Chapter 9
A Recent Advance in the Determination of Scission and Cross-Linking Yields of Gamma-Ray Irradiated Polymers
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1
†
David J. T. Hill, K. A. Milne , James H. O'Donnell , and Peter J. Pomery Polymer Materials and Radiation Group, Department of Chemistry, University of Queensland, Brisbane, Queensland 4072, Australia Scission of main chain bonds and crosslinking between polymer molecules, the two major effects of ionizing radiation on polymers, are discussed. The historical development of the theoretical treatments for radiation induced scission and crosslinking is reviewed. In particular, scission and crosslinking yields, G(S) and G(X) respectively, may be determined by observing the variation of the number average, weight average and z-average molecular weights with gamma radiation dose. The interpretation of the experimental results depends on the initial polymer molecular weight distribution. The molecular weight versus dose relationship is considered for the simple case of an initial most probable (random) distribution. For the more general case of a Schulz-Zimm (or Poisson) distribution, the equation proposed by Inokuti and Dole(1), is now able to be solved exactly by numerical methods(2). Precisely determined values of G(S) and G(X) for various experimental systems are presented and compared with the values determined by approximate methods. During the early studies of the physical and chemical effects of ionizing radiation on materials the special case of high polymers became clear. In these polymers, not only the usual radiation chemical and physical events take place, but by virtue of the large number of monomer units composing the polymers, two events, scission of the polymer backbone and crosslinking of two or more polymer molecules, have great importance for the physical properties of these polymers. Depending on the specific structure of the polymer, other radiation-induced events include the formation of gaseous products, the reduction of existing unsaturation, and the production of new unsaturation(3). If oxygen is present, peroxy species are formed and undergo further reactions depending on the polymer composition(J). The term scission is defined as any event which results in the breakage of one polymer molecule into two parts. It may occur as a result of ionization of the irradiated 1
Corresponding author †
Deceased 0097-6156/96/0620-0130$12.00/0 © 1996 American Chemical Society In Irradiation of Polymers; Clough, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
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9. HILL ET AL.
Scission & Cross-Linking Yields of Irradiated Polymers
131
polymer molecule and a direct rearrangement of the backbone into two separate entities, or the ionized polymer molecule may undergo loss of sidegroups and consequent rearrangement, which still has the final result of dividing the molecule into two separate entities, e.g. beta scission in poly(methyl methacrylate)(4). Sometimes scission is called degradation since the breakage of the long polymer molecules results in loss of structural strength and plasticity and the physical properties are degraded or diminished. Crosslinking occurs when two polymer molecules join to form one large molecule. It may occur when hydrogen is abstractedfromtwo neighbouring polymer molecules leaving two radicals in close proximity, which may then react to form a crosslink, such as an H-link, endlink or Y-link. The present work discusses H-links where a crosslink is formed between sites on the polymer molecules which are not at the ends of the molecules, thus forming a H shaped structure where the horizontal line in the H represents the crosslink. Crosslinking may result in improvement of the mechanical properties of a material since the molecular weight increases. For example, crosslinking causes a reduction of solubility, elimination of the melting point, and increased resistance to corrosive attack, all desirable material properties. The present work does not distinguish between the mechanisms of scission or the mechanisms of crosslinking. The important physical fact is that one scission causes one molecule to become two, and one crosslink causes two molecules to become one. It is generally accepted that during irradiation of a polymer, both scission and crosslinking events are occurring randomly and simultaneously. The predominance of scission over crosslinking depends on the polymer structure, temperature, crystallinity, and the presence of air. In the case of scission, the peroxy radicals prevent the geminate recombination of the radical chain ends(3/ If scission predominates, then degradation of physical properties occurs, and the polymer may become unusable. If crosslinking predominates, then gelation will eventually occur at high enough doses. The gel point is defined as the point where there is at least one crosslinked monomer unit in each molecule. Charlesby(5) reviewed the development of mathematical models for the effect of random simultaneous scission and crosslinking on polymer properties. The germinal idea was to use the average molecular weights of the polymer samples to measure the amount of scission or crosslinking. The effects of radiation dose depend critically on the initial molecular weight distribution of the polymer i.e. the molecular weight distribution before irradiation. Charlesby assumed a random or initial most probable distribution, thus simplifying the problem. His equations for this case are used at the present time although his symbolism ofp and q for the representation of scissions and crosslinked units has been replaced with the symbols G(S) and G(X) where G(S) represents the number of scissions formed and G(X) is the number of crosslinks formed per lOOeV of energy absorbed by the polymer sample. Note that Charlesby defined q as the proportion of crosslinked monomer units. Since there are two crosslinked monomer units for every crosslink formed, it follows that the proportion of crosslinks is q/2. Thus G(X) must be directly compared with q/2. This point has caused some confusion in the past. For the general case of an arbitrary initial molecular weight distribution, no general treatment was proposed by Charlesby, but he indicated that for large radiation doses, where the number of scissions per average initial molecule is greater than about three, the initial distribution no longer affects the final result and may be replaced by a random distribution of the same number average molecular weight. His contribution was noteworthy as it formed the basis of the ensuing determinations of p and q. Later workers were, however, to develop more precise theories which did not depend on the assumption of an initial random distribution.
