Polymer degradation by elimination—II. The probability—theoretical treatment of the elimination assuming a priori given polymer sequences

European Polymer Journal, 1969, Vol. 5, pp. 617-627. Pergamon Press. Priated in England.

POLYMER DEGRADATION BY ELIMINATION--II THE P R O B A B I L I T Y - T H E O R E T I C A L T R E A T M E N T OF THE E L I M I N A T I O N A S S U M I N G A P R I O R I G I V E N P O L Y M E R SEQUENCES T. KELEN,G. GALAMBOS,F. T/3D6S and G. B.~LINT Central Research Institute for Chemistry of the Hungarian Academy of Sciences, Pusztaszeri fit 57/69, Budapest, Hungary.

(Received 18 October 1968) A~tract--The treatment of the model concerning the thermal degradation of PVC based upon the probability theory has been performed with the assumption of particular polymer sequences. The treatment has been extended to both the case of no limitations being made to the values of the rate constants and the case simplified by the assumption/3 >> a. In the case of the simplified model the probability-theoretical treatment has led to the same results as the kinetic treatment, Besides approaching the problem with the aid of other methods, the application of the probability theory permits calculation of the extent of elimination also at arbitrary values of the rate constants.

1. I N T R O D U C T I O N IN PART [ of this paper c1) our experimental results concerning the thermal degradation of P V C have been reported. The model constructed for the interpretation of the process on the basis of the experimental results has been reported. The main features of the model are: (a) The initiating step of the degradation process is a random reaction, which is unimolecular with respect to the polymer molecule. Its rate constant is designated as a. (b) The chain propagation is carried out in a series of activated unimolecular reaction steps: the rate constant of the reaction is designated as 13. The activation propagates only in one direction determined by the head-to-tail linkage. (c) The chain termination takes place: (i) either by the reaction chain reaching the end of the a priori given polymer sequences (it is determined by the length distribution of the polymer sequences, which can be described with the aid of a probability factor a), (ii) or by a reaction step which interrupts the propagation of activation i.e. a reaction step takes place to give a product which has no activating effect on the neighbouring monomeric unit (it is determined by a probability factor 3, which characterizes the occurrence of this kind of decomposition step). In Part I of the paper the treatment of the model, based on reaction kinetics has been reported. It has been accomplished by assuming the apriori given polymer sequences, with the simplifying limitation fl ~ a. In this part of the paper, the treatment of the model by probability theory will be reported. It has been accomplished on the basis of assuming the a priori given 617

618

T. KELEN. G. GALAMBOS,F. TC,,.tD(3Sand G. B,6.LINT

sequences of the polymer, both without limitations to the values of rate constants and by applying the simplifying conditions. The relationships of probability calculus, forming the basis of this treatment (in the Appendix) and the discussion based on the probability calculus with the assumption of the reaction step interrupting the activation propagation also will be reported in Part III of the series. Thorough information on the topic of the methods and fundamental relations of the probability theory can be found in the special literature (see for example Ref. 2). 2. TREATMENT OF THE MODEL WITHOUT LIMITATIONS TO THE VALUES OF THE RATE CONSTANTS (a) According to the basic assumptions of our model, the changes occurring in a given sequence of the polymer (for example, the decomposition) are independent of the processes taking place in other sequences. It is reasonable, therefore, to choose the polymer sequences instead of polymer molecules or monomeric units as basic units for the kinetic treatment of the process, and to investigate the states of sequences, being formed during their various transformations. The various transformation states are further called configurations. The designations used for describing the configurations are: The monomeric units forming the sequence are numbered in the direction of activation. The monomeric units are designated by their own positional number, and by some marks characterizing the 'destiny' of the monomer unit; for example the possible states of the monomer unit in the k-tb position are designated as follows: = undecomposed monomeric unit, being under no activating effect, ~* = the same, being under activating effect, k = decomposed monomeric unit, decomposition of which has proceeded without activation, k* = monomeric unit decomposed in activated state. Applying these designations, the possible configurations of polymer sequences having various length and arranged into a scheme characterizing the course of the process, are as follows: "=~ T a...~

