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Kinetics of Bimolecular Termination
12 Kinetics of Bimolecular Termination KENNETH F. O'DRISCOLL University of Waterloo, Ontario, Canada 12.1
INTRODUCTION
12.2
kt AT LOW CONVERSIONS
161 162
12.3 kt AT INTERMEDIATE CONVERSIONS: THE GEL EFFECT
165
12.4
167
kt AT VERY HIGH CONVERSIONS
12.5 TERMINATION IN COPOLYMERIZATION
168
12.6 CONCLUSIONS
170
12.7
170
12.1
REFERENCES
INTRODUCTION
Before two polymeric free radicals react to terminate their chain growth, they must first undergo a series of diffusional steps. These diffusional steps can be separated into two different types: the translational diffusion, during which the two chains enter into collision with each other and the segmental diffusion, during which the two chain ends approach each other and ultimately enter into a small volume element in which reaction can take place. This process, shown schematically in Figure 1, was first described by Benson and North 1 and has come to be recognized as a necessary basis for describing the complex behavior of the termination process in free radical polymerization. When the concentration of polymer chains, propagating or terminated, is low, the segmental diffusion process will control the rate of termination. As the polymer concentration increases, translational diffusion becomes rate controlling. Thus, as a polymerization advances to moderate conversion, the increase in rate often observed (variously called the Trommsdorf2, Norrish-Smith 3 or gel effect) can be ascribed to the decrease in the rate constant for termination, caused by the change in the diffusion process which controls the rate. Finally, at very high conversion, propagating chains are quite immobile and their termination occurs by a process termed 'reactive diffusion', a unique form of translational diffusion where the addition of monomer units serves to bring two (otherwise immobile) chain ends into reactive proximity. Reactive diffusion is discussed in more detail below. Regardless of which of these processes is rate controlling, the overall process occurring when two radicals (P) terminate each other can be mechanistically described by k
k
k
p + p — u { p p j — ^ (pp) —s~+ p - p
(i)
Chemical reaction
Figure 1 Schematic representation of two polymeric free radicals diffusing together, undergoing segmental motion and then terminating by combination
161
Free-radical Polymerization
162
where the rate constants subscripted T, S and C refer, respectively, to the net rate constants for the translational and segmental diffusion processes, and for the chemical reaction step. The encounter collision of the two coils is {P P}, the encounter collision of the two radical chain ends is (PP) and the dead polymer is P-P. (The latter is treated as though the chains had combined, but it could equally have been treated as though a disproportionation reaction had occurred.) The overall rate constant for the termination reaction, fct, can be defined by either the appearance of the product (P-P) or disappearance of the two propagating radicals. Following conventional kinetic treatment -(l/2)d[P]/dr = d[P-P]/df = /c t [P] 2
(2)
- d [ P ] A k = 2fct[P]2
(3)
whence The overall rate constant for termination has been defined according to equation (3) by many authors (mostly North American). However, the reader is advised that many authors have defined the rate constant by -d[P]/df =
fct[P]2
(4)
Clearly the rate constant defined by equation (4) is just twice the numerical value of that defined by equation (3) and one must make allowance for this in using literature values for the overall termination rate constant and numbers derived from it. In this chapter kt is defined by equation (4). Although kt is referred to as a rate 'constant', it must be appreciated that it can and does change during the course of a polymerization, even if it is isothermal. As a result of the existence of the three sequential diffusion processes outlined above, the whole description of the termination kinetics can become quite complicated for a polymerization taking place over a wide range of polymer concentrations. To simplify this description somewhat, the following discussion is broken down into three separate categories: very low conversion (ca. 0 to 10% polymer in solution), the gel effect region (ca. 20 to 80%) and very high conversion (ca. >90%). As will be seen, this division is quite arbitrary and the exact boundaries between these regions depends on the molecular weight of the polymer and on those reaction parameters which affect polymer mobility — solvent, chain stiffness, temperature, etc.
12.2
kt AT LOW CONVERSIONS
The ratio of rate constants k^/kt appears frequently in any analytical description of free radical polymerization kinetics and can often be determined from experimental information on steady state polymerizations where, for example, rate or molecular weight is measured. Typically, kt values of 10 7 ± 1 1 mol" 1 s" 1 are observed. However, to determine the absolute value of kt it is necessary to carry out an analysis of a non-steady state polymerization. In such an experiment the average lifetime of a propagating chain, T, can be measured and related to the ratio kp/kt. This numerical value can, in turn, be used with the separately measured rate of initiation, R{, or /Cp//ct to obtain separately the numerical values of kp and kt. A number of techniques have been used to conduct experiments under non-steady state con ditions. Perhaps the best known is the rotating sector technique,4 which depends on intermittent photochemical initiation of polymerization in which the illumination period is short compared to the time necessary to obtain steady state conditions. In the rotating sector experiment, the lifetime is given by T= *P[M]/M?P
(5)
An analogue of the rotating sector experiment, called spatially intermittent polymerization,5 provides a sample adequate for both conversion and molecular weight measurements. This technique is conducted in a continuous flow tubular reactor, in which the monomer and photoinitiator pass alternately through dark and light regions. The large sample which can be collected obviates the need for a separate experimental determination of R{ or k\jkv The equations describing this experiment are formally identical to those of the rotating sector, with the period of rotation being replaced by the ratio of linear flow velocity to length. Other techniques which have been used include non-steady state measurements of viscosity,6 or of volume changes by dilatometry 7 and steady state measurements of emulsion polymerization rate. 8 The latter determines kp when the number of particles is known and the number of radicals per
163
Kinetics of Bimolecular Termination
particle is assumed to be 1/2. Recent work which measures this quantity directly by electron spin resonance (ESR)9 is discussed below. The non-steady state experiments have, collectively, given data which exhibit enormous scatter, even though they have been restricted to conversion levels below 1%. Figure 2 shows one such collection 10 of rate constants for styrene. It is probable that the scatter is caused, at least in part, by the existence of a chain length dependence of kv The lifetime determination is done under one set of conditions and the k\jkx measurement under another, where the chain length is different. This experimental problem is circumvented in the spatially intermittent polymerization technique, by which it was unequivocally shown that a chain length dependence of kt does exist as illustrated in Figure 3. 5 1000
100
10
J
3.0
I
3.2
I
3.4-
L
3.6
3.8
1/TxlO 3
Figure 2
Arrhenius plot of kt data as collected in ref. 10
160
350 120
h
300
80
H250
40
2
3
log Xn
Figure 3
Effect of chain length on kp and kt for methyl methacrylate
A theoretical description of the chain length dependence of kx can be found in the work of Mahabadi and O'Driscoll. 11 Using the model embodied in equation (1) and considering chain flexibility, chain size, excluded volume effects and polymer-solvent interactions, they showed that the average termination rate constant, defined by *t= I
I (*.c».*,[PJ[PJ/[P])
(6)
164
Free-radical Polymerization
could be described as a product of two functions ^F,
(7)
where F 2 is a function of the segmental friction coefficient, £, and temperature, and F 2 is a function of the linear chain expansion coefficient, a, and the lengths of the two reacting chains, N{ and N^ (8) F 2 =(a i a j )- 1 - 3 [l -0.37(a i a j )-°- 3 7 x {1
-4(2/3n)ll2(N0)l'2(aiai)-ll2(NiNiy114}^
(9)
Assuming N{ = N^ = Xn (the average chain length), these equations predict a chain length dependency *,-*:
(10)
where a is predicted to be —0.5 for short chains (length ca. 100 or less), falling off to —0.1 for large chains (length > 1000). They also predict an effect of segment size and solvent viscosity (as they affect £) and solvent quality (as expressed by a) which have been experimentally verified. The chain length dependency is not noticed in most polymerizations at low conversions and high degrees of polymerization because the —0.1 power is so small. However, when dealing with low molecular weights (as for example in the case of measuring chain transfer constants or the rate constant for primary radical termination, or when doing precise work at ordinary molecular weights) the chain length effect can be large enough to distort experimental results and conclusions unless it is explicitly considered. Yasukawa and Murakami 12 showed by computer simulation that a — 0.1 power dependence of kt on chain length would produce an apparent order of polymerization, with respect to initiator, of 0.45 instead of the expected 0.5. This was verified by Stickler,13 who also showed that the termination rate constant for a monomeric radical was 50 times greater than the average for the polymeric radicals. As another example of the possible importance of chain length dependence, Figure 4 1 4 shows the primary radical termination as calculated for styrene, with and without consideration for chain length dependence of kt. The curvature in the plot where chain length dependence was neglected was originally attributed to the possibility of disproportionation of styrene in the termination reaction. (It had previously been accepted from a great deal of evidence that styrene terminates only by combination.) The plot without curvature shows that, when chain length dependence is included, disproportionation need not be invoked, and the termination of styrene is indeed by combination.
