Polymer Degradation and Stability 91 (2006) 1598e1605www.elsevier.com/locate/polydegstab

Oxidation of polyethylene under irradiation at low temperature andlow dose rate. Part II. Low temperature thermal oxidation

N. Khelidj a,b, X. Colin a,*, L. Audouin a, J. Verdu a,C. Monchy-Leroy b, V. Prunier b

a Laboratoire de Transformation et de Vieillissement des Polymeres, ENSAM,

151 boulevard de l’Hopital, 75013 Paris, Franceb EDF R&D, site des Renardieres, Ecuelles, 77818 Moret-sur-Loing, France

Received 21 July 2005; accepted 7 September 2005

Available online 7 November 2005

Abstract

This paper deals with the kinetic modelling of unstabilised polyethylene thermal oxidation, particular attention being paid to the domain oflow temperatures, typically below 80 �C. Experimental data show that the temperature dependence of the induction time ti and the steady staterate of oxygen absorption rS display a discontinuity at 80 �C. A model based on the hypothesis that this discontinuity concerns only the PO2

�

bimolecular combination processes and is essentially explained by the competition between terminating and non-terminating PO2�

C PO2�

reac-tions, was proposed. With pertinent values of the Arrhenius parameters of the elementary reactions under consideration, the model fits well theexperimental data (in the 40e200 �C temperature range) and is consistent with previously analysed results of radiochemical ageing. Accordingto this model, 35e40% of the bimolecular PO2

�combinations would not be terminating at 45 �C and this proportion would increase with the

temperature. Concerning terminations, the relative fraction of coupling processes, leading to peroxide bridges, would decrease relatively tothe disproportionation processes when the temperature increases.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Polyethylene; Thermal oxidation; Kinetics; Modelling; Rate constants

1. Introduction

This series of papers deals with the lifetime predictionof polyethylene (PE) subjected to low dose rate (I ) irradiation(typically I ! 10�4 Gy s�1), at low temperature (T ! 80 �C),for very long periods (several years). The first step of thisstudy consists of establishing a mechanistic scheme for the un-stabilised polymer, to determine the elementary rate constantsand their temperature dependence. The existence of twoasymptotic regimes corresponding, respectively, to I / N(regime R) and I / 0 (regime H) has been previously shown[1]. In regime R, initiation is exclusively due to polymer radi-olysis, hydroperoxides being considered as stable. In regime

* Corresponding author.

E-mail address: xavier.colin@paris.ensam.fr (X. Colin).

0141-3910/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.polymdegradstab.2005.09.012

H, initiation is exclusively due to hydroperoxide decomposi-tion, radical production from polymer radiolysis being consid-ered as negligible.

The first part of this series, devoted to regime R, was basedon the detailed analysis of a paper published more than 30 yearsago by Decker et al. [2]. This analysis led us to distinguish threeprocesses of peroxyl radical (PO2

�) bimolecular combination:

ðVI� 0Þ PO�2 C PO

�2 / ½PO

� �OP�cage C O2 ðk60Þ

ðVI� 1Þ ½PO� �

OP�cage / PeOeOeP ðk61Þ

ðVI� 2Þ ½PO� �

OP�cage / P]O C PeOH ðk62Þ

1599N. Khelidj et al. / Polymer Degradation and Stability 91 (2006) 1598e1605

ðVI� 3Þ ½PO� �

OP�cage / 2P�C products ðk63Þ

Quantitative data obtained at 45 �C under gamma irradia-tion can be briefly summarised as follows:

(i) Termination is not very efficient: 35e40% of the combina-tion events are non-terminating reactions (VI-3). Alkoxyradicals escaping from the cage are presumably the mainprecursors of chain scissions according to the followingprocess:

½PO� �

OP�cage / 2PO�

PO�C PH / PeOH C P

�

PO�/ P]O C s C P

� ðb scission; s ¼ chain scissionÞ

The above reaction VI-3 is the balance of these three ele-mentary processes.

(ii) Among termination events, coupling (reaction VI-1)predominates over disproportionation (reaction VI-2):k61=k62z3:2 at 45 �C.

