Application of the theory of branching processes (cascade theory) to polymer degradation and crosslinking: Postgel stage

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Radiat. Phys. Chem. Voi.28, No.5/6, pp.479-486, 1986

Copyright(~1986. PergamonJournals Ltd

Int. J. Radiat. Appl. Instrum. Part C Printed in Great Britain. All rights reserved

APPLICATION OF THE THEORY OF BRANCHING PROCESSES (CASCADE THEORY) TO POLYMER DEGRADATION AND CROSSLINKING: POSTGEL STAGE Karel Du~ek* and Milo~ Demjanenko Institute of Macromolecular Chemistry, Czechoslovak Academy of Sciences, 162 06 Prague 6, Czechoslovakia G.V.Akimov State Research Institute of Materials Protection, 250 97 Prague 9, Czechoslovakia

ABSTRACT The theory of branching processes (cascade theory) has been extended to cover degradation and crosslinking of primary chains. It has been applied to the postgel stage and different degree of polymerization distributions of primary chains. Relations have been derived for the sol fraction, fraction of crosslinked units in the gel, concentration of elastically active network chains, cycle rank and average functionality of elastically active crosslinks as a function of the degree of degradation and crosslinking.

KEYWORDS Polymer chains; degradation; crosslinking; sol fraction; elastically active network chains; cascade theory.

INTRODUCTION Degradation and crosslinking are typical structural changes induced in macromolecular systems by radiation and ageing. These structural alterations manifest themselves in the changes of molecular weight distribution and possibly in gelation and formation of crosslinked structures. In turn, the experimental quantitative characterization of the depth of these changes serves for elucidation of the mechanism of crosslinking and degradation and determination of their extents. A theory relating the above mentioned structural changes (manifested e.g. by the changes in molecular weights, gel points, sol fractions and effective degree of crosslinking) to the extent of degradation and crosslinking and parameters of the initial systems is the necessary tool. The theories published so far are based on statistical or kinetic approaches, and degradation and crosslinking are considered to be statistically independent processes. Within this approximation, irrespective of the real situation, degradation can be treated first and the degraded system is subjected to crosslinking in the next step. The problem of combined non-random degradation and crosslinking can be well treated using Smoluchowski equations or percolation in n-dimenslonal space. The approaches used for treatment of degradation and crosslinking were reviewed in monographs by Charlesby (1960) and Dole (1972). The kinetic theory by Salto (1972) and others (Inokuti, 1963; Kimura, 1964) has been used most widely. As stated earlier, (Demjanenko and Du~ek, 1980), this theory is applicable in the region of small extents of degradation 8 and crosslinklng ~, i.e. 8 << I and ~ ~ I. Moreover, the use of continuous distributions instead of discrete distributions in this theory limits the applicability of the theory to the average degree of polymerization pO m I. In the limit of e ~ I, 8 ~ I and po m I, the results of the kinetic and statistical theories are identical, provided degradation and crosslinking are random processes. Recently, the application of the theory of branching processes (cascade theory) was extended by us to cover degradation and joint degradation and crosslinklng (Demjanenko and Du~ek, 1980). The theory of branching processes (TBP) is the most versatile and elaborated statistical method for description of branching processes in macromolecular systems; it is based on the Flory-Stockmayer tree-like model with uncorrelated circuit closing in the gel (Gordon, 1962; Gordon and Ross-Murphy, 1975; Du~ek, 1979, 1986). This method uses cascade substitution and probability generating functions (pgf) as tools for generating branched structures. This approach is not limited by the requirements = ~ I, 8 ( I, and pO m I. The proper construction of rooted trees from primary chains undergoing random degradation and crosslinking up to the gel point has been explained by Demjanenko and Du~ek (1980). In this communication, this approach is extended beyond the gel point. Network parameters such as sol and gel fractions, concentration of elastically active network chains and cycle rank of the network have been calculated. The calculation has been illustrated using a few selected distributions of primary chains.

