Nonlinear Chain-Length Distributions in Free-Radical Polymerization

Ind. Eng. Chem. Res. 1997, 36, 1283-1301

1283

Nonlinear Chain-Length Distributions in Free-Radical Polymerization Stefano Fiorentino, Alessandro Ghielmi, Giuseppe Storti,† and Massimo Morbidelli* Laboratorium fu¨ r Technische Chemie LTC, ETH Zentrum, CAB C40, Universita¨ tstrasse 6, CH-8092 Zu¨ rich, Switzerland

By using results previously reported in the literature, the role and interactions of various termination mechanisms in determining the molecular weight distribution of nonlinear polymer chains produced by different processes are discussed. The analysis is focused on the instantaneous polydispersity of nonlinear chain distributions generated by chain transfer to polymer, cross-linking, and terminal double-bond propagation. Cumulative properties are considered, and criteria for the occurrence of polymer gelation for each of the above reactions are developed. Finally, the role of active radical compartmentalization, which is peculiar of emulsion polymerization, is discussed. 1. Introduction In the last 30 years, many studies have been devoted to the analysis of polymerization systems in which nonlinear polymeric chains are produced. The modeling of these systems is of fundamental importance, and a reliable prediction of the polymer chain microstructure (such as molecular weight distribution or tri- and tetrafunctional branch-point distribution) is very helpful in process design and optimization, since it directly affects the end-use properties of the material. Following the pioneering works of Flory (1941, 1953) and Stockmayer (1943, 1944) on stepwise polymerization, several statistical approaches have been developed and applied to addition or chain polymerizations, which are characterized by a chain lifetime much smaller than the reaction time (Gordon, 1963; Gordon and RossMurphy, 1975; Tiemersma-Thoone et al., 1991; Scranton and Peppas, 1990; Macosko and Miller, 1976; Miller and Macosko, 1976; Durand and Bruneau, 1982; Dotson et al., 1988; Dotson, 1992). However these methods, which provide a good characterization of the molecular architecture of a single chain, cannot be used directly to describe the kinetics of the MWD formation (Galina, 1990). For this reason, several models based on the classical kinetic approach, which naturally accounts for the dependence of polymer properties on the reaction conditions, have been developed. In particular, these models are based on population balance equations which describe the time evolution of each individual of the population in terms of the polymer chain length. Therefore, the model results in a countable system of ordinary differential equations whose dimension is directly related to the largest length that a polymer chain can reach in the reactor. Since in practical applications this can easily be as large as 104-105, a direct solution of the ordinary differential equations is impractical. Accordingly, several numerical techniques have been adopted to compute approximate solutions of these population balance equations: the method of moments (Tobita and Hamielec, 1988, 1989a,b; Zhu and Hamielec, * To whom correspondence should be addressed. Telephone: +41-1-6323034. Fax: +41-1-6321082. e-mail: [email protected]. † Dipartimento di Ingegneria Chimica e Materiali, Universita` degli Studi di Cagliari, Piazza d’Armi, 09123 Cagliari, Italy. S0888-5885(96)00484-8 CCC: $14.00

1992, 1993; Xie and Hamielec, 1993), Monte Carlo simulation (Tobita and Hamielec, 1989c, 1990a,b, 1992; Tobita, 1993a,b), and discrete weighted residual methods, such as Galerkin, collocation, and the more recent Galerkin h-p (Deuflhard and Wulkow, 1989; Canu and Ray, 1991; Wulkow, 1996). This is an important aspect, since, as we will see later, the availability of appropriate numerical techniques has limited in the past our ability to model the nonlinear MWD in free-radical polymerizations. The population balance approach to describe the kinetics of formation of linear and nonlinear chains in free-radical polymerizations has now reached a level of maturity. At least this is the case of what we will define as ideal polymerizations, where all intramolecular crosslinking reactions, such as internal cyclizations, and all the reactions yielding more than one growing center on a polymer chain (polyradicals) are neglected. The present analysis will be limited to this case. In particular, it is possible to derive simple equations for the instantaneous characteristics of the molecular weight distribution (MWD), which account for the presence of several reactions such as propagation, chain transfer to monomer, to chain-transfer agent, and to polymer, bimolecular termination by combination and by disproportionation, terminal double-bond propagation, crosslinking, and so on. Indeed, this result can be largely attributed to Prof. Archie Hamielec and his co-workers, who have developed over the years an impressive amount of knowledge in this area (cf. previously referred papers). The aim of this work is to start from the above results to revisit the mechanisms of formation, growth, and termination of polymer chains and their effect on the final MWD. This includes the analysis of three different mechanisms of nonlinear chain formation: long-chain branching by chain transfer to polymer, cross-linking, and propagation to terminal double bond. The interactions between each of these reactions and various termination mechanisms are discussed in detail. This analysis allows one to understand the role and the interplay of each termination and branching/crosslinking mechanism in determining the characteristics of the instantaneous MWD. These peculiar features are identified and elucidated on a solid physical ground, providing guidelines about the characteristics of the MWD of the instantaneously formed polymer which are not discussed elsewhere. © 1997 American Chemical Society

1284 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

The integral or cumulative MWDs have also been studied. Due to the large polydispersity values often encountered in nonlinear polymerizations, the numerical techniques usually adopted for solving the population balance equations are not adequate. A typical example is given by the numerical singularity found with the method of moments close to polymer gelation, due to the divergence of the higher order moments. These problems have been overcome using the recently developed numerical fractionation technique (Teymour and Campbell, 1994). By means of this technique, it has been possible to integrate the population balance equations also when gelation occurs. This allows one to analyze the role of the various termination and branching reactions on the final MWD of the polymer. Moreover, the prediction of the critical gelation point is obtained, thus allowing one to compare these results with the gelation criteria developed earlier in the literature. Finally, the role of active chain compartmentalization, which differentiates emulsion polymerization from all other polymerization processes, is discussed. In particular, the attention is focused on the effect of compartmentalization on the MWD produced in both linear and nonlinear free-radical polymerizations. 2. Modeling of Nonlinear Molecular Weight Distribution 2.1. Kinetic Scheme. The reactions characterizing the adopted kinetic scheme and the corresponding frequencies, f (given by the pseudo-first-order rate constant with respect to the active chain concentration), are reported in the following:

termination by combination ktc

R•n + R•m 98 Pn+m

f ) ktcRm

where R•n indicates a radical chain of length n, Pm a dead polymer of chain length m, Pd m a dead polymer of chain length m with a terminal double bond formed by either chain transfer to monomer or termination by disproportionation, M the monomer, T the chain transfer agent, and I the initiator. Two variables identify the double-bond distribution in the system: R, defined as the ratio between the number of double bonds in a terminated chain (responsible for cross-linking through internal double-bond addition) and that of its monomer units, and γ, defined as the fraction of terminated chains with one terminal double bond. In the present analysis both these parameters are assumed constant in time and independent of chain length, even though the precise calculation of both these quantities could be obtained by appropriate balances. As can be seen from the above kinetic scheme, the chain transfer to monomer mechanism considered in this analysis leaves a terminal double bond on the terminated polymer chain, while the resulting radical grows without terminal unsaturations. This is the case, for instance, for vinylidene fluoride and vinylidene chloride polymerizations (cf. Moad and Solomon, 1995). The terminal double bonds formed through this reaction are the same as those produced by termination by disproportionation, and, therefore, they are assumed to have the same reactivity. Besides this one, another chain transfer to monomer mechanism is possible, i.e., the hydrogen abstraction (cf. Flory, 1953)

initiation kI

I 98

R•n + M 98 R•nd + Pn 2R•1

propagation kp

• R•n + M 98 Rn+1

f ) kpM

chain transfer to monomer kfm

• R•n + M 98 Pd n + R1

f ) kfmM

chain transfer to modifier kfT

R•n + T 98 Pn + R•1

f ) kfTT

chain transfer to polymer kfp

R•n + Pm 98 Pn + R•m

f ) kfpmPm

cross-linking k* p

• R•n + Pm 98 Rn+m

f ) k* pRmPm

propagation to terminal double bond kd p

• R•n + Pd m 98 Rn+m

f ) kd p γPm

termination by disproportionation ktd

R•n + R•m 98 Pd n + Pm

f ) ktdRm

(1)

where R•1d is a monomeric radical with one double bond. This results in a terminated chain with no terminal insaturations and in a growing radical with a terminal double bond. Although, in general, an a priori discrimination is difficult, it is likely that these terminal double bonds have a reactivity different from the ones produced through the other mechanism (cf. Tobita et al., 1994). With respect to the kinetic model to be developed in the following, it is important to note that the chain transfer to monomer reaction (1) may lead to a number of terminal double bonds per terminated chain equal to or even larger than two and that these macromolecules can act as macro-cross-linkers. In this case the distributions for active and terminated chains should account not only for the length but also for the number of double bonds per chain. This approach has already been developed by Zhu and Hamielec (1994), but it is not adopted in the present work. Similarly, in the case where the chain transfer to monomer reaction leaves the terminal double bond on the terminated chain, the reaction conditions can produce again more than one terminal double bond per terminated chain, if chain transfer to polymer is also present. Accordingly, in the present work, we neglect both the case where these two reactions are simultaneously present in the system and the case where chain transfer to monomer occurs through the mechanism (1). With these limitations, the definition of γ as given above is justified, since a maximum of one terminal double bond per polymer chain is present. The selected kinetic scheme is a general one in bulk free-radical polymerization of nonlinear chains. Three

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1285

different branching mechanisms are accounted for: chain transfer to polymer, cross-linking, and propagation to terminal double bond. Moreover, as mentioned above, the assumption of ideal nonlinear chains is implicit in the kinetic scheme. This approximation holds as long as moderately branched or cross-linked chains are considered. The onset of gelation, where infinite-sized chains (usually with large rings) are formed, will also be considered, while keeping in mind that the kinetic model reliability is greatly reduced as soon as gelation is approached. About the reaction frequencies, it is worth noting that chain transfer to polymer can occur at any position along the terminated chain (usually it is the result of a hydrogen atom removal) and its frequency is therefore proportional to the degree of polymerization of the dead chain. Similarly, the cross-linking reaction can take place on any double bond along a terminated chain. Accordingly, the frequency must be proportional to the density of double bonds in the polymer chain, R. For instance, in the case of butadiene polymerization, if the double-bond consumption by cross-linking is negligible, there is one double bond per added monomer unit, i.e., R ) 1. With reference to the selected kinetic scheme it is possible to derive the population balance equations for active and terminated polymer chains. The mathematical derivation and the solution of these equations have been confined to the Appendix. In the following only the final results in terms of the main MWD instantaneous properties, i.e., number-average degree of polymerization, Mn, weight-average degree of polymerization, Mw, and polydispersity, Pd, are reported:

dµ1 ) dµ0

Mn )

Mw )

1 β d τ + - c* p - cp 2

(2)

dµ2 D D ) 2(1 + c* pMn Pd + dµ1 D D d D 1 + (cp + c* p)Mn Pd + cp Mn + τ + β + cp

D cd p Mn )

[

β

Pd )

(

]

(3)

]}

(4)

D D d D 1 + (cp + c* p)Mn Pd + cp Mn τ + β + cp

){

2

Mw β d D D ) τ + - c* 2(1 + c* p - cp pMn Pd + Mn 2 D D d D 1 + (cp + c* p)Mn Pd + cp Mn + τ + β + cp

D cd p Mn )

[

β

D D d D 1 + (cp + c* p)Mn Pd + cp Mn τ + β + cp

2

where µi (i ) 0, 1, 2) indicates the ith-order moment of the terminated polymer distribution, MD n the numberaverage chain length of the terminated polymer distribution, and PD d its polydispersity ratio. In order to better understand the analysis reported in the following section, which is aimed at a physical interpretation of the role of the reactions involved in determining the MWD, it is convenient to briefly comment on the above equations. It is evident that the main characteristics of the instantaneous MWD are entirely determined by only five dimensionless

parameters:

τ)

kfm kfTT ktdλ0 + + kp kpM kpM for monomolecular termination (5) β)

ktcλ0 kp M

for bimolecular termination

(6)

cp )

kfpµ1 kpM

for chain transfer to polymer

(7)

c*p )

cd p )

k* pRµ1 kpM

for cross-linking

(8)

kd p γµ0 kpM for propagation to terminal double bond (9)

where λ0 is the 0th-order moment of the active chain distribution. The above parameters have a straightforward physical meaning, since they represent the ratios between the frequency (or, equivalently, the inverse of the characteristic time) of each of the different termination and branching reactions and the propagation reaction. It is worth noting that the first two, i.e., τ and β, account for the termination mechanisms. In particular, τ includes the effect of what we will refer to as monomolecular termination, i.e., chain transfer to monomer or to chain transfer agent and disproportionation, whereby each terminating radical retains its character (i.e., its length) after termination. The other one, β, accounts for those termination events where the two radicals lose their individuality, i.e., termination by combination, which we refer to as bimolecular termination. Of course, since large values of one of these parameters would lead to very short chains, it is expected that in practical applications, where rather long chains are produced, these parameters should have values much smaller than 1. The remaining three parameters account for rebirth reactions which lead to the formation of nonlinear chains, since they involve a polymer chain previously formed. Accordingly, each of them includes some dependence on the MWD of the dead polymer present in the system at the time instant under consideration. 2.2. Instantaneous Molecular Weight Distribution. From eq 4, it can be seen that the instantaneous polydispersity ratio, Pd, depends upon the five kinetic parameters τ, β, cp, c*p, and cd p , accounting for monomolecular termination, bimolecular termination, chain transfer to polymer, cross-linking, and terminal doublebond propagation, respectively, and upon the characteristics of the preexisting terminated chains, i.e., cumulative number-average degree of polymerization and polydispersity ratio. The effect of the different termination mechanisms, i.e., mono- and bimolecular, and of the MWD of the terminated polymer on the instantaneous polydispersity in the case where a chain rebirth reaction is active, i.e., nonlinear chains are formed, is analyzed in the following. Before discussing the results obtained, a relevant remark has to be made. The term “instantaneous property” has a unique and clear meaning when dealing with linear chains, where it represents the molecular