In Irradiation of Polymers; Clough, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
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132
IRRADIATION OF POLYMERS
Inokuti(tf) considered the problem of simultaneous crosslinking and scission where the crosslinking predominates over the scission and gel formation occurs. His work included the case of the generalized Poisson-type initial distribution. As an extension of this work, Inokuti and Dole(7) derived equations for the dependence of the weight- and zaverage molecular weights on radiation dose for a generalized Poisson initial distribution (also called a Schulz-Zimm distribution). The assumptions of Inokuti and Dole(7) are (i) random crosslinking and scission occur simultaneously (ii) G(S) and G(X) are constant with radiation dose (iii)scission and crosslinking may be treated separately and their effects added (iv) not more than one crosslink connects any two polymer molecules (v) no cyclic structures are produced (vi) molecular weight changes due to other effects (such as decay of vinyl unsaturation) are absent and (vii) a generalized Poisson or Schulz-Zimm initial molecular weight distribution. Their equations are given later in the present work, and are the basis of later attempts(7,#) to utilize the theory to interpret experimental data. Due to the mathematical complexity of the equations, the task of solving the equations to give values of G(S) and G(X) has been formidable, and various approaches have been used. O'Donnell, Smith and Winzor(7) use a binomial expansion and discard the cubic and higher terms. O'Donnell, Winzor and Winzor(tf) also use a binomial expansion discarding the squared and higher terms. Both of these approaches, while correct in their context, do not give a universal approach to solving the equations and in particular to making the solutions accessible for efficient interpretation of the experimental data. The work of Saito(9)has parallelled that of Charlesby(i) and Inokuti and DoIe(7,6). Indeed, some of Saito's equations have been the starting point for the deductions of Inokuti and Dole(/). Saito(9) has summarized his contribution to the field, but since he also has been concerned mainly with gelation, he has not provided an efficient way of interpreting experimental molecular weights to give values of G(S) and G(X). The aim of the present work is to present the essential points of a recent advance in the interpretation of experimental molecular weight/dose relationships, to show that computer methods may now be utilized to give quick reliable numerically accurate determinations of G(S) and G(X), and to apply these new methods to some experimental results which have been published elsewhere to show that the new methods give agreement with results calculated by other means. THEORY The Equations The full set of general equations(i) for the dose dependence of each of the molecular weights, M (number average), M (weight average), and M (z average) are given below: n
w
z
M (D)
....(1)
n
(1 + ( f / i 2 M (0) n
l)uxD) *,(iirD, 1
l
^(ufD,*) = ufD - 1 + [l-KufD/cr)]""*
....(4a)
a+1
0 (ufD,(r) = 1 + [l+(ufD/a)]-< ) - (2/nfD){l - [l+(ufD/a)3"^}....(4b) 2
D denotes the radiation dose in grays, u is the number—average degree of polymerization (u = M (0)/M where M is the molecular weight of the monomer R