io.oh
3.0
3.2
3.4
r ' x I O 3 (KH) Figure 4
Calculated variation of the characteristic constant for primary radical termination with temperature in experiments done at variable (O) and constant ( • ) chain lengths
Kinetics of Bimolecular Termination
165
A particular effect of solvent quality on kt is caused by the influence of polymer concentration on polymer size at low polymer concentrations. Rosen 15 showed experimentally that small amounts of polymer in solution would actually increase the value of kt. He reasoned that the polymer in solution would cause the polymeric radical to find itself in a thermodynamically poorer solvent compared to the same radical in pure solvent. Coils would be smaller because of this. Therefore, two colliding chains would have a smaller volume to explore by segmental diffusion before reacting, and would take less time to react. Mahabadi and O'Driscoll 16 expanded the theoretical treatment of a to show quantitatively that the rate enhancement observed at ca. 1 to 10% polymer in solution was consistent with this reasoning. Horie and Mita 17 went on to show the limitation of this approach as the solvent quality becomes sufficiently poor to make it difficult for the chains to explore their combined coils. Experimental observation of this phenomenon in bulk polymerization has been made, both with respect to rate and degree of polymerization.18 It should be noted that the effect is small, seldom amounting to more than a 5 or 10% change in the expected rate. The most significant conclusions to be drawn from this theoretical treatment are: (i) any factor which decreases the segmental friction coefficient will increase kt; (ii) any factor which decreases the reacting coil size will increase /ct; and (iii) any careful kinetic study which changes the chain length or solvent character must take into consideration the effects embodied in the previous two points. While equation (7) gives an exact relation between the important physical parameters and the experimentally observable average termination rate constant, it is an awkward expression of the chain length dependence. Numerous workers have suggested and used an equation of the form K(n,m) = Ko(nmyii
(11)
where ki{nm) is the rate constant for termination between two radicals of lengths m and n and kt0 is a constant. Mahabadi 19 has shown that, if the exponent a is treated as a constant, the relation between the observed kt and Xn is /ct = ^ 0 g ( a ) X M - 2 a
(12)
where g(a) involves the gamma function of a. Experimental verification of the latter equation has often been found.
12.3 fct AT INTERMEDIATE CONVERSIONS: THE GEL EFFECT The gel effect is commonly accepted as beginning in that portion of a bulk or solution polymerization where the reaction rate increases and the molecular weight of the polymer formed also increases. Any complete model of the gel effect must describe not only the increased rate (and molecular weight) during the gel effect, but also a smooth transition between the low conversion rate and the gel effect. To that end it is useful to consider the question: when does the gel effect begin? Some authors have used simple conversion-time plots, and taken the upward curvature from linearity to indicate the onset of the gel effect; others have used the deviation from linearity of a plot of ln(K p )/[M]. Given the decrease in rate caused by the presence of polymer, it has also been suggested that the minimum in rate which can be observed defines the onset of the gel effect. As will be seen in the following discussions, most models employ an empirical or theoretical means of specifying the onset of the gel effect in order to trigger a 'switch' from low conversion kinetics to gel effect kinetics. Burnett and Duncan 20 were the first to postulate a set of equations to describe the gel effect. They suggested the existence of a critical concentration of polymer, which is dependent on the polymer molecular weight, and above which the propagating chains are immobile and incapable of ter minating each other. This approach gave a simple set of equations which were in accord with experimental observations. The equations demanded the existence of a 'switch', i.e. a concentration at which chains switched from being mobile and capable of terminating to being immobile and incapable of terminating. Furthermore, no allowance was made for the fact that small growing chains might terminate before they became immobile. Fifteen years later, Cardenas and O'Driscoll 21 postulated that two populations of propagating chains existed in a solution at a given concentration: those large enough to be entangled and those small enough to be freely mobile. The transition between the two chain sizes was established with a concept, 22 useful in viscosity measurements, that a critical constant, X c , exists, which is equal to the product of polymer concentration and chain length, and which determines the minimum chain size for entanglement. Assigning a rate constant kt to reaction between the entangled species, the
Free-radical Polymerization
166
0
20
40
60
80
100
Time (min)
Figure 5
Bulk polymerization of methyl methacrylate at 70°C (A, A) and 90°C (O, • ) ; solid lines are the model 21 predictions
normal kt to reaction between the unentangled and a geometric mean of the two for reaction between the unentangled and the entangled chains, it was shown that a two parameter model (Kc and kjkue) accurately predicted rate and molecular weight behavior over a wide range of conversions, initiator concentrations and temperature for methyl methacrylate (MMA) in bulk. This is illustrated in Figure 5 for MMA at two temperatures and two initiator concentrations. In addition to the successful description of the rate and molecular weight to be expected in polymerization, the model also led to a simple understanding of the reason why some monomers show a stronger gel effect than others. Put simply, it depends on the ratio k\jkv A monomer having a large value of such a ratio and polymerizing at a given rate will produce, in a given time, the same amount of polymer but with a higher molecular weight than will a different monomer, polymerizing at the same rate, with a lower value for that ratio. This explains why, for example, styrene has such a small gel effect compared to MMA. In spite of its ability to describe bulk polymerization, the Cardenas and O'Driscoll model failed to adequately describe polymerization in solution and, most importantly, was not intended to describe very high conversion. The success of the model coupled with its limitations led to several other models which took different physical approaches. One such approach to a useful description of the gel effect was taken by Hamielec and Marten 23 who introduced the concept of free volume into a semi-empirical model, They postulated a relation between the weight average molecular wight, M w and the free volume of the reacting system, Vf kt^M-115e\p(-A/V{)
(13)
The exponent of molecular weight was empirically chosen, A is a constant and the concentration dependence of kt is contained in Vf. This approach and the approach of O'Driscoll and Cardenas fitted the same data equally well. In a series of papers, Soh and Sundberg 24 combined the entanglement and free volume concepts so that, using a switch at some critical chain length, the value obtained was **,.») = K0{XJXnX2A
exp(-X/K f )
(14)
This approach gave a good fit to bulk polymerization data for six different monomers. At the same time, the work of de Gennes with scaling and reptation concepts 25 was showing great promise for describing physical phenomena of polymers in solution, particularly diffusion. Tulig and Tirrell 26 developed a model for kt using the concepts of reptation and a critical concentration, C*, above which chains interpenetrate, and a second critical concentration, C**, above which reptative behavior dominates. For chains of length N, C** is defined as KJN0,5, and above that con centration K = *,.min + * r (N* W(C**/C) 175 (15) where the ** superscripts refer to values at the transition concentration and kt min is a minimum value for the rate constant. This works quite well at concentrations ranging up to ca. 70%, but in that range another switch had to be introduced (C***) and another formulation for the rate constant was needed (see below).
