Generalized kinetic master plots for the thermal degradation of polymers following a random scission mechanism

1  GENERALIZED KINETIC MASTER PLOTS FOR THE THERMAL DEGRADATION OF POLYMERS FOLLOWING A RANDOM SCISSION MECHANISM. Ped o E. Sánchez-Jiménez, Luis A. Pé ez-Maqueda, An onio Pe ejón and José M. C iado. Ins i u o de Ciencia de Ma e iales de Se illa, C.S.I.C.-Uni e sidad de Se illa, C. Amé ico Vespucio nº49, 41092 Se illa, Spain Abs ac In his pape , he (α) con e sion unc ions o andom scission mechanisms ha e been p oposed in o de o allow o he cons uc ion o gene alized mas e plo s sui able o hese kind o mechanisms. The mas e plo s ha e been alida ed by i s applica ion o simula ed da a and o he he mal deg ada ion o polybu ylen e eph ala e, polye hylene and poly e a luo oe hylene. Keywo ds: Kine ics, Mas e plo s, Random scission, Polyme deg ada ion, Mechanisms   Co esponding au ho . Tel +34954489548 Fax +34954460665 e-mail add ess: ped o.en [email protected] 2  1. In oduc ion The kine ic modelling o solid s a e eac ions keeps aising a b oad in e es in ma e ials science and enginee ing. A eliable e alua ion o he kine ic pa ame e s ha go e n a p ocess p o ides a aluable insigh in o he mechanism ollowed by he eac ion. A p ope kine ic analysis calls o he de e mina ion o he kine ic iple , namely, he ac i a ion ene gy, E; he p e-exponen ial ac o , A; and he kine ic model, (α). This la e pa ame e , also known as con e sion unc ion, is an algeb aic exp ession ha is associa ed wi h he physical model ha desc ibes he kine ics o a eac ion.1-2 A numbe o di e en me hods ha e been de eloped o e he yea s o ex ac ing kine ic in o ma ion om expe imen al da a, many equi ing he expe imen al da a o be ob ained unde ce ain expe imen al condi ions.3-12 Thus, iso he mal, non iso he mal, o he ecen combined analysis me hods ha e been p oposed. Recen ly, he so called “model- ee” me hods, ha allow de e mining he ac i a ion ene gy o a p ocess as a unc ion o he deg ee o con e sion wi hou any p e ious assump ion o he kine ic model, ha e a ained g ea popula i y.12-19 Howe e , hey do no di ec ly yield he eac ion kine ic model, al hough me hods o i s e alua ion ha e been de eloped wi h he use o mas e plo s. 20-23 Mas e plo s a e e e ence heo e ical cu es ha depend on he kine ic model bu a e independen o he kine ic pa ame e s, E and A. Expe imen al da a can easily be ans o med in o expe imen al mas e plo s and compa ed wi h he heo e ical ones de e mined o he di e en kine ic models. Kine ic Analysis is equally impo an in he ield o he mal s abili y o polyme s. The de elopmen o wo kable models able o desc ibe he decomposi ion p ocesses and o de e mina e ma e ials esponse o di e en he mal condi ions has been he conce n o many au ho s and ex ensi e wo k has been p oduced du ing las decades.24 Howe e , deg ada ion o polyme s is a complex phenomenon and despi e he g ea deal o 3  esea ch pe o med on his subjec , a high con o e sy s ill emains. The di icul y o de e mining he p ope kine ic model o polyme deg ada ion eac ion has p omp ed ha mos o he wo ks ound in he li e a u e eso o model- ee me hods, 19,25-28 o jus assume i s o “n-o de ” kine ic models wi hou epo ing a gumen s ha suppo his assump ion.29-37 Howe e , a ecen wo k has p o en ha he mal deg ada ion o polyme s do no necessa ily ake place h ough i s o “n-o de ” kine ics and o he mechanisms such as