479

480

THEORY The construction of rooted m o l e c u l a r trees has already been explained (Demjanenko and Du~ek, 1980) Consider first a set of monodisperse chains of degree of polymerization x. The links between the repeat units (or m a i n chain atoms - the selection of the size of the unit in t~e chain depends on the m e c h a n i s m of degradation) are considered to be split w i t h probability B and each repeat unit to be crosslinked w i t h another repeat unit W i t h the p r o b a b i l i t y a. Each chain of the set is considered to be rooted by any of its repeat units with the same probability I/x (Fig. I).

.

.

.

.

.

.

.

.

.

.

.

.

g=5

. . . . . . . . . . . . . . . .

:24

.... '"'%S

Fig.

I.

Scheme of rooting and crosslinking

-,:;'

of p r i m a r y

chains

= 0(I - y + yUpl)(l

- y + yUql)(l

Upl

= 8(I - y + yUp2)(1

- a + au)

Up,i-2

ffi 0(I - y + yUp,i_1)(1 =

0(I

-

Uq 1

= e(1

a

Uq,x_ i = e(1

can be described

by the following

(1)

- a + au)~ OaxibxiCxi

- ~ + au)

- a + au)

+ yUq,x_i)(1

- a + au)

- a + .u)

_1 x u = x

of d e g r a d a t i o n

au)

+

- y + yUq2)(1

U q , x _ i _ l = 0(1 - y

for treatment

the situation

Wxi(O)

Up,i-1

o__,

.....

.....

If the chain is rooted by its i-th segment, weight fraction generating function Wxi(O) ,

g=4

x

Z 0 (I - y + yUpl)(1 i=I

- y + yUql)

(2)

= x-IZ @a .b . i=I xl xl

and y = I - 6 The resulting

weiBht

Wx(e)

fraction

=

function Wx(@)

reads

=~i=i w.O i=~ x-lWxi (o) i i=I

where w x is the weight u

generating

Wx(O)/(1

It was shown 8 that Eqs

Wx(O) =

fraction

of the x-mer~

(3)

so that (4)

- ~ + au) (I) yield

m

{x(1 _y)2 + xx2(1 _~)2(y~)x-1

x(1 - y~)2

+

2 y ( I . y)(1 - ~ ) ( I - (T~)x)} (s) I

-

yLo

where e = @(I - ~ + au) For a system of polydisperse w(o)

= ~ w ° W (o) x=i x x

chains (6)

481

where w ° is the weight fraction of the x-mer in the primary distribution before degradation and .x . crossllnklng. Likewise, for a continuous distribution it holds W(O) = ~ w°(X)Wx(O)dx

(7)

The weight-average degree of polymerization,

Pw' is obtained from W(@) by differentiation

Pw = (3W(@)/~O)0=I P

w

(8)

can also be written as e w

= p(B) I + e w I -a(p(B) w

(9) I)

w h e r e p(B) i s t h e w e i g h t - a v e r a g e degree of polymerization ,w ° crossllnklng. At the gel point, Pw diverges, i.e. I -

s(P(B)-

w

1)

of chains

after

degradation

O

=

and b e f o r e

(10)

Post-Gel Stage For the treatment of degradation and crosslinking in the post-gel stage we will proceed similarly as in the case of exclusive crosslinking (Dobson and Gordon, 1965; Du~ek, 1973). However, the primary distribution used in the treatment will not be the initial distribution before degradation and crosslinking (identified later by superscript(~but the distribution after degradation identified by superscript (6) (Thus, W(O) of Eq.(6) goes over to W~'(O) putting ~ = O. The form of WL~)(@) for a particular W °)(O) will be given in the next Section. 8) For calculating the network parameters not only W (B) (0) but also the number fractions (nx) N ( (@) are necessary. N(B)(@) can be derived from W(B)(O) as follows: It holds generally that 0 W(O) N(O) E ~ n G x = P ~ dO (11)) x n 6 From Eqs.

