Maurice Morton
Institute of Polymer Science Tlie University of Akron Akron, Ohio 44325
Polymerization as a Model Chain Reaction
The polymerization reaction is easily grasped by the chemistry student as the type of process whereby many small molecules are combined into a very large one, usually a long chain-like molecule. When first introduced, usually as a minor aspect of organic chemistry, its importance is related only to the utility of the products, i.e., synthetic materials such as plastics, fibers, rubbers, etc. The process itself does not "turn on" the student, since it is brushed off as merely a repetitive form of well-known chemical reactions mentioned earlier in the course. This may be true in the case of polymerizations involving such reactions as the formation of polyesters, polyamides, polyurethanes, etc., i.e., reactions of difunctional molecules. However, it is far from being true in the case of the chain reaction polymerizations. As a matter of fact, these offer the best opportunities for demonstrating and visualizing the principles of chain reactions, quite apart from the fascinating assortment of synthetic materials which are their end result. The chain reaction, which is probably the most basic one in the universe (cf. nuclear reactions, combustion) receives relatively little attention in chemistry teaching, generally showing up as one of the minor topics in a chemical kinetics course. Here, too, it is simply classified as a type of reaction which leads to large yields of product from a single initial step, by a repetitive process. This type of process, however, can he demonstrated in a most striking and vivid fashion by choosing polymerization as the example, since the product of the reaction, i.e., the long-chain molecule, serves as a permanent "photograph" of the process itself. In this case, it is not the yield of product so much as the type of product which tells the story; and this is generally the whole story, describing the relative rates of all the individual steps of initiation, propagation, termination and transfer. Furthermore, the chain length distribution provides direct evidence for the statistical nature of these chain reactions, something which is difficult, if not impossible to demonstrate in the usual treatment of chemical reactions. In addition to providing an excellent demonstration of chain reaction kinetics, polymerization can also he used to show the effect of the different mechanisms on such chain reactions, since there is a substantial body of knowledge now available on free radical, anionic, and cationic polymerizations. The discussion that follows will describe the special features of each of these mechanisms of chain addition polymerization, and how these can he deduced from a study of the resulting chain molecule.
The Free Radical Mechanism Kinetic Considerations Chain addition polymerization of vinyl monomers by the free radical mechanism probably offers the widest scope for study. It represents the classical case of a chain reaction, involving a series of very rapid repetitive steps, as shown by the series of equations in Tahle 1. The symbol R- represents the initiating free radical moiety, generally obtained by homolytic scission of a weak covalent bond, e.g., peroxide, disulfide, etc. The rate constants for the various individual steps are identified by their subscripts; kt, and k,a representing, respectively, termination by combination and disproportionation, the two wellknown paths for mutual termination of free radicals. Thus Tahle 1 presents a relatively simple picture of a polymerization chain reaction, the polymer chain growing very rapidly by successive additions of monomer units (free radicals being very unstable, reactive species which react vigorously with a-bonded electrons) during the short interval between the initiation and termination of the chain. In the absence of any side reactions, the kinetic chain length is given by the value of x (or y ) , which also represents the number of monomer units in the chain molecule. The actual lifetime of a growing chain radical is only of the order of a few seconds, hut this interval is sufficient for the successive addition of thousands of monomer units. Thus the free radical is a short-lived but very reactive species, leading to formation of long chains almost instantaneously. As usual for chain reactions, the values of the individual rate constants are not readily accessible. However, the overall characteristics of the reaction can he deduced by means of the very useful "steadystate" assumption, i.e., that the rate of initiation becomes equal to the rate of termination very soon after the reaction starts. Thus the kinetics can be expressed as shown in Tahle 2. This leads to a relatively simple rate equation, as shown in eqn. (I), despite the complexity of the three steps involved, and this can be easily verified experimentally. The kinetic chain length, or the number of monomer units consumed per free radical, can also he deduced by this means. Thus Kinetic Chain Length (KCL) = R J R ; = h,[~]/Z(k,k,)"'~"' ( 2 ) As stated previously, in the absence of any side reactions, the kinetic chain length also represents the actual number