(iii) Assuming that k3 can be estimated from the Arrheniusparameters resulting from previous model compoundstudies [3], it was possible to determine orders ofmagnitude for the rate constants of PO2

�bimolecular com-

binations at 45 �C [4]: k60z4:0!106 l mol�1 s�1, k61z2:0!106 l mol�1 s�1, k62z6:4!105 l mol�1 s�1, andk63z1:6!106 l mol�1 s�1.

(iv) There are only second order differences between low andhigh density PE, which confirms the results of a recentcompilation on thermal oxidation kinetic data [5].

This compilation will be used, here, to study the regime H(initiation exclusively by hydroperoxide decomposition).However, literature data relative to PE thermal oxidation atlow temperature are very scarce. Moreover, a discontinuityat 80 �C can be observed in the temperature dependence ofkinetic parameters so that there is no way to predict the PEbehaviour at T ! 80 �C from (abundant) data obtained atT O 80 �C, at least with the common kinetic models.

PE oxidation kinetics can be characterized by two keyquantities: the induction time ti and the steady state rate rS. In-deed, there is no way to determine ti, at low temperature, ina short time. In contrast, this is possible for rS, using thefollowing procedure based on the fact that ti is a decreasingfunction of the initial hydroperoxide concentration [POOH]0:the samples are first ‘‘pre-aged’’ at high temperature Tb

for a time slightly shorter than the induction time ti (Tb).Then, they are exposed at the temperature Ta of interest: theinduction period is considerably shortened so that the steadystate can be observed in a reasonable time as schematised inFig. 1.

2. Theory

We have chosen to use a mechanistic scheme in whichpropagation and PO2

�combination elementary events are the

same whatever the initiation process: POOH decompositionor PE radiolysis or both. Certain authors have chosen distinctoptions, for instance the hypothesis that radiolysis at low doserate could involve specific termination mechanisms. But, ourstrategy consists to choose always the simplest scheme andto give it up only in case of failure of the corresponding kineticmodel.

It is now clear that, for PE in regime H, hydroperoxidedecomposition is predominantly bimolecular, this tendency be-ing accentuated at low temperatures since the activation energyis lower for bimolecular than for unimolecular decomposition[5]. In numerical calculations, both initiation processes canbe taken into account to obtain more rigorous results. In analyt-ical calculations, however, unimolecular decomposition will beneglected for the sake of simplicity.

Mathematically speaking, the problem can be schematisedas follows: a system of differential equations is derived from thechosen mechanistic scheme, this latter being composed of theelementary processes identified by Decker et al. [2] minusthe polymer radiolysis plus the hydroperoxide decomposition,plus the non-terminating PO2

�C PO2

�combination (VI-3).

The aim of our work is to solve the inverse problem, i.e. todetermine the kinetic parameters from the experimental data.The solution must fulfil the following conditions:

(i) Be consistent with all the experimental data relative to PEthermal oxidation in the 40e200 �C temperature intervalin non-diffusion controlled conditions (thin films).

(ii) Simulate the discontinuities observed in the temperaturedependences of ti and rS.

(iii) Predict the quantitative data obtained by Decker et al. [2]in their study of PE radiochemical ageing at 45 �C.

tta

Qox(a)(b) (exp)

rSrS

Fig. 1. Schematisation of the accelerated test to determine the steady state

oxidation rate at low temperature. The temperature is Tb at 0 ! t ! ta and

Ta at t O ta. Qox is the conversion of the oxidation process. (exp) is the exper-

imental curve. (b) is obtained during a full exposure at Tb. (a) is obtained

during a full exposure at Ta. rS is the same for curves (exp) and (a).

1600 N. Khelidj et al. / Polymer Degradation and Stability 91 (2006) 1598e1605

The starting mechanistic scheme is thus:

ðIuÞ POOH / 2P� ðk1uÞ

ðIbÞ POOH C POOH / P�C PO

�2 ðk1bÞ

ðIIÞ P�C O2 / PO

�2 ðk2Þ

ðIIIÞ PO�2 C PH / POOH C P

� ðk3Þ

ðVI� 0Þ PO�2 C PO

�2 / ½PO

� �OP�cage C O2 ðk60Þ

ðVI� 1Þ ½PO� �

OP�cage / POOP ðk61Þ

ðVI� 2Þ ½PO� �

OP�cage / P]O C PeOH ðk62Þ

ðVI� 3Þ ½PO� �

OP�cage / 2P� ðk63Þ

This scheme leads to the following set of differentialequations:

d½P��dt¼2k1u

�POOH

�Ck1b½POOH�2�k2½O2�

�P��

Ck3

�PH�½PO

�2�C2k63½PO

�2�

2 ð1Þ

d½PO�2�

dt¼ k1b½POOH�2Ck2½O2�

�P��� k3

�PH�½PO

�2�

� 2k60½PO�2�

2 ð2Þ

d½POOH�dt

¼�k1u½POOH� � 2k1b½POOH�2Ck3½PH�½PO�2� ð3Þ

d½PO� �OP�cage

dt¼ k60½PO

�2�

2�ðk61Ck62Ck63Þ½PO� �

OP�cage ð4Þ

rox ¼�d½O2�

dt¼ k2½O2�½P�� � k60½PO

�2�

2 ð5Þ

Simulations will be based on numerical solutions of thissystem with no simplifying hypotheses. However, in a firstapproach, the classical analytical analysis, based on the hy-pothesis of steady state for radical and hydroperoxide concen-trations [6], is interesting, at least to illustrate the globaltrends. In this analysis, the unimolecular initiation (Iu) whichhas been found to be negligible [5], is suppressed.

The steady state hypothesis leads to the following expres-sion for the whole rate of oxygen absorption rox:

rox ¼k2

3½PH�2

2k60

�1C

k63

kt

�1

1Cb expð �KtÞ

!

�1C

k63

2kt

1

1Cb expð �KtÞ

�ð6Þ

where kt ¼ k61Ck62 is the true termination rate;

b¼�½POOH�N�½POOH�0½POOH�0

¼ k3½PH�2ðk1bk60Þ1=2½POOH�0

�1C

k63

kt

�1=2

�1

and

K ¼ k3½PH��

k1b

k60

�1=2�1C

k63

kt

�1=2

The steady state rate rS is thus:

rS ¼k2

3 ½PH�2

2k60

�1C

k63

kt

��1C

k63

2kt

�¼ k2

3½PH�2

2kT app

ð7Þ

where

kT app ¼k60�

1Ck63

kt

��1C

k63

2kt

� ð8Þ

kT app is the apparent termination rate constant, using for rS theexpression found for a ‘‘bimolecular’’ scheme in which all thePO2

�bimolecular combinations are terminating [6].

Since, in the case of bimolecular initiation, the autoacceler-ation is very strong at the end of the induction period, the du-ration ti of this latter can be estimated taking the followingcriterion: ti is the time at which the oxidation rate rox is halfof its steady state value:

rox

rS

¼ 1

2ð9Þ

By substituting in Eqs. (6) and (7), one obtains:

ti ¼1

KLn

b

R� 1ð10Þ

with R ¼�1C

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1C2gð1CgÞ

p ��ð1CgÞ and g ¼ k63=2kt.

R is a slowly decreasing function of g: R¼ 2 for g¼ 0 andR¼ 1.62 for g¼ 1, so that ti increases when g increases.

3. Experimental

Most experimental data were collected from the literature.They are relative to oxygen absorption or IR spectrophotometry

1601N. Khelidj et al. / Polymer Degradation and Stability 91 (2006) 1598e1605

measurements performed on unstabilised or extractedpolyethylene samples. There was no selection respective topolymerisation mode, branching degree, crystallinity ratio,etc. The temperature range was 40e200 �C.

A limited number of isothermal ageing tests were carriedout in air at a temperature Ta ranging from 100 �C to 40 �Cusing ventilated ovens regulated at G1 �C, according to theaccelerated ageing procedure described in Fig. 1 (Tb¼100 �C), essentially in order to increase the number of exper-imental data in the low temperature range, in particular in the40e80 �C temperature range.

The experiments were conducted with 50, 100 and150 G 2 mm thick press-moulded films of two industrial poly-ethylenes: a linear-low (0.92) and a medium density polyethyl-ene (0.94). Additives initially present in the polymer as well asother soluble impurities were extracted in a Soxhlet apparatuswith a 4:1:1 hexane/chloroform/ethanol mixture for 48 h; afterextraction, the films were dried under vacuum for 48 h at 40 �C.