(5, 6) and (11) we have N(B)(~) =P(B)r~W°Wn ~ x X x (O)dO-~= p(B)nx=1 ~ Wx 0~ {X;1~B(yO)i-118x-6ii=1 x

+ I +¥]+ (¥o)x-1}dO

(12)

After integration and summation one gets N(B)(@) =

p(6) n 2 {B2(1-y@) +B7 ( 2 - @ - y @ ) / P ~

}

(13)

P°O n

(I - ye)

where

y(1 -0)2N(°)(y0) +

pO

p(B)

n

n

I + B(P ° - I)

and N (°) is the number fraction gf. In the calculation of the post-gel parameters, the whole primary degraded chains are routinely placed in the root and higher generations, in contrast to the procedure used for the treatment of degradation, where the chains were rooted by their repeat units. The new situation is depicted on Fig. 2.

a~

bj

ACU

ACU

ACU

ACU

continuation to intinity

Fig. 2.

Scheme of rooting of(degrade~ primary chains for calculation of the extinction probability, sol fraction and concentration of elastically active network chains.

482

The probability generating function (pgf) for the number of bonds issuing from a primary chain in the root is F0(D) =

Znx(1 - a +a@) x = N(B)(U)

(14)

where

The pgf for the number of bonds issuing from a primary chain on generation g > 0 to a chain in generation g + I, F(O) is obtained by differentiation with respect to 0 and renormalization

F(8)

=

~ W x ( 1 - a + ae) x-1 X= I

W(6)(~)

(15)

The extinction probability, v, is the key quantity of the post-gel stage. It is defined as a conditional probability that, given a bond exists, this bonds starts a finite sequence of bonds. Alternatively, I - v is the probability that this sequence is infinite. The probability v is defined by v

=

W(~) (z) F(v) = - z

Z

=

I - ~ +

=

Zw

x

(I - ~ ÷~v) x-~

(16)

where GV

The sol is composed of units in which all issuing bonds have only finite continuation; each unit from which the trees are constructed, i.e., each primary chain satisfying the above condition, contributes to the sol fraction w by its oT~-nweight. Thus, s

ws =

I - wg

ZWx(1-

~v) x

W(B)(z)

(17)

In conjunction with Eq.(7), an equation of universal validity can be obtained ws =

v(1 - ~ + ~v)

(18)

The structural parameters characterizing the rubber elasticity of the network are derived from the pgf Fog(D) for the units (primary chains) in the gel

Fog(Dv*D1- v) =

F0(Dv'@1 v ) -F0(Dv'@1 v =0) _ I - F0(I,0)

F0(ev,e1_ v) =

F0(I - ~ + a(ve v + (I - v)e1_v))

(19)

where

The cycle rank ~- an important quantity in the rubber elasticity theories (Flory, 1976) - c a n be derived using this pgf. The cycle rank is defined as the minimum number of cuts necessary to convert a graph with cycles into a tree. Thus, to join X building units (primary chains) into a tree,,_X-1 bonds are necessary. The actual number of bonds connecting X chains in the gel is equal to X P ~ ) a B , where P ~ ) is the number-average degree of polymerization of primary chains in the gel and ~ is the fractlon of crossllnked units in the gel. g The cycle rank per primary chain is thus

=

XP (B) a /2 - ( X - I ) ng X

(20)

which for large X gives =

(21)

e(B)a / 2 - 1 ng g

The cycle rank per repeat unit ~' is then ~, =

(22)

p(B)ng = ~ g / 2 - I/Png

The product p ( B ) is obtained by differentiation ng g ng

g

[~F°g (Dv' @ l-v)

~F°g (Dv' O l-v)]

(23)

483

p(8)(I -v2)a n 1 - F0(v)

p(B)a ng g

(24)

The degree of polymerization of primary (degraded) chains in the gel, p(8) is given by ng ' Exn x - ~Xnx(1 - a + a v ) x ~

p(8) = ng

Pn[1 -v(1 - e + a v ) ] =

I - Fo(v)

P(B)w n g

I - Fo(V)

(25)

I - Fo(V)

The number of elastically active network chains (EANC), NA, can be calculated independently or from the cycle rank and average functzonalzty of elastzcally actlve crosslznks, f~. An EANC s is a sequence of bonds between two elastically active crosslinks. An elastically active crosslink is such erosslink from which at least three paths issue to infinity. As can be seen from the drawing in Fig. 2, each crosslink is composed on two crosslinked units since the bond connecting

l.O w= O.B

06

0.4

02

]

I

~d4

Fig. 3.