(l)

n=2

..~..T 2 a t z T~'~a i g'-P-B i 2"

n=3 -Tf_ 3

j / ---~

a

\ a~k

~'-~.

a.~.,,J 23 a-~---123 e a'~'(I ~'3), -(I 2"3)~ ~12~" ~,(t23%

B'~.T 2 3. • .9..(I 2"3)2

~'1

2"3"

a (I 2 3"}z ~ (J 2"3)2

/~, I 2"3"

(2)

(3)

Polymer Degradation by Elirninatiort--lI

619

Similar schemes can be written for values of n > 3 ; the total number of configurations in the case of an n long sequence being' n--I

N("~= 1 +n!

~

--'i!l

(4)

i=0

Since within 0.1 per cent accuracy from n > 5

(5)

~ e (4) can be written as follows (n > 5):

(6)

N ~"~ ~ n!e.

The probability of any configuration can be directly obtained by using the method described in the Appendix (see Part III) if the order of succession determined unambiguously by the scheme characterizing the process is taken into account. The product of the probability calculated in this way and of the number of sequences having the given length gives the concentration of configuration involved. The results of these operations are summarized in the following table, for the cases n = 1, 2, 3. The designations used in the table are q, = e - ~'

(7)

qa = e-B'

(8)

and N1, N2, N3 . . . . , iV,. . . . designates the number of 1, 2, 3 , . . . n . . . . long sequences in the polymer (for further information concerning the latter and the length distribution of polymer sequences the reader is referred to Part I). TABLE I

1{ 24

Designation

Concentration

of the configuration

of the configuration

i

Nt q~

1

N t (1 -- q~)

I ~

N2q~

I 2

N2 q~ (1 -- q:t)

1 ~*

N2 ~---S--~a(q~ -- qa)

12

N2{(I -- q,) -- ½(1 --q])}

I 2*

N2

O.

2

(1 -- q])

fl- --- 2a (1 - q#)}

620 Table I

T. KELEN, G. G A L A M B O S , F. TIODOS and G. B,/~LINT

(continued) Designation of the configuration

Concentration of the configuration

i23 I23 c1

3

I23"

N3 ~--'-Z-~_a(q=-- q~q~)

2*J

N~ ~_--Z-~_~(q~-- q~qo)

I

I23

¢~

3

N3{q=(1

--

q,)-

~. q~,(1 - -

2

%)

)

(I 2"3h 12~*

N3 {(fl

--

a ~ -- 3 0 2a)(t3

(q: -- qB) - ~ 17.

i23" (1 2*3)2

(qt~-- q,qa)}

N~

(/3_

2,z) (/3 - - 3,~) q ~ - - qB) - - / 3 - - 2 a aft

3

12*3*

N3((~-

123

N3{½(I - - q x ) - - ½(1 --q~) + ](1 --q~}

(12"3h

J N3 (fl _ 2a) (fl -- 3a) (1 --

2a)(/3- 3a)(q=--q~)--~----~a

/3 2(/3 2 a - ---~ )

qo) +

(qtJ-q:'qt~)} (1 -- q~) --

3(13 - 3a) (123")~

N3 3(,S -- 2,~) (,~ -- 3,~)(l -- q~) -- ~

(1 2 3*)2

Ns ½ (1 -- q~)

6(fl -- 2ct) (I -- q:,) +

" (,~ + ~ (/~ -- 2a) (1 -(I 2"3)z

aft N~ 3 ( / 3 - 2 ~ ) 6 8 - 3 ~ ) ( 1 - q ~ ) a/3

+ ( £ + / 3 ) (/3 - 1 2*3*

N~

0 -- qB) +

2a)

q~q~)