Kinetics of Bimolecular Termination
167
In several publications, Ito 2 7 has also used scaling and reptation concepts and has proposed equations similar to those of Tulig and Tirrell. De Gennes has discussed bimolecular diffusioncontrolled reactions in general and the termination reaction of polymerization in particular in a pair of papers. 28 He concluded that kt should vary inversely with the square of chain length when reptation controls the process. In all the gel effect modeling described above, the authors sought to explain the deviations from the behavior expected for classical free radical polymerization kinetics by changes in kt alone. In fact, it has long been recognized that initiator efficiency and even the propagation rate constant, /cp, are reduced by increasing viscosity. Based on a series of papers on the kinetics of emulsion poly merization, it has been shown 9 that it is possible to use ESR measurements to obtain chain radical concentrations in emulsion particles. This enables the determination of rate constants with few assumptions. In this manner, it has been demonstrated 9 that kt can be adequately described by the Soh and Sundberg model. 24 This experimental development is particularly interesting since the technique involves seeded emulsion polymerizations, and it is quite possible to experimentally generate a given average chain length radical in the presence of a different chain length polymeric matrix. A similar approach in bulk or solution polymerization might be possible using the pulsed laser technique pioneered by Buback and co-workers 29 in high pressure ethylene polymerization. These could be powerful tools for testing models, especially when it is recognized that most models have used an average molecular weight of the polymerizing system, which drifts with conversion, to describe the gel effect on the rate constants of a separate distribution of radical chain lengths.
12.4
kt AT VERY HIGH CONVERSIONS
The description of a polymerization approaching very high conversion, ca. 90% or more, is of great industrial importance. However, judging by the published literature, it has (until recently) received very little detailed attention in a kinetic sense. Any description of the polymerization process at very high conversion will necessarily include, not only diffusion in viscous media, but also diffusive behavior in a glass, the validity of steady state assumptions and the possibility of depropagation becoming kinetically important. Such considerations imply that one cannot consider the termination rate constant in isolation, but rather that it is necessary to consider it together with the propagation rate constant, the initiator rate constant and its efficiency, and with the relation between the polymerizing systems' glass transition temperature, Tg, the conversion level and the temperature of polymerization as well as the Gibbs free energy of polymerization (i.e. reversibility). The most coherent approach to this problem has been made by Stickler and his colleagues30 who have utilized a combination of the free volume 23 approach for kp with the reactive diffusion concept of Schulzfor fct.31 A theoretical consideration of the course of a bulk polymerization carried out at a temperature below the polymer's Tg suggests that a polymerizing mixture would become glassy before reaching 100% conversion. The glass temperature of the reacting mixture may be calculated from free volume considerations to be determined by a weighted sum, computed from the volume fractions of each species in the reaction mixture and their glass transition temperatures. In this glassy state, it has been proposed that polymerization will cease, because monomer is unable to diffuse to a live chain end. Wunderlich and Stickler have presented 32 experimental evidence to show that MMA does not follow this simple idea. Figure 6 shows that the limiting conversion' observed at 80 °C, well below the Tg of PMMA, was a moderately strong function of initiator concentration. Further study revealed that polymerization continued, albeit at a very low rate, without cessation after the monomer-polymer reaction mixture had become glassy. From a practical, as opposed to a theoretical, point of view it is reasonable to ignore this very slow rate of polymerization (which is not understood) in order to model the industrially important final stage of polymerization. The following treatment does so, and explicitly includes kp as well as /ct, but does not include depolymerization or the influence of high conversion on the initiation process. Under conditions where the polymer concentration is very high, it might be expected that the reduced mobility of chains described in the gel effect becomes so great that they may be considered truly immobile. If that is the case, ordinary bimolecular reactions as illustrated by equation (1) can no longer occur. Instead, we must recognize that the active ends of two propagating chains can move through space by adding monomer, and so come into the same small volume element where they can terminate each other. This is termination by reactive diffusion and the rate constant for the process
Free-radical Polymerization
168 100
[ I ] 0 = 5.4xl0" 2 [I]0=5.3xl0"3
100
150
t Imin) Figure 6
Influence of initiator concentration on limiting conversion of methyl methacrylate at 80 °C; solid lines are the model predictions 32
can be written as proportional to the diffusion coefficient, DR CDR
(16)
where the constant C contains known geometric factors and (17)
where /0 and ns are the length and number of monomers in a statistical segment of polymer. Because the termination process is, under these conditions, defined in terms of the propagation rate constant, a precise description of kp at high conversion is needed. At low conversions, the propagation reaction is quite fast and chemically controlled with a rate constant fcp of the order of magnitude of 102 to 103. As the polymerization becomes more and more viscous, it is not unreasonable to suppose that the addition of monomer might be limited by its diffusivity to the immobile chain end. The rate constant is then given, in terms of the free volume theory, by kP^Voexp[-K*(l/^-l/Kf,0)]
(18)
where V* is an empirical constant and V{ 0 and Vi are the free volumes of monomer and reacting mixture, respectively. The latter can be calculated from the work of Kelley and Bueche.33 The empirical constant V* may be regarded as a critical fraction of the free volume required for monomer to diffuse to the chain end. Fitting of data to the above equations gave K* = 0.35, independent of temperature and initiator type or concentration. This model was combined with that of Hamielec and Marten 23 to produce a comprehensive description of the kinetics of bulk polymerization over a wide conversion range, extending to very high conversion. The major additions to the equations presented above were the introduction of two 'switches', Kf c r l and Kf cr2 — critical free volumes of the reaction mixture at which the gel effect and the very high conversion equations are used. V{>crl replaces VUQ in equation (18) above and V* was assigned the value 1.0. With four 'fixed' parameters and two adjustable ones this then gave an excellent fit to both rate and molecular weight data. Other approaches, such as using reptation theory, have been less successful in describing very high conversion. So it must be regarded as an important area in which the lack of general knowledge and insufficient data combine to make it necessary that more work be done. 12.5 TERMINATION IN COPOLYMERIZATION In copolymerization, the termination reaction is often discussed in terms of the three different reactions possible when two different chain ends combine PA + PA PB + PB PA + PB
