Polymer Degradation and Stability 29 (1990) 155-163

Kinetics of Polymer Degradation Involving the Splitting off of Small Molecules: Part l Basic Concepts

Peter ~imon

Department of Physical Chemistry, Faculty of Chemical Technology, Slovak Technical University, Radlinsk6ho 9,

812 37 Bratislava, Czechoslovakia

(Received 10 July 1989; accepted 27 July 1989)

A BS TRA C T

A method of derivation of the kinetic equations Jor polymer degradation which involves the splitting off of low-molecular weight compounds based on the stoehastic approach is suggested. The idea is that the degradation occurs via the zip mechanism and that the initiating step, i.e. the .formation of an olefinic site in the polymer chain, occurs at random. Assuming that the polyenes formed in the course of the zip reaction can be considered as partitions dividing the poO,mer chain into non-dehydrochlorinated sequences, it has been shown that the lengths of sequences are distributed according to the most probable distribution. The method is tested on the mechanism o['P VC degradation via the dehydroehlorination of a priori given sequences in o.v.vgen-free conditions.

INTRODUCTION

Degradation of some polymers takes place via the elimination of low- molecular compounds where the length of the macromolecules does not change. It is mainly vinyl polymers which exhibit this feature, polyvinyl chloride (PVC) being a typical representative. We will therefore discuss PVC dehydrochlorination although the theory will also finally be applied to the degradation of other polymers.

PVC decomposes at elevated temperatures, dehydrochlorination being almost the only reaction up to 250°C. Almost all authors accept the view that dehydrochlorination occurs via the so-called zip mechanism which involves

155 Polymer Degradation andStabili O' 0141-3910/90/$03'50 ~* 1990 Elsevier Science Publishers Ltd, England. Printed in Great Britain

156 Peter Simon

three steps. ~ The first is the formation of an olefinic site, which is described as initiation. Owing to allylic activation, the double bond present in the polymer chain labilizes monomeric units in its vicinity. The result is that rapid growth of the polyene sequence takes place; this is the zip propagation. The final reaction step is termination of the polyene growth.

Since PVC is one of the most commonly used polymers, papers dedicated to experimental study of its dehydrochlorination are countless. When studying the dehydrochlorination up to high conversions, two types of kinetic curves are generally encountered. The first resembles a first order reaction curve, but the 'rate constant' is not constant over the whole extent of dehydrochlorination. Such kinetic curves are obtained mainly during dehydrochlorination of PVC in an inert atmosphere. The other type, namely, autocatalytic kinetic curves, are observed during PVC dehydro- chlorination in an atmosphere of oxygen, air, HC1, HBr or halogens. In contrast to the abundance of experimental studies, there have been surprisingly few attempts to develop a theory covering experimental data and describing the kinetics of the decomposition over wide ranges of temperature and reaction times. The first extensive analysis of the problem was made by Kelen and his coworkers. 2 - 4 Danforth et al. suggested a theory based on the zip mechanism with an estimation of the termination reaction s and tested it for PVC dehydrochlorination in inert and HC1 atmospheres. 6 Finally, a statistical model of PVC dehydrochlorination in an inert atmosphere has been proposed. ~

In the present series of papers we shall propose a method for deriving the kinetic equations for polymer degradation which involves the splitting offof low molecular compounds, based on the stochastic approach. In order to obtain kinetic parameters for the degradation, the theory will be applied to experimental data. This paper deals with the length distribution of non- dehydrochlorinated sequences in the chain and the basic concepts of the theory are presented by analyzing the PVC degradation which involves dehydrochlorination of a pr ior i given sequences. As this case has already been analyzed in detail, 2 it serves us also as a test of our method.

