A Nonempirical Model for the Lifetime Prediction of Polymers Exposed

Chapter 11

A Nonempirical Model for the Lifetime Prediction of Polymers Exposed in Oxidative Environments

Downloaded by COLUMBIA UNIV on July 30, 2012 | http://pubs.acs.org Publication Date: January 1, 2009 | doi: 10.1021/bk-2009-1004.ch011

X. Colin, B. Fayolle, L. Audouin, and J. Verdu CIVIPOL, LIM (UMR 8006), ENSAM 151, Boulevard de l'Hôpital, 75013 Paris, France

This paper presents a non-empirical kinetic model for the lifetime prediction of polymers exposed in their normal use conditions. After having described the different components (the core, the optional layers) of the model, its efficiency is demonstrated for polyethylene in large temperature and γ dose rate ranges. Future developments are briefly presented.

Introduction Since the end of the 90's, our group has been developing a non-empirical kinetic model, named K I N O X A M , for the lifetime prediction of polymers and polymer matrix composites in their use conditions. The model is totally open. It is composed of a core, common to all types of polymers, derived from the now well-known "closed-loop" mechanistic scheme (/). Around this core, various optional layers can be added according to the complexity of oxidation mechanisms and the relationships between the structural changes taking place at the molecular scale and the resulting ones at larger scales (the macromolecular and macroscopic scales). The efficiency of the kinetic model has been demonstrated for many substrates: polypropylene (2), polyethylene (3-4), poly(ethylene terephthalate) (5), polyisoprene (6), sulphur vulcanized polyisoprene (7), polybutadiene (8-9), amine crosslinked epoxy (10) and polybismaleimide (11) in large temperature, γ dose rate and oxygen pressure ranges. © 2009 American Chemical Society

In Polymer Degradation and Performance; Celina, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2009.

121

122 In the following sections, particular attention is paid to polyethylene radiothermal oxidation.

Core of the Mode! The core of the kinetic model describes the thermal oxidation at low temperature (typically at Τ < 200°C) at low conversion ([PH] = [PH] = constant) and in oxygen excess of unfilled and unstabilized saturated hydrocarbons. It is derived from the "closed-loop" mechanistic scheme ( C L M ) of which the main characteristic is that radicals are formed by the thermal decomposition of its main propagation product: the hydroperoxide group P O O H (12). This closed-loop character explains the sharp auto-acceleration of oxidation at the end of an induction period (Figure 1).


0

Figure 1. Typical shape of kinetic curves of0 absorption. Determination of oxidation induction time (if) and maximum oxidation rate (r^). 2

In its simplest form (in oxygen excess) the C L M involves five elementary reactions: POOH -> 2P° + γ ^ Ο + (1 - γ ^ Ρ - Ο Η (k ) Initiation 2POOH -> P° + P 0 ° + γ ι Ρ = 0 + (I γ,)Ρ-ΟΗ (k ) Initiation P° + 0 -> P 0 ° (k ) Propagation P 0 ° + POOH -> POOH + P° (k ) Propagation P0 ° + P0 ° -> inactive products + 0 (k ) Termination lu

2

2

lb

2

2

2

2

3

2

2


6

123 The only assumption, rarely justified and developed in the literature, is the uniqueness of the reactive site P H . P H corresponds to the most abstractable hydrogen atom in the monomer repeat unit: As an example, it corresponds to methyne C - H bonds in PP ([PH] ~ 20 mol.l ) and methylene C - H bonds in PE (60 mol.l ). The C L M leads to a system of three differential equations (SED), constituting the core of the kinetic model: 1

0

1

^P dt

2

0

= 2k [POOH] + k [POOH] - k [ 0 ] [ P ° ] + k [PH][P0 ] lu

l b

2

2

3


2

^ 5 * ! ! = k [ P O O H ] + k [ 0 ][P°] - k [ P H ] [ P 0 °] - 2 k [ P 0 ° ] dt I b

d[POOH] dt

2

=

_

k

|

u

[

P

0

0

H

2

]

_

3

k

[

b

[

P

0

0

H

2

]

2

+

Μ

Ρ

Η

(1)

2

6

2

(2)

2

][Ρ0 °] 2

(3)