Figure 1. Instantaneous polydispersity values as a function of the bimolecular termination by combination parameter β (model D parameter values: τ ) c*p ) cd p ) 0 and various cp values, Mn ) 1.35 × 104, and PD d ) 2).

property of the terminated polymer at a given time instant. On the other hand, in the case of nonlinear chains, where previously terminated chains are continuously reactivated, there is an exchange between active and (temporarily) terminated macromolecules. Therefore, the instantaneous properties of the dead polymer evaluated through eqs 2-4, which account for all these positive and negative contributions, may lead to anomalous results. As an example, from eq 2, it is apparent that negative values of Mn are possible when d τ + β/2 is less than c* p + cp . Of course, this is not acceptable from a physical point of view, and we have to recall that eq 2 is nothing but the ratio between eqs 50 and 49 reported in the Appendix. Thus, a negative value of the denominator means that more terminated chains are reactivated by cross-linking or terminal double-bond propagation than produced by the various termination mechanisms, thus leading to a negative net rate of terminated polymer formation. This is possible when the reaction producing chain nonlinearities is extremely active, in contrast with the main model assumption of moderately branched or cross-linked polymer. Hence, this is not the case dealt with here, d and the constraint c* p + cp < τ + β/2 is assumed verified in the rest of this work. Finally, note that, for this same reason, the amount of terminated polymer is always significantly larger than that of the active one, thus justifying the presence of only the moments of the terminated polymer in eqs 2-4, while neglecting the contribution of the growing chains. Thus, the analysis of the instantaneous properties reported below has to be examined by keeping in mind that it applies to systems with a moderate degree of chain nonlinearity. In this case the instantaneous average properties calculated through eqs 2-4 represent the characteristics of most of the terminated polymer actually formed. 2.2.1. Chain Transfer to Polymer. The case where cross-linking and terminal double-bond propagation (c*p ) cd p ) 0) are absent, and chain transfer to polymer is the only reaction allowing for chain rebirth is first considered. In Figure 1, the behavior of the instantaneous polydispersity, Pd, as a function of the bimolecular termination parameter, β, and without any monomolecular

Figure 2. Instantaneous polydispersity values as a function of the monomolecular termination parameter τ (model parameter D 4 values: β ) c*p ) cd p ) 0 and various cp values, Mn ) 1.35 × 10 , and PD d ) 2).

termination (τ ) 0) is shown for various values of the chain transfer to polymer parameter, cp, while assuming D constant values for both PD d and Mn , i.e., with the same preformed polymeric matrix. Note that the same values D for PD d and Mn are used in Figures 1 and 2 in order to facilitate the comparison between the results obtained. It can be observed that, as expected, Pd ) 1.5 for linear chains, while, in the presence of branching (cp > 0), Pd increases sharply with β, reaching an asymptotic value. Thus, in this case, if a sufficient termination by combination is present, the instantaneous polydispersity ratio becomes a function of chain branching only. The value of the polydispersity corresponding to the asymptotes in Figure 1, P∞d , can be readily obtained from the general expression (4) for β . cp:

P∞d )

1 + cpMD w (3 + cpMD w) 2

(10)

D D where MD w ) Mn Pd is the weight-average chain length of the preexisting terminated polymer. In Figure 2, the same results are shown in the case where only monomolecular termination is present, while termination by combination is absent (β ) 0, τ * 0). In this case the value Pd ) 2 is obtained in the absence of chain branching, while, again, polydispersity increases sharply with τ when cp > 0. Similarly to the case of termination by combination, the curves reach an asymptotic value for monomolecular termination when τ . cp, which is given by:

P∞d ) 2(1 + cpMD w)

(11)

Thus, it can be concluded that in both cases, when chain branching increases, the instantaneous MWD becomes broader. Moreover, as soon as mono- or bimolecular termination becomes significant, the instantaneous polydispersity becomes independent of these and it is entirely determined by chain branching only. However, it is remarkable that the effect of branching on Pd is determined by the underlying termination mechanism, i.e., mono- or bimolecular. This is clearly observed in parts a and b of Figure 3, where the asymptotic values of polydispersity, P∞d , have been


Figure 4. Instantaneous polydispersity values as a function of parameter β in the case of bimolecular termination by combination with a constant τ ) 1.56 × 10-11 (model parameter values: c*p ) cd p ) 0 and various cp values).

Figure 3. Instantaneous asymptotic polydispersity values as a function of parameter cp in the case of (a) bimolecular termination d -3 by combination (τ ) c* p ) cp ) 0; β ) 1.56 × 10 ) and (b) d -3 monomolecular termination (β ) c* p ) cp ) 0; τ ) 1.56 × 10 ) at various values of the cumulative weight-average degree of polymerization of the preformed polymer.

plotted as a function of the branching parameter, cp, for various MD w values in the case of bi- and monomolecular termination, respectively. A comparison between the two figures shows an essential difference in the behavior of the two systems: branching when coupled to bimolecular termination affects polydispersity much more than when coupled to monomolecular termination. This can be explained considering that the chain transfer to polymer reaction allows for successive rebirths more frequently the longer the dead chain is. Therefore, its effect increases when a combination event occurs, because the involved active radicals yield longer dead polymer chains, i.e., with a length given by the sum of the two colliding radicals. A particular feature in the behavior of the instantaneous polydispersity can be observed when monomolecular (by chain transfer or disproportionation) and bimolecular (by combination) terminations are present simultaneously. In Figure 4, the values of the instantaneous polydispersity are shown as a function of the bimolecular termination parameter, β, in the presence of a constant monomolecular termination, τ, and for various values of chain branching, cp. In the case of linear chains, cp ) 0, Pd decreases, as expected, from 2