153

~62

~dI

100

¢

The sol fraction as a function of the degree of crosslinking in dependence on the degree of polTmerization distribution of primary chains and degree of degradation, B

Designation of curves: curve:

I

2

3

4

5

6

primary distribution

7

SZ

MP

M,P

SZ

MP

M,P

SZ,M,P,MP

pO

10 3

10 3

10 3

102

10 2

10 2

10 2 _ 10 3

10- 6

10. 6

10- 6

10- 6

10. 6

10. 6

n

B

10-1

SZ Schulz-Zimm (P°/P°=5), MP most probable, M monodisperse chains, w ~ P Poisson the two units is considered very short compared to an EANC. As can he seen from Fig. 25, each primary chain can contribute to the number of EANC's if it contains two or more active erosslinked units (ACU). An ACU is a crosslinked unit from which two or three paths issue to infinity. Thus, a primary degraded chain having i Z 2 ACU's has two end ACU's and i-2 inner ACU's. Each inner ACU is a part of an elastically active crosslink of effective functionality fe equal to 3 or 4 and contributes to the number of EANC's by 2x(I/2) per primary chain. Each end ACU contributes by I/2 to the number of EANC's if the primary chain in the first generation is bound to this end ACU by its inner ACU (Fig. 2b). If the chain in the first generation happens to be bound by its end ACU, the effective functionality of this crosslink is 2 and the contribution to the number of ACU's is zero (cf. also Dobson and Gordon, 1965; Du~ek 1973). The distribution of primary chains with respect to the number of ACU's can be described by the per T(e) defined as _

~

i

T(8) =iE=otie = F 0 ( v + (I -v)8) = N (8)(I - a + a ( v +

(I -v)O))

(26)

484

Thus, N

=

~ ( i - 2 ) t i + (I/2)x2( ~ t.) i~2ti i-2 i-2 i i~2iti

e

(27)

or i~2ti Ne

= ~ (i - 2)t i i=2

I +

(28)

i~2iti

It is easy to find that i~ 2 it i = T'(1) -T'(O)

i=2

t

= ~(I -v)2Pn

= T(1) - T(O) - T'(0) = I - N(B)(I - ~ + av) - P ~ B ) a v ( I - v )

i

The pgf T(O) can also serve for derivation of the expression for the average functionality of the elastically effective crosslink which issues three or four bonds to infinity. Employing Fig. 2b, one can see that f is given by e 6 Z t i ~ ( i - 2 ) t i + Z ( i - 2 ) t i [ 4 Z ( i - 2 ) t i +6Zti] f

=

(29)

e

2 Z tiZ(i-2)t i + Z(i-2)ti[Z(i-2)t i+2Zti]

After rearrangements one gets 4 Z(i+ I) t. l

f e

(30)

z(i + 2) t. i

The sums in Eqs.(29) and (30) extend from 2 toginfinity. The sums in Eq.(30) can be obtained from the values of derivatives of pgf's 8T(@) and O"T(0). The number of EANC's is related to the number-average building unit, i.e. to the degraded pri mary chain. The rubber elasticity theories require, however, the concentration of EANC's per unit weight or volume. Thus, the concentration of EANC's per unit weight of the gel ~eg' is given by N

=

~eg

e

M(B)w n g

(31)

where M° n I + 8(P~- I)

M(8) n

and M ° is the number-average molecular weight of primary chains. n The relation between the cycle rank ~' per repeat unit in the gel (Eq~22)) and the concentration of EANC's per unit weight of the gel, ~eg' is as follows ~,

= ~

M(G) f e -

eg

n

2

~e

(32)

Thus, all calculated parameters depend explicitly on the fraction of crosslinked units ~, on ~ N (~) (0) and extinction probability v and on the distributions of degraded primary c h ains, given oy W(8)(0). The extinction probability is not independent; it is determined also by ~,B and the primary distribution.