It is assumed that 0 dissolution is instantaneous and obeys Henry's law: 2

[0 ]-[0 ]o = S x P 2

2

(4)

0 2

where S is the coefficient of 0 2 solubility in the polymer and P 0 2 the 0 2 partial pressure in the surrounding atmosphere. Initial conditions are: at t = 0, [P°] = [P02°] = 0 and [POOH] = [POOH]0. The numerical resolution of the SED gives access to the changes (with time) of concentrations of the reactive species: [02], [P°], [P02°] and [POOH] = f(t). A n interesting quantity, which can be directly determined from the SED without the use of additional assumptions (or additional adjustable parameters), is 0 2 absorption: 0

2

Qo2 = | ( k [ 0 ] [ P ° ] - k [ P 0 ] ) d t 2

2

6

2

(5)

The changes (against time) of Q have been calculated for PE at different temperatures, with the parameter values reported in Table I, and compared to literature data (13). At temperatures higher than 100°C, good agreement has been obtained between theory and experiment (Figure 2). However, at temperatures lower than 100°C, deviations have been observed: the core of the model overestimates the oxidation induction time tj and underestimates the maximum oxidation rate r , both deviations increasing when the temperature decreases. A n additional layer has been added around the core in order to reduce both deviations. 0 2

s

Layer 1: Non-terminating Bimolecular Combination A detailed kinetic analysis of a paper published more than 30 years ago by Decker et al. (14) led us to distinguish three distinct processes of bimolecular


124

Table I. Parameters used for Kinetic Modeling. Rate constant (s' or l.mor'.s' ) 1


1

Pre-exponential factor (s' or l.mor'.s" ) 1

1

145

7

86

2.0

10

kn,

1.9

10

k

10

2

0

8

1.5 10'°

3

K

5.6

1

12

kiu k

Activation energy (kJ.mor )

10

8

73 0

Figure 2. Examples of 0 absorption curves for PE at T> 100°C in oxygen excess (P >2 10 Pa). Points: Experimental data. Continuous lines: kinetic model based on equations 1-5. 2

4

Q2


125 combination of peroxy radicals: Complex form. P 0 ° + P 0 ° -> [PO° ° O P ] +0 ( M Termination [PO° ° O P ] -> POOP ( M Termination [PO° ° O P ] ->P=0 + P - O H ( M Non-termin. [PO° ° O P ] -> 2P° + 2 γ , Ρ = 0 + 2(1 - γΟΡ-ΟΗ ( M Quantitative data obtained at 45°C by these authors indicate clearly that termination (by coupling or disproportionation) is not very efficient since a significant fraction of alkoxy radicals (typically 35-40%) escape from the cage to initiate new oxidation radical chains. These four processes have been added to the C L M . The resulting mechanistic scheme leads to a new SED: 2

2

cage

2

cagc

cage


cagc

2

= 2k [POOH] + k [POOH] - k [ 0 ] [ P ° ] + k [PH][P0 °] lu

l b

2

2

3

2

dt +2k [PO° °OP] 63

^lu dt d

= k [POOH] l b

P

I ^2Îil

_

k l u [

diPO° °OPl -i ^ * i

=

=

2

(6)

cage

o

+

k [O ][P°]-k [PH][PO ]-2k [PO °] 2

2

3

poOH]-k [POOH]

2

l b

2

6 0

2

2

+ k [PH][P0 °] 3

(7)

(8)

2

dt

2

k [PO °] -(k 6 0

2

6 1

+k

o o

62

+ k )[P0 0P] 6 3

(9)

c a g e

Initial conditions are: at t = 0, [P°] = [ P 0 ° ] = [PO° ° O P ] = 0 and [POOH] = [POOH] . The changes (with time) of Q have been calculated for PE at different temperatures from equation 5 with the parameter values reported in Table II. Values of tj and r have been determined graphically as shown in Figure 1, over a large temperature range (typically between 40 and 200°C) and compared to literature data (4). Good agreement has been obtained between theory and experiment (Figures 3 and 4). The model explains, without any additional assumptions, the discontinuity observed at about 110-120°C in the temperature dependence of both quantities. This discontinuity is due to the fact that the apparent termination rate constant ke does not obey Arrhenius law (Figure 5). 2

cage

0

0 2

s

Layer 2: Polymer Radiolysis In the case of radio-oxidation, a second important source of radicals is the polymer radiolysis, i.e. the breakdown of lateral bonds in the polymer chain. P E radiolysis leads to the formation of very reactive H° radicals which recombine rapidly by hydrogen abstraction. Thus, the balance reaction can be written as follows:


126

Table II. Parameters used for kinetic modeling.