to 1.5 as bimolecular terminations become dominant with respect to monomolecular terminations. The Pd behavior changes qualitatively when chain branching is present. In fact, in this case, polydispersity goes through a maximum, in the presence of moderate branching, while it increases monotonically with β to an asymptotic value, when branching is significant. This is a rather interesting feature and indicates that, for linear chains, termination by combination has the effect of narrowing the MWD with respect to monomolecular termination. An opposite behavior is found when chain branching is present: in this case bimolecular termination produces a MWD broader than the one obtained for monomolecular termination. On physical grounds, this can be explained by considering that combination leads to more uniform and longer chains by coupling together pairs of active radicals. On the other side, when branching is present, since this acts preferentially on the longest chains, the resulting MWD becomes broader. The dependence of the instantaneous asymptotic polydispersity ratio, P∞d , upon the preexisting polymer is evidenced in Figure 5. Here P∞d is shown as a function of the cumulative number-average chain length of the terminated polymer assuming constant its polydispersity value, PD d ) 2, and a constant chain transfer to polymer rate parameter, cp. Note that, in this case, since the total number of polymerized monomer units is constant, increasing values of MD n mean that the number of chains forming the existing polymer is decreasing. The two curves in Figure 5 represent the case where bi- (a) and monomolecular (b) terminations are present. Note that, since the polydispersity ratio of the preexisting polymer is kept constant (PD d ) 2), these curves allow one to compare the instantaneous asymptotic polydispersity, P∞d , with that of the preexisting polymer. It can be seen that, for increasing values of the number-average molecular weight of the preexisting polymer, the system produces MWDs significantly broader than the one of the polymer matrix. For instance, from a chain length distribution with MD n ) 60 000 the instantaneous polydispersity ratio is about 3-5 times larger than that of the preexisting polymeric matrix, depending upon the termination mechanism.


Figure 5. Instantaneous asymptotic polydispersity values as a function of the cumulative number-average degree of polymerization of the preformed polymer (model parameter values: cp ) d D -3 2.17 × 10-5; c* p ) cp ) 0; Pd ) 2; (a) τ ) 0, β ) 1.56 × 10 ; (b) β ) 0, τ ) 1.56 × 10-3).

This behavior can be explained on physical grounds by considering that in a polymerization system where a rebirth mechanism is present it can be assumed that the instantaneous MWD results from the sum of two contributions. One is the distribution of the linear chains which start with negligible chain length and propagate until a termination event occurs. The other is the nonlinear polymer distribution formed by the reactivation of the preexisting polymer, with numberaverage degree of polymerization MD n . The former distribution is directly originated by the radical initiator fragments or by a transfer to monomer or to transfer agent reaction, while the latter is originated from a polymer distribution with a number-average degree of polymerization significantly larger than zero. It can be expected that the more these distributions are far from one another, i.e., for increasing values of MD n , the more polydisperse is the distribution of the instantaneous polymer, resulting from the superimposition of the linear and the branched ones. A comparison between the curves in Figure 5 for mono- and bimolecular termination shows that the increase of the polydispersity ratio is stronger when termination by combination is active. This can be explained following the same arguments reported for parts a and b of Figure 3. It is worth noting that this conclusion does not apply for low MD n values, where the P∞d values obtained in the case where monomolecular termination is dominant are larger than those corresponding to combination. This happens because a smaller MD n value means that the reactivated chains are shorter. Hence, the chains instantaneously produced in these conditions are slightly branched or even linear, thus approaching the limit cases of Pd ) 2 for monomolecular and Pd ) 1.5 for bimolecular terminations. As soon as the rebirth mechanism becomes influent, i.e., for larger MD n values, this characteristic is overwhelmed by the increase of the polydispersity ratio due to the coupled effect of a mechanism leading to longer chains (combination) and a rebirth mechanism more effective on the longer polymer chains (chain transfer to polymer). This causes the cross of the two curves in the figure.

Figure 6. Instantaneous polydispersity values as a function of the cumulative polydispersity of the preformed polymer (model d D parameter values: cp ) 2.17 × 10-5; c* p ) cp ) 0; Mn ) 1.35 × 104; (a) τ ) 0 and various β values; (b) β ) 0 and various τ values).

In order to go deeply into the role of the molecular structure of the preformed polymer, in parts a and b of Figure 6 the dependence of the instantaneous polydispersity ratio upon the broadness of the MWD of the terminated polymer, i.e., its PD d , is reported in the cases of bi- and monomolecular termination, respectively. In both cases constant values of the numberaverage degree of polymerization, MD n , and of the chain transfer to polymer rate parameter, cp, have been considered. The dotted lines in the two figures represent the case where the instantaneous polymer has the same polydispersity as the preexisting one, i.e., P d ) PD d . In general, the chain transfer to polymer reaction is expected to produce a broadening of the MWD, since it further separates long chains from shorter ones. In both figures it can be observed that a limiting curve is reached for increasing values of the corresponding termination parameter. This is expected from the results of Figures 1 and 2, where an asymptotic behavior was found at large values of the termination parameter. In Figure 6a the polydispersity values obtained in the case where bimolecular termination is dominant are shown. It can be seen that, in the presence of chain branching, combination has the effect of producing rather heterogeneous chain-length distributions. Thus,


Figure 7. Instantaneous polydispersity values as a function of the bimolecular termination by combination parameter β (model D parameter values: τ ) cp ) cd p values, Mn ) p ) 0 and various c* 1.35 × 104, and PD ) 2). d

Figure 8. Instantaneous polydispersity values as a function of the monomolecular termination parameter τ (model parameter D 4 values: β ) cp ) cd p values, Mn ) 1.35 × 10 , p ) 0 and various c* D and Pd ) 2).

for sufficiently large β values, the polydispersity of the instantaneous polymer is always larger than that of the polymer already present in the system. As β decreases, the situation is reversed. This confirms the conclusion that large polydispersity values are obtained when combining the two independent events, i.e., termination by combination and chain transfer to polymer. In the case of monomolecular termination, shown in Figure 6b, the behavior is different. When the MWD of the terminated polymer becomes broad enough, the system, in which the branching reaction is not supported by a scale change in the polymer chain size, as in the previous case, is not able to maintain such a large heterogeneity. Accordingly, the instantaneous polymer is less polydisperse than the one already present in the system. 2.2.2. Cross-Linking. The case where chain transfer to polymer and terminal double-bond propagation reactions (cp ) cd p ) 0) are absent, the cross-linking reaction being the only kinetic mechanism responsible for chain rebirth, is now considered. Figure 7 shows the behavior of the instantaneous polydispersity ratio Pd as a function of the bimolecular termination rate parameter (β * 0, τ ) 0), for various values of the cross-linking kinetic parameter c*p and D constant PD d and Mn . The behavior is analogous to the one discussed above in the case of chain transfer to polymer (Figure 1): all the curves reach an asymptotic value as soon as bimolecular termination becomes significant. Therefore, in the case of moderately crosslinked polymers, Pd may be considered independent of the termination by combination rate constant and the corresponding asymptotic value has the following analytical expression, obtained from eq 4 for β . c* p:

cal expression for P∞d is readily obtained from eq 4 for τ . c* p:

3 D 2 P∞d ) (1 + c* pMw) 2

(12)

Also in this case, for linear chains, the expected Pd value of 1.5 is obtained. In Figure 8, similar results are reported in the case where monomolecular termination is present alone (β ) 0, τ * 0). The curves show again the characteristic asymptotic behavior; accordingly, the following analyti-