3 _ (t --

a ~ (

-q~)+

q:,q~)}

3(~ - - 2,~) (/3 - - 3 , 0 (1 - - q , ) - -

" (,~ +/~) (/3 -- 2,~) (I

1

--q:'qe)}

(1 - - qB) +

Polymer Degradation by Elimination--II

621

The concentration of configurations can be similarly obtained in the cases where n>3. (b) On the basis of the configuration concentrations, the concentration of potyenes (formed during the process and consisting of various number of double bonds) can be calculated. It is obvious that the concentration of polyenes consisting of conjugated double bonds of number k can be calculated by summing the concentrations of all the configurations which contain conjugated double bonds of number k in isolated form. In this relation, it should be mentioned that a given configuration can contain polyenes of various length on the one hand, and several polyenes of the same length, on the other. Let nRk be the concentration of the k long polyenes formed from n long sequences. Then, according to the schemes (I)-(3) these concentrations can be expressed for the cases n = I, 2, 3 respectively as follows: IR L =

NIP[l]

2R~ = N2{P[1 2] '-- P[1 2"]} 2R2 = N2{[1 2] + P[1 2"1}

(9)

aR~ = Ns{P[-f 2 3] -i- P[I 2 3"1 + P[1 2"31 q- 2P[(I 5_'3)~] + 2P[(1 -~'3)2]} aR2 = N3{P[i 2 31 + P[1 2 3"1 + P[i 2 3"1 + P[1 2"3"]} aR 3

=

Na{P[I 2 31 + P[(1 2"3)d + P[(1 2 3")~1 + q- P[(1 2 3*)2] + P[1 2"3)21 + P[I 2*3*]}

where, for example, P[I 2] is the probability of the T 2 configuration. The concentration of the k long polyenes can be obtained with the aid of these expressions in the form: Rk =

Z "Rk.

"I0)

It is advisable to calculate the concentrations of polyenes per monomeric unit : R, y~ = - - .

(11)

mo

The conversion of the process, i.e. the quantity of the decomposed monomeric units per initial concentration of monomeric units, can be expressed as ¢ta

----- Z kyk.

(12)

k=l

The determination of configurations and the calculation of their probability and concentration can be performed, as can be seen for the above presented values of n. The actual performance of these calculations is, however, almost impossible because of the enormous number of configurations. As can be concluded from (6) the value of this number--considering the greatest sequence length (n = 16-17) to be expected on the basis of our experiments and of the literature--is of order 101~-1015 so that the

622

T, KELEN, G. GALAMBOS, F. TODOS and G. BALINT

application of the configuration probabilities in resolving this problem is unreal even with use of modern computers. (c) The treatment based on probability theory provides a possibility to determine the probability of the decomposition or of the remaining 'intact' of any arbitrarily chosen k-th monomeric unit of the polymer sequence. The importance of the above method lies in the fact that it can be done without using the configuration probabilities the application of which is relatively difficult. As will be shown, these probabilities are identical for each k-th monomeric unit irrespective of how long a sequence the monomeric unit belongs to. Calculating the probability of remaining unaffected or of being decomposed for the k-th monomeric unit, the previous 'destiny' of each of the monomeric units with positional numbers 1 to (k -- 1) irt the polymer sequence will be considered but the monomeric units having positional number greater than k will not be taken into account. Introducing the designation Qk for the probability of the k-th monomeric unit remaining unaffected, i.e. P[]~] = Q~ (13) then its value can be expressed as follows [see the relationship (A.10) of the Appendix in Part III]: Q~ = P [ ( k - - l)~] q- P [ ( k - - 1),~]. (14) Taking into account that the (k -- l)-th monomeric unit being undecomposed, the k-th monomeric unit is under no activating effect, the value of the first term in the righthand side of the expression is: P[(k --

1) ,~] = Q~_~e-".

(15)

Applying the corresponding formulae and methods described in the Appendix (see Part III) the value of the probability P [ ( k - - i)~] in (14) can be expressed in the following form: t

P[(k --

1)~] = -- e-B'j

e(B-"dr.

