (X ) are calculated as follows: X ¼Y =Y 1 ð16aþ X 0 ¼Y0 Y 0 1 ð16bþ X ¼ Q =Q 1 ð ¼ 1; 2; 3;...Þ ð16cþ where ¼ 1 ad 2 correspod to the umber- ad weightaverage chai legths, respectively. Ivokig the steadystate approximatio to Equatio (12) (13), the momets of the active polymer distributios are evaluated as follows: Thiyl radical momets: dy 0 Carbo radical momets: dy ¼ k i½aš½mšþk p1 X ¼ k i ½AŠ½SHŠþk tr ½RŠW 0 fk p1½mš þ k t ðy 0 þ Y 0 0 ÞgY0 ffi 0 ¼0 Y 0 W X þ k p ¼0 fk p ½MŠþk tr ½SHŠþk t ðy 0 þ Y 0 0 ÞgY ffi 0 ð17þ Y W ð18þ Assumig that the cosumptio of the fuctioal groups by iitiatio is egligible relative to propagatio, ad sice the radical termiatio rate i thiol-ee polymerizatios is egligible compared to the rates of propagatio or chai trasfer, [12 14] from Equatio (17) (18), the momets of the active radicals are expressed as: y ¼ X ¼0 y 0 w þða 1Þ X 1 y w ¼0 y 0 ¼ w0 ð19þ ð20þ Y0 0 ¼ð1 jþ R 0:5 I ð21þ k t Y 0 ¼ j R 0:5 I ð22þ k t where y 0 ¼ Y0 Y 0 0, y ¼ Y =Y 0, w ¼ W =W 0, w 0 ¼ W 0 W 0 0, R I is the rate of iitiatio, R I ¼ k i ½AŠð½MŠþ ½SHŠÞ, j is the mole fractio of carbo radicals withi the total radical species, k p1 ½MŠ j ¼ ð23þ k p1 ½MŠþk tr ½SHŠ ad a is the ratio of the cosumptio rates of viyl to thiol fuctioal groups, a d½mš d½shš ¼ 1 þ k p½mš ð24þ k tr ½SHŠ Note that the cosumptio of viyl fuctioal groups via homopolymerizatio does ot occur i typical thiol-ee systems, [5] so that a remais uity throughout the reactios ad the secod term of the right had side of Equatio (19) vaishes. Homopolymerizatio occurs, however, for example i thiol-acrylate systems, which leads to higher viyl group cosumptio relative to thiol cosumptio. [4] Furthermore, for moofuctioal thiol moomers, the first term of the right had side of Equatio (19) is uity due to the fact that the chai legth of the thiyl radicals caot be greater tha oe ðy 0 ¼ 1Þ. Usig the assumptios made above, the material balace for iactive molecules give by Equatio (14) yields the rate equatio for the th momet of the polymer distributios as follows: dq ¼ k tr ½RŠ½SHŠðy aw w 0 Þ; Q ð0þ ¼½P 1 Š 0 ð25þ where [P 1 ] 0 is the iitial moomer cocetratio, i.e., [P 1 ] 0 ¼ [M] 0 /f 1 þ [SH] 0 /f 2. Equatio (25) describes the distributio of all species icludig the ureacted moomer molecules. Momets of Fuctioal Group Distributios Let DM ad DSH be the umber of reacted viyl ad thiol groups at a give reactio time or fuctioal group coversio. If there is o homopolymerizatio, that is, if k p ffi 0, the DM/DSH ratio remais uity durig the etire polymerizatio (Equatio (24)). Thus, uder this coditio, every treelike molecule, whatever its size, bears (r 1) reacted thiol ad (r 1) reacted viyl groups (Figure 2). However, if k p 6¼ 0, the ratio of DM/DSH becomes larger tha uity; i this case, every molecule will still have (r 1) reacted viyl groups, but the average umber of reacted thiol groups is reduced to (r 1)/(DM/DSH). I additio, cyclizatio reactios may also occur durig thiol-ee reactios, reducig further the umber of pedat fuctioal groups. Lettig x be the fractio of reacted fuctioal groups i cycles, the umbers of reacted viyl ad thiol