T H E O R E T I C A L

Length distribution of non-dehydrochlorinated sequences

One must deal with randon quantities when building up a kinetic model based on the stochastic approach. Let us analyze which they are in the case of PVC dehydrochlorination. We suppose that the initiation step, i.e. the formation of olefinic sites, occurs at random throughout the chain. This

Kinetics o f polymer degradation o[ small molecules: Part 1 157

assumption does not conform with the widespread opinion that dehydrochlorination starts at defective structures. However, the content of defects in the chain is relatively low, 8 so they could cause dehydro- chlorination up to a few percent only; after the defects have decayed, dehydrochlorination should stop. Of course, this prediction is diametrically opposed to experimental facts. In addition, in recent years it has been widely accepted that initiation of thermal dehydrochlorination can also occur at normal CHz--CHC1 repeat links. 9- 13 If one deals with dehydrochlori- nation up to high conversions, HC1 elimination due to defective structures should cause only a small systematic error.

After the olefinic site is formed, zip propagation takes place. The polyenes that arise in the course of the zip reaction, are described as zips. The zip reaction can pass along only a relatively short section of the polymer chain: the most frequent zip length is between 5-10 double bonds. This indicates that a regularity exists in the zip length, and that it can hardly be treated as a random quantity. But, the length of non-dehydrochlorinated sequences should be a random quantity since the initiating step occurs at random. Hence, the theory ofPVC dehydrochlorination should be constructed on the basis of the manipulation of non-dehydrochlorinated sequences and, consequently, the distribution of their lengths is required.

Further, it is assumed that the PVC sample is formed of one, practically infinite polymer chain containing N monomeric links. Zips can be understood as partitions dividing the polymer chain into non- dehydrochlorinated sequences. The probability of finding a partition between two non-dehydrochlorinated links is given as:

p = s / M I1)

where s is the total number of partitions (i.e. the number of zips) and M is the number of non-dehydrochlorinated links. It is obvious that M and N are linked by the relationship:

M = N(1 --x) (2)

where x is the extent ofdehydrochlorination defined as the ratio of the moles of HCI evolved from the sample to the moles of monomeric links in the sample. If we denote:

z = s / N (3)

the probability p can also be expressed as:

p = - / (1 - x) (4)

The existence of an isolated non-dehydrochlorinated monomeric link requires the occurrence of partitions at both sides of the link; therefore, the

158 Peter ~imon

probabili ty of the event is p2. As the number of non-dehydrochlor ina ted links is M, the number of non-dehydrochlor ina ted sequences of length 1 links, sl, is:

s 1 = M p 2

If a sequence with a length of two non-dehydrochlor ina ted links occurs, the pair of links is flanked by partitions, but there is no part i t ion between them. Thus, the probabil i ty of the event is p2(1 - p ) . As the number of all pairs is M - 1, the number of non-dehydrochlor ina ted sequences with a length of two links is:

s z = (M - 1)p2(1 --p)

Expansion of this procedure leads to the result that the number of non- dehydrochlor ina ted sequences with a length of l monomer ic links is:

st = ( g - l + 1)p2(1 _ p ) t - 1 (5)

As the value of N is practically infinite (e.g. 1 g of PVC contains about 1022 links), the value of M mus t also be very great. Consequently, after a very short t ime f rom the beginning of dehydrochlor inat ion, M is much greater than the length of any non-dehydrochlor ina ted sequence and eqn (5) can be reduced to the form:

s l = Mp2(1 -- p)t - 1 (6)

Parti t ions (represented by zips) and non-dehydrochlor ina ted sequences alternate in the chain. Hence, it is a mat ter of course that the total number of non-dehydrochlor ina ted sequences is equal to the number of zips:

~ Sl = M p = s (7)

1=1

The last par t of this equat ion is consistent with eqn (1). The frequency of occurrence of a non-dehydrochlor ina ted sequence with

a length of / monomer ic links can be defined as:

rl = Sl/S = p(1 - p)t- a (8)

This is the so-called mos t probable distr ibution often used in polymer chemistry.