(16)

0

Using (15) and (16) and taking into account that for the first monomeric unit Q1 = e - "

(17)

the following system of integral equations can be written to determine the Q~ probabilities: k =

1

Ql

=

e -z' t

k > 1 Qk

--

Qk-le -~' - - e _a,~'dQ~-t j - - g - - e(a_~,d~..

(18)

0

This can be transformed into the following system of differential equations: d k=

1

k > I

dt at

= (fl -

a) e - ~ ' -

d d t Q~ = (fl - - a ) e - ~ ' Q ~ - t

flQt

-flQ~"

(19)

PobTner Degradation by Elimination--II

623

This system can also be written in a matrix form and solved by the methods of matrix analysis :

{I

~,

/3 - o(i

Qk=

e_~,

ct

,=1

_-e::,)]'-'] .

(k - - i)!

k

+

)

(20)

i--i

(k--i)!

/3-- (r + 1)cz J J

I_(i-- 1 - - r ) ! r !

i=l

r=O

Applying the Qk probability the conversion of the process can be calculated as follows: 1

~=1

s.

ti

~ .

(21)

It should be noted that the differential equation system (19) can be transformed with the aid of (21) into a linear differential equation relating direct to the conversion: d__~q- [~ _ (/3 _ a) (1 -- a) e-='l~ = /3 -- (/3 -- ~) e-='. dt

(22)

The direct evaluation of the conversion without calculating the values of Qk according to (20) becomes possible by solving (22):

~=l--

k

.exp

(1 -- e) (1 -Ct

G,

J

]

--/3t .

(23)

In the above formula the meaning of S(t) is: /3-°(1--~)

S(t) =

ct

,=o

i!

1--

exp { -- [i ~ -- (,8 -- ~)lt}

.

(124)

i-~ ~ (-~ Z ~)

The (23) relationship obtained for the conversion of the process, in contrast to the (12) relationship obtained on the basis of configuration probabilities, can be applied with no computational difficulty. The treatment performed without limitations to the values of the rate constants yields a practically applicable formula only for the calculation of the conversion leaving the problem of the time-dependence of the polyene concentration (on account of computational reasons) unsolved for the time being.

3. T H E T R E A T M E N T OF T H E SIMPLIFIED M O D E L (a) Let us assume that the rate of the activated reaction step following a nonactivated (random) reaction step is so great that the reaction chain passes over all the

624

T. KELEN, G. GALAMBOS, F. TI]DOS and G. B./~LINT

monomeric units up to the end of the sequence before another random reaction step occurs. Then the number of activated reaction steps, following a random step, is only a function of the sequence len~h and of the position of the randomly decomposed monomeric unit along the polymer sequence. In this case, the rate of the overall elimination process is determined by the lower rate of the non-activated (random) reaction step. This condition is equivalent to the statement

>> i

¢z

and the essentially new feature of the simplified model formed in this way is that there may be only one polyene at a time in one polymer sequence. (b) The possible configurations arising from the polymer sequences of various length which are arranged in a scheme characterizing the course of the process, are as follows:

(26)

,,--I T aLJ n=2 _G..-i 2 ct. s 2 i 2..a....s 2"

(27)

n=3

__

.~..7 a3 a I 2"3

y

ct. 12 3

T ~ ~ L-.q,--T 2 3.-..%.~f 2 3• ~I

2"3"

TABLE 2 Designation of the configuration

f

Concentration of the configuration

I

N~ q

1

N~ (I -- q)

I 2 I 2

N2 q: N2 q(l -- q)

12

N,{0

12*

N2½(I --q2)

I~3 i~3

Naq2(l--q)

I 23

N~{q(1 -- q) - -

I 23*

N3 ½q(1 -- q2)

123 12*3 ! 23"

N3{½ (1 - - q ) - - ½(1 _ q 2 ) + { ( 1 _ q 3 ) } N3{½(I -- q2) _ } ( 1 -- q3)} N3{½ ( 1 - - q) -- ~(1 - - q 3 ) }