6 272 O. Okay, C. N. Bowma Usig Equatio (24) (25) ad (28), they ca be calculated as follows: dx SH ¼ 1 ð1 xþ k tr½ršð1 x SH Þ ð30þ dx M ¼ ar 0 dx SH ð31þ where r 0 is the stoichiometric imbalace for the reactio system, Figure 2. Schematic represetatio of a polymer molecule formed from tetrathiol ad triviyl moomer uits. The total umber of uits is 10. If there is o homopolymerizatio, the umber of reacted thiol or viyl groups is 9. The dashed lies show the possible homopolymer blocks. The arrows show the cycle formatio routes. groups i a cyclic molecule become (r 1)/(1 x) ad (r 1)/{(1 x)(dm/dsh)}, respectively. Thus, the cocetratios of pedat viyl ad thiol groups i rmers are give by: ½M r Š¼r½P r ŠF 1;r f 1 ðr½p r Š ½P r ŠÞ=ð1 xþ ð26þ ½SH r Š¼r½P r Šð1 F 1;r Þf 2 ðr½p r Š ½P r ŠÞ= fð1 xþðdm=dshþg ð27þ (r > 1) where F 1,r is the mole fractio of viyl moomer i rmer molecules. Assumig a homogeeous distributio of the pedat fuctioal groups alog the polymer molecules, i.e., F 1;r ¼ F 1, the th momet of the viyl ad thiol group distributios is writte as follows: W 0 ¼ X1 W ¼ X1 r ½M r Š¼½MŠ 0 þ F 1 f 1 ðq þ1 Q 1 Þ ðq þ1 Q Þ=ð1 xþ r ½SH r Š¼½SHŠ 0 þð1 F 1 Þf 2 ðq þ1 Q 1 Þ ðq þ1 Q Þ=fð1 xþðdm=dshþg ð28þ ð29þ It is worth otig that the cyclizatio parameter x i large molecules (r >> 1) should scale with r by r 3/2, while for small molecules (r ¼ ), it should maily deped o the chemical ature of the fuctioal groups. [17,18] Sice i thiol-ee systems the chai legth of the molecules is rather small (X 1 < 10 1 ), we eglect the r depedece of x i the preset aalysis. Coversio Thiol ad viyl group coversios are defied as x SH ¼ 1 [SH]/[SH] 0 ad x M ¼ 1 [M]/[M] 0, respectively. r 0 ¼½SHŠ 0 =½MŠ 0 Chai Legth Averages ð32þ Equatio (25) together with Equatio (28) (29) give a geeral momet expressio, which yields simple solutios for several of the momets. For example, the equatio for the zeroth momet Q 0 correspodig to the total umber of molecules is: dq 0 ¼ ak tr ½RŠ½SHŠ Q 0 ð0þ ¼½P 1 Š 0 ð33þ while the first momet Q 1 equals the total umber of moomer uits i the reactio system, ad it is time ivariat: dq 1 ¼ 0 Q 1 ð0þ ¼½P 1 Š 0 ð34þ Moreover, the rate equatio for the secod momet Q 2 is obtaied from Equatio (25) as follows: dq 2 ¼ 2ðk p1 Y 0 1 þ k py 1 ÞW 1 Q 2 ð0þ ¼½P 1 Š 0 ð35þ Solutio of Equatio (33) (34) for the umber-average chai legth X 1 gives: X 1 ¼ 1 þ r 0 1 þ r 0 x M f avr ð1 xþ ð36þ where f avr is the average fuctioality of the moomer mixture, i.e., f avr ¼ 1 þ r 0 f1 1 þ r 0 f2 1 ð37þ Equatio (36) is valid for all thiol-ee reactios with or without homopolymerizatio. The two extreme cases of Equatio (36) are give by x ¼ 0 ad x ¼ 1, correspodig to the rig-free ad cyclo thiol-ee polymerizatios, respectively. For the weight-average chai legth X 2, such a geeral equatio caot be derived from the momet equatios of the kietic model. However, for rig-free thiol-