dead polymer
(19)
Kinetics of Bimolecular Termination
169
where P A and P B represent active chains having terminal units of monomers A and B respectively. Rate constants for the three steps are usually designated /ctAA, /ctBB and ktAB respectively. The problem with this approach to describing the termination reaction in copolymerization is that the diffusion-controlled processes which operate in homopolymerization also operate in copolymerization. Designating rate constants in terms of the end unit of the chain is, therefore, the equivalent of assuming that that unit determines the speed of the rate-controlling diffusion process. This is physically incorrect, but, so far, no one has given a quantitative treatment which is correct as an alternative in all cases. What follows then is a brief summary of those treatments which have been given. The subject has been reviewed by Chiang and Rudin. 34 The well-known 40 factor', defined by « = W(*.AA*.BB)0-5
(20)
was originally introduced to describe rates of copolymerization.35 It can be determined from rates of copolymerization as a function of composition. Often the values are much greater than unity. Since (p is the ratio of the cross termination to the geometric mean of the homotermination reaction rate constants, values of 0 that are very different from unity imply that chemical effects (e.g. electrostatic interactions between groups of differing polarity) cause a marked increase in the cross termination reaction. Since the reaction is diffusion controlled, chemical changes cannot be expected to cause an increase in the rate. For this reason, the factor must be regarded as an empirical parameter which may be useful for describing the rate of copolymerization, but which does not have any physical significance at the molecular level. An extension of the 0 factor has been developed by Russo et a/.,36 who have attributed the segmental diffusion which is rate controlling to the two end units of the reacting chain, i.e. the last four carbon atoms. This leads to what is essentially a penultimate unit effect in the termination step, and necessitates the determination of a larger number of parameters than is usually justified by the data. It has, however, been successful in describing some systems,36 but not others. 37 Treating the termination process as the average of reactions between chains which have many different diffusive speeds was suggested first by Atherton and North, 38 who postulated the simple expression *, = F,* l A A + F2fclBB
(21)
proposing that the average kt was merely the average of the homotermination kt values weighted by the copolymer compositions ¥x and F 2 . This was extended by Ito and O'Driscoll,37 who averaged the segmental friction coefficients. All of the above has been done on systems of two monomers at low conversion. No one has yet treated binary copolymerizations taken to high conversion where the gel effect is important, or multicomponent polymerizations with three or more monomers, Also, all these treatments have assumed that the simple Mayo-Lewis model of copolymerization propagation is applicable. Recent work by Fukuda et a/.39 has demonstrated, by experimental determination of kt in copolymerization (using the rotating sector technique), that the Mayo-Lewis model may be inapplicable for styrenemethyl methacrylate. The apparent large maximum in kt as composition is varied may be an artifact 2
0
if -2 Ac"
O _
-4
-6 c,
C2
C 3 CE 100
Conversion (%)
Figure 7
Schematic variation of the termination rate 'constant1 with conversion in a bulk polymerization. The symbols are defined in the text
Free-radical Polymerization
170
of the model being assumed; they suggested a penultimate unit effect in copolymerization for this system might make the experimental kt data easily understandable.
12.6
CONCLUSIONS
From the work described above, it would appear that calling the parameter k{ a rate 'constant' is a gross abuse of the meaning of the word. Not only does kt begin to increase as soon as polymer appears in the reaction mixture, but at about 10% conversion it changes from being controlled by segmental diffusion, /cs, to being controlled by translational diffusion, kT. As the polymer con centration increases further, the combined effect of more polymer and higher molecular weight polymer further depresses /cT, until it is so slow that reactive diffusion, /cR, becomes dominant. This is shown schematically in Figure 7 where the numbers on the ordinate are very approximate. The conversion level C1 is the onset of the gel effect, C 2 the onset of reactive diffusion and C 3 the limiting conversion, at which the system becomes glassy and beyond which little polymerization is observed. Should C 3 be higher than the equilibrium monomer concentration, C E , the schematic representation would need revision to indicate that the apparent value of kt would go to zero at CE
12.7 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.
REFERENCES
S. W. Benson and A. M. North, J. Am. Chem. Soc, 1962, 84, 935. E. Trommsdorf, H. Kohle and P. Lagally, MakromoL Chem., 1948, 1, 169. R. G. W. Norrish and R. R. Smith, Nature (London), 1942, 150, 336. G. M. Burnett and H. W. Melville, Nature (London), 1945, 156, 661. K. O'Driscoll and H. K. Mahabadi, J. Polym. ScL, Polym. Chem. Ed., 1976, 14, 869. M. J. S. Dewar and C. H. Bamford, Nature (London), 1946, 158, 380. S. W. Benson and A. M. North, J. Am. Chem. Soc., 1958, 80, 5625. M. Morton, P. Saltiello and H. Landfield, J. Polym. ScL, 1952, 8, 111. M. J. Ballard, R. G. Gilbert, D. H. Napper, P. J. Pomery, P. W. O'Sullivan and J. H. O'Donnell, Macromolecules, 1986,19, 1303. R. Korus and K. O'Driscoll, in 'Polymer Handbook', ed. J. Brandrup and E. Immergut, Wiley, New York, 1975, p. 11. H. K. Mahabadi and K. O'Driscoll, J. Polym. ScL, Polym. Chem. Ed., 1977, 15, 283. T. Yasukawa and K. Murakami, Polymer, 1980, 21, 1423. M. Stickler, MakromoL Chem., 1986, 187, 1765. H. K. Mahabadi and K. O'Driscoll, MakromoL Chem., 1977, 178, 2629. W. A. Ludwico and S. L. Rosen, J. Polym. ScL, Polym. Chem. Ed., 1976, 14, 2121. H. K. Mahabadi and K. O'Driscoll, Macromolecules, 1977, 10, 55. K. Horie and I. Mita, Macromolecules, 1977, 11, 1175. J. Dionisio, H. K. Mahabadi and K. O'Driscoll, J. Polym. ScL, Polym. Chem. Ed., 1979, 17, 1891. H. K. Mahabadi, Macromolecules, 1985, 18, 1319. G. M. Burnett and G. L. Duncan, MakromoL Chem., 1962, 51, 154. J. Cardenas and K. O'Driscoll, J. Polym. ScL, Polym. Chem. Ed., 1976, 14, 883. R. Porter and J. Johnson, Chem. Rev., 1966, 66, 1. F. L. Marten and A. E. Hamielec, J. Appl. Polym. ScL, 1982, 27, 489. S. K. Soh and D. C. Sundberg, J. Polym. ScL, Polym. Chem. Ed., 1982, 20, 1299, 1315, 1331, 1345. P. G. de Gennes, 'Scaling Concepts in Polymer Physics', Cornell University Press, Ithaca, NY, 1979. T. Tulig and M. Tirrell, Macromolecules, 1981, 14, 1501. K. Ito, Polym. J., 1980, 12, 499. P. G. de Gennes, J. Chem. Phys., 1982, 76, 3316, 3322. M. Buback, H. Hippler, J. Schweer and H. Vogele, MakromoL Chem., Rapid Commun., 1986, 7, 261. M. Stickler, D. Panke and A. E. Hamielec, J. Polym. ScL, Polym. Chem. Ed, 1984, 22, 2243. G. V. Schulz, Z. Phys. Chem. (Frankfurt Main), 1956, 8, 290. W. Wunderlich and M. Stickler, Polym. Prepr., Am. Chem. Soc, Div. Polym. Chem., 1984, 25, (2), 7. F. N. Kelley and F. Bueche, J. Polym. ScL, 1961, 50, 549. S. S. M. Chiang and A. Rudin, J. Macromol. ScL, Chem., 1975, A9, 237. H. W. Melville, R. Noble and W. F. Watson, J. Polym. ScL, 1947, 2, 229. G. Bonta, B. M. Gallo and S. Russo, J. Chem. Soc, Faraday Trans. I, 1975, 71, 1727. K. Ito and K. F. O'Driscoll, J. Polym. ScL, Polym. Chem. Ed., 1979, 17, 3913. J. N. Atherton and A. M. North, Trans. Faraday Soc, 1962, 58, 2049. T. Fukuda, Y. Ma and H. Inagaki, Macromolecules, 1985, 18, 17.