Rate constant (S- or L mol s ) -1

1

_l

Activation energy ( k J mol ) 140 105 0 73 80 0 5 50

Pre-exponential factor (s" or L mol' s" ) 8.0 10 2.8 10 10 1.5 10 4.9 10 2.0 10 1.2 10 8.0 10 1

1

1

1

12

kiu

9

kib

8

k

2

k

3

10

19

6

k6i

6

k62


12

k*3

Initiation PH + hv -> P° + V4H (n) where r is the initiation rate (mollis* ) proportional to dose rate D (Gy.s* ): 2

1

1

x

7

r «10" GiD

(10)

{

G, being the radical yield expressed in number of radicals P° per 100 eV absorbed. From a detailed kinetic study (5), it was found that Gj ~ 8 for PE. This reaction has been added to the CLM, which leads to a modification of equation 6: ^P at

2

= r +2k [POOH] + k [POOH] - k [ 0 ] [ P ° ] + k [PH][P0 °] f

lu

lb

2

+2k [PO° °OP] 63

2

3

2

(6b)

cage

Steady radiochemical yields have been calculated for each (reactive and inactive) species Y at different temperatures and y dose rates:

o -ii«a

do

v

D dt and compared to literature data (J). Good agreement has been obtained between theory and experiment (Table III).

Layer 3: Oxygen Diffusion/Reaction Coupling In many cases (except for PE), the critical 0 pressure P , beyond which the assumption of oxygen excess can be considered valid, is far from atmospheric pressure. As an example, in the case of PP, P ~ 1.2 MPa (2). Thus, reactions involving P° radicals cannot be neglected: 2

c

c



127

Figure 3. Arrhenius plot of the oxidation induction time for PE in oxygen excess (P02 > 2 10 Pa). Points: Experimental data. Continuous line: kinetic model based on equations 5—9. 4

Figure 4. Arrhenius plot of the maximum oxidation rate for PE in oxygen excess (P02 >2 10* Pa). Points: Experimental data. Continuous line: kinetic model based on equations 5-9.



128

Table III. Radiochemical yields determined by Decker et al. at 45°C in oxygen excess (P02 > 2 10 Pa) (14). Comparison to theoretical ones predicted with equations 6-11. 4

type Experiment Modeling Experiment Modeling Experiment Modeling Experiment Modeling

1

I (Gy.s )

ldPE

0.108

hdPE

0.108

ldPE

0.292

hdPE

0.292

G02

GpoOH

16 16.0 12 12.0 12 12.0 9.7 9.7

9.2 9.2 5.6 5.6 5.5 5.5 3.5 3.5

GpoH

12.5 9.3 8.0 6.3

Gpoop

Gpo

3 3.0 3 3.0 3.2 3.2 3 3.0

4.2 4.2 3.2 3.2 4.1 4.1 3.6 3.6


129 Termination P° + P° - » inactive products (k ) Termination P° + P 0 ° -> (1 - γ )ΡΟΟΗ + inact. prod (k ) Both reactions have been added to the C L M . The resulting mechanistic scheme leads to a new SED: 4

2

5

5

d[P°] . , _ = r, + 2 k [POOH] + k [ P O O H ] ' - k [ 0 ] [ P ° ] + k [ P H ] [ P 0 ° ] dt l u

I b

2

2

3

2k [P°]' - k [ P ° ] [ P 0 ° ] + 2k [PO° °OP] 4

5

2

63

2

(12)

cage

2

= k [POOH] + k [0 ][P°]- k [PH][P0 °]-k [P°][P0 °] l b

2

2

3

2

5

2


dt -2k

6 0

[PO °]

2

(13)