APPLICATION TO SELECTED DISTRIBUTIONS In order to calculate the network parameters for a given system, it is necessary to specify the primary distribution. Several types have been treated in the literature. For illustration of the application of the procedure described above, we have selected the case of a monodisperse system, the most probable distribution, Poisson distribution and S c h u l z - Z i ~ distribution. The pgf's for the distribution of primary (undegraded and uncrosslinked chains) N(°)(O), W(°)(@) and W(~)(@) are given below. The pgf N(8)(@) is given by Eq. (13) Monodisperse Chains:

N(°)(O)

=

W (°) (@)

= Ox

@x

W (~)(8) = cf. Eq. (5)

485

Most Probable Distribution: I N(O)(8) = (I -q)O

eO =

I -q8

n

( I - q)20

=

w(O) (8)

pO w

(I -qO) 2

W(6) (O)

I -q

(I _yq)2 ( I - yqe) 2

I+q I -q

y=

I - 6

Poisson Distribution

N(°)(e)

= eexp[y(e-1)]

pO = ~ + i

w(O)(8)

= e(1+YG)exp[~(O_1) ]

pO = w

n

1+y

~2+3~ v+

+ I I

W(6) (0) = 0{(I -y)2(I -y0)(I +~) +2(I -y)y(1 -0) + exp [-v(1 - 3"0)] [y2 (I - e) 2 (I - ye) (I + ~yO) 23,2(I -y)(1 - 0 ) 0 ] } ( ~ + I)-I(I -yO) -3 Schul z-Zimm Distribution a n

pO = p O O + 1 w

n

u

I

w(6)(e) . . . . . . . 2y(1-y)(1-@)

,D2÷ ~ -((1-e) 2 [I - (

o _

I -

ye

÷

o-~lnye

)

°i}

o - P°lnyO n

The examples of calculated dependences

of w

s

and ~

eg

are shown in Figs.3 and 4. It can be seen

~d2

~g

~6e

I

~-4

Fig. 4.

I

~4

10-2

10-I

I .

~

~0 °

Concentration of elastically active network chains in the gel, ~eg plotted against the degree of crosslinking in dependence on the degree of polymerization distribution of primary chains and degree of degradation, 8. Designation of curves as in Fig. 3.

486

that considerable differences existing between dlstributions of different width and type disappear as degradation proceeds. This is due to the fact that the distribution of degraded chains approaches the most probable distribution when B increases. The application of the results of the cascade theory is simple and straightforward and, unlike the Saito-lnokuti kinetic treatment, is not limited by the assumptions a ~ I, 8<> I" A n solution analogous to those given above is expected to be found for any continuous dlstrlbution f(x,S) for which an inverse Laplace transform F(O,B) exists, since F(@,8) = f~(x,B)exlnOdx = WB(@) o

REFERENCES Charlesby, A. (1960). Atomic Radla~ion and Polymers. Pergamon Press, Oxford. Demjanenko, M., and K. Du§ek (1980). Macromolecules, 13, 571. Dobson, G.R. and M. Gordon (1965). J. Chem. PhYs.,' 43, 705. Dole, M. (1972). The Radiation chemistry of MaeromOlecules. Academic Press, New York. Du~ek, K. (1973). J. Polvm~ Sci.. P~olvm'. Svmmosla, 42, 701. Du~ek, K. (1979). Makromol, Chem. SuDDI. 2, 80, 35. Du§ek, K. (1986). Adv.Polvm. Sci., 78, I. Flory, P.J. (1976). Proc. Roy Soc. London, A351, 351. Gordon, M. (1962) Proc, R~7 ,Soc. London, A268, 240. Gordon, M., and S.B. Ross-Murphy (1975) Pure ADD1. Chem,, 43, I. Inokuti, M. (1963) J. Chem. Phys., 38, 1174, 2999. Kimura, T. (1964) J. Phys. Soc. Jpn., 19, 777. Saito, O. (T972) in The Radiation Chemistry o~ Macromolecule@, (M. Dole, Editor). Academic Press, New York.