INTRODUCTION
12.2
kt AT LOW CONVERSIONS
161 162
12.3 kt AT INTERMEDIATE CONVERSIONS: THE GEL EFFECT
165
12.4
167
kt AT VERY HIGH CONVERSIONS
12.5 TERMINATION IN COPOLYMERIZATION
168
12.6 CONCLUSIONS
170
12.7
170
12.1
REFERENCES
INTRODUCTION
Before two polymeric free radicals react to terminate their chain growth, they must first undergo a series of diffusional steps. These diffusional steps can be separated into two different types: the translational diffusion, during which the two chains enter into collision with each other and the segmental diffusion, during which the two chain ends approach each other and ultimately enter into a small volume element in which reaction can take place. This process, shown schematically in Figure 1, was first described by Benson and North 1 and has come to be recognized as a necessary basis for describing the complex behavior of the termination process in free radical polymerization. When the concentration of polymer chains, propagating or terminated, is low, the segmental diffusion process will control the rate of termination. As the polymer concentration increases, translational diffusion becomes rate controlling. Thus, as a polymerization advances to moderate conversion, the increase in rate often observed (variously called the Trommsdorf2, Norrish-Smith 3 or gel effect) can be ascribed to the decrease in the rate constant for termination, caused by the change in the diffusion process which controls the rate. Finally, at very high conversion, propagating chains are quite immobile and their termination occurs by a process termed 'reactive diffusion', a unique form of translational diffusion where the addition of monomer units serves to bring two (otherwise immobile) chain ends into reactive proximity. Reactive diffusion is discussed in more detail below. Regardless of which of these processes is rate controlling, the overall process occurring when two radicals (P) terminate each other can be mechanistically described by k
k
k
p + p — u { p p j — ^ (pp) —s~+ p - p
(i)
Chemical reaction
Figure 1 Schematic representation of two polymeric free radicals diffusing together, undergoing segmental motion and then terminating by combination
161
Free-radical Polymerization
162
where the rate constants subscripted T, S and C refer, respectively, to the net rate constants for the translational and segmental diffusion processes, and for the chemical reaction step. The encounter collision of the two coils is {P P}, the encounter collision of the two radical chain ends is (PP) and the dead polymer is P-P. (The latter is treated as though the chains had combined, but it could equally have been treated as though a disproportionation reaction had occurred.) The overall rate constant for the termination reaction, fct, can be defined by either the appearance of the product (P-P) or disappearance of the two propagating radicals. Following conventional kinetic treatment -(l/2)d[P]/dr = d[P-P]/df = /c t [P] 2
(2)
- d [ P ] A k = 2fct[P]2
(3)
whence The overall rate constant for termination has been defined according to equation (3) by many authors (mostly North American). However, the reader is advised that many authors have defined the rate constant by -d[P]/df =
fct[P]2
(4)
Clearly the rate constant defined by equation (4) is just twice the numerical value of that defined by equation (3) and one must make allowance for this in using literature values for the overall termination rate constant and numbers derived from it. In this chapter kt is defined by equation (4). Although kt is referred to as a rate 'constant', it must be appreciated that it can and does change during the course of a polymerization, even if it is isothermal. As a result of the existence of the three sequential diffusion processes outlined above, the whole description of the termination kinetics can become quite complicated for a polymerization taking place over a wide range of polymer concentrations. To simplify this description somewhat, the following discussion is broken down into three separate categories: very low conversion (ca. 0 to 10% polymer in solution), the gel effect region (ca. 20 to 80%) and very high conversion (ca. >90%). As will be seen, this division is quite arbitrary and the exact boundaries between these regions depends on the molecular weight of the polymer and on those reaction parameters which affect polymer mobility — solvent, chain stiffness, temperature, etc.
12.2
kt AT LOW CONVERSIONS
The ratio of rate constants k^/kt appears frequently in any analytical description of free radical polymerization kinetics and can often be determined from experimental information on steady state polymerizations where, for example, rate or molecular weight is measured. Typically, kt values of 10 7 ± 1 1 mol" 1 s" 1 are observed. However, to determine the absolute value of kt it is necessary to carry out an analysis of a non-steady state polymerization. In such an experiment the average lifetime of a propagating chain, T, can be measured and related to the ratio kp/kt. This numerical value can, in turn, be used with the separately measured rate of initiation, R{, or /Cp//ct to obtain separately the numerical values of kp and kt. A number of techniques have been used to conduct experiments under non-steady state con ditions. Perhaps the best known is the rotating sector technique,4 which depends on intermittent photochemical initiation of polymerization in which the illumination period is short compared to the time necessary to obtain steady state conditions. In the rotating sector experiment, the lifetime is given by T= *P[M]/M?P
(5)
An analogue of the rotating sector experiment, called spatially intermittent polymerization,5 provides a sample adequate for both conversion and molecular weight measurements. This technique is conducted in a continuous flow tubular reactor, in which the monomer and photoinitiator pass alternately through dark and light regions. The large sample which can be collected obviates the need for a separate experimental determination of R{ or k\jkv The equations describing this experiment are formally identical to those of the rotating sector, with the period of rotation being replaced by the ratio of linear flow velocity to length. Other techniques which have been used include non-steady state measurements of viscosity,6 or of volume changes by dilatometry 7 and steady state measurements of emulsion polymerization rate. 8 The latter determines kp when the number of particles is known and the number of radicals per
163
Kinetics of Bimolecular Termination
particle is assumed to be 1/2. Recent work which measures this quantity directly by electron spin resonance (ESR)9 is discussed below. The non-steady state experiments have, collectively, given data which exhibit enormous scatter, even though they have been restricted to conversion levels below 1%. Figure 2 shows one such collection 10 of rate constants for styrene. It is probable that the scatter is caused, at least in part, by the existence of a chain length dependence of kv The lifetime determination is done under one set of conditions and the k\jkx measurement under another, where the chain length is different. This experimental problem is circumvented in the spatially intermittent polymerization technique, by which it was unequivocally shown that a chain length dependence of kt does exist as illustrated in Figure 3. 5 1000
100
10
J
3.0
I
3.2
I
3.4-
L
3.6
3.8
1/TxlO 3
Figure 2
Arrhenius plot of kt data as collected in ref. 10
160
350 120
h
300
80
H250
40
2
3
log Xn
Figure 3
Effect of chain length on kp and kt for methyl methacrylate
A theoretical description of the chain length dependence of kx can be found in the work of Mahabadi and O'Driscoll. 11 Using the model embodied in equation (1) and considering chain flexibility, chain size, excluded volume effects and polymer-solvent interactions, they showed that the average termination rate constant, defined by *t= I
I (*.c».*,[PJ[PJ/[P])
(6)
164
Free-radical Polymerization
could be described as a product of two functions ^F,
(7)
where F 2 is a function of the segmental friction coefficient, £, and temperature, and F 2 is a function of the linear chain expansion coefficient, a, and the lengths of the two reacting chains, N{ and N^ (8) F 2 =(a i a j )- 1 - 3 [l -0.37(a i a j )-°- 3 7 x {1
-4(2/3n)ll2(N0)l'2(aiai)-ll2(NiNiy114}^
(9)
Assuming N{ = N^ = Xn (the average chain length), these equations predict a chain length dependency *,-*:
(10)
where a is predicted to be —0.5 for short chains (length ca. 100 or less), falling off to —0.1 for large chains (length > 1000). They also predict an effect of segment size and solvent viscosity (as they affect £) and solvent quality (as expressed by a) which have been experimentally verified. The chain length dependency is not noticed in most polymerizations at low conversions and high degrees of polymerization because the —0.1 power is so small. However, when dealing with low molecular weights (as for example in the case of measuring chain transfer constants or the rate constant for primary radical termination, or when doing precise work at ordinary molecular weights) the chain length effect can be large enough to distort experimental results and conclusions unless it is explicitly considered. Yasukawa and Murakami 12 showed by computer simulation that a — 0.1 power dependence of kt on chain length would produce an apparent order of polymerization, with respect to initiator, of 0.45 instead of the expected 0.5. This was verified by Stickler,13 who also showed that the termination rate constant for a monomeric radical was 50 times greater than the average for the polymeric radicals. As another example of the possible importance of chain length dependence, Figure 4 1 4 shows the primary radical termination as calculated for styrene, with and without consideration for chain length dependence of kt. The curvature in the plot where chain length dependence was neglected was originally attributed to the possibility of disproportionation of styrene in the termination reaction. (It had previously been accepted from a great deal of evidence that styrene terminates only by combination.) The plot without curvature shows that, when chain length dependence is included, disproportionation need not be invoked, and the termination of styrene is indeed by combination.