2

2

^ = -k [POOH]-k [POOH] +k [PH][P0 °] l u

dt

l b

3

2

o

+(l-y )k [P°][P0 ] 5

d[PO° ° O P ]

5

(14)

2

cage

= [PO " (VkH l "+^k6 2 ^^63 + k XAP O ° ° O PJ cage ] - ^k 60L ^ 2 ° ]J ~ 2

~

6 0

1 V

2

6 1

6 2

1

63

w

W

1

(15)

cage

Initial conditions are: at t = 0, [P°] = [ P 0 ° ] = [PO° ° O P ] = 0 and [POOH] = [POOH] . In the case of thick samples (typically few mm thick), oxidation is restricted to superficial layers. As a result, 0 concentration in an elementary sublayer, located at a depth χ beneath the sample surface, is all the more so small since this sublayer is deeper. The spatial distribution (in the sample thickness) of 0 concentration has been predicted from a balance equation expressing that [0 ] variation in an elementary sublayer is equal to the 0 supply by diffusion (predicted by the classical Fick's second law) minus its consumption by the chemical reaction: 2

cage

0

2

2

2

2

2

dt

dx

where D is the coefficient of 0 diffusion into the polymer. In the superficial elementary layer (at χ = 0 and L), the boundary conditions are given, at any time, by equation 4. The numerical resolution of the new SED composed of equations 12-16 gives access to the spatial distribution (in the sample thickness) of concentrations of the reactive species and their changes (against time): [0 ], [P°], [ P 0 ° ] , [POOH] and [PO° ° O P ] = f(x, t). The local 0 absorption has been calculated for PE at different temperatures and γ dose rates in atmospheric air from equation 16, with the parameter values 0 2

2

2

cage

2


2

130 reported in Table III, and compared to literature data (13, 15). In some cases, from the spatial distribution of Q 2 along x, a thickness of oxidized layer T O L has been determined according to an arbitrary criterion, as schematized in Figure 6. Good agreement has been obtained between theory and experiment in the case of both pure thermal oxidation, for instance at 100°C (Figure 7), and radio-thermal oxidation at ambient temperature in a large dose rate range (typically between 10" and 10 Gy.s* ) (Figure 8). In the case of radio-thermal oxidation, the model describes, without the use of additional assumptions, the negative curvature observed at intermediary dose rates and the horizontal asymptote reached when the dose rate tends towards natural radioactivity value (1.5 10" Gy.s" ). 0

10

1


10

1

Q02/Q02 s

χ TOL Figure 6. Typical shape of normalized oxidation profiles into thick PE samples (typically few mm thick). Determination of the thickness of oxidized layer using an arbitrary criterion.

•

The negative curvature is due to the fact that: Pure radio-oxidation predominates at high dose rates (typically higher than about 0.5 Gy.s" ) and is characterized by a T O L quasi-proportional to the reciprocal dose rate. Pure thermal oxidation predominates at very low dose rates (typically lower than about 10" Gy.s" ) and is characterized by a large T O L , typically 6 mm. At intermediate dose rates, no one initiation process can be neglected relative to the other one. The model covers all of these domains. 1

•

7

•

1

Layer 4: Lifetime Prediction Finally, we have tried to use the model to predict the lifetime of relatively thin P E samples (typically lower than 1 mm thick) (75). As a first approach, the lifetime t has been determined starting from 2 assumptions: F



131

Figure 7. Spatial distribution of oxidation products for PE after 207 and 255 hours of exposure at 100°C in atmospheric air. Points: Experimental carbonyl profiles determined by microspectrophotometry. Continuous line: kinetic model based on equations 12—16.

Figure 8. Plot in logarithmic co-ordinates of the thickness of oxidized layer for PE at room temperature in atmospheric air versus ydose rate. Points: Literature experimental data. Continuous line: kinetic model based on equations 12-16.


132 1. First, it has been considered that embrittlement occurs when the endlife criterion is reached in the superficial elementary layer. For polymers predominantly undergoing chain scission during their oxidation, embrittlement corresponds to a critical number of chain scissions s (16). But, for polymers undergoing both chain scission and crosslinking, such as PE, the problem remains totally open. F


2.

