Quantitative Confirmation of Simple Theoretical ... - ACS Publications

Chapter 28

Quantitative Confirmation of Simple Theoretical Models for Diffusion-Limited Oxidation

Downloaded by EAST CAROLINA UNIV on September 22, 2015 | http://pubs.acs.org Publication Date: November 12, 1991 | doi: 10.1021/bk-1991-0475.ch028

Kenneth T. Gillen and Roger L. Clough Organic Materials Division (Org. 1811), Sandia National Laboratories, Albuquerque, NM 87185

Theoretical modelling of diffusion-limited oxidation leads to oxidation profile shapes governed by two parameters α and β plus a simple relationship relating these parameters to the oxygen permeability coefficient and the oxygen consumption rate underlying the oxidation conditions. To quantitatively test the predicted profile shapes and the governing theoretical relationship, a commercial EPDM rubber material of two thicknesses was radiation-aged (Co-60) in air. Experimental oxidation profiles were obtained by monitoring the density changes caused by oxidation. The experimental profile shapes and their dependence on sample thickness could be accurately fit with the theory, yielding values for α and β. Comparison of these results with the independently measured oxygen permeability coefficient and the oxygen consumption rate allowed a quantitative confirmation of the theoretical relationship. Exposure of polymers to air during aging (radiation, thermal, UV) often results in inhomogeneously oxidized samples, a complication which impacts attempts both to understand the oxidation process and to extrapolate accelerated exposures to long-term conditions. The most important such complication involves diffusion-limited oxidation which can occur if the rate of oxygen consumption in a material is greater than the rate at which oxygen can be resupplied by diffusion processes from the surrounding air atmosphere. This scenario will usually lead to a heterogeneity in the oxidation across the material, with equilibrium oxidation (e.g., corresponding to air-saturated conditions) occurring at the sample surfaces and reouced or non-existent oxidation in the interior. The importance of this effect depends on material geometry coupled with the oxygen consumption rate and the oxygen permeability coefficient 0,2). Frequently, the percentage of the sample oxidized under shorter-term (e.g., hours to months) accelerated aging conditions is substantially lower than under longer-term application conditions (2-8). To extrapolate accelerated simulations to long-term, air-aging conditions, one must be able to monitor and quantitatively understand diffusion-limited oxidation effects. 0097-6156/91/0475-0457S06.00/0 © 1991 American Chemical Society

In Radiation Effects on Polymers; Clough, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1991.

458

RADIATION EFFECTS ON POLYMERS

Experimental techniques for monitoring diffusion-limited oxidation profiles have recently been reviewed (2-11). By comparing experimental and theoretical profiles, the theories can be verified and then confidently used to predict the importance of diffusion effects prior to the initiation of aging tests. This paper summarizes what we believe is the first rigorous quantitative experimental confirmation of diffusion-limited oxidation theories.


Experimental Material. The commercial EPDM material used for these studies was supplied by Parker Seal Group and designated as E740-75. It was obtained as compression-molded sheets (15 cm by 15 cm) in several thicknesses. The formulation is described as a peroxiae-cured terpolymer of ethylene and propylene in approximately equal amounts plus -3% hexadiene; it contains —35% by weight carbon black and a free-radical scavenger antioxidant. Radiation Aging. Combined radiation-thermal exposures were carried out in an underwater cobalt-60 facility using water-tight aging cans (volume of ~ 1 liter), an arrangement that facilitated long-term exposures. By selecting positions relative to the cobalt-60 pencils, dose rates ranging from -0.1 kGy/h to 7 kGy/h (10 krad/h to 700 krad/h) were available for the current experiments. Uncertainties in the dose rates are estimated to be ±10%. Can temperatures were kept at 70±0.5° C during the radiation exposures. A slow, steadyflow(-30 cc/min) of either air or nitrogen was supplied to each can throughout the experiment. A detailed description of the aging facility has been published (12). Density Profiling. Details of the density profiling technique have been published previously (6). The technique is based on the use of a density gradient column to obtain the density of successive thin slices across a sample. It depends on the fact that oxidation reactions often lead to substantial and easily measurable increases in polymer density. For the current material (nominal unaged density -1.12 g/cc), gradient columns were made with a density range from -1.10 to 1.15 g/cc using Ca(N03)2-H20 solutions. The experimental uncertainty for a single density measurement is estimated to be less than ±0.0004 g/cc. Oxygen Consumption Measurements. In order to determine the amount of oxygen consumed by the EPDM sample as a function of the radiation dose, measured quantities of material were sealed with known amounts of oxygen in glass containers. The containers were then aged at 70° C in combination with a selected dose rate ranging from 0.12 kGy/h to 2.08 kGy/h. After exposure, a Tracor MT150g gas chromatograph, equipped with a molecular sieve column and a thermal conductivity detector, was used to determine the amount of oxygen remaining in the tube. The oxygen was quantified using primary standards. In general, the oxygen consumption rate will depend on the oxygen partial pressure surrounding the material. Since the analysis below will require an estimate of the consumption rate under air-aging conditions (-13.2 cmHg oxygen partial pressure in Albuquerque), and since the experimental procedure requires a measurable drop in oxygen pressure, attempts were made to select the ratio of sample weights to cell volumes such that the average of the initial and final oxygen pressures was reasonably close to the desired 13.2 cmHg value. For the seven reported consumption runs, the initial and final oxygen partial pressures in the cells averaged 16.1 cmHg and 6.4 cmHg, respectively.