D 2 P∞d ) 2(1 + c* pMw)

(13)

which, for linear chains, leads to P∞d ) 2. Depending upon whether cross-linking or chain transfer to polymer is present, a significant qualitative difference can be observed. In particular, here, the asymptotic polydispersity value in the presence of monomolecular termination is always larger than that in the presence of bimolecular termination for all considered values of c* p. The reason for this lies in the particular effect of the cross-linking reaction on the polymer chain structure. When cross-linking occurs between an active and a terminated polymer chain, the length of the resulting cross-linked growing chain is given by the sum of the units of the two original chains. Hence, cross-linking itself is a rebirth reaction combined with a mechanism allowing for an increase in the chain dimensions, thus resulting in a significant MWD broadening. This effect is contrasted by bimolecular termination by combination, which, although increasing the chain sizes, results in narrowing the MWD by adding together pairs of active chains. In parts a and b of Figure 9 the asymptotic Pd values are shown as a function of the cross-linking parameter D c* p for various values of Mw in the case of bi- and monomolecular termination, respectively. This allows one to analyze the important role of the weight-average degree of polymerization of the preformed polymer. It appears that the broadness of the instantaneous MWD depends upon the molecular weight of the preexisting polymer through the rebirth reaction, in this case crosslinking. The comparison between the values of the two figures confirms that monomolecular terminations lead always to larger polydispersity ratios with respect to bimolecular terminations. The dependence of the instantaneous polydispersity ratio upon the length of the preexisting polymer has been evidenced in Figure 10, where P∞d is shown as a function of the cumulative number-average chain length of the terminated polymer with constant polydispersity ratio, PD p. Similarly to the case of d , and constant c*


Figure 10. Instantaneous asymptotic polydispersity values as a function of the cumulative number-average degree of polymerization of the preformed polymer (parameter values: c*p ) 2.17 × D -3 10-5; cp ) cd p ) 0; Pd ) 2; (a) τ ) 0, β ) 1.56 × 10 ; (b) β ) 0, τ ) 1.56 × 10-3).

Figure 9. Instantaneous asymptotic polydispersity values as a function of parameter c* p in the case of (a) bimolecular termination -3 by combination (τ ) cp ) cd p ) 0; β ) 1.56 × 10 ) and (b) -3 monomolecular termination (β ) cp ) cd p ) 0; τ ) 1.56 × 10 ) at various values of the cumulative weight-average degree of polymerization of the preformed polymer.

chain transfer to polymer, the effect of the numberaverage chain length is relevant: larger average molecular weights provide broader instantaneous MWDs. Again, it appears that, when bimolecular termination is dominant, polydispersity ratios smaller than when monomolecular termination is dominant are obtained. This behavior is opposite to that observed in the presence of chain branching (cf. Figure 5). The effect of the polydispersity of the preexisting polymer on the instantaneous polydispersity is analyzed in parts a and b of Figure 11, in the case of bi- and monomolecular termination, respectively. Here, constant values for the number-average molecular weight, MD p, have been n , and for the cross-linking parameter, c* considered. Similarly to the case of branching by chain transfer to polymer, a limiting curve is approached at large values of the corresponding termination parameter. However, differently from the previous case, all the curves lie above the diagonal for both the termination mechanisms. This means that in a polymerization process where cross-linking is active, as a result of the simultaneous presence of a size increase and a rebirth mechanism through the same reaction, a continuous broadening of the MWD is observed, the instantaneous

Figure 11. Instantaneous polydispersity values as a function of the cumulative polydispersity of the preformed polymer (model D parameter values: c*p ) 2.17 × 10-5; cp ) cd p ) 0; Mn ) 1.35 × 104; (a) τ ) 0 and various β values; (b) β ) 0 and various τ values).

polydispersity ratio always being larger than that of the preformed polymer. Also in this case the limiting curve


for bimolecular termination lies always below the one for monomolecular termination. 2.2.3. Terminal Double-Bond Propagation. The case of terminal double-bond propagation is qualitatively very close to that of the last examined case. In fact, the behavior of the instantaneous polydispersity ratio as a function of the termination parameters is similar to the one discussed above in Figures 7 and 8, for cross-linking. The asymptotic values for this case are obtained from eq 4 for β and τ . cd p . Therefore, when bimolecular termination is active, the following equation holds:

3 D 2 P∞d ) (1 + cd p Mn ) 2

(14)

while, when monomolecular termination is dominant: D 2 P∞d ) 2(1 + cd p Mn )

(15)

Notice that, as in the cross-linking case, the terminal double-bond propagation is a rebirth reaction combined with a mechanism allowing for a size increase in the chain dimensions. The effect of bimolecular termination by combination is therefore the same as that observed in the cross-linking case: it mixes up the chain lengths, giving an instantaneous MWD narrower than the one obtained in the case of monomolecular termination. This is evident when comparing the analytical expressions of the asymptotic values. The important difference between the cross-linking and terminal double-bond propagation mechanisms is that, while the rate of the former is proportional to the number of monomer units in the terminated chains, the rate of the latter is simply proportional to the number of terminated chains. In this case the polymer properties depend only upon the number-average degree of polymerization of the preexisting polymer and not upon the broadness of its distribution, i.e., its polydispersity ratio. In fact, if the rebirth reaction is not selective with respect to chain length, the reaction occurs with the same probability on short and long chains and the broadness of the MWD of the instantaneous polymer is not affected at all. On the other side, the effect of the number-average degree of polymerization of the preexisting polymer is the same as that discussed earlier: increasing values of MD n mean more difference in size between the linear and the branched chains, thus resulting in a broader instantaneous molecular weight distribution. 3. Cumulative Properties The main difficulty encountered in modeling polymerization processes involving branching or cross-linking reactions is the possible formation of nonlinear chains with practically infinite size, the so-called polymer gel fraction. One of the characteristic properties of these materials is solvent insolubility. From the application point of view, possible objectives are to produce very highly cross-linked materials or, else, to completely avoid the formation of such a polymer fraction. Therefore, a key parameter for a proper design of the optimal reaction path is the so-called gel point, i.e., the conversion or time value corresponding to the onset of the polymer gelation. In the following, several qualitative criteria for estabilishing the occurrence of gel formation depending upon the operating reaction mechanism are discussed. Then,

quantitative relationships for estimating the gel point are reported. Finally, in the case of cross-linking, a comparison is made between the gel criterion obtained from the kinetic model described above and the one provided by the statistical approach used by Flory (1947). 3.1. Criteria for the Occurrence of Gelation. From a mathematical point of view, the gel point can be identified as the time or conversion value at which the molecular weight distribution moments of order larger than 1 diverge. In fact, the polymer is constituted by an extremely broad distribution of chains, ranging from the relatively low degree of polymerization of the sol polymer fraction to the huge chain lengths of the gel fraction. In these conditions, the second and higher order moments reach values so large as to result in a practical divergence. This behavior may be exploited to identify the gel point, which may be estimated as the time or conversion value where the second-order moment goes to infinity. In the following, some gel criteria are derived, taking advantage of the reported moment divergence following the procedure reported by Zhu and Hamielec (1992). For this, the definitions of the dimensionless parameters for chain transfer to polymer, cross-linking, and terminal double-bond propagation given by eqs 7-9, respectively, have been modified so as to extract the moments of the terminated polymer from the corresponding rate parameters:

cjp )

cp kfp ) µ1 kpM

cjp* )

jcd p )

for chain transfer to polymer

c* k*pR p ) µ1 kpM

(16) for cross-linking

(17)

cd kd p pγ ) µ0 kpM for propagation to terminal double bond (18)