Then, since, at the end of the induction period, [POOH] is sufficiently close to its steady state value [POOH]^ obtained in the case of pure thermal oxidation, oxidation accelerates sharply and the end life criterion is rapidly reached. As a result, it has been considered that embrittlement occurs for a critical hydroperoxyde concentration [POOH] such as: F

[POOH] =

[

P

Q

Q

H

L

F

(17)

q corresponding to the onset of rapid autoacceleration, q being an arbitrary number, higher than unity but not by very much (typically q ~ 13).

Figure 9. Plot in logarithmic co-ordinates of the lifetime for PE at room temperature in atmospheric air versus ydose rate. Points: Literature experimental data. Continuous line: kinetic model based on equations 12-16 plus assumptions 1 and 2.


133 Here also, a good agreement has been obtained between theory and experiment (Figure 9). The model describes the negative curvature observed at intermediate dose rates and the horizontal asymptote reached when the dose rate tends towards natural radioactivity value. The corresponding ceiling lifetime is about 18.5 years.


Conclusions A non-empirical kinetic model was developed for the lifetime prediction of polymer parts in their normal use conditions. This model gives access to the spatial distribution (in the sample thickness) of the structural changes at the different scales and the resulting changes of normal use properties. Its efficiency was demonstrated for many substrates in large temperature and dose rate ranges. Here, we have paid special attention to PE radio-thermal oxidation. Its main advantage is to be based on experimentally checkable physicochemical parameters strictly obeying Arrhenius' law. These parameters can now be used for extrapolation. The next stage will introduce the stabilization reactions and the stabilizer transport phenomena into the kinetic model, and determining the corresponding rate constants and other physical parameters.

References 1.

Achimsky, L . ; Audouin, L . ; Verdu, J.; Rychly, J.; Matisova-Rychla, L . Polym. Degrad. Stab. 1997, 58(3), 283. 2. Richaud, E.; Farcas, F.; Fayolle, B.; Audouin, L.; Verdu, J. Polym. Degrad. Stab. 2006, 91(2), 398. 3. Khelidj, N.; Colin, X . ; Audouin, L . ; Verdu, J.; Monchy-Leroy, C.; Prunier, V . Polym. Degrad. Stab. 2006, 91(7), 1593. 4. Khelidj, N.; Colin, X . ; Audouin, L.; Verdu, J.; Monchy-Leroy, C.; Prunier, V . Polym. Degrad. Stab. 2006, 91(7), 1598. 5. Assadi, R.; Colin, X.; Verdu, J. Polymer 2004, 45(13), 4403. 6. Colin, X.; Audouin, L.; Verdu, J. Polym. Degrad. Stab. 2007, 92(5), 886. 7. Colin, X.; Audouin, L.; Verdu, J. Polym. Degrad. Stab. 2007, 92(5), 898. 8. Coquillat, M.; Verdu, J.; Colin, X.; Audouin, L . ; Nevière, R. Polym. Degrad. Stab. 2007, 92(7), 1334. 9. Coquillat, M . ; Verdu, J.; Colin, X.; Audouin, L . ; Nevière, R. Polym. Degrad. Stab. 2007, 92(7), 1343. 10. Colin, X.; Marais, C.; Verdu, J. Polym. Degrad. Stab. 2002, 78(3), 545. 11. Colin, X.; Verdu, J. Comp. Sci. Technol. 2005, 65, 411.


134 12. Audouin, L . ; Achimsky, L.; Verdu, J. Handbook of Polymer Degradation , 2 edition; Marcel Dekker Inc.: New-York, 2000; pp. 727. 13. Colin, X.; Audouin, L . ; Verdu, J. Polym. Degrad. Stab. 2004, 86, 309. 14. Decker, C.; Mayo, F. R.; Richardson, H . J. Polym. Sci.: Polym. Chem. Ed. 1973, 11, 2879. 15. Colin, X . ; Khelidj, N . ; Monchy-Leroy, C.; Audouin, L . ; Verdu, J. Nucl. Instr. Meth. Phys. Res. Β 2007, 265, 251. 16. Fayolle, B.; Audouin, L.; Verdu, J. Polym. Degrad. Stab. 2000, 70, 333.


nd


A Nonempirical Model for the Lifetime Prediction of Polymers Exposed

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