di usion o andom scission can con ol he decomposi ion eac ion.38 The use o mas e plo s in he mal deg ada ion o polyme s would help o disc imina e he kine ic model wi hou he p e ious assump ion o a pa icula con e sion unc ion ha does no gua an ee he p ope desc ip ion o he deg ada ion mechanism. While a se o mas e plo s applicable o expe imen al da a eco ded unde any hea ing p o ile has al eady been p oposed in his jou nal,23 hey canno be applied as p oposed o andom scission kine ic models due o he impossibili y o exp ess (α) as a unc ion o he eac ed ac ion in a closed o m. As andom scission is one o he mos usual mechanisms in deg ada ion o polyme ic ma e ials, 36,39-45 he ex ension o he mas e plo s in o de o co e hese si ua ions is o he u mos in e es . In his wo k, he o iginal Simha-Wall equa ion o depolyme isa ion p ocesses 46 has been e o mula ed in such a way ha he eac ion a e can be di ec ly exp essed as a unc ion o (α), ime o empe a u e. Then, making use o he new equa ions, he gene alized mas e plo s 23 ha e been ex ended o andom scission mechanisms, The e alua ion o he p oposed andom scission kine ic model and i s co esponding mas e plo has been ca ied ou by simula ed and expe imen al cu es alike. The la e we e ob ained om he he mal deg ada ion o h ee comme cial polyme s: polybu ylen e eph ala e (PBT), polye hylene (PE) and poly e a luo oe hylene (PTFE). 4  2. P oposal o a new kine ic model o andom scission mechanisms The eac ion a e, dα/d , can be desc ibed by he ollowing equa ion:     RTEAk d d exp (1), whe e A is he A henius p e-exponen ial ac o , R is he gas cons an , E he ac i a ion ene gy, α he eac ed ac ion, T is he p ocess empe a u e and (α) he kine ic model, which accoun s o he eac ion a e dependence on α. Table 1 shows he unc ions co esponding o he mos common models in he li e a u e. Decomposi ion o a polyme by andom scission implies a andom clea age o bonds along he polyme chains, p oducing agmen s o p og essi ely sho e leng h ha will e en ually e apo a e when he size is small enough.24 Acco ding o Simha-Wall 46, he clea age o bonds ollows i s o de kine ics and he ollowing exp essions hold ue: )1()1( xAexk d dx RT E   (2)              N LLN xx L1 111 1  (3), whe e x, N and L a e he ac ion o bonds b oken, he ini ial deg ee o polyme iza ion and he minimum leng h o he polyme ha is no ola ile, espec i ely. As L is usually negligible in compa ison o N, Eq. (3) can be simpli ied o:   )1(111 1 Lxx L  (4) 5  Mos he mal deg ada ion s udies a e ca ied ou by he mog a ime y because he mass los can be easily ela ed o he con e sion. Howe e , in he case o andom scission mechanisms only he b oken bonds ha p oduce agmen s small enough would ac ually e apo a e and he e o e be de ec ed as mass loss. Tha p oblem is sol ed by Eq. (4), which es ablishes a ela ionship be ween he de ec ed mass loss and he ac ual eac ed ac ion in e ms o ac ion o bonds b oken. This ela ionship is shown g aphically in Figu e 1, assuming L alues anging om 2 o 8. Howe e , as x canno be measu ed by con en ional echniques and L is e y di icul o ob ain expe imen ally, he applica ion o Eq. (4) has been se e ely limi ed. Ne e heless, by di e en ia ing Eq. (4), and inco po a ing Eq. (2) we ge : )1()1()1( 2xkxxLL d dL   (5) This way, aking in o accoun Eq. (1), we can ge om Eq. (5) he con e sion unc ion (α) which is cha ac e is ic o a andom scission model: 