Polymer Degradation and Stability 35 (1992) 45-48

Kinetics of polymer degradation involving the splitting-off of small molecules: Part 4.

Autocatalytic mechanism with immediate zip growth

Peter ~imon Department of Physical Chemistry, Faculty of Chemical Technology, Slovak Technical University, Radlinsk~ho 9, 812 37

Bratislava, Czechoslovakia

(Received 2 November 1990; accepted 23 November 1990)

The kinetic equations for the autocatalytic thermal dehydrochlorination of PVC have been derived on the assumptions that the formation of olefinic sites in the polymer chain occurs stochastically both for non-catalyzed and catalyzed activations, that the zip growth is immediate, and that zip termination takes place after a certain zip length is reached. The features of the kinetic model are discussed.

INTRODUCTION

In the degradation of polymers via the elimination of low-molecular-weight compounds, two types of kinetic curves are generally encountered. The first resembles first-order reaction curves and occurs mainly in the dehydrochlorination of polyvinyl chloride (PVC) in inert atmospheres. We have analyzed this case in Parts 1-3 of this series. 1-3 The other type, autocatalytic kinetic curves, is observed more often. It occurs not only in PVC de- hydrochlorination in atmospheres of HCI, oxy- gen, air, HBr and halogens, but, also in the degradation of polyvinyl bromide, polyvinyl esters and polyvinylidene chloride. Since PVC is a typical representative of this group of polymers, we will discuss PVC dehydrochlorina- tion although the theory will also finally be applied to the degradation of other polymers.

Under constant external conditions (in particu- lar the composition of the dehydrochlorination atmosphere), the only explanation for the autocatalytic course of the dehydrochlorination is that it is caused by entities formed in the polymer chain. These are polyene sequences. In principle,

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45

two possibilities exist as to which step of the zip reaction can bring about the autocatalysis: (i) C----C bonds participate in the formation of olefinic sites in non-dehydrochlorinated se- quences of the polymer chain, i.e. the initiation of the zip reaction is autocatalytic. The zip growth is considered to be immediate. (ii) The rate of zip propagation is comparable with the rate of initiation, i.e. the zip growth is gradual. Zip termination is absent or delayed. As the dehydrochlorination proceeds, the number of zips producing HC1 increases until they become interconnected. The initiation may or may not be autocatalytic.

This paper is dedicated to the derivation of the autocatalytic curves for polymer degradation occurring via the zip mechanism. The first type of mechanism above is analyzed assuming an autocatalytic initiation step followed by the immediate propagation and the termination of the zip reaction.

THEORETICAL

Model of dehydrochlorination

As previously, 1'2 it is assumed that the PVC sample is formed by one, practically infinite,

46 Peter ;~imon

polymer chain containing N monomeric links. Dehydrochlorination occurs through a set of successive decomposition cycles. A cycle starts with activation of a monomeric link where the activation consists of the attainment of energy equal to or greater than the activation energy for elimination of HCI from a PVC chain. The probability of activation is the same for all the links inclusive of the dehydrochlorinated ones. The activation of a non-dehydrochlorinated link is identical with the initiating step of the zip reaction. In this case the decomposition cycle can continue by one-sided zip propagation and then by zip termination. It is supposed that the propagation rate is much greater than the rate of activation so that the zip growth is considered to be immediate. The difference between the previous models 1,2 and this one is that the activation is assumed to take place both at random and in an autocatalytic way. Both activations are assumed to occur stochastically which means that each link has the same probability of being activated.

When deriving the length distribution of non-dehydrochlorinated sequences, 1 stochasticity of the activation was the only assumption. Accordingly, eqns (6) and (8) from Part 1 hold also for autocataltytic degradation mechanisms. 1 The probability p is expressed by eqn (4) of Part 1 for an ideal PVC chain 1 or by eqn (18) of Part 2 for PVC containing inert impurities. 2

Number of zips

For zip propagation followed by zip termination, the change in the zip number during a decomposition cycle is given by eqn (2) Part 2. z A decomposition cycle brings about a very small, practically infinitesimal, change of the state of the polymer chain? The change in the zip number given by eqn (2) Part 2 can thus be expressed by the derivative:

ds dn ( 1 - x ) [ ( 1 - p ) " - p ] (1)

The symbols in eqn (1) have the same meaning as previously. 1-3

The transformation from the number of decomposition cycles (n) to time has been given by eqn (13) of Part 1.1 In order to involve the autocatalysis by the initiation step in the kinetic equation, the transformation should be modified

tO

dn/N = B(1 + fix) dt (2)

where fl is the number of reaction cycles induced by the autocatalysis per random cycle, or in other words, it is the ratio of the rate constants of autocatalytic and random formation of olefinic sites in the polymer chain. Combination of eqns (1) and (2) yields

dz dt=B(l + f lx) (1-x)[(1-p)m-p] (3)

where z is given by eqn (3) of Part 1.1 If the presence of inert impurities is considered, eqn (3) takes the form

dz dt= B(l + f lx)(xm-x)[(1-p)m-p] (4)

Kinetic equation

The average number of HCI molecules elimin- ated in a decomposition cycle, h, is given by eqn (4) of Part 2. 2 After performing the sums in the equation, one can obtain

h = (1 - x ) [ 1 - (1 __p)m]/p (5)

Following the procedure in Part 1, combination of eqns (2) and (5) gives

dx d t = B(1 + fix)(1 - x)[1 - (1 - p)m]/p (6)

In the case of the presence of inert impurities, the kinetic equation for the dehydrochlorination is

dx d--t = B(1 + fx)(x m - x ) [ 1 - - (1 -p ) ' ] /p (7)

Combination of eqns (21) of Part 2, (4) and (7) (or eqns (3), (6) of this paper and (18) of Part 1) leads to the kinetic equation for the probability p:

d__p_p= B(1 + fix)(1 - p ) (8) dt

Equations (6) and (8) represent the set of kinetic equations describing the kinetics of de- hydrochlorination of an ideal PVC chain. When the presence of inert impurities is considered, the kinetics are described by eqns (7) and (8).