In order to obtain analytical expressions for the gel criteria, all dimensionless quantities, τ, β, cjp, cjp*, and cjd p have been assumed constant during the polymerization. This is, of course, an approximation because of the continuous variation of the reaction conditions, such as monomer and radical concentration. However, the resulting criteria are believed to give a significant physical insight on the role of the various rebirth and termination mechanisms in determining the occurrence of gelation. Moreover, it has been shown that, at least when gelation occurs at low conversion values (1020%), the above approximation leads to rather reliable and accurate criteria (Tobita and Hamielec, 1988). 3.1.1. Chain Transfer to Polymer. In this case, the second-order moment depends upon the extent of the branching reaction and of the dominant termination mechanism. Therefore, applying the LCA, eq 51 of the Appendix reduces in this case to:

[

(

1 + cjpµ2 dµ2 1 + cjpµ2 ) Rp 2 +β dt τ + β + cjpµ1 τ + β + cjpµ1

)] 2

(19)

while the integration of the first-order moment equation provides µ1 ) Rpt. In order to consider separately the effects of the different termination mechanisms combined with the chain transfer to polymer reaction, eq


19 has been integrated in the two limiting cases of β . τ and vice versa. When monomolecular termination is the dominant mechanism, the cumulative expression for µ2 becomes

µ2 )

[(

cjpRp 1 1+ cjp τ

) ] 2

-1

(20)

As is clearly shown by this equation, the second-order moment has no possibility to diverge. This behavior strengthens that already stated in section 2.2.1: when the chain transfer to polymer is not coupled to a termination mechanism able to increase the dead chain dimensions, the broadness of the MWD remains finite, and it can be concluded that branching alone is not capable of causing gelation. The situation is completely different when a chain-connecting mechanism is operating. In particular, in the case of bimolecular termination by combination, the following expression for µ2 is obtained:

[

(

)

cjpRp 2 t 1+ 1 β µ2 ) -1 cjp cjpRp t 1β

]

(21)

By imposing the divergence of the second-order moment, the gel point in this case can be estimated as:

tc )

β cjpRp

(22)

where tc is the critical time for gelation. This same expression has already been reported by Zhu and Hamielec (1992) in terms of conversion. As expected, systems with a higher tendency to branching reach the gel point at shorter times, while a stronger termination mechanism, which reduces chain length, delays gelation. The difference between these two cases once more confirms the key role of a chain-connecting mechanism in determining gelation when combined with branching reactions; in other words, we may say that this is a necessary condition for gelation. 3.1.2. Cross-Linking. In the previous section it has been evidenced that a polymerization system where a chain-length proportional rebirth mechanism is coupled to a chain-connecting reaction may lead to gelation. The cross-linking reaction has both these characteristics: it is a rebirth mechanism whose rate is proportional to chain length and it connects polymer chains. Therefore, analyzing the behavior of the second-order moment, a divergence point is expected in both cases where bi- and monomolecular terminations are dominant. In fact, combination is no longer the only mechanism allowing for a scale increase in the chain size. The corresponding moment equation in this case can be written as:

(

dµ2 1 + cjp*µ2 ) Rp(2τ + 3β) dt τ+β

)

2

(23)

The integration of eq 23 in the case of monomolecular termination provides

µ2 )

2Rp/τ 2cjp*Rp t 1τ

and the critical time for gel formation results:

(24)

tc )

τ/2 jcp*Rp

(25)

As expected, at large values of the cross-linking rate parameter, a very rapid gelation is possible. Moreover, this result confirms that, since cross-linking is itself a connecting reaction, gelation can occur also in the case of monomolecular termination. Finally, when considering the case of termination by combination, by integrating eq 23, the following expression for the cumulative second-order moment is obtained:

µ2 )

3Rp/β 3cjp*Rp t 1β

(26)

and, for the gel point:

tc )

β/3 jcp*Rp

(27)

Note that both eqs 25 and 27 have been reported by Zhu and Hamielec (1992) in terms of conversion. 3.1.3. Propagation to Terminal Double Bond. The case of terminal double bond is particularly interesting to point out the importance that a rebirth mechanism independent of chain length has on gel formation. The rate of the terminal double-bond propagation reaction is simply proportional to the concentration of the terminated chains ending with a double bond, and it does not depend upon the length of the terminated chain. This means that long and short polymer chains have the same probability to undergo rebirth through terminal double-bond propagation. In a system where propagation to a terminal double bond is the dominant branching mechanism, the secondorder moment equation reduces to:

(

1 + cjd dµ2 p µ1 ) Rp(2τ + 3β) dt τ+β

)

2

(28)

Notice that, differently from eq 23, the time derivative of µ2 is proportional to µ1 because of the chain-length independence of the terminal double-bond propagation reaction. As in the case of chain transfer to polymer, this equation can be readily integrated under the same simplifying assumptions. When monomolecular termination is the dominant chain stoppage mechanism, the following expression is obtained:

µ2 )

2Rpt 2 [1 + cjd jd p Rpt + (c p Rpt) ] τ

(29)

Accordingly, it is apparent that the second-order moment increases during the reaction but never diverges. Thus, we can conclude that gelation never occurs in such a system. It is useful to notice that, as discussed earlier, because of the transfer to the monomer mechanism chosen, the systems under consideration can produce at most one terminal double bond per terminated chain. In other words, no macro-cross-linker (a polymer chain with two or more terminal double bonds) is present in the reaction system. However, if chain transfer to polymer were also present, then more than one terminal double bond per chain would be possible even with the


chain transfer to monomer reaction chosen and, therefore, the formation of these macro-cross-linkers may eventually lead to gelation (Zhu and Hamielec, 1994). A result similar to the previous one is obtained in the case of bimolecular termination by combination, where the second-order moment is given by:

µ2 )

3Rpt 2 [1 + cjd jd p Rpt + (c p Rpt) ] β

(30)