1 )1()1()(   L xxLL  (6) Many kine ic analysis me hods in ol e he i ing o expe imen al da a o a ce ain kine ic model. This equi es he (α) unc ions o he di e en models o be p e iously known. Thus, i andom scission mechanisms a e o be used in his way, (α) mus be de e mined. Howe e , a symbolic solu ion can only be eached o L=2. In his la e case, om Eq. (1) and Eq. (6) we ob ain: )(2 2/1    k d d (7) The e o e, (α) mus be: 6  )(2)( 2/1   (8) Taking in o accoun he ela ionship be ween x and α as es ablished in Eq. (4), o any gi en L and assigning alues o α, om Eq. (4) and (6) i is possible o calcula e nume ically he co esponding (α) con e sion unc ions, which a e plo ed agains α in Figu e 2a. Fo he sake o compa ison, Figu e 2 also includes he (α) con e sion unc ions co esponding o he mos common kine ic models in li e a u e: “n o de ” (2b), di usion con olled (2c) and nuclea ion and g ow h kine ic models (2d). Random scission unc ions ha e a cha ac e is ic shape which is qui e di e en om he o he models. Since he esul s o a kine ic analysis a e hea ily dependen on he kine ic model conside ed, andom scission d i en eac ions could ne e be adequa ely desc ibed by o he models, and in pa icula by “n-o de ” models, as i is o en done in li e a u e, and doing so will only esul in ob aining inco ec kine ic pa ame e s. Howe e , he andom scission kine ic model could be desc ibed by he modi ied Ses ak- Be gg en exp ession ha was p oposed as a i ing equa ion o he combined kine ic analysis p ocedu e. 42,47 3. Gene alized Mas e Plo s In a p e ious pape he gene alized kine ic equa ions in oduced by Ozawa 48 was used o he p oposal o uni e sal mas e plo s ha we e alid o expe imen al da a eco ded unde any hea ing p o ile.23 Thus, i he gene alized ime is de ined as 49: d RT E       0exp  (9), 7  whe e, conside ing he in eg al o Eq. (1), i is clea ha θ ep esen s he ime needed o each a ce ain α alue a in ini e empe a u e. By di e en ia ing Eq. (9) he ollowing equa ion can be ob ained:       RT E d dexp  (10) The combina ion o Eq. (1) and Eq. (10) leads o: )(    A d d (11), which can also be exp essed in he ollowing way:       RT E d d d dexp    (12), dα/dθ being he gene alized eac ion a e ha , acco ding o Eqs. (1), (11) and (12), ep esen s he eac ion a e ex apola ed a in ini e empe a u e as p e iously shown by Ozawa.49 Since he p e ious knowledge o he ac i a ion ene gy allows o he ex apola ion o in ini e empe a u e o expe imen al da a eco ded unde any hea ing p o ile, Eq. (12) should be alid o he analysis o any da a, independen ly o he empe a u e p o ile unde which hey we e ob ained. F om Eq. (11) and aking α = 0.5 as a e e ence we ge :  )5.0( )( / / 5.0 dd dd      (13) As (0.5) is cons an o a ce ain kine ic model, Eq (13) indica es ha o a gi en α, he educed-gene alized eac ion a e, (dα/dθ)/(dα/dθ)α=0.5, would be equi alen o (α)/ (0.5) when he p ope (α) is selec ed o desc ibe he p ocess. F om Eq. (12) and 8  Eq. (13), he ela ionship be ween he gene alized eac ion a e and he expe imen al da a can be es ablished:    )/exp( /exp / / / / 5.0 5.05.0 RTE RTE d d d d dd dd       (14), whe e T0.5 ep esen s he empe a u e co esponding o α = 0.5. In he case o expe imen al da a ob ained unde iso he mal condi ions, he exponen ial e m o he second hal in Eq. (14) cancels and he equa ions becomes:  5.05.0 / / / / d d d d dd dd       (15) On he o he hand, o