Dehydrochlorination of a priori given sequences

Kelen et al. 2 have treated the kinetics of polymer degradat ion by el imination in oxygen-free conditions. Their theory is based on the assumpt ion that in

Kinetics of polymer degradation of small molecules: Part 1 159

the process of polymerization, monomeric units are linked in two different ways, namely, those on which the adjacent double bond can exert an activating effect and those which cannot be activated. The activation propagates only in the direction of allylic chlorine. The zip propagation is so rapid that the zip growth can be considered immediate which means that the zip propagation is completed before a new activation. The initiating step is the random formation of an olefinic site in the polymer chain and the zip terminates when it reaches the end of an a priori given sequence. The length distribution of a priori given sequences is expressed by eqn (8).

In our treatment of the problem it is supposed that the dehydro- chlorination occurs through a set of successive decomposition cycles. A cycle starts with activation of a monomeric link. The activation consists of the attainment of energy equal to or greater than the activation energy' for random elimination of HCI from the link. We suppose that the probability of activation is the same for all the links (inclusive of the dehydrochlorinated ones). After activation ofa dehydrochlorinated link no kinetic process takes place. Activation ofa non-dehydrochlorinated link leads to the formation of an olefinic site in the polymer chain, i.e., this activation is identical with the initiating step of the zip reaction. In this case the decomposition cycle can go on by propagation and termination of the zip. The ends of a priori given sequences represent barriers for zip propagation; accordingly, they can be treated as partitions present in the polymer chain before degradation where Po is the probability of their occurrence. Like Kelen et al., 2 we also assume that the length distribution of a priori given sequences is expressed by eqn (8) and that the zip propagation is one-sided and immediate.

In order to find the number of HC1 molecules split off in one decomposition cycle, one must distinguish between the HC1 molecules eliminated in the initiation step and those split off in the course of zip propagation. The probability of activation of a non-dehydrochlorinated link is 1 - x; this is, in fact, the probability of occurrence of the initiating step of the zip reaction. As initiation gives rise to one eliminated HC1 molecule, the average number of HC1 molecules eliminated in the initiating step during one decomposition cycle is 1 - x . After the initiating step zip propagation would follow. However, the activation of an arbitrary non-dehydro- chlorinated link does not lead to propagation. The problem is represented in Scheme 1 where a part of the polymer chain is shown. The arrow indicates the direction of zip propagation, the digits represent non-dehydro- chlorinated links, X represents a dehydrochlorinated link and Y represents the end of an a priori given sequence:

. . . 0 2 Y I O O 2 X X X X Y I O 0 0 0 0 0 2 Y100... •

Scheme 1

160 Peter ~imon

If a link 2 is activated, no zip propagation occurs due to the presence of barriers in the chain represented by the zips or by the ends of a pr ior i given sequences. The number of links on the right side of non-dehydrochlorinated sequences is s and, hence, the probability that zip propagation occurs in a decomposition cycle is [ N ( 1 - x ) - s ] / N . The olefinic site formed in the initiation step also represents a partition in the polymer chain where the lengths of non-dehydrochlorinated sequences in its vicinity are distributed according to eqn (8). The zip propagation passes along all the non- dehydrochlorinated sequences at the right side of the site; thus, the average number of HC1 molecules eliminated in the propagation step is ~T= llrv The average number of HC1 molecules split off in one decomposition cycle, h, is then expressed as:

h = 1 - x + N(1 -NX) - s ~ , lr l (9)

/=1

where the term 1 - x gives the number of HC1 molecules split off in the initiating step and the other term the number of those eliminated in the zip propagation. After performing the sum in eqn (9) and some manipulations, one can obtain:

h = (1 -- x ) /p (10)

The number of HC1 molecules eliminated in one decomposition cycle is, in relation to the conversion of dehydrochlorination, h = N A x , where Ax is the increase in conversion in one decomposition cycle. The number is of the magnitude of the average zip length, which is usually 5-10 double bonds. As the number of links in the polymer chain is infinite, one decomposition cycle causes only a very small, practically infinitesimal, change in the state of the polymer chain. Accordingly, the increase in conversion in one decompo- sition cycle can be replaced by the derivative:

h = N A x = N d x / d n (11)

where n represents the number of cycles. Equation (10) can then be rewritten in the form:

N d x / d n = (1 - x) /p (12)