Table 1. Free Radical Mechanism of Polymerization of Vinyl Monomers Initiation k R' + CHs=CHX R-CHz-CHX Pmpn&vztion kn R-CH2-CHX + CHFCHX -. . . Termination
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RfCH2--CHX],-CH2-CHX
Table 2. Kinetics of Free Radical Polymerization
Initiation Rote: Ri = dlMlldt = 2kJIl ~
-
. - ~ .,--, ~~
~olyrnerEzationRote: R, = k,(k,/k;) Definitions: I = initiator M = monomer M = propagating radical of any size of units per chain (x,). Hence eqn. (2) above is also subject to experimental verification, and has been so used,. This brought to light the fact that there usually are side reactions which render the actual chain size smaller than the KC1, i.e., transfer reactions. These can be described as follows Transfer to monomer: M; Transfer to solvent: M;
+ M !?+ M, + M' + S !k% M, + S ' k
-
S f M A S M 4 ... As can be seen, this transfer reaction, which generally involves the abstraction of a hydrogen atom, or similar moiety, by the growing radical, leads to termination of the growing chain and the initiation of a new chain. It should be noted that the new radical must he capable of initiating a new chain without delay, otherwise the polymerization will be retarded, or even inhibited. Taking into account the possibility of such transfer reactions, it is possible to derive an expression for the actual chain size, as follows Rate of propagation X" = Rate of termination and transfer -
kdM'ILM1
(k,,
+ 2h,d)[M'I2 + k,,,[M'lIM] + h,,.[M'][SI
(3)
Equation (3) takes into account both types of termination (combination and disproportionation), as well as transfer reactions to monomer and solvent. It is also possible to include other transfer reactions, such as with the initiator Itself or with impurities. However, it is not possible to treat any transfer to the polymer in this manner, since this would lead to branched chains, and not the initiation of new chains. Equation (3) is actually most generally applicable. It can be simplified by using the relation in Table 2 to eliminate the troublesome parameter [M 1, leading to the relation shown in eqn. (4).
constant (kt,,/kp) is obtained from a plot of 1/x, against [S]/[M], and such a plot is shown in the figure for styrene polymerization. The slopes of the lines demonstrate the marked difference in the values of the transfer reaction rate for these three solvents. Actually such transfer constants are really a measure of the reactivity of various compounds toward a given free radical, in this case the substituted benzyl radical. Presumably the transfer reaction here involves the abstraction of a labile hydrogen from the solvent by the polystyryl radical, and the slopes of the lines in the figure are a measure of the relative reactivity of phenyl hydrogens, primary henzyl hydrogens, and tertiary benzyl hydrogens. There are, of course, other compounds which are even more reactive toward radicals, and some of these are listed in Table 3. The marked difference in reactivitv between the hydrogens in the hydrocarbons, the bromine in carbon tetrabromide and the hydrogen in the sulfhydryl group is obvious. This demonstrates how simple measurements of chain lengths in polymerizations can yield interesting and important infairnation about the reactivity of compounds in free radical reactions. Statistical Considerations A unique feature of polymerization chain reactions, which is not readily apparent in ordinary chain reactions, is the concept of statistics. Thus the fact that the kinetic chain length of a chain reaction actually represents an average value is not considered either important or relevant, because it reallv is not. However. in the case of nolvmer~ - - ~ ization, a distri:bution of kinetic chain lenGhs actually results in a distribution of molecular weights of the nroduct. which is indeed a very relevant fact of life. ~ h u s ' t h esta: tistical distribution of such molecular weights becomes an important new parameter to be considered. Furthermore, the possibility of expressing more than one type of mathematical average of such a distribution introduces additional new concepts. The distribution of molecular weights resulting from polymerization depends on the statistics of the reaction. Since the various steps involved in a free radical polymerization are governed by random statistics, in homogeneous media, so are also the resulting molecular weights. Such distributions then have the following characteristics A~
~
.