A Bruker IFS 28 IR spectrophotometer, with a minimal res-olution of 4 cm�1, was used to determine the average concen-tration of carbonyl species at z1740e1710 cm�1 in the films.The whole carbonyl concentration was calculated using theclassical BeereLambert equation and the usual molar absorp-tivity for ketone groups 3K z 200 l mol�1 cm�1.

4. Results and discussion

Two characteristics of a given chemical specie Y were in-vestigated: the induction time ti and the maximum (steadystate) rates rYS. Both quantities were graphically determinedfrom the kinetic curves of oxygen absorption and carbonylbuild-up.

The results are presented in Fig. 2 (Arrhenius plot of ti),Fig. 3 (Arrhenius plot of the maximum oxidation rate rS)and Fig. 4 (Arrhenius plot of the maximum carbonyl build-up rate rP]O, S). For each characteristic, the least squaresmethod gave the Arrhenius parameters reported in Table 1.

It appears clearly that there is a change in the kinetic regime(change in the slope of the Arrhenius plots) around 80 �C.

Basically, there are two ways to explain the existence ofa discontinuity in the temperature dependence kinetics:

(a) There is no discontinuity in the temperature dependence ofelementary rate constants. The observed discontinuitycomes from the competition between several elementaryprocesses having different activation energies.

(b) There is a discontinuity in the temperature dependence ofelementary rate constants. The observed behaviour resultsfrom a transition between a regime where the reaction rateis governed by the intrinsic reactivity of the reactants anda regime where the reaction rate is governed by the mobil-ity of the reactants in the polymer matrix.

Let us first examine, from this point of view, the elementaryreactions in the steady state (see Table 2). The reactive speciesare characterized by their concentration, which leads us to distin-guish three categories: the relatively high concentration species(code F for PH and O2), the medium concentration species (codeM for POOH) and the low concentration species (V for radicalsP�and PO2

�). It is obvious that if diffusion control (case (b) above)

operates, it must first concern the PO2�

combination processeswhich involve an encounter between two rare species.

The reactions are also characterized by the possible coexis-tence of several competitive processes. Such a situation canbe envisaged for initiation and for PO2 combination processes.

Considering the analytical expressions of rS (Eq. (7)) and ti(Eq. (10)), one sees that, in the case of rS, the discontinuityprobably comes from a peculiarity of the temperature variationof kT app, whereas in the case of ti, both kT app and k1b could beinvolved.

Let us first consider the case of rS. Eq. (7) can be rewritten:

kT app ¼k2

3½PH�2

2rS

ð11Þ

4

6

8

10

12

14

16

18

20

22

2,0 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 2,9 3,0 3,1 3,2 3,3 3,4

1000/T (K-1

)

Ln

ti (s)

Fig. 2. Arrhenius plot of the induction oxidation time ti. Points: experimental data. Continuous line: kinetic model.

1602 N. Khelidj et al. / Polymer Degradation and Stability 91 (2006) 1598e1605

-19

-17

-15

-13

-11

-9

-7

-5

-3

Ln

rS (m

ol.l-1.s

-1)

2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 2,9 3,0 3,1 3,2 3,3 3,41000/T (K

-1)

Fig. 3. Arrhenius plot of the steady state oxidation rate rS. Points: experimental data. Continuous line: kinetic model.

Direct determinations of k3 are difficult to envisage, butwhat is important in kinetic modelling, is to take the rightvalue of k2

3=kT app. Here, it has been decided to use, in a firstapproach, a k3 value derived from model compound studies:

k3 ¼k30 exp��E3=RT

�with k30 ¼ 1:5!1010 l mol�1 s�1

and E3 ¼ 73 kJ mol�1

In PE, [PH] is the methylene concentration in the amor-phous phase:

�PH�z60 mol l�1

kT app was calculated from experimental values of rS usingEq. (11). The Arrhenius plot of kT app is shown in Fig. 5. Theseresults call for the following comments:

(i) The apparent activation energy of kT app tends to almostzero above 80 �C.