Hence, the second-order moment has the same expression as in the previous case and the polymerizing system cannot reach the gel point. This is a rather interesting result. In fact, although the terminal double-bond propagation is a rebirth and a chain-connecting mechanism at the same time, its rate depends only upon the concentration (and not the mass) of the terminated polymer and gelation cannot occur. This result has already been achieved through a more detailed approach by Zhu and Hamielec (1994). 3.2. Comparison with Flory’s Gel Criterion. A treatment for gel formation was developed and a gel criterion was obtained in the case of a cross-linking system by Flory (1947). Flory’s criterion for gel formation assumes that gel occurs as soon as the polymer chains reach a certain value of cross-linking density. In particular, defining the cross-linking density F as the ratio between the cross-linked, ν, and the polymerized monomer units, µ1, gelation occurs when:

FMwP ) 1

(31)

where MwP is the weight-average chain length of the primary chains, i.e., the weight-average chain length that would result if all cross-linkages were severed. Under the assumptions used in the previous section (β, τ, c*p ) constant), this criterion leads to the same results obtained in section 3.1.2. In fact, since the crosslinked polymer units are twice the number of crosslinkages, ν ) 2nC, from eq 54 of the Appendix the moles of cross-linked units as a function of time are given by: 2 ν ) (k* pRλ0Rp)t

(32)

Since the first-order moment can be expressed as µ1 ) Rpt, the cross-linking density becomes

F ) k* pRλ0t

(33)

For example, when considering the case of monomolecular termination, for which the weight-average chain length of the primary chains is 2/τ, Flory’s criterion provides

tc )

τ/2 τ/2 ) k*pRλ0 cjp*Rp

(34)

which is identical to eq 25. Similarly, in the case of termination by combination, Flory’s criterion leads to the same expression (27) obtained above. The convergence between the gelation criteria derived through the two examined approaches is indeed a significant result, since the two treatments are very different. In Flory’s treatment, in fact, the gel criterion has been obtained only through “topological” considerations, whatever is the reaction path. Nevertheless, the final results are in complete agreement with the kinetic treatment.

Figure 12. Gel fraction as a function of conversion for various values of the cross-linking rate coefficient (parameter values as in Table 1 but cp ) cd p values). Dotted line: p ) 0 and various k* Flory’s model. Solid line: kinetic model.

Finally, a comparison has been made also in terms of a gel polymer fraction as a function of conversion. According to Flory’s approach, the gel polymer fraction (i.e., the weight fraction of the gel polymer with respect to the overall sol and gel polymer amount) can be expressed as: ∞

wg ) 1 -

wp(r) (1 - Fwg)r ∑ r)1

(35)

where wp(r) is the weight chain-length distribution of the primary chains, evaluated assuming no cross-linking in the system. This has been compared with the gel fraction predicted by numerically solving the kinetic model using the numerical fractionation technique (Teymour and Campbell, 1994). Accordingly, the polymer chain distribution is partitioned into several classes or generations. The fractionation rules are chosen so as to guarantee a suitable selection of chains depending upon their size. Thus, each generation is expected to have a much narrower MWD than the overall polymer. Moreover, a finite number of generations, NG, is used, typically between 5 and 10, being generation NG + 1 constituted by so large chains that it can be defined as the gel polymer. The molecular weight properties of the sol polymer are evaluated as weighed sums of the same properties of the individual NG generations. If the fractionation rules are effective, the MWD of each generation can be reliably reconstructed using the first three moments only. Moreover, the problem of the numerical divergence of the moments of the overall polymer is encompassed, because the calculations are restricted to the moments of the sol fraction of the polymer, the gel fraction being evaluated as the difference between the overall amount of polymer and the sum of the first-order moments of each sol generation. In this work, the fractionation rules have been selected by extending the same approach described by Teymour and Campbell (1994) to the case of cross-linking reactions. In all examined cases, no more than seven generations were required. The comparison between the two model results is shown in Figure 12, where the gel weight fraction has been plotted as a function of conversion. The parameter values used in the calculations are reported in Table 1.

1294 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 Table 1. Numerical Values of the Model Parameters Used for the Cumulative Calculations in Homogeneous Polymerization parameter

value

parameter

value

(mol/cm3)

8.43 × 1.9 × 10-5 9.07 1.18 × 10-6

[cm3/(mol

2.59 × 106 0 5.97 × 109

M0 I0 (mol/cm3) kfm [cm3/(mol s)] kI (1/s)

10-3

kp s)] ktd [cm3/(mol s)] 3 ktc [cm /(mol s)]

The solid line is the prediction of the kinetic model, while the dotted line is the behavior predicted by Flory’s theory. In the last case, the onset of the gel fraction is predicted by eq 31, while in the kinetic model the gel point is identified as the conversion corresponding to a calculated gel fraction equal to 0.001. The agreement is excellent for low gel point conversion, but, for larger values, the kinetic model underestimates the value predicted by Flory’s treatment. This can be explained by noting that the different microstructures of chains produced at different conversion values are properly accounted for by the kinetic approach, while it is not by a statistical treatment such as Flory’s, where the polymeric matrix is always characterized in terms of time-average properties. As a consequence, the polymer heterogeneity is underestimated and the gel point erroneously delayed. 4. The Case of Emulsion Polymerization The treatment reported in the previous sections applies to all homogeneous and to some heterogeneous polymerization processes, such as suspension. However, in the case of emulsion polymerization, some peculiarities arise, which should be taken into account when describing the MWD of linear and nonlinear polymers. In particular, these include (i) active chain compartmentalization and (ii) a large polymer to monomer ratio in the reaction locus throughout most of the polymerization. In emulsion polymerization the reaction is carried out inside very small particles (diameter of the order of 0.1 µm). The number of active chains inside each particle is usually very low, particularly in the so-called zeroone systems, where no more than one radical chain is present in any particle. This results in the segregation of the active chains, usually referred to as chain compartmentalization, which affects the chain life and, consequently, its molecular weight. Since compartmentalization reduces the probability of active radicals to meet each other, all bimolecular termination rates are significantly decreased so that polymerization rates and molecular weights significantly higher than in the other polymerization processes are obtained. Note that, in general, no segregation is observed for the monomer and the terminated polymer chains. In fact, since the monomer has high diffusivity and the number of dead polymer chains accumulated in each particle is large (usually between 100 and 1000), the same concentration values for both monomer and terminated polymer can be used for all polymer particles in the system. Only in the case of highly cross-linked chains, the limiting condition of chains with size comparable to that of the particle can be reached (Tobita and Yamamoto, 1994a), and, then, compartmentalization of terminated polymer should be considered. This, however, is certainly not the case for the ideal nonlinear polymerizations considered in this work. A second aspect refers to the polymer to monomer ratio in the polymer particles. During the first part of