expe imen al da a eco ded unde non iso he mal condi ions, he p e ious knowledge o he ac i a ion ene gy is equi ed in o de o cons uc he expe imen al mas e plo s. By plo ing oge he e sus α he gene alized eac ion a e, as calcula ed om Eq. (14) (o Eq. (15) o iso he mal condi ions), and he ac ion (α)/ (0.5), co esponding o di e en heo e ical kine ic models, i is possible o deduce by compa ison he kine ic model ollowed by he p ocess. I mus be no ed ha , acco ding o Eq. (14), o non iso he mal expe imen s a single ac i a ion ene gy alue is assumed. The e o e, o his analysis p ocedu e o be alid, he s udied p ocess mus obey single s ep kine ics. He e esides he impo ance o he p e ious isocon e sional analysis, checking ha he ac i a ion ene gy does no a y wi h alpha in a signi ican way. 9  4. Random Scission Mas e Plo s The educed-gene alized eac ion a e o andom scission models is de i ed om Eq. (5) by aking α = 0.5 as a e e ence:   )1( )1( )1( )1( 5.0 2 5.05.0 2 5.0 xk xk xx xx d d d d L L         (16) Acco ding o Eq (15), in he case o iso he mal condi ions, (dα/dθ)/(dα/dθ)α=0.5 and (dα/d )/ (dα/d )α=0.5 a e equi alen and Eq (16) becomes:   1 5.05.0 1 5.0 )1( )1(     L L xx xx d d d d   (17) The gene alized mas e plo s o andom scission mechanisms can now be cons uc ed nume ically by gi ing alues o x and plo ing Eq (17) agains α. The esul ing cu es a e plo ed in Fig 3 o di e en alues o L and compa ed wi h he mas e plo s co esponding o he es o he kine ic models in Table 1. As i can be clea ly no iced, andom scission mas e plo s could be dis inguished easily due o he maximum hey show a α alues o a ound 0.275. Di usion and “n-o de ” mas e plo s p esen no maximum while he mas e plo s co esponding o nuclea ion and g ow h laws ha e i a α = 0.4 o highe . 5. 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bounda y con olled eac ion (con ac ing a ea) R2 21 )1(   Phase bounda y con olled eac ion (con ac ing olume) R3   32 )1(   Random nuclea ion ollowed by an ins an aneous g ow h o nuclei. (A ami-E o ee eqn. n =1) F1)1(   Random nuclea ion and g ow h o nuclei h ough di e en nuclea ion and nucleus g ow h models. (A ami-E o ee eqn ≠1.) An   n n11 )1ln()1(     Two-dimensional di usion D2       1ln1  Th ee-dimensional di usion (Jande equa ion) D3         3/1 3/2 112 )1(3    Th ee-dimensional di usion (Gins ling-B ounsh ein equa ion) D4    112 3 31     Random Scission L=2 L2       21 2 Random Scission L>2 Ln  Nosymbolicsolu ion  19   TABLE 2. Ac i a ion ene gy alues a di e en alues o con e sion and hei co ela ion coe icien s, ob ained by he F iedman isocon e sional analysis o he he mal decomposi ion o poly e a luo oe hylene (PTFE), polye hylene (PE) and poly(1,4-bu ylen) e eph ala e (PBT).   PFTE  PE PBT α REa (kJmol‐1) α REa (kJmol‐1) α REa (kJmol‐1) 0.10.998299±60.10.997246±80.10.999182±4 0.21.000286±30.20.996246±160.21.000182±4 0.31.000282±40.30.999259±100.31.000180±4 0.40.999286±60.40.998256±130.41.000180±4 0.50.999285±50.50.998255±130.51.000181±3 0.61.000282±40.60.999259±90.61.000182±3 0.70.999280±40.70.998254±110.71.000183±2 0.80.998291±80.80.998255±100.81.000185±2 0.90.998296±90.90.997260±110.90.999195±5 20  Figu e Cap ions Figu e 1: Rela ionship be ween he ac ual ac ion o bonds b oken (x) and he con e sion, α, o di e en andom scission kine ic unc ions, acco ding o Eq. (4). The igh mos cu e was plo ed assuming L=2 while he le mos o