io.oh
3.0
3.2
3.4
r ' x I O 3 (KH) Figure 4
Calculated variation of the characteristic constant for primary radical termination with temperature in experiments done at variable (O) and constant ( • ) chain lengths
Kinetics of Bimolecular Termination
165
A particular effect of solvent quality on kt is caused by the influence of polymer concentration on polymer size at low polymer concentrations. Rosen 15 showed experimentally that small amounts of polymer in solution would actually increase the value of kt. He reasoned that the polymer in solution would cause the polymeric radical to find itself in a thermodynamically poorer solvent compared to the same radical in pure solvent. Coils would be smaller because of this. Therefore, two colliding chains would have a smaller volume to explore by segmental diffusion before reacting, and would take less time to react. Mahabadi and O'Driscoll 16 expanded the theoretical treatment of a to show quantitatively that the rate enhancement observed at ca. 1 to 10% polymer in solution was consistent with this reasoning. Horie and Mita 17 went on to show the limitation of this approach as the solvent quality becomes sufficiently poor to make it difficult for the chains to explore their combined coils. Experimental observation of this phenomenon in bulk polymerization has been made, both with respect to rate and degree of polymerization.18 It should be noted that the effect is small, seldom amounting to more than a 5 or 10% change in the expected rate. The most significant conclusions to be drawn from this theoretical treatment are: (i) any factor which decreases the segmental friction coefficient will increase kt; (ii) any factor which decreases the reacting coil size will increase /ct; and (iii) any careful kinetic study which changes the chain length or solvent character must take into consideration the effects embodied in the previous two points. While equation (7) gives an exact relation between the important physical parameters and the experimentally observable average termination rate constant, it is an awkward expression of the chain length dependence. Numerous workers have suggested and used an equation of the form K(n,m) = Ko(nmyii
(11)
where ki{nm) is the rate constant for termination between two radicals of lengths m and n and kt0 is a constant. Mahabadi 19 has shown that, if the exponent a is treated as a constant, the relation between the observed kt and Xn is /ct = ^ 0 g ( a ) X M - 2 a
(12)
where g(a) involves the gamma function of a. Experimental verification of the latter equation has often been found.
12.3 fct AT INTERMEDIATE CONVERSIONS: THE GEL EFFECT The gel effect is commonly accepted as beginning in that portion of a bulk or solution polymerization where the reaction rate increases and the molecular weight of the polymer formed also increases. Any complete model of the gel effect must describe not only the increased rate (and molecular weight) during the gel effect, but also a smooth transition between the low conversion rate and the gel effect. To that end it is useful to consider the question: when does the gel effect begin? Some authors have used simple conversion-time plots, and taken the upward curvature from linearity to indicate the onset of the gel effect; others have used the deviation from linearity of a plot of ln(K p )/[M]. Given the decrease in rate caused by the presence of polymer, it has also been suggested that the minimum in rate which can be observed defines the onset of the gel effect. As will be seen in the following discussions, most models employ an empirical or theoretical means of specifying the onset of the gel effect in order to trigger a 'switch' from low conversion kinetics to gel effect kinetics. Burnett and Duncan 20 were the first to postulate a set of equations to describe the gel effect. They suggested the existence of a critical concentration of polymer, which is dependent on the polymer molecular weight, and above which the propagating chains are immobile and incapable of ter minating each other. This approach gave a simple set of equations which were in accord with experimental observations. The equations demanded the existence of a 'switch', i.e. a concentration at which chains switched from being mobile and capable of terminating to being immobile and incapable of terminating. Furthermore, no allowance was made for the fact that small growing chains might terminate before they became immobile. Fifteen years later, Cardenas and O'Driscoll 21 postulated that two populations of propagating chains existed in a solution at a given concentration: those large enough to be entangled and those small enough to be freely mobile. The transition between the two chain sizes was established with a concept, 22 useful in viscosity measurements, that a critical constant, X c , exists, which is equal to the product of polymer concentration and chain length, and which determines the minimum chain size for entanglement. Assigning a rate constant kt to reaction between the entangled species, the
Free-radical Polymerization
166
0
20
40
60
80
100
Time (min)
Figure 5
Bulk polymerization of methyl methacrylate at 70°C (A, A) and 90°C (O, • ) ; solid lines are the model 21 predictions
normal kt to reaction between the unentangled and a geometric mean of the two for reaction between the unentangled and the entangled chains, it was shown that a two parameter model (Kc and kjkue) accurately predicted rate and molecular weight behavior over a wide range of conversions, initiator concentrations and temperature for methyl methacrylate (MMA) in bulk. This is illustrated in Figure 5 for MMA at two temperatures and two initiator concentrations. In addition to the successful description of the rate and molecular weight to be expected in polymerization, the model also led to a simple understanding of the reason why some monomers show a stronger gel effect than others. Put simply, it depends on the ratio k\jkv A monomer having a large value of such a ratio and polymerizing at a given rate will produce, in a given time, the same amount of polymer but with a higher molecular weight than will a different monomer, polymerizing at the same rate, with a lower value for that ratio. This explains why, for example, styrene has such a small gel effect compared to MMA. In spite of its ability to describe bulk polymerization, the Cardenas and O'Driscoll model failed to adequately describe polymerization in solution and, most importantly, was not intended to describe very high conversion. The success of the model coupled with its limitations led to several other models which took different physical approaches. One such approach to a useful description of the gel effect was taken by Hamielec and Marten 23 who introduced the concept of free volume into a semi-empirical model, They postulated a relation between the weight average molecular wight, M w and the free volume of the reacting system, Vf kt^M-115e\p(-A/V{)
(13)
The exponent of molecular weight was empirically chosen, A is a constant and the concentration dependence of kt is contained in Vf. This approach and the approach of O'Driscoll and Cardenas fitted the same data equally well. In a series of papers, Soh and Sundberg 24 combined the entanglement and free volume concepts so that, using a switch at some critical chain length, the value obtained was **,.») = K0{XJXnX2A
exp(-X/K f )
(14)
This approach gave a good fit to bulk polymerization data for six different monomers. At the same time, the work of de Gennes with scaling and reptation concepts 25 was showing great promise for describing physical phenomena of polymers in solution, particularly diffusion. Tulig and Tirrell 26 developed a model for kt using the concepts of reptation and a critical concentration, C*, above which chains interpenetrate, and a second critical concentration, C**, above which reptative behavior dominates. For chains of length N, C** is defined as KJN0,5, and above that con centration K = *,.min + * r (N* W(C**/C) 175 (15) where the ** superscripts refer to values at the transition concentration and kt min is a minimum value for the rate constant. This works quite well at concentrations ranging up to ca. 70%, but in that range another switch had to be introduced (C***) and another formulation for the rate constant was needed (see below).