the reaction, unreacted monomer droplets are present in the reacting system and maintain the reacting particles under monomer saturation conditions. Accordingly, in the reaction locus, the polymer to monomer volume ratio is constant and of the order of unity even since the beginning of the polymerization. Only after monomer droplets disappearance, the reaction proceeds, consuming the residual monomer inside the particles and thus increasing the polymer to monomer ratio similarly to what happens in other polymerization processes. Therefore, emulsion polymerization is characterized, since the beginning of the reaction, by polymer concentration values in the reaction locus higher than in all other polymerization processes. This, obviously, has a strong effect in promoting all reactions involving the terminated polymer chains, such as chain transfer to polymer and cross-linking. In order to compute the MWD of polymers produced in emulsion, the two aspects discussed above need to be accounted for. This significantly complicates the population balances describing the evolution of the polymer chains, particularly because of the compartmentalization which forces us to follow the growth of the chain while keeping track of the number of radicals which at any given time are present in the same particle. It is not possible to introduce at this point the models which can be used for describing these processes, since they require a heavy mathematical treatment. Only the physical concepts on which such models are based will be discussed here, while the interested reader can refer to Lichti et al. (1980) and Ghielmi et al. (1996) for further details. The main idea for treating compartmentalization, introduced by Lichti et al. (1980) for linear chains, is based on the formulation of the population balance equations of two active chain distributions, the so-called singly and doubly distinguished particles. The latter is needed to properly compute the length of chains terminated by bimolecular termination by combination, which requires one to follow the life of both active chains growing inside the same particle which will eventually terminate by combining each other. This same concept has been extended to the case of nonlinear chains, where a new internal coordinate of the distribution has been introduced: the chain prelife (Ghielmi et al., 1996). This accounts for the number of monomer units added by a polymer chain before its last rebirth. In order to obtain the overall degree of polymerization, chain prelife has to be added to the length of the last born branch, i.e., the current life. This approach can be applied to the case of chain nonlinearities due to chain transfer to polymer, terminal double-bond propagation, and crosslinking. 4.1. Instantaneous Properties. Let us now analyze the behavior of MWDs produced in emulsion by focusing on its main peculiarity, i.e., active chain compartmentalization. In particular, we analyze the behavior of the instantaneous polydispersity as a function of the average number of active chains per particle, n j , which in our computations has been changed by considering different frequencies of the radical entry in the polymer particles. It is worth noting that for large values of n j , typically above 5, the number of active radicals per particle is so large that compartmentalization has no more effect, and the emulsion behaves like a noncompartmentalized (or pseudobulk) polymerization. On the other hand, for n j approaching zero, the total number of radicals in the system is so small that

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1295 Table 2. Numerical Values of the Model Parameters Used for the Instantaneous Calculations in Emulsion Polymerization parameter

value

parameter

value

(mol/cm3)

5.7 × 1.3 × 10-3 10 2.6 × 105 1.16 × 109 1.16 × 109

(cm3)

1.21 × 10-15 1 2.38 × 10-7 3.22 × 10-3 105.3

Mp k (1/s) kfm [cm3/(mol s)] kp [cm3/(mol s)] ktc [cm3/(mol s)] ktd [cm3/(mol s)]

10-3

VP R µ0 (mol/cm3) µ1 (mol/cm3) µ2 (mol/cm3)

chain compartmentalization does not play any role, since bimolecular termination is negligible in these conditions. Thus, we can conclude that the effect of compartmentalization vanishes in limits of both very small and very large values of the average number of active radicals per particle. Of course, any deviation from the behavior at these limiting n j values which arises in the range 0.5 < n j < 1 is due to compartmentalization. The adopted numerical values of the model parameters are summarized in Table 2, unless explicitly otherwise stated. The following dimensionless quantities, representing the ratio between the frequencies of branching and cross-linking, respectively, and the sum of combination and disproportionation frequencies have been introduced:

q)

kfpµ1 c

(36)

q* )

k*pRµ1 c

(37)

Figure 13. Instantaneous polydispersity values as a function of the average number of active chains per particle at various q values and q* ) 0: (a) bimolecular termination by combination only; (b) bimolecular termination by disproportionation only (model parameter values as in Table 2).

where c is defined as c ) (ktc + ktd)/(2NAVP), NA being Avogadro’s number and VP the particle volume. 4.1.1. Chain Transfer to Polymer. The calculated instantaneous polydispersity as a function of n j is shown in Figure 13a for a system where chain rebirth is due to chain transfer to polymer only and the dominant termination is by combination. Two sets of curves are reported: the solid lines correspond to the complete model, while the dotted lines are obtained by neglecting the active chain compartmentalization, i.e., by forcing the bulk model, discussed in the previous sections, to describe the molecular weight of a polymer produced in emulsion. As discussed above, these two curves are expected to converge for both low and large n j values, where the compartmentalization effect vanishes. By inspection of the curve for q ) 0, i.e., the case of linear chains, it is evident that, as the number of active chains per particle increases, the instantaneous polydispersity decreases from 2, the typical value for monomolecular termination, to 1.5, which is instead typical of bimolecular terminations. This is quite reasonable since, as the number of active radicals per particle increases, bimolecular mechanisms become dominant with respect to monomolecular ones. It is worth noting that, also in the case where only bimolecular terminations are present, as n j decreases, the prevailing termination mechanism is always monomolecular. The explanation of this in the case of emulsion polymerization requires some elaboration. In the first place it is to be noted that the average number of active chains per particle approaches zero when a significant desorption of the active radicals back in the aqueous phase is present in the system. This phenomenon is typical of emulsion polymerizations involving relatively water-soluble monomers, and it is usually associated with a chain-transfer reaction to

monomer followed by the diffusion of the newly formed radical in the aqueous phase. Accordingly, when radical desorption is important, i.e., n j tends to zero, the instantaneous polydispersity for linear chains equals 2, as is also apparent in Figure 13a. As n j increases toward 0.5, the events of termination by combination become more frequent and they are indeed dominant for n j = 0.5. This value could, in fact, be obtained in the absence of radical desorption with an infinitely fast termination by combination. However, from Figure 13a we see that the instantaneous polydispersity (continuous curve) remains equal to 2 and not to 1.5, as expected for termination by combination. This is explained by considering that for n j equal to about 0.5 we have substantially a zero-one system, and if a new radical enters a particle which already contains one radical, the two immediately terminate. Therefore, the new radical has no time to grow and the length of the produced terminated chain is substantially equal to that of the longer radical, just like in a monomolecular termination, even though the actual termination is by combination. Only for values of n j larger than 0.5, the active chains can accumulate in the particle and the dead chains are produced by the combination of two radicals which have been growing together in the same polymer particle. This is the typical mechanism of termination by combination and then, as shown in Figure 13a, as n j increases, the instantaneous polydispersity approaches the expected value of 1.5. When comparing the results of the compartmentalized and pseudobulk systems (continuous and dotted curves, respectively), it is seen that the two curves representing the polydispersity values are identical, but the one for the compartmentalized system is shifted toward larger n j values. As discussed above, this is because in a compartmentalized system