7 Kietic Modelig of Thiol-Ee Reactios with Both Step ad Chai Growth Aspects 273 ee systems without homopolymerizatio (k p ¼ 0), oe obtais from Equatio (19) (20), (28) (29), (34) (35): 1 þ x M r0 X 0:5 2 ¼ 1 x M r0 0:5 fðf 1 1Þðf 2 1Þg 0:5 ð38þ Equatio (36) ad (38) are typical equatios of stepgrowth polymerizatio. [19 21] Thus, eglectig homopolymerizatio reactios, that is, for a ¼ 1, the kietic model derived for thiol-ee systems is idetical to that of the stepgrowth reactios. However, if a > 1, that is, if homopolymerizatio occurs, the differetial equatios of the kietic model produce a differet sceario. I this case, the primary molecules (that is the molecules, which would result if all cross-liks were cut) with chai legth r >> 1 start to appear i the reactio system so that the kietic model yields equatios closer to those for chai-growth polymerizatios. [22 24] Calculatios To simulate the thiol-ee reactios with time as the idepedet variable, accurate predictio of the iitiatio rate, R I, is ecessary. Equatio (39a) predicts the iitiator cocetratio as a fuctio of time, allowig R I to vary as a fuctio of time ad assumig that each absorbed photo leads to the decompositio of oe iitiator molecule. The, R I is determied by Equatio (39b). [25] d½iš ¼ 2:303e I 0l N A hc ½IŠ ½IŠð0Þ ¼½IŠ 0 ð39aþ R I ¼ f d½iš ð39bþ where f is the umber of radicals formed per absorbed photo that iitiate polymerizatio, i.e., the iitiator efficiecy, e is the molar absorptivity coefficiet, I 0 is the icidet light itesity, l is the wavelegth, h is the Plack s costat, ad c is the speed of light. For calculatios, we set e ¼ 150 L/(mol cm), l ¼ 365 m, f ¼ 0.1, I 0 ¼ 2 mw/cm 2, ad [I] 0 ¼ 0.02 M. Simultaeous solutios of the differetial Equatio (7) (10) ad (25) give the fuctioal group cocetratios ad the momets of polymer distributios as a fuctio of the reactio time. Whe the secod momet or higher goes to ifiity, the oset of gelatio occurs. The idepedet variable time, t, ca also be replaced with the fractioal coversios of either viyl or thiol groups by use of Equatio (30) or (31). After this trasformatio, the system specific parameters required to solve the model are: a) iitial cocetratios ad fuctioalities of the moomers, b) the ratio of homopolymerizatio to chai trasfer rate costat k p /k tr, ad c) the cyclizatio parameter x. Results ad Discussio Data are preseted first for the effect of the homopolymerizatio o the gelatio kietics i thiol-ee systems. This behavior is importat i thiol-acrylate polymerizatios where the ratio of the homopolymerizatio to the chai trasfer kietic costat (k p /k tr ) is o the order of 1. Calculatios i this sectio were for a reactio system cosistig of diviyl ad trithiol moomers with equivalet feed ratios of thiol ad viyl groups (r 0 ¼ 1). Figure 3 shows the weight average chai legth X 2 vs. the reactio time ad viyl group coversio (x M ), whe the k p /k tr ratio varies from 0 (curves 1) to 4 (curves 6). For Figure 3. Weight-average chai legth X 2 i thiol-ee systems with r 0 ¼ 1 plotted agaist the reactio time (A) ad viyl group coversio, x M (B). f 1 ¼ 2, f 2 ¼ 3. Symbols were calculated usig Equatio (38). Calculatios were for six differet k p /k tr ratios: k p /k tr ¼ 0 (1), ¼ (2), ½ (3), 1/1 (4), 2/1 (5), ad 4/1 (6), [I] 0 ¼ 0.02 M,[M] 0 ¼ 2 M, k p1 ¼ k tr ¼ L mol 1 s 1, k tc ¼ k td ¼ 10 6 L mol 1 s 1, x ¼ 0.

