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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
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 Co esponding au ho . Tel +34954489548 Fax +34954460665
e-mail add ess: ped o.en [email protected]
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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
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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).
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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)
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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:

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
)(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),
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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
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
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.
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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. Expe imen al
Comme cial Polybu ylene e eph ala e (Ald ich, p oduc numbe 435147),
poly e a luo oe hylene (Ald ich, p oduc numbe 182478) and polye hylene (Ald ich,
p oduc numbe 332119, medium densi y d=0.940) we e used in his wo k.
16

(16)Mamlee ,V.;Bou bigo ,S.;LeB as,M.;Le eb e,J.Th eemodel‐ eeme hods
o calcula iono ac i a ionene gyinTG.Jou nalo The malAnalysisandCalo ime y2004,
78,1009.
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o  a ia ionin heac i a ionene gy.Jou nalo Compu a ionalChemis y2001,22,178.
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s imula edp ocessesinpolyme s.Mac omolecula RapidCommunica ions2006,27,1515.
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kine icanalysiso non‐iso he malda a.The mochimicaAc a1989,147,377.
(21)C iado,J.M.;Pe ez‐Maqueda,L.A.;Go o ,F.J.;Malek,J.;Koga,N.Auni ied
heo y o  hekine icanalysiso solids a e eac ionsunde any he malpa hway.Jou nalo 
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disc imina ing hekine icmodelo solids a e eac ions omasinglecons an ‐ a e he mal
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eac ions:Theuni e sali yo mas e plo s o analyzingiso he malandnoniso he mal
expe imen s.Jou nalo PhysicalChemis yA2000,104,10777.
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(25)Habibi,A.;DeWilde,J.Kine icmodelingo  he he maldeg ada iono 
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me hod o e alua iono ac i a ionene giesinmodula ed he mog a ime yandanalysiso 
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(27)Vyazo kin,S.;Sbi azzuoli,N.Isocon e sionalanalysiso calo ime icda aon
noniso he malc ys alliza iono apolyme mel .Jou nalo PhysicalChemis yB2003,107,
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(28)Vyazo kin,S.;D anca,I.;Fan,X.W.;Ad incula,R.Deg ada ionand elaxa ion
kine icso polys y ene‐Claynanocomposi ep epa edbysu aceini ia edpolyme iza ion.
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o poly(me hylme hac yla e)s udiedwi h heassis anceo  he ac ionalcon e siona  he
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17
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(36)Lu,L.;Yu,H.Y.;Wang,S.F.;Zhang,Y.The malDeg ada ionBeha io o 
S y ene‐Bu adiene‐S y eneT i‐BlockCopolyme /Mul iwalledCa bonNano ubesComposi es.
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Deg ada iono Polys y eneunde Mul ipleEx usions.Polime os‐CienciaETecnologia2008,
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model o  hekine icanalysiso  he maldeg ada iono polyme sd i enby andomscission.
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Enginee ingChemis yResea ch2008,47,6828.
(44)Liu,Y.;Wei,R.Q.;Wei,J.;Liu,X.N.The malDeg ada iono Poly(L‐Lac ide)
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a eac i eex ude .Polyme Deg ada ionandS abili y2007,92,1721.
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analysiso solid‐s a e eac ions:Apowe ul ool o  hesimul aneousde e mina iono kine ic
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(49)Ozawa,T.Noniso he malkine icsand hegene alized imeThe mochimica
Ac a1986,100,109.
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De elopmen o auni e salcons an  a e he malanalysissys em o beingusedwi hany
he moanaly icalins umen .Jou nalo The malAnalysisandCalo ime y2007,87,297.
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C.De elopmen o anew he mog a ime icsys em o pe o mingcons an  a e he mal
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o a he mobalance o con olling he eac ion a eo solid‐s a e eac ions.Re iewo 
Scien i icIns umen s2004,75,2620.
(53)Pe ez‐Maqueda,L.A.;C iado,J.M.;Go o ,F.J.;Malek,J.Ad an ageso 
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PhysicalChemis yA2002,106,2862.

18

TABLE1. ()kine ic unc ions o  hemos widelyusedkine icmodels,including henewly
p oposed andomscissionmodel.

Mechanism Symbol ()
Phase 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