DISCUSSION

The kinetic equations for autocatalytic PVC dehydrochlorination are derived on the assump-

. 8

tion that the dehydrochlorination is a chain zip reaction which involves an initiating step, zip propagation and zip termination. Of these reaction steps, initiation is responsible for autocatalysis. Autocatalysis is taken into con- sideration in eqn (2) and the mechanism of catalytic action is not analyzed in more detail. The proportionality between the number of catalyzed initiations and the conversion, x, expressed by eqn (2) does not mean that each C-----C bond exerts the catalytic action. The catalytic effect can involve arbitrarily long sequences of C-----C bonds the number of which is proportional to conversion. Another possibility of autocatalytic action also exists, namely the assumption that the catalytic action is propor- tional to the number of zips, z. In this case the transformation from the number of decomposi- tion cycles to time would be expressed as

dn/N = B(1 + flz) dt (9)

Integration of the differential equations obtained by combination of eqns (1), (5) and (9) gives autocatalytic kinetic curves, but, the assumption leading to eqn (9) does not seem to be reasonable. Due to secondary reactions, 4 the zips formed during dehydrochlorination are cut into shorter polyene sequences, the number of which should be proportional rather to the conversion than to the number of zips.

Kinetic curves obtained by integration of eqns (6) and (8) by the Runge-Kut ta method are shown in Fig. 1. The curves become steeper as m or fl increase. For any value of m, autocatalysis is exhibited for fl > 1. Figure 2 demonstrates that the dependences of the probability of p on time are S-shaped curves. Curves 2 and 3 in Fig. 1 practically coincide for x > 0-7, but, curves 2 and 3 in Fig. 2 are considerably different. This is obviously due to the existence of longer non-dehydrochlorinated sequences for greater values of m (for the same value of x).

The curves of conversion x versus probability p depend parametrically on the zip length m, but, they do not depend on ft. This can easily be explained by dividing eqns (6) and (8) (or eqns (5) and (6) from Part 2). Integration of the differential equation so formed for the initial condition x = 0 for p -- 0 is

x -- 1 - (1 - p ) exp - i = 1 - (10)

As can be seen, eqn (10) does not involve the

,6

.4

!

. 2

Kinetics of polymer degradation: Part 4 47

• 0 2 . 0 4 . 0 6 • 0 8

- 7 Fig. 1. Kinetic curves for various values of fl and m (xm = 1, p o = 0 , r = B t ) ; (1) f l = 1 0 , m = 5 ; (2) f l = 2 0 , m = 5 ; (3)

f l = 1 0 , m = 1 0 .

parameter ft. The consequence of eqn (10) is that the number of zips calculated by eqn (4) of Part 1 does not depend on ft. Figure 3 shows that z reaches a maximum, the position of which is shifted towards higher values of x with increasing m. The lower values of m lead to higher values of Z.

Experimental results are quite often presented in the form of the dependence of the reaction rate dx/dt on the conversion. In the case of autocatalytic mechanisms, a maximum reaction rate is found and it is observed that the position

.8

.6

2

. 2

0 k I K i I

. 0 2 .04. .06 . 0 8

P V

Fig. 2. Dependence of the probability p on time for various values of fl and m. Description as in Fig. 1.

48 Peter ~imon

18

12

4

1

• . 4 . 8 . 8

Fig. 3. Dependences of the number of zips on conversion for various zip lengths: (1) m = 2; (2) m = 5; (3) m = 10.

Table 1. Dependence of the conversion on the values of ~ and m

f l m

1 2 5 10

5 0.400 0 .335 0 .295 0-239 10 0-450 0.390 0-355 0.351 15 0-467 0.409 0-375 0.363 20 0-475 0 .421 0-382 0.371 50 0.490 0-435 0-399 0.387

maximum rate is encountered , ~, is presented as a function of m. With increasing rn, the value of

tends to lower values. Conversely, higher values of fl give rise to a shift of ~ towards higher values. The value of ~ = 0.5 cannot be exceeded and can be reached only in the case of a simple autocatalytic mechanism when m = 1.

of the maximum depends on tempera ture . Equat ion (2) of Part 3 shows that the zip length depends on tempera ture ; 3 hence, it can be deduced that the posit ion of the maximum depends on the zip length. This is demons t ra ted in Table 1 where the conversion at which the

R E F E R E N C E S

1. ~imon, P. Poly. Deg. and Stab., 29 (1990) 155. 2. Simon, P. & Valko, L. Poly. Deg. and Stab., 30 (1990)

253. 3. Simon, P., Gatial, A. & Valko, L. Poly. Deg. and Stab.,

30 (1990) 263. 4. IvY.n, B., Kelen, T. & Tiid6s, F. In Degradation and

Stabilization of Polymers, Vol. 2, ed. H. H. G. Jellinek. Elsevier, 1989.