(ii) An interpolation of this graph at 45 �C would give a valueof kT app close to the one determined from the results ofDecker et al. [2]. This seems to justify the hypothesisthat propagation and termination processes should bethe same in thermal and radiochemical oxidation.

Can the discontinuity on the temperature dependence ofkT app be explained only by a competition between the PO2

bimolecular combination reactions?Elements of answer can be derived from Eq. (8) expressing

kT app versus elementary rate constants. Eq. (8) can be rewritten:

kT app ¼2k60k2

t

ðktCk63Þð2ktCk63Þ¼ 2k60k2

t

k263

1�1C

kt

k63

��1C

2kt

k63

�

ð12Þ

The apparent activation energy of kT app can be zero only if:(i) E60, E63 and Et¼ 0.

-20

-18

-16

-14

-12

-10

-8

-6

-4

Ln

rc=

o, s (m

ol.l-1.s

-1)

2,2 2,3 2,4 2,5 2,6 2,7 2,8 2,9 3,0 3,1 3,2 3,3 3,41000/T (K

-1)

Fig. 4. Arrhenius plot of the steady state carbonyl build-up rate rP]O, S. Points: experimental data compiled from literature (A) and determined in the laboratory

(-). Continuous line: kinetic model.

1603N. Khelidj et al. / Polymer Degradation and Stability 91 (2006) 1598e1605

But in this case, how to explain the temperature dependenceof kT app at T ! 80 �C?(ii) E60z2ðE63 � EtÞ thus E63 O Et, i.e. 2kt=k63/0 whenT / N.In this case, the apparent activation energy of kT app would

be z0 above and zE60 below 80 �C. In fact, kt is the sumof two components: kt ¼ k61Ck62. Thus, kt does not obeyArrhenius’ law but is expected to behave as the highest activa-tion energy component at high temperature. It is possible todistinguish between both elementary termination processesfrom a comparison of oxygen absorption data and carbonylgrowth data. As a matter of fact, the carbonyl yield is zerofor termination VI-1 and unity for termination VI-2. Then,this analysis leads to:

E62OE61

Coupling predominates at low temperature whereas dispro-portionation (according to Russell [7]) predominates at hightemperature. Thus:

E60z2ðE63�E62Þ and E63OE62OE61

Finally, it was tempted to simulate the oxidation kineticcurves, at various temperatures, using the set of kinetic param-eters listed in Table 3 with ½POOH�0¼ 5!10�2 mol l�1 asa boundary condition. Firstly, ti and rS were determined on ki-netic curves of oxygen absorption resulting from the numericalresolution of the system of Eqs. (1)e(5). Then, the Arrheniusplots of ti and rS were built (see Figs. 2 and 3). The theoreticalcurves coincide with experimental ones within experimental

Table 1

Arrhenius parameters of the different characteristics under study below and

above the discontinuity at 80 �C

ti rS rP]O, S

T R 90 �C Pre-exponential factor

(s or mol l�1 s�1)

2.7 ! 10�10 5.5 ! 1013 1.3 ! 1017

Activation energy

(kJ mol�1)

�108 138 160

T % 80 �C Pre-exponential factor

(s or mol l�1 s�1)

1.5 ! 10�7 1.2 ! 105 1.8 ! 102

Activation energy

(kJ mol�1)

�89 75 62

Table 2

Characteristics of the elementary processes susceptible to be involved in the

discontinuity of the temperature dependence of oxidation kinetics

Reaction Type Existence of several competitive processes

POOH C POOH M C M Yes (unimolecular, bimolecular,

catalysed decomposition) [6]

P�C O2 V C F No

PO2�

C PH V C F No

PO2�

C PO2�

V C V Yes (VI-0, VI-1, VI-2 and VI-3)

incertitudes. The transition at 80 �C is well simulated, aswell for rS as for ti showing that there is no need of additionaladjustable parameters.