12"3"

N3 ½ (1 -- q')

-- q) -- ½ (1 -- q')}

N3 q 3

½q(l -- q2)}

(28)

Polymer Degradation by Elimination--II

625

(It should be noted that in Part I other designations have been used: instead of~, k and k* the symbols O, 1, Z irrespective of the positional number of the monomeric unit involved.) Similar schemes can be written for the case n > 3; the number of configurations arising from an n long sequence is: N (") = 2"

(29)

i.e. considerably smaller than that resulting from (6). The probabilities and the concentrations of configurations can be obtained by applying the methods described in the Appendix (see Part III); the results obtained are summarized in Table 2 for the cases n = 1, 2, 3. In the table the meaning o f q is: q = e -~'.

(30)

The concentration of configurations can be similarly obtained for the cases n > 3. (c) The concentrations of the polyenes arising from the polymer sequences of a given length are to be obtained according to (26)-(28) as follows: aR1 = NaP[l] = Xl(1 - - q) 2Ra = N2PtT 21 = N2q(1 -- q) ZR2 = N2{P[1 21 + P[1 2"1} = N2(1 -- q) 3R~ = N3P[i 2 3] = N3q2(l - - q)

(31)

3R2 = N3(P[T 2 3] + P[i 2 3*]} = U3q(1 -- q) aR3 = N3{P[I 2 3] + P[1 2*3] + P[1 2 3*] + P[I 2*3*]} = N3(1 -- q). Generalizing these results, the concentrations of polyenes arising from an n long polymer sequences are: "R1 = N,q"- 1(1 -- q)

(32)

"Rk = N.q"-k(1 -- q) "R. =

W.(1 -- q).

The total concentration of the k long polyenes is to be obtained by the following summation :

Rk=2"Rk=~N.q"-k(1--q). rl~k

(33)

rl~k

By application of the length distribution of polymer sequences (see part I) the following result can be obtained: (1 - - q)a R, = N,

S,q"-*(l - - q)

=

Skm 0 1 -- (1 -- a)q

(34)

n=k

or the value per monomeric unit : Y*

Rk mo

(1 -- q)~r

S, 1 -- (1 -- Oq

(35)

626

T. KELEN, G. GALAMBOS, F. TI.J'DfS and G. BALINT

The conversion of the process is: 1 -

=

q

.

k;/~= l--(1--cr)q

(36)

k=l

From (35) and (36) the relationship between the conversion and the relative polyene concentration is as follows: ~,~ = S~cr~:.

(37)

One can recognize that these results are in agreement with those of the reactionkinetic treatment (see Part I) based on the same basic assumptions. (d) The probability considerations concerning the probability that decomposition of the k-th monomeric unit (Q~) does not occur can be applied to the simplified model also. The first term P [ ( k - - 1)/~] on the fight-hand side of (14) is determined by (15). The value of the probability P [ ( k - - 1)/~] is equal to zero, since according to the assumptions of the simplified model the decomposition of the (k -- 1)-th monomeric units is instantly followed by the activated decomposition of the k-th monomeric unit so the joint occurrence of (k -- 1)/~ is an impossible event: e[(k -- 1)~1 = 0.

(38)

Thus the following algebraic equation system can be obtained from (14) instead of the integral equation system (18): k=

1

k > 1

Q1 = q Q~ = Qk-lq.

(39)

The successive application of the recursion formula leads to the following expression for Q~: Qk = q~.

(40)

Substituting (40) into (21) the conversion of the process is as follows:

= 1

12(

n

1--q"~

S, ql_q/-----

1--q 1--(1--cr)q

(41)

It is to be noted that the expression (41) obtained by evaluation of the probability of the non-decomposition of the k-th monomeric unit is equivalent to (36) formed on the basis of the configuration probabilities. (e) It is obvious that only the first approximation of the true polyene concentration is #ven by (35) and (37) since the polyenes formed in the neighbouring sequences can be interconnected, supposing that the decomposing parts are next to other. This, however, can occur only when one of the neighbouring sequences has been completely converted into polyene. It can be pointed out that taking into account the interconnection of the polyenes, the value of the relative polyene concentration is instead of 7~: Ck = =q(1 -- =q)'- 1.eq.se.