L=8. Cu es assuming L=3 o L=7 lie in-be ween hem. Figu e 2: The (α) con e sion unc ions o : (a) he newly p oposed andom scission model, plus he di e en kine ic models mos commonly used in li e a u e, (b) di usion con olled, (c) “n o de ” and (d) nuclea ion and g ow h kine ic models. Figu e 3: Gene alized mas e plo s co esponding o he di e en kine ic models in Table 1 as cons uc ed om Eq. (13) and (17). (a) Random scission models; (b) di usion con olled models; (c) “n o de ” models and (d) nuclea ion and g ow h models. Figu e 4. Cu es simula ed assuming a andom scission L2 model, E = 150kJmol-1, A=1011 s-1 and he ollowing hea ing p o iles: (a) linea hea ing a e o 1, 2 and 5 K min- 1 and (b) con olled a e o 0.06 min-1. Figu e 5. Compa ison be ween he gene alized mas e plo s cons uc ed o he di e en simula ed cu es included in Figu e 4 (symbols) and he mas e plo s co esponding o some o he ideal kine ic models included in able 1 (solid lines). Figu e 6. Expe imen al cu es ob ained o he he mal decomposi ion o poly e a luo oe hylene unde he ollowing expe imen al condi ions: (a) linea hea ing a e o 1, 2 and 5 K min-1 and (b) sample con olled deg ada ion a e o 5 10-4 min-1. 21  Figu e 7. Compa ison be ween he gene alized mas e plo s co esponding o he expe imen al cu es in Fig 6 wi h he heo e ical mas e plo s cons uc ed om he ideal kine ic models in Table 1. Figu e 8. Expe imen al cu es ob ained o he he mal decomposi ion o polye hylene unde he ollowing expe imen al condi ions: (a) linea hea ing a e o 1, 2 and 10 K min-1 and (b) sample con olled deg ada ion a e o 1.6 10-4 min-1. Figu e 9. Compa ison be ween he gene alized mas e plo s co esponding o he expe imen al cu es in Fig 9 wi h he heo e ical mas e plo s cons uc ed om he ideal kine ic models in Table 1. Figu e 10. Expe imen al cu es ob ained o he he mal decomposi ion o poly(1,4- bu ylene) e eph ala e unde he ollowing expe imen al condi ions: (a) linea hea ing a e o 1, 2 and 5 K min-1 and (b) sample con olled deg ada ion a e o 1.4 10-5 min-1. Figu e 11. Compa ison be ween he gene alized mas e plo s co esponding o he expe imen al cu es in Fig 10 wi h he heo e ical mas e plo s cons uc ed om he ideal kine ic models in Table 1. 22  Figu e 1: Rela ionship be ween he ac ual ac ion o bonds b oken (x) and he con e sion, α, o di e en andom scission kine ic unc ions, acco ding o Eq. (4). The igh mos cu e was plo ed assuming L=2 while he le mos o L=8. Cu es assuming L=3 o L=7 lie in-be ween hem. 23  Figu e 2: The (α) con e sion unc ions o : (a) he newly p oposed andom scission model, plus he di e en kine ic models mos commonly used in li e a u e, (b) di usion con olled, (c) “n o de ” and (d) nuclea ion and g ow h kine ic models. 24  Figu e 3: Gene alized mas e plo s co esponding o he di e en kine ic models in Table 1 as cons uc ed om Eq. (13) and (17). (a) Random scission models; (b) di usion con olled models; (c) “n o de ” models and (d) nuclea ion and g ow h models. 25  Figu e 4. Cu es simula ed assuming a andom scission L2 model, E = 150kJmol-1, A=1011 s -1 and he ollowing hea ing p o iles: (a) linea hea ing a e o 1, 2 and 5 K min-1 and (b) con olled a e o 0.06 min-1. 32  Figu e 11. Compa ison be ween he gene alized mas e plo s co esponding o he expe imen al cu es in Fig 10 wi h he heo e ical mas e plo s cons uc ed om he ideal kine ic models in Table 1. 33  TABLE OF CONTENTS IMAGE