Kinetics of Bimolecular Termination
167
In several publications, Ito 2 7 has also used scaling and reptation concepts and has proposed equations similar to those of Tulig and Tirrell. De Gennes has discussed bimolecular diffusioncontrolled reactions in general and the termination reaction of polymerization in particular in a pair of papers. 28 He concluded that kt should vary inversely with the square of chain length when reptation controls the process. In all the gel effect modeling described above, the authors sought to explain the deviations from the behavior expected for classical free radical polymerization kinetics by changes in kt alone. In fact, it has long been recognized that initiator efficiency and even the propagation rate constant, /cp, are reduced by increasing viscosity. Based on a series of papers on the kinetics of emulsion poly merization, it has been shown 9 that it is possible to use ESR measurements to obtain chain radical concentrations in emulsion particles. This enables the determination of rate constants with few assumptions. In this manner, it has been demonstrated 9 that kt can be adequately described by the Soh and Sundberg model. 24 This experimental development is particularly interesting since the technique involves seeded emulsion polymerizations, and it is quite possible to experimentally generate a given average chain length radical in the presence of a different chain length polymeric matrix. A similar approach in bulk or solution polymerization might be possible using the pulsed laser technique pioneered by Buback and co-workers 29 in high pressure ethylene polymerization. These could be powerful tools for testing models, especially when it is recognized that most models have used an average molecular weight of the polymerizing system, which drifts with conversion, to describe the gel effect on the rate constants of a separate distribution of radical chain lengths.
12.4
kt AT VERY HIGH CONVERSIONS
The description of a polymerization approaching very high conversion, ca. 90% or more, is of great industrial importance. However, judging by the published literature, it has (until recently) received very little detailed attention in a kinetic sense. Any description of the polymerization process at very high conversion will necessarily include, not only diffusion in viscous media, but also diffusive behavior in a glass, the validity of steady state assumptions and the possibility of depropagation becoming kinetically important. Such considerations imply that one cannot consider the termination rate constant in isolation, but rather that it is necessary to consider it together with the propagation rate constant, the initiator rate constant and its efficiency, and with the relation between the polymerizing systems' glass transition temperature, Tg, the conversion level and the temperature of polymerization as well as the Gibbs free energy of polymerization (i.e. reversibility). The most coherent approach to this problem has been made by Stickler and his colleagues30 who have utilized a combination of the free volume 23 approach for kp with the reactive diffusion concept of Schulzfor fct.31 A theoretical consideration of the course of a bulk polymerization carried out at a temperature below the polymer's Tg suggests that a polymerizing mixture would become glassy before reaching 100% conversion. The glass temperature of the reacting mixture may be calculated from free volume considerations to be determined by a weighted sum, computed from the volume fractions of each species in the reaction mixture and their glass transition temperatures. In this glassy state, it has been proposed that polymerization will cease, because monomer is unable to diffuse to a live chain end. Wunderlich and Stickler have presented 32 experimental evidence to show that MMA does not follow this simple idea. Figure 6 shows that the limiting conversion' observed at 80 °C, well below the Tg of PMMA, was a moderately strong function of initiator concentration. Further study revealed that polymerization continued, albeit at a very low rate, without cessation after the monomer-polymer reaction mixture had become glassy. From a practical, as opposed to a theoretical, point of view it is reasonable to ignore this very slow rate of polymerization (which is not understood) in order to model the industrially important final stage of polymerization. The following treatment does so, and explicitly includes kp as well as /ct, but does not include depolymerization or the influence of high conversion on the initiation process. Under conditions where the polymer concentration is very high, it might be expected that the reduced mobility of chains described in the gel effect becomes so great that they may be considered truly immobile. If that is the case, ordinary bimolecular reactions as illustrated by equation (1) can no longer occur. Instead, we must recognize that the active ends of two propagating chains can move through space by adding monomer, and so come into the same small volume element where they can terminate each other. This is termination by reactive diffusion and the rate constant for the process
Free-radical Polymerization
168 100
[ I ] 0 = 5.4xl0" 2 [I]0=5.3xl0"3
100
150
t Imin) Figure 6
Influence of initiator concentration on limiting conversion of methyl methacrylate at 80 °C; solid lines are the model predictions 32
can be written as proportional to the diffusion coefficient, DR CDR
(16)
where the constant C contains known geometric factors and (17)
where /0 and ns are the length and number of monomers in a statistical segment of polymer. Because the termination process is, under these conditions, defined in terms of the propagation rate constant, a precise description of kp at high conversion is needed. At low conversions, the propagation reaction is quite fast and chemically controlled with a rate constant fcp of the order of magnitude of 102 to 103. As the polymerization becomes more and more viscous, it is not unreasonable to suppose that the addition of monomer might be limited by its diffusivity to the immobile chain end. The rate constant is then given, in terms of the free volume theory, by kP^Voexp[-K*(l/^-l/Kf,0)]
(18)
where V* is an empirical constant and V{ 0 and Vi are the free volumes of monomer and reacting mixture, respectively. The latter can be calculated from the work of Kelley and Bueche.33 The empirical constant V* may be regarded as a critical fraction of the free volume required for monomer to diffuse to the chain end. Fitting of data to the above equations gave K* = 0.35, independent of temperature and initiator type or concentration. This model was combined with that of Hamielec and Marten 23 to produce a comprehensive description of the kinetics of bulk polymerization over a wide conversion range, extending to very high conversion. The major additions to the equations presented above were the introduction of two 'switches', Kf c r l and Kf cr2 — critical free volumes of the reaction mixture at which the gel effect and the very high conversion equations are used. V{>crl replaces VUQ in equation (18) above and V* was assigned the value 1.0. With four 'fixed' parameters and two adjustable ones this then gave an excellent fit to both rate and molecular weight data. Other approaches, such as using reptation theory, have been less successful in describing very high conversion. So it must be regarded as an important area in which the lack of general knowledge and insufficient data combine to make it necessary that more work be done. 12.5 TERMINATION IN COPOLYMERIZATION In copolymerization, the termination reaction is often discussed in terms of the three different reactions possible when two different chain ends combine PA + PA PB + PB PA + PB
dead polymer
(19)
Kinetics of Bimolecular Termination
169
where P A and P B represent active chains having terminal units of monomers A and B respectively. Rate constants for the three steps are usually designated /ctAA, /ctBB and ktAB respectively. The problem with this approach to describing the termination reaction in copolymerization is that the diffusion-controlled processes which operate in homopolymerization also operate in copolymerization. Designating rate constants in terms of the end unit of the chain is, therefore, the equivalent of assuming that that unit determines the speed of the rate-controlling diffusion process. This is physically incorrect, but, so far, no one has given a quantitative treatment which is correct as an alternative in all cases. What follows then is a brief summary of those treatments which have been given. The subject has been reviewed by Chiang and Rudin. 34 The well-known 40 factor', defined by « = W(*.AA*.BB)0-5
(20)
was originally introduced to describe rates of copolymerization.35 It can be determined from rates of copolymerization as a function of composition. Often the
(21)
proposing that the average kt was merely the average of the homotermination kt values weighted by the copolymer compositions ¥x and F 2 . This was extended by Ito and O'Driscoll,37 who averaged the segmental friction coefficients. All of the above has been done on systems of two monomers at low conversion. No one has yet treated binary copolymerizations taken to high conversion where the gel effect is important, or multicomponent polymerizations with three or more monomers, Also, all these treatments have assumed that the simple Mayo-Lewis model of copolymerization propagation is applicable. Recent work by Fukuda et a/.39 has demonstrated, by experimental determination of kt in copolymerization (using the rotating sector technique), that the Mayo-Lewis model may be inapplicable for styrenemethyl methacrylate. The apparent large maximum in kt as composition is varied may be an artifact 2
0
if -2 Ac"
O _
-4
-6 c,
C2
C 3 CE 100
Conversion (%)
Figure 7
Schematic variation of the termination rate 'constant1 with conversion in a bulk polymerization. The symbols are defined in the text
Free-radical Polymerization
170
of the model being assumed; they suggested a penultimate unit effect in copolymerization for this system might make the experimental kt data easily understandable.