(42)

Polymer Degradation by Elimination--II

627

O t h e r w i s e , it is c o n c e i v a b l e t h a t the c o n c e n t r a t i o n of" the k l o n g p o l y e n e s c a l c u l a t e d a c c o r d i n g to (35) or (37) is i n c r e a s e d by the i n t e r c o n n e c t i o n o f the p o l y e n e s s h o r t e r t h a n k a n d d e c r e a s e d by the i n t e r c o n n e c t i o n o f the k l o n g p o l y e n e s with a polyerle o f w h a t e v e r length. It c a n be s h o w n t h a t the effect o f the c o r r e c t i o n e x p r e s s e d by (42) is negligible in t h e initial part o f the process so t h a t in this p a r t , u s i n g the linearized f o r m o f (36) : Ck ~ yk ~ Sk ctt.

(43)

REFERENCES (1) T. Kelen, G. B:ilint, G. Galambos and F. Tfid6s, Europ. Polym. J. 5, 597 (1969). (2) A. R6nyi, Wahrscheinlichkeitsrechnung, mit einem Anhang ~iber Informationstheorie, VEB Deutscher Verlag der Wissenschaften, Berlin (1962). R~sum6--On a trait~ le module de la d~gradation thermique du poly(chlorure de vinyle), bas6 sur une th6orie de probabilitY, en supposant l'existence de sequences polym~riques particuli~res. Le traitement a 6t6 ~tendu au cas oct aucune limitation n'est impos~e quant aux valeurs des constantes de vitesse et au cas simplifi6 dans lequel on suppose ~ >~, ~. Dans le cas du mod61e simplifi6, le traitement th6orique par probabilit6 a conduit aux mSmes r~sultats que le traitement cin~tique. En m~me temps qu'elle permet d'approcher le probl6me au moyen d'autres m~thodes, l'application de la th6orie de probabilit6 permet de calculer l'avancement de l'~limination pour des valeurs arbitraires des constantes de vitesse. Sommario---ll trattamento del modello relativo alia degradazione termica del PVC basato sulla teoria delle probabilit/l 6 stato eseguito sulla premessa di sequenze reticolari polimeriche. I1 trattamento 6 stato esteso sia al caso in cui non sono posti limite ai valori delle costanti di velocitY, che al caso semplificato da: 13>~, ct Nel caso del modello semplificato il trattamento basato sulla teoria delle probabilit/t ha portato agli stessi risultati conseguiti col trattamento cinetico. Oltre ad impostare il problema con l'ausilio di altri metodi, l'applicazione della teoria delle probabilit5, consente di calcolare la portata dell'eliminazione anche con valori arbitrari delle costanti di velocit/~. Zusammenfassung--Es wurde die Behandlung der thermischen Degradation von PVC, mit der Annahme yon a priori gegebenen Polymersequenzen, auf Grund der Wahrscheinlichkeitsrechnung durchgeftihrt, sowohol ohne Beschrtinkungen der Werte yon Geschwindigkeitskonstanten, wie auch in dem mitder Annahme fl >~, a vereinfachten Falle. Im Falle des vereinfachten Modells fBhrten die Behandlungen, begrtindet an der Wahrscheinlichkeitsrechnung, bzw. der Reaktionskinetik, zu denselben Ergebnissen. Ausser der Anntiherung des Problems mit einer anderer Methode ermSglichte die Anwendung der Wahrscheinlichkeitsrechnung auch die Berechnung der Konversion der Elimination sogar bei beliebigen Werten der Geschwindigkeitskonstanten.

POLYMER 5 / 5 ~