Complications appear when ‘‘secondary’’ products such aschain scissions or carbonyls are considered. Let us first considerthe case of chain scissions (S). They are supposed to resultfrom b scission of PO

�radicals. Thus, they are expected to

result only from initiation (Iu and Ib) or non-terminating PO2�

bimolecular combination (VI-3):

ðIuÞ POOH / 2P�C g1S ðk1uÞ

ðIbÞ POOH C POOH / P�C PO

�2 C g1S ðk1bÞ

ðVI� 3Þ ½PO� �

OP�cage / 2P�C 2g1S ðk63Þ

where g1 is the yield of chain scission per alkoxy radical.Thus, the number of moles of chain scissions per mass unit

is given by:

ds

dt¼ g1k1u½POOH�Cg1k1b½POOH�2C2g1k63½PO

� �OP�cage

ð13Þ

Since literature values of s and QOX (the number ofoxygen molecules consumed per polymer mass unit) areavailable [4,8e14], it seemed to us interesting to plot theratio Qox/s, i.e. the number of oxygen molecules absorbedper chain scission, against temperature (see Fig. 6). Despitethe existence of a noticeable (but not surprising) scatter, itseems clear that the curve displays a discontinuity in themelting region (100e120 �C) since Qox/s z 2 at T !100 �C and Qox/s z 5 at T R 120 �C. In other words, theyield of chain scission is significantly higher in solid statethan in liquid state.

To have a good fit of experimental data, one needs g1 val-ues varying with temperature as in Fig. 7: g1 z 1 in solid state

Table 3

Rate constants used for the simulations

Rate constant

(s�1 or l mol�1 s�1)

Pre-exponential factor

(s�1 or l mol�1 s�1)

Activation energy

(kJ mol�1)

k1u 8.0 ! 1012 140

k1b 2.8 ! 109 105

k2 108 0

k3 1.5 ! 1010 73

k60 4.9 ! 1019 80

k61 2.0 ! 106 0

k62 1.2 ! 106 5

k63 8.0 ! 1012 50

1604 N. Khelidj et al. / Polymer Degradation and Stability 91 (2006) 1598e1605

13

14

15

16

17

18

19

20

21

22

23

Ln

k

Tap

p (l.m

ol-1.s

-1)

2,22,1 2,3 2,4 2,5 2,6 2,7 2,8 2,9 3,0 3,1 3,2 3,3

1000/T (K-1

)

Fig. 5. Arrhenius plot of the apparent termination rate constant kT app. Points: experimental data determined from Eq. (14). Continuous line: kinetic model.

and z0.2 in liquid state. Chain scission is in competition withhydrogen abstraction by PO

�radicals:

The above result can be thus explained as follows: chainscission is a unimolecular rearrangement not very dependenton the macromolecular mobility and thus, insensitive to melt-ing. In contrast, hydrogen abstraction is a bimolecular processeventually dependent on the macromolecular mobility, itsyield could thus increase significantly from the solid to theliquid state whereas globally, there is no strong discontinuityon the whole oxidation rate.

The case of carbonyls is much more complicated, first be-cause there are many kinds of carbonyl groups: aldehydescoming from b scissions, ketones coming from the terminating

1

2

3

4

5

6

20 40 60 80 100 120 140 160 180 200T (°C)

Qo

x/s (m

ole/m

ole)

Fig. 6. Variation of the steady state ratio Qox/s with the temperature. Points:

experimental data. Continuous line: kinetic model.

CH

O

+ PH

+CH2

CH2

CH2

CH2

C

O

H

CH

O

+ PCH

OH

disproportionation of PO2�

radicals (VI-2), and ‘‘tertiary’’products such as acids coming from the oxidation of eCH2

�

primary radicals resulting from b scissions, or from aldehydesoxidation, etc. A rigorous investigation in this domain wouldneed first a careful deconvolution of the carbonyl IR band,but it seems difficult to obtain an unquestionable solution ofthis problem. In a first approach, it seemed to us interestingto distinguish only between two types of carbonyls: thosecoming from PO

�b scissions (reactions Iu, Ib and VI-3), char-

acterized by a yield g2, and those coming from Russell’s ter-mination (VI-2), characterized by a yield of unity. Thus:

ðIuÞ POOH / 2P�C g1S C g2P]Ob ðk1uÞ

ðIbÞ POOHCPOOH / P�CPO

�2 C g1S Cg2P]Ob ðk1bÞ

ðVI� 2Þ ½PO� �

OP�cage / P]OK C PeOH ðk62Þ

ðVI� 3Þ ½PO� �

OP�cage / 2P�C 2g1S C 2g2P]Ob ðk63Þ

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

γ1

γ2

i

20 40 60 80 100 120 140 160 180 200T (°C)

Fig. 7. Theoretical variation of the yields of chain scission g1 and carbonyl

build-up g2 with the temperature.