12.6
CONCLUSIONS
From the work described above, it would appear that calling the parameter k{ a rate 'constant' is a gross abuse of the meaning of the word. Not only does kt begin to increase as soon as polymer appears in the reaction mixture, but at about 10% conversion it changes from being controlled by segmental diffusion, /cs, to being controlled by translational diffusion, kT. As the polymer con centration increases further, the combined effect of more polymer and higher molecular weight polymer further depresses /cT, until it is so slow that reactive diffusion, /cR, becomes dominant. This is shown schematically in Figure 7 where the numbers on the ordinate are very approximate. The conversion level C1 is the onset of the gel effect, C 2 the onset of reactive diffusion and C 3 the limiting conversion, at which the system becomes glassy and beyond which little polymerization is observed. Should C 3 be higher than the equilibrium monomer concentration, C E , the schematic representation would need revision to indicate that the apparent value of kt would go to zero at CE
12.7 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.
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S. W. Benson and A. M. North, J. Am. Chem. Soc, 1962, 84, 935. E. Trommsdorf, H. Kohle and P. Lagally, MakromoL Chem., 1948, 1, 169. R. G. W. Norrish and R. R. Smith, Nature (London), 1942, 150, 336. G. M. Burnett and H. W. Melville, Nature (London), 1945, 156, 661. K. O'Driscoll and H. K. Mahabadi, J. Polym. ScL, Polym. Chem. Ed., 1976, 14, 869. M. J. S. Dewar and C. H. Bamford, Nature (London), 1946, 158, 380. S. W. Benson and A. M. North, J. Am. Chem. Soc., 1958, 80, 5625. M. Morton, P. Saltiello and H. Landfield, J. Polym. ScL, 1952, 8, 111. M. J. Ballard, R. G. Gilbert, D. H. Napper, P. J. Pomery, P. W. O'Sullivan and J. H. O'Donnell, Macromolecules, 1986,19, 1303. R. Korus and K. O'Driscoll, in 'Polymer Handbook', ed. J. Brandrup and E. Immergut, Wiley, New York, 1975, p. 11. H. K. Mahabadi and K. O'Driscoll, J. Polym. ScL, Polym. Chem. Ed., 1977, 15, 283. T. Yasukawa and K. Murakami, Polymer, 1980, 21, 1423. M. Stickler, MakromoL Chem., 1986, 187, 1765. H. K. Mahabadi and K. O'Driscoll, MakromoL Chem., 1977, 178, 2629. W. A. Ludwico and S. L. Rosen, J. Polym. ScL, Polym. Chem. Ed., 1976, 14, 2121. H. K. Mahabadi and K. O'Driscoll, Macromolecules, 1977, 10, 55. K. Horie and I. Mita, Macromolecules, 1977, 11, 1175. J. Dionisio, H. K. Mahabadi and K. O'Driscoll, J. Polym. ScL, Polym. Chem. Ed., 1979, 17, 1891. H. K. Mahabadi, Macromolecules, 1985, 18, 1319. G. M. Burnett and G. L. Duncan, MakromoL Chem., 1962, 51, 154. J. Cardenas and K. O'Driscoll, J. Polym. ScL, Polym. Chem. Ed., 1976, 14, 883. R. Porter and J. Johnson, Chem. Rev., 1966, 66, 1. F. L. Marten and A. E. Hamielec, J. Appl. Polym. ScL, 1982, 27, 489. S. K. Soh and D. C. Sundberg, J. Polym. ScL, Polym. Chem. Ed., 1982, 20, 1299, 1315, 1331, 1345. P. G. de Gennes, 'Scaling Concepts in Polymer Physics', Cornell University Press, Ithaca, NY, 1979. T. Tulig and M. Tirrell, Macromolecules, 1981, 14, 1501. K. Ito, Polym. J., 1980, 12, 499. P. G. de Gennes, J. Chem. Phys., 1982, 76, 3316, 3322. M. Buback, H. Hippler, J. Schweer and H. Vogele, MakromoL Chem., Rapid Commun., 1986, 7, 261. M. Stickler, D. Panke and A. E. Hamielec, J. Polym. ScL, Polym. Chem. Ed, 1984, 22, 2243. G. V. Schulz, Z. Phys. Chem. (Frankfurt Main), 1956, 8, 290. W. Wunderlich and M. Stickler, Polym. Prepr., Am. Chem. Soc, Div. Polym. Chem., 1984, 25, (2), 7. F. N. Kelley and F. Bueche, J. Polym. ScL, 1961, 50, 549. S. S. M. Chiang and A. Rudin, J. Macromol. ScL, Chem., 1975, A9, 237. H. W. Melville, R. Noble and W. F. Watson, J. Polym. ScL, 1947, 2, 229. G. Bonta, B. M. Gallo and S. Russo, J. Chem. Soc, Faraday Trans. I, 1975, 71, 1727. K. Ito and K. F. O'Driscoll, J. Polym. ScL, Polym. Chem. Ed., 1979, 17, 3913. J. N. Atherton and A. M. North, Trans. Faraday Soc, 1962, 58, 2049. T. Fukuda, Y. Ma and H. Inagaki, Macromolecules, 1985, 18, 17.