Number (or mole) fraction of x-mers = f,,(x) = pX-'(1 - p )
(6)
Table 3. Transfer Constants (kt,/k,) in Radical Polymerization of Styrene at 60%
This relation can be used to determine experimentally such interesting parameters as kt,,/kp and k[,,/k,, i.e., the relative rates of transfer to propagation, usually referred to as the "transfer constants" for solvent and monomer, respectively. The monomer transfer constant k / k , which is usually very small, can thus be obtained by carrying out polymerizations in undiluted monomer a t different initiator concentrations, and noting the relation between x , and R,. The solvent transfer constant, on the other hand, which is usually more important, can he obtained from uncatalyzed polymerizations a t different initial monomer concentrations. Under such circumstances, it is usually found that R, = [MI2, SO that the first two terms on the right side of eqn. (4) are constant, and independent of monomer concentration, thus representing the value of l/x, for undiluted monomer. Equation (4) can then be written as
Benzene
Toluene Triphenylmethane Carbon tetrachloride Carbon tetrabromide n-Butanethiol
'"
,"
where x,, = number of units per chain for polymerization of undiluted monomer. The value of the solvent transfer
I.[ /I . [ Chain transfer in styrene polymerization at 10O'C
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Weight fraction of x-mers fJx)
Table 4. The Cationic Mechanism
= =
xpZ-'(1 - p)2 (i)
where p = probability of propagation and 1 - p = probability of termination (or transfer). Equation (6) arises from the fact that an x-mer, which contains x cbain units, must have (x - 1) "linkages" between such units, the probability of each linkage (propagation) being "p," while the single end group in such a chain results from a termination (or transfer) step, having a probability of (1 - p). Equation (7) is easily derived from eqn. (6). These expressions are readily recognized as representing the well-known binomial or "most probable" distributions. The two distrihution functions shown in eqns.' (6) and (7) can then be used to yield two types of average values of x, i.e. The Number Average I., The Weight Average x ,
= =
Zxf,,(x) = 1/(1- p) Zxf,,(x) = (1 p)/(l
+
(8)
- p)
b Initiation: BFrH?O + CH2=C(CHd2 (CH3)&+[BF3.0H] kn Propagotron: (CH~JC+[BFQOH]+ CH2=C(CH&- - . (cH)&-- - ~ ( c H ~ [ B F ~ D H ] Termination: kc (cH~)~c---~(cH~)~[BF~.oH~CHs
--
-
Transferto Monomer
(CHdaC-
- -C=CH2 I
+ BF3.Hz0
+ CHZ=C(CH&
(CHa)aC---C+(CH3)z[BF3.OH]-
CHa I
Table 5. Kinetics of Cationic Polvmerization
(9)
The expressions shown in eqns. (8) and (9) result, of course, from a summation of the infinite series that arise from an expansion of the expressions in eqns. (6) and (7). As a matter of fact, eqn. (8) is obvious from inspection, since, by definition, the average number of units per chain is given by the ratio of the probabilities of propagation to termination, i.e., p / ( l - p ) , which is equivalent to 1/(1 p ) when p is close to unity, as it is for these macromolecules. It is convenient to express such distributions of chain sizes by the ratio of weight- to number-average values, i.e.
.
.
,
.. .
-
-
No. of unitslchaii= x,.. ih, + k,lk,lMl , . = k,,,lk, .. .., ., . ~efinitions:'~ = initiator M = monomer M + = ~ r o ~ a e a t i chain n e of anvsize
Table 6. Mechanism of Anionic Polymerization Initiation:
RLi + CH2=CHCeHs
k
RCH~~HL~+
b Propagation: R C H ~ $ H L ~++CHZ=CHC~H~ -. . .
Equation (10) is, therefore, characteristic for the "most probable" distribution of molecular weights. It should be noted, however, that it applies only in the case of termination by disproportionation or transfer, and not for termination by combination. In the latter case, the coupling of a random assortment of cbain lengths leads to a narrowing of their distribution, such that
At this point i t is well to realize that all the expressions derived for values of chain lengths and their distributions apply only to "instantaneous" values of these parameters. During the course of the polymerization, such instantaneous values can be expected to change with changes in the concentration of monomer and initiator, so that the "cumulative" values will actually be a summation of the values of a series of increments. Hence the distribution of chain lengths in the accumulated polymer can he expected to he considerably broader than indicated by eqns. (6) to(11). The Cationic Mechanism Vinyl monomers may also be susceptible to polymerization initiated by cations rather than radicals. Strong acids, either of the protic or Lewis type, often act as initiators of this type of reaction, by heterolytic attack on a electrons. A typical mechanism is shown in Table 4 for the polymerization of isobutene by boron trifluoride hydrate. These quations show why this mechanism is referred to as "cationic," since the positive charge resides on the carbenium ion at the tip of the growing chain. Actually, no assumptions are inherent in this mechanism about the state of dissociation into free ions, hence the counteranion is included in all the equations. In the mechanism shown here, the initiator is actually a hydrogen ion, and propagation occurs in the interval between the addition of the initiating proton to the monomer and the final transfer of a proton from the cbain end either to the anion (termination) or to the monomer (transfer). 742
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kc,,
-
This mechanism, like that of free radical polymerization, involves a very reactive short-lived species, the carbenium ion, which is capable of adding thousands of monomer units almost instantaneously. The kinetic treatment in this case is therefore analogous to that used in the free radical case, i.e., involving the steady state assumption. This is shown in Table 5. Thus eqn. (12) can be verified experimentally to test the validity of the mechanism, while a plot of eqn. (13), i.e., l/x, versus 1/[M] permits the evaluation of the ratios kl,,/kD and kt/kD. As in the case of the free radical mechanism, the individual steps in this polymerization, as shown in Table 5, are governed by purely statistical considerations. Hence the chain length distribution can be expected to he "most probable" in this case, too, and eqns. (6) to (9) would describe the increments of nolvmer formed durine the nolv" merization. The cumulative molecular weight distribution would aeain be broadened over that described in ems. . (6) . . and (I)," depending on the rate of depletion of monomer and initiator.