unit), t and x e the respective probabilities per gray of scission and crosslinking of Downloaded by UNIV OF ARIZONA on November 23, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0620.ch009
ar
a single monomer unit, and a = l/[(M (0)/M (0)) - 1] and is a measure of the width of the initial molecular weight distribution. w
n
Strategiesforsolution (a) Initial most probable distribution (o = 1)
For the case of the initial most probable distribution (a = 1), the general equations become simplified and are given below: M (0)/M (D) = 1 + ( f / i - l)uiD n
....(la)
n
M (0)/M (D) = 1 + (t/x - 4)uxD w
....(5)
w
M (0)/M (D) = (1 + ufD - 4uxD)2/(l + ufD) ....(6) These equations are then amenable to solution as simultaneous equations in pairs by some of the more popular symbolic mathematics computer programs e.g. M A T H E M A T I C A L ) and MACSYMA(li). The solutions are given below: z
z
Considering M and M n
f
=
x
=
4C - A 3¥D C - A 3uD
3 7
-< *> ...(7b)
Considering M and M w
A r=
z
2
— B BuD A — AB 4BuD
....(8a)
s
*~
8b
-•( )
Considering M and M n z
r=
-(48BC -(48BC
X=
1
+ B*) /*
+ 24C + B -
J^JJJ
....(9a)
1
4- B 2 ) / 2 ^
18
+
6C + B
In Irradiation of Polymers; Clough, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
....(9b)
IRRADIATION OF POLYMERS
134 Where
M (0) A=— M (D)
M (0) M (0) ,B= — , and C = — ....(resp. 10a,10b,10c ) M (D) M (D)
w
z
=
n
=
w
z
n
There are two sets of solutions for M and ML The negative roots of the function n
2
1
2
(48BC 4- B ) / are accepted since they provide physically significant solutions and the positive ones do not. f and \ related to G(S) and G(X) by the equations: a r e
G(S) = 9.65x109 uf/M (0) ; G(X) = 9.65x109 ux/M (0) Downloaded by UNIV OF ARIZONA on November 23, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0620.ch009
n
....(resp. lla,llb)
n
(6) The general case (cr # 1)
In this case, equations (1),(2) and (3) were taken in pairs and solved simultaneously: Considering M and M n w
From equation (2), by simple rearrangement, 2
(ufD) M (D) - 4uxD0 M (D) - 20 M (O) = 0 w
t
w
t
....(12)
n
From equation (1) by rearrangement and multiplication of both sides by M (D), i
w
40 M (D) + 4 1 1 ^ 0 ^ ( 0 ) - 4 0 M ( D ) M (0)/M (D) = 4uxD0 M (D) ....(13) t
w
t
w
n
n
t
w
Equation (13) may then be used to eliminate the term in x in equation (12) to give, 2
(ufD) M (D) - 2 0 ^ ( 0 ) - 4^M (D) - 4ufD0 M (D) w
w
t
w
+ 40 M (D)M (O)/M (D) = 0 1
Considering M
w
w
n
n
....(14)
and M
z
Squaring both sides of equation (2) and dividing by equation (3) and rearranging eliminates x & d gives, n
2
30 (ufD)4 [M (0)/M (D)] [M (D)/M (0)] - 40,3 = 0 2
n
z
w
n
....(15)
Considering M and M n z
Equation (3) may be written as 3 M (0) ^ n
M (D) = z
0 /0 2
t
....(16)
where X = 1 - 4 * ^ 7 * 0 From rearrangement of equation (1), 4i uf»D
....(16a)
4
4
(ufD)*
ufD
4 M (0) n
(ufD)'
M (D)
In Irradiation of Polymers; Clough, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
( 1 7 )
9. HILL ET AL.
Scission & Cross-Linking Yields of Irradiated Polymers 135
Substituting equation (17) into equation (16a) gives 4*
40!
X = 1 (ufD)2 and equation (16) becomes
+ u
M (0) (ufD) M (D) £
n
....(18)
2
fD
n
X2M (D) - 3 ( ^ ) 5 ^ ( 0 ) = 0 ....(16b) where X is defined by equation (18) and