28. GILLEN & CLOUGH

Models for Diffusion-Limited Oxidation

459

Oxygen Permeability Coefficient Measurements. Unaged sheets of the EPDM material were sent to Mocon Modern Controls, Inc. of Minneapolis where oxygen permeability coefficients were measured at 23 °C and 60° C. The instrument is based on the ASTM standard test method D-3985-81. To avoid detector saturation, the measurements were conducted with a test gas containing 0.5% oxygen.


Diffusion-Limited Oxidation Theories In order to quantitatively model diffusion-limited oxidation profiles, one must combine expressions for oxygen consumption with diffusion equations. To derive oxygen consumption rates, detailed knowledge of the kinetics of oxidation must be available. This is a difficult requirement given the complexities of oxidation processes in polymers, especially those containing antioxidants (13). Most modelling attempts assume that the following scheme (14,15) represents a reasonable first order approximation for the oxidation of organic materials in various environments (thermal, mechanical, UV light, ionizing radiation, etc.).

PROPAGATION

1

Polymer

INITIATION

{

R.

(1)

R0 . k' R0 » + RH 3-+ ROOH + R» R. + 0

(2)

2

2

K

(3)

2

k

R» + R» BIMOLECULAR

k

R0 « + R»

4

PRODUCTS

(4)

5

PRODUCTS

(5)

6

PRODUCTS + 0

2

TERMINATION k

R0 . + R0 . 2

2

k

UNIMOLECULAR TERMINATION

R0 . 2

{

7

2

PRODUCTS

k

(6) (7)

(8) R* 8 PRODUCTS 9 RO. + ·ΟΗ ROOH BRANCHING (9) Environmental differences occur mainly in the details of the free-radical producing initiation step. It is particularly easy to control the initiation rate in the gamma-initiated case, since Ri is typically independent of aging time and linear with the radiation dose rate. In addition, branching reactions, which may complicate the oxidation processes occurring at the high temperatures used for thermal oxidation studies, are often unimportant for lower-temperature, relatively short-term gamma-initiated oxidation. Any growth in importance of branching reactions as degradation proceeds will cause the oxidation rate to increase with time, further complicating attempts at quantitative modelling. For these reasons, gamma-initiated oxidation is the best choice for initial attempts at quantitatively modelling diffusion-limited oxidation effects. Complications, such as time-dependent oxidation rates, may then be addressed once confidence exists in models derived for the simpler situations. k


460

RADIATION EFFECTS ON POLYMERS

For gamma-initiated oxidations involving moderate temperatures, we will therefore assume that the chemistry is adequately represented by reactions 1-8. The scheme involving the bimolecular termination steps (reactions 1-6), originally derived for the oxidation of organic liquids, has been invoked for many years for explaining the oxidation of polymers. Using a steady-state analysis and assuming long kinetic chain lengths (many propagation cycles compared to termination reactions) and k$2 = 4k4kg, the oxygen consumption rate is given by (14) ]


d[0

dt

[O

c

*

1

-

D

]

Z

(10) 1

'2b

where the constants Cib and C2b» appropriate to the bimolecular theory, are given by k

C

9

R.

k-

= ———-z— _ lb )0.5

C = \ * 2b ^ 0 . 5 ^

and

1 K

o u

( 2 k 4

(11)

and k3 = k3'[RH]. For many polymeric materials, unimolecular termination reactions are often found to be dominant. For instance, in the presence of sufficient radicalscavenger antioxidant, the radical species (Κθ2· and R« ) can terminate in pseudo first-order reactions, yielding an oxidation scheme consisting of reactions 1-3 plus 7 and 8. The rate of oxidation (oxygen consumption rate) is then given by (2) d [ 0

d

t

2

]

C

2

_

1

[ 0

]

lu 2 1U 2 2u[° ] +

C

(

1

2

)

2

This expression is identical in form to the bimolecular result given in equation 10, except that the constants C i and C2u from the unimolecular analysis are given by u

c

i

«

—

a

n

d

S u • vv'v

respectively), it is clear that identical results hold for unimolecular kinetics if C i and C2u are used. The boundary conditions now become θ = 1 at X = 0 and X = 1 and de/dX = 0 at X = 0.5. Numerical methods lead to a family of solutions for the relative (e.g., normalized to the surface value) oxygen concentration versus spatial position. If ej refers to the relative oxygen concentration at point i in the sample, combining either equation 10 or equation 12 with the definitions of α and β leads to the following expression for the oxidation rate at point i u


462

RADIATION EFFECTS ON POLYMERS aDe [0 ] i

Rte.]

2

i (

2

2

1

)

2

L (i+/}e.)

Similarly, the oxidation rate at the surface (Rs), which is identical to the equilibrium oxygen consumption rate, is given by


R

= —

2

s

(22)

Thus the relative oxidation at point i is given by R[e.]

θ.(1+/?)

We have used the above analysis to derive theoretical oxidation profiles (plots of Ri versus normalized cross-sectional position) and the corresponding profiles of the relative oxygen concentration (oj versus position) (2). Representative results for three values of β and various values of a are given in Figures 1-3, where Ρ gives the percentage of the distance from one oxygen-exposed surface of the sheet to the opposite oxygen-exposed surface. For small values of β, the oxidation is proportional to the oxygen concentration, yielding "U-shaped" profiles. When β is large, oxidation is insensitive to oxygen concentration until the latter has dropped significantly, resulting in "step-shaped" profiles. For intermediate values of β, profile shapes of intermediate character result. By rearranging equation 22 and noting that D[0 ] = pP (24) 2s * ox ' where Ρ χ is the oxygen permeability coefficient through the material, the following very useful theoretical relationship is obtained L

o

J

x

0

2

R L / ( p P ) =

Quantitative Confirmation of Simple Theoretical ... - ACS Publications

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