8 274 O. Okay, C. N. Bowma calculatios, typical values of the rate costats ad cocetratios were used: [12 14] k p1 ¼ k tr ¼ L mol 1 s 1, k tc ¼ k td ¼ L mol 1 s 1, ad [M] 0 ¼ 2 M. These costats are typical for thiol-ee photopolymerizatio of dithiol diallyl ad tetrathiol-viyl silazae comoomer pairs. [12 14] For k p /k tr ¼ 0, i.e., whe o homopolymerizatio was allowed, the depedece of X 2 o x M is predicted usig Equatio (38), as show i the figure by the symbols. The asymptotic limit i X 2 was used to predict the critical time, t cr, ad the critical coversio, x cr, at the gel poit. The gelatio times decrease from 34.5 s to 0.9 s as the k p /k tr ratio is icreased from 0 to 4. The critical coversio also decreases from to with icreasig k p /k tr ratio. Thus, the gel poit i particular is very sesitive to the value of k p /k tr, ad it shifts toward lower coversios or shorter reactio times as k p /k tr is icreased. For the same reactio system, the solid curve i Figure 4A shows the umber-average chai legth X 1 plotted agaist the viyl group coversio, x M. As predicted by Equatio (36), X 1 vs. x M plots are idepedet of the k p /k tr ratio; X 1 slightly icreases with icreasig coversio from 1 to 6. It should be oted that X 1 is the average chai legth of all molecules preset i the reactio system, icludig the ureacted moomers. If the moomers are ot take ito accout, the resultig umber-average chai legth, X 1;P, ca be calculated from the momets of molecules with r > 1, deoted by Q. The rate equatio for Q relates to Q usig the followig: dq ¼ dq d½p 1Š Q ð0þ ¼0 ð40þ The dashed curves i Figure 4A show X 1;P vs. x M plots for various k p /k tr ratios. It is see that X 1;P is k p /k tr depedet; the higher the k p /k tr, the higher the average chai legth of the polymer molecules. Over the whole rage of k p /k tr ratios preseted here, the average chai legths durig the thiolee reactios remai below Figure 4B shows viyl ad thiol group coversios, x M ad x SH, respectively, plotted agaist the reactio time up to the oset of gelatio. Icreasig the k p /k tr ratio also icreases the coversio rate of the viyl groups, while that of the thiol groups slightly decreases. This more rapid disappearace of viyl groups is due to the icreasig cotributio of the homopolymerizatio reactios (Equatio (4b)) to the gel formig system. The results as show i Figure 4B have bee observed experimetally i thiol-acrylate ad thiol-methacrylate systems with ad without photoiitiators. [4,5,26] The disadvatage of more rapid viyl group cosumptio is the icomplete coversio of the thiol groups after the thiol-ee reactios. To obtai equivalet coversio of both fuctioal groups, oe has to alter the feed ratio of thiol ad viyl groups, r 0. The critical coditio for obtaiig equivalet coversios of both fuctioal groups is obtaied from Equatio (30) (31) as follows: r 0 ¼ 1 k p ð41þ k tr The dashed ad dotted curves i the iset to Figure 4B are x M ad x SH vs. time plots, respectively, for a reactio system with r 0 ¼ 1 ad k p /k tr ¼ 0.5. It is see that the viyl groups are cosumed much more rapidly tha the thiol groups. Figure 4. (A) Number-average chai legth of all molecules X 1 (solid curve) ad, of those molecules with r > 1, X 1;P (dashed curves) plotted agaist the viyl group coversio, x M. (B): x M (solid curves) ad x SH (dashed curves) vs reactio time plots. Calculatios were for six differet k p /k tr ratios: k p /k tr ¼ 0 (1), ¼ (2), ½ (3), 1/1 (4), 2/1 (5), ad 4/1 (6). All other kietic parameters are the same as give i the captio for Figure 3. The dashed ad dotted curves show i the iset to Figure 4B are x M ad x SH vs. reactio time plots, respectively, calculated for r 0 ¼ 1 ad k p /k tr ¼ 0.5. Chagig r 0 from 1 to 0.5, accordig to Equatio (41), results i equivalet coversio rates of both fuctioal groups, also show i the iset by the solid curve.