1605N. Khelidj et al. / Polymer Degradation and Stability 91 (2006) 1598e1605

where P]Ob is a carbonyl coming from PO�

b scission andP]OK is a ketone.

Thus, the global rate of carbonyl build-up is given by:

rP]O ¼d½P]O�

dt¼ g2k1u

�POOH

�Cg2k1b½POOH�2

Ck62½PO� �

OP�cageC2g2k63½PO� �

OP�cage ð14Þ

The physical meaning of g2 can be understood on the basisof a simplified model in which we consider that the measuredIR absorbance results from the sum of three contributions, re-spectively, relative to ketones (K), aldehydes (A) and carbox-ylic acids (C). It can be written:

A1720 ¼ ð3K½K�C3A½A�C3C½C�Þl ð15Þ

where 3i is the molar absorptivity at 1720 cm�1, of the speciesi, and l is the sample thickness.

It has been chosen to determine a so-called ‘‘average’’ car-bonyl concentration using an ‘‘average’’ molar absorptivityequal to the ketone one (3K). Thus:

½P]O� ¼ A1720

3K l¼ ½K�C3A

3K

½A�C3C

3K

½C� ð16Þ

These quantities are related to the mechanistic scheme by:

½P]OK� ¼ ½K� ð17Þ

and

½P]Ob� ¼3A

3K

½A�C3C

3K

½C� ð18Þ

The temperature variation of the yield g2 of b scission car-bonyls (absorbing at 1720 cm�1) is shown in Fig. 7 and callsfor the following comments: since an aldehyde is formed perchain scission, the contribution of aldehydes to the absorptionat 1720 cm�1 ðð3A=3KÞ½A�Þ is expected to vary according to anaffine curve of g1 ¼ f ðTÞ and thus, to be lower in liquid statethan in solid state. The fact that g2 varies in the opposite wayshows that the acidic component ðð3C=3KÞ½C�Þ is largely pre-dominant at high temperature, that is not surprising if one sup-poses that acids result from aldehydes oxidation. The acidconcentration is not necessarily higher than the aldehydesone, but no doubt at 1720 cm�1:

3C

3K

[3A

3K

:

As a matter of fact, the maximum absorbance of the stretch-ing C]O vibration in carboxylic acids lies in the 1710e1720 cm�1, whereas the aldehydes peak lies at a noticeablyhigher wave number: 1750e1740 cm�1.

These results show the relative complexity of the interpre-tation of kinetic data relative to carbonyl build-up.

Literature values of the maximal (steady state) rate rP]O, S

of carbonyl build-up for unstabilised polyethylene sampleshave been compiled and plotted in Fig. 4. Values determinedin our laboratory, according to the acceleration procedure de-scribed in Fig. 1, have been superimposed in this graph.

g2 values presented in Fig. 7 have been used to obtaina good fit of experimental data. Here also, it can be observedthat the transition at 80 �C is well simulated (see Fig. 4).

5. Conclusions

A mechanistic scheme and the corresponding kinetic modelhave been proposed for the thermal oxidation of unstabilisedpolyethylene in the 20e200 �C temperature range, in non-diffusion controlled regime (thin samples).

The mechanistic scheme is based on predominating bimo-lecular hydroperoxide decomposition for initiation, couplingand disproportionation of caged PO

�radicals resulting from

PO2�

C PO2�

combinations for terminations, these reactions co-existing with a non-terminating PO2

�C PO2

�combination.

The kinetic model predicts the correct shape of the curvesof oxygen absorption, carbonyl, peroxide and hydroperoxidebuild-ups in the 40e200 �C temperature range. The Arrheniusplots of the oxidation induction time and the steady stateoxidation rate, which display a more or less marked disconti-nuity at 80 �C are also well simulated without the recourse toadjustable parameters. Except for initiation, all the other ele-mentary processes and their rate constant values remain validfor radiochemically-initiated oxidation.

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