Experimentally, the conversion is observed as a function of time. In order to compare theory with experiment, time must be introduced into eqn (12). The transformation is easily feasible if one realizes that, at a certain moment, the number of links with energy equal to or greater than the activation energy of random HC1 elimination is given by an energy distribution law which relates the fraction of links having sufficient energy to temperature. Hence, the

Kinetics qf polymer degradation of small molecules: Part 1 161

number of random activations per unit of time is a function of temperature only and, for a constant temperature, it should be:

d n / N = B d t (13)

where B is a constant of proportionality and t is time. Combination of eqns (12) and (13) gives:

d x / d t = B(1 - x ) /p (14)

Equation (14) involves the probabilityp, the time dependence of which is not yet known. The relationship between p and s is expressed by eqn (1); thus, it is sufficient to find time dependence of s. Of course, s here stands for the total number of partitions in the polymer chain, i.e. it includes both the ends of a p r i o r i given sequences and the zips formed by dehydrochlorination. If a link 0 or 2 is activated, the number of partitions remains unchanged (see Scheme 1). The number of partitions changes only ifa link 1 from the left side of the a p r i o r i given sequences is activated; in this case two ends of a pr io r i

given sequences (or the end of the a p r i o r i given sequence and the zip) interconnect giving one partition and the number of partitions decreases by one. The change in the number of partitions in the course of one decomposition cycle is:

As = -- s i N (15)

because the number of links on the left side of non-dehydrochlorinated sequences is s. Again, the difference in the zip number in one decomposition cycle can be replaced by the derivative and allowing for the transformation given by eqn (13) one can get:

d z / d t = - B z (16)

or, taking into account eqns (1) and (2) eqn (17) can be obtained

d z / d t = Bp(1 - x) (17)

Differentiating eqn (4) we have

dp l(d d t - 1 - x \ d t + P d-[ <18)

Combinat ion of eqns (14), (17) and (18) gives:

d p / d t = B(1 --p) (19)

Equations (14) and (19) represent the system of kinetic equations describing the process. For the initial condition p = Po at time t = 0 the integrated form of eqn (19) is:

p = 1 - ( 1 - p o ) e -Bt (20)

162 Peter ~irnon

Integration of eqn (14) after introducing eqn (20) for the initial condition x = 0 for t = 0 leads to the result:

1 - e - B t

x = 1 - ( 1 - p o ) e -B' (21)

This kinetic equation is identical with that previously presented. 2

DISCUSSION

The cornerstones of the method of deriving the kinetic equations for polymer degradation via splitting off of small molecules are represented by eqns (6) and (8). The length distribution of non-dehydrochlorinated sequences given by eqn (8) has been obtained after the simplification leading from eqn (5) to eqn (6). Consistence of eqns (1) and (7) shows that the simplification is admissible. Other crucial points are the transfer from differences to derivatives expressed by eqns (11) and (16) and the transformation from the number of decomposition cycles to time given by eqn (13). However, the procedure carried out here has provided the same kinetic equation for PVC degradation by dehydrochlorination of a priori given sequences (eqn (21)) as was obtained by Kelen et al. 2 Not only the kinetic equations coincide but the expressions for the probabilityp (eqn (20)) as well. 3 This indicates that the assumptions accepted here are reasonable.

To describe the kinetics of PVC dehydrochlorination, it has been necessary to derive two differential equations, one for the conversion and the other for the dependence of s (or p) on time. Generally, the probability p will certainly appear in all kinetic equations of this kind and, consequently, it will always be necessary to solve the system of at least two differential equations. The method of deriving the kinetic equations described here is very simple, much simpler than the method offered previously, z -4 Due to its simplicity it is suitable for the derivation of kinetic equations in cases of more complex degradation mechanisms.

REFERENCES

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629. 4. Kelen, T., Galambos, G., TiJd6s, F. & Bfilint, G., Europ. Polym. J., 6 (1969) 127. 5. Danforth, J. D. & Takeuchi, T., J. Polym. Sci. Polym. Chem. Ed., 11 (1973) 2083,

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Kinetics of polymer degradation of small molecules: Part 1 163

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