9 Kietic Modelig of Thiol-Ee Reactios with Both Step ad Chai Growth Aspects 275 Figure 5. Number-average chai legth of carbo radicals X 1 (solid curves) ad thiyl radicals X 1 0 (dashed curves) plotted agaist the viyl group coversio, x M. Calculatios were for six differet k p /k tr ratios: k p /k tr ¼ 0 (1), ¼ (2), ½ (3), 1/1 (4), 2/1 (5), ad 4/1 (6). All other kietic parameters are the same as give i the Figure 3 captio. However, accordig to Equatio (41), if the same reactio is carried out at a feed ratio r 0 ¼ 0.5, exactly equivalet coversio of both fuctioal groups ca be obtaied, as see by the solid curve i the figure. Figure 5 shows umber-average chai legth of thiyl radicals, X 0 1 (dashed curves), ad carbo radicals, X 1 (solid curves), plotted agaist x M for various k p /k tr ratios. Whe o homopolymerizatio was allowed (curves 1), X 0 1 ffi1 ad X 1 ffi2 over a wide rage of coversios, idicatig that the moomeric thiyl radicals together with the dimeric carbo radicals cosistig of oe thiol ad oe ee moomer uits are active species i the pre-gel regime of thiol-ee systems. Icreasig the extet of homopolymerizatio also icreases the chai legth of the active species. As the reactio system approaches the gel poit, both X 0 1 ad X 1 diverge due to the formatio of giat molecules with more tha oe radical ceter, as also reported for free-radical cross-likig copolymerizatios. [16,22] Accordig to Equatio (35) ad (38), all the kietic parameters of a thiol-ee system, except the homopolymerizatio rate costat, k p, do ot affect the gel poit coversio. However, these parameters do affect the polymerizatio rate ad thus, the gelatio time. For example, icreasig the iitiatio rate R I or decreasig the termiatio rate costat k t icreases the total cocetratio of the radicals so that gelatio occurs more rapidly. A importat kietic parameter i thiol-ee systems is the ratio of the thiyl radical propagatio costat to the chai trasfer rate costat, k p1 /k tr. The relative cocetratio of carbo radicals j i the reactio system is proportioal to this ratio Figure 6. Effect of the k p1 /k tr ratio o the gelatio time i thiolee systems at three differet feed ratios, r 0, of thiol ad viyl groups. r 0 ¼ 0.5 (solid curve), 1 (dashed curve), ad 2 (dotted curve). ad, for a equivalet feed ratio of thiol ad viyl groups, k p1 /k tr is equal to the cocetratio ratio of carbo to thiyl radicals (Equatio (23)). Figure 6 shows the variatio of the gelatio time, t cr, with k p1 /k tr at three differet feed ratios of thiol ad viyl groups. Calculatios were for f 1 ¼ f 2 ¼ 3, x ¼ 0, ad k p ¼ 0, which leads to, accordig to Equatio (38), the gel poit coversios x SH,cr ¼ 0.707, 0.500, ad 0.354, for r 0 ¼ 0.5, 1.0, ad 2.0, respectively. Figure 6 shows that gelatio occurs earlier as k p1 /k tr is icreased up to a critical value; thereafter, t cr becomes early idepedet of k p1 /k tr. Oe may explai this behavior with the predomiat attack of thiyl radicals o the viyl groups ad formatio of more stable carbo radicals as the k p1 /k tr ratio is icreased. The gelatio times t cr, i terms of the mole fractio of carbo radicals j are calculated from Equatio (22) ad (30) as: t cr ¼ t cr;1 ð42þ j where t cr,1 is the limitig value of the gelatio time at j ¼ 1, i.e., t cr;1 ¼ lð1 x SH;crÞ k tr ðr I =k t Þ 0:5 ð42aþ Thus, accordig to Equatio (22) ad (42), if k p1 / k tr >> r 0, the fractio of carbo radicals j approaches to uity so that t cr becomes idepedet of k p1 ktr.this coditio is approached at k p1 /k tr > 10 0 ad 10 1 for r 0 ¼ 0.5 ad 1, respectively (Figure 6). However, if k p1 /k tr r 0, j is much smaller tha uity; o gel forms withi a reasoable period of time.

10 276 O. Okay, C. N. Bowma Figure 7. Weight- average chai legths X 2 i thiol-ee polymerizatios with f 1 ¼ 2 ad f 2 ¼ 3 (solid curves) ad f 1 ¼ f 2 ¼ 5 (dotted curves) plotted agaist the reactio time (A) ad viyl group coversio, x M (B). Calculatios were for various cyclizatio parameters, x. The kietic parameters used i the calculatios are the same as give i the Figure 3 captio. r 0 ¼ 1, k p ¼0. x ¼ 0 (1), 0.10 (2), 0.15 (3), ad 0.20 (4). Figure 7 shows the weight-average chai legths X 2 i thiol-ee reactios with r 0 ¼ 1 plotted agaist the reactio time ad viyl group coversio, x M. Calculatios were for various values of the cyclizatio parameter, x. The fuctioalities of the moomers are take as f 1 ¼ 2, f 2 ¼ 3 (solid curves) ad f 1 ¼ f 2 ¼ 5 (dotted curves). The average molecular weights, X 2, i the pre-gel regime are below 10 1 up to about 80% coversio. At a give reactio time or coversio, X 2 slightly decreases with icreasig x. The gel poit is, however, very sesitive to x, particularly if the moomer fuctioalities are low; the gel poit shifts toward higher coversios or loger reactio times as x is icreased. The depedece of the critical coversio x cr o x is also illustrated i Figure 8A for the two thiol-ee systems. It is see that, for f 1 ¼ f 2 ¼ 5, x cr chages oly slightly with x up to 0.4 due to the large umber of pedat fuctioal groups available i this system for gelatio. However, for a reactio system with f 1 ¼ 2 ad f 2 ¼ 3, eve a small degree of cyclizatio rapidly icreases the gel poit coversio, x cr. For x > 0.16 or x > 0.60, gelatio is ot attaied for the comoomer pairs with f 1 ¼ 2, f 2 ¼ 3 ad f 1 ¼ f 2 ¼ 5, respectively. The lack of gelatio occurs because the cocetratio of pedat fuctioal groups becomes isufficiet to reder all moomer uits joied ito a three-dimesioal etwork. To illustrate the effect of the moomer fuctioality o the Figure 8. (A) The gel poit coversios, x cr, show as a fuctio of the cyclizatio parameter, x. The moomer fuctioalities are idicated i the figure. (B) The critical x value, x cr, is show as a fuctio of f 2. f 1 ¼ 2. The kietic parameters used i the calculatios are the same as give i the Figure 3 captio.

11 Kietic Modelig of Thiol-Ee Reactios with Both Step ad Chai Growth Aspects 277 critical value of x (x cr ), i Figure 8B, x cr is plotted agaist f 2 for a fixed viyl moomer fuctioality f 1. x cr is a icreasig fuctio of the moomer fuctioality. This result arises from the fact that, icreasig fuctioality of the moomers also icreases the cocetratio of the pedat fuctioal groups i the reactio system. As a result, the effect of the fuctioal groups lost i cycles o the oset of gelatio becomes weaker at high moomer fuctioalities. For multifuctioal thiol moomers with f 2 10, gelatio occurs eve after cosumptio of more tha 40% of the fuctioal groups by the cyclizatio reactios. Coclusio A kietic model was developed for thiol-ee photopolymerizatio reactios. The model predicts the molecular weight distributio of active ad iactive species i the pregel regime of thiol-ees, as well as the gel poits depedig o the sythesis parameters. It is show that, whe o homopolymerizatio is allowed, the average molecular weights ad the gel poit coversio are give by the typical equatios valid for the step-growth polymerizatio. Icreasig the extet of homopolymerizatio also icreases the average molecular weights ad shifts the gel poit toward lower coversios or shorter reactio times. It is also show that the ratio of thiyl radical propagatio to the chai trasfer kietic parameter (k p1 /k tr ) affects the gelatio times, t cr. Gelatio occurs earlier as k p1 /k tr is icreased up to a critical value; thereafter t cr becomes isesitive to k p1 /k tr. This behavior was explaied with the predomiat attack of thiyl radicals o the viyl groups ad formatio of more stable carbo radicals as the k p1 /k tr ratio is icreased. It was also foud that the gel poit is very sesitive to the extet of cyclizatio. I particular, if the moomer fuctioalities are low, the gel poit shifts toward sigificatly higher coversios or loger reactio times as the fractio of fuctioal groups lost i cycles icreases. Ackowledgemets: The authors ackowledge a NSF Tie Grat ( ) ad Istabul Techical Uiversity Research Fud (11_04_102, 11_04_158) for fudig this work. [1] A. F. Jacobie, i: Radiatio Curig i Polymer Sciece ad Techology III, Polymerizatio Mechaisms, J. D. Fouassier, J. F. Rabek, Eds., Elsevier Applied Sciece, Lodo 1993, Chapter 7, p [2] J. G. Woods, i: Radiatio Curable Adhesives i Radiatio Curig: Sciece ad Techology, S. P. Pappas, Ed., Pleum, New York 1992, p [3] C. E. Hoyle, T. Y. Lee, T. Roper, J. Polym. Sci., Part A: Polym. Chem. 2004, 42, [4] N. B. Cramer, C. N. Bowma, J. Polym. Sci., Part A: Polym. Chem. 2001, 39, [5] N. B. Cramer, J. P. Scott, C. N. Bowma, Macromolecules 2002, 35, [6] M. S. Kharasch, W. Nudeberg, G. J. Matell, J. Org. Chem. 1951, 16, 524. [7] M. S. Kharasch, J. Read, F. R. Mayo, Chem. Id. (Lodo) 1938, 57, 752. [8] K. S. Aseth, C. M. Wag, C. N. Bowma, Macromolecules 1994, 27, 650. [9] K. S. Aseth, I. M. Klie, T. A. Walker, K. J. Aderso, C. N. Bowma, Macromolecules 1995, 28, [10] W. Fuke, O. Okay, B. Joos-Muller, Adv. Polym. Sci. 1998, 136, 139. [11] O. Okay, Prog. Polym. Sci. 2000, 25, 711. [12] N. B. Cramer, T. Davies, A. K. O Brie, C. N. Bowma, Macromolecules 2003, 36, [13] N. B. Cramer, S. K. Reddy, A. K. O Brie, C. N. Bowma, Macromolecules 2003, 36, [14] S. K. Reddy, N. B. Cramer, A. K. O Brie, T. Cross, R. Raj, C. N. Bowma, Macromol. Symp. 2004, 206, 361. [15] H. Galia, J. B. Lechowics, Adv. Polym. Sci. 1998, 137, 135. [16] S. Zhu, A. E. Hamielec, Macromolecules 1993, 26, [17] H. Jacobso, W. H. Stockmayer, J. Chem. Phys. 1950, 18, [18] G. Ercolai, L. Madolii, P. Mecareli, Macromolecules 1988, 21, [19] P. J. Flory, Priciples of Polymer Chemistry, Corell Uiversity Press, Ithaca, New York 1953, Chapter VIII. [20] G. Odia, Priciples of Polymerizatio, 3 rd editio, Joh Wiley & Sos, Ic. New York [21] H. R. Kricheldorf, G. Schwarz, Macromol. Rapid Commu. 2003, 24, 359. [22] H. Tobita, A. E. Hamielec, Makromol. Chem., Macromol. Symp. 1988, 20/21, 501. [23] H. Tobita, A. E. Hamielec, Macromolecules 1989, 22, [24] O. Okay, Polymer 1994, 35, 796. [25] T. M. Lovestead, K. A. Berchtold, C. N. Bowma, Macromol. Theory Simul. 2002, 11, 729. [26] L. Lecamp, F. Houllier, B. Youssef, C. Buel, Polymer 2001, 42, 2727.