.
. .
The Anionic Mechanism In contrast to the cationic mechanism, anionic polymerization results from the action of strong bases on vinyl monomers. The initiators are generally organometallic compounds involving carbanionic bases strong enough to interact with the n electrons of the double hond (or with the polar hond of a heterocyclic). Organoalkali compounds are typical initiators, since they are generally soluble in polar solvents, such as ethers. The organolithium initiators are the most versatile, since they are soluble in a variety of solvents. Table 6 depicts the mechanism of polymerization of styrene by an organolithium initiator, RLi, where R is generally an alkyl group, such as butyl.
The special feature of the anionic mechanism, as depicted in Table 6 is, of course, the ahsence of any termination (or transfer) step. This is due to the fact that the alkali metal "salts" of carbanionic acids are relatively stable, in the ahsence of any protic impurities, oxygen, etc. Hence, unlike the cationic systems, the anionic mechanism is often characterized by the ahsence of any "side reactions," i.e., termination or transfer. This, then, gives rise to a unique chain reaction in which the growing chains are relatively long-lived; in fact, within the time scale of the experiment (days, weeks, months) they do not lose their reactivity. As a matter of fact, under such conditions where the propagation reaction does not "die," these systems have heen euphemistically nicknamed as "living" polymers! The kinetics of these anionic chain reactions are therefore markedly different from those of the other two mechanisms. The ahsence of any termination step means that there is no "steady state" as long as there is any initiation of new chains. On the other hand, i t also means that these reactions follow a simple stoichiometry whereby each initiator molecule generates one chain. Hence, once the initiator is all consumed, the polymerization rate corresponds to the propagation rate, and it is possible to thus isolate k,, the propagation rate constant (as well as hi,of course). The really interesting aspects of these polymerizations, however, arise from a consideration of the molecular weights of the polymer. Thus, if the initiation step is not too slow (relative to propagation), all the chains will he initiated a t an early stage of the reaction, and each chain will continue to grow as long as any monomer remains. Under such conditions, where all chains have equal probability of growth, they will all tend to attain a similar size, and the distrihution of chain lengths will he very narrow. Statistics predict that the chain length distribution can he defined by eqn. (14) while the numher-average chain length is given by eqn. (15).
, , / x.
=
+
1/x, moles monomer/moles initiator
= 1
(14) (15)
It can be seen that, even when x , is no more than 100 units, the ratio xwlx, will be 1.01. This type of narrow distribution is known as the Poisson Distribution. Thus, in anionic polymerization the situation differs sharply from that prevailing in the free radical and cationic sys-
tems. Instead of short-lived species which rapidly attain a high molecular weight and then terminate, we have very long-lived chains each of which continues to grow (slowly or rapidly) throughout the course of the reaction. If initiation is fast relative to propagation, then all the chains will he growing simultaneously, leading to a very narrow molecular weight distrihution. If, on the other hand, initiation is very slow compared to propagation, then there will he a continuous generation of new chains, and the molecular weight distrihution will tend to broaden suhstantially. Hence it is clear that the anionic mechanism of polymerization comes closest of all to the ideal of synthesizing a uniform array of macromolecules. I t should be noted, too, that it is this type of organometallic system that has been found to lead t o "stereospecific" polymerization, where the monomeric units in the chain all (or nearly all) have the same stereoisomeric form. Conclusion It should he apparent from the foregoing discussion that the building of long chain macromolecules by means of chain reactions offers the best possible opportunity for the study of the nature of such chain reactions. This is selfevident from the fact that the final product of the reaction offers a permanent record of the preceding events. Furthermore, the size of the final macromolecules, and the distrihution of such sizes are a hallmark of the mechanism of their formation. Thus in the case of the free radical or cationic mechanism, the growing chain is a short-lived species which rapidly attains a very large size within a few seconds, and then terminates. Since all of these events are governed purely by chance, the distrihution of chain sizes of each increment of polymer will be random, or "most probable," while the accumulation of such increments of polymer will generally have a broader distribution. On the other hand, in the case of the anionic mechanism, the growing chains are long lived ("living" polymer) and each chain continues to grow until the monomer is depleted. Hence the molecular weights increase continuously with time (and conversion) and will generally tend to attain the same value, leading to a very narrow distrihution. In this way polymerization can he used to illustrate the statistics of chain reactions in a direct and vivid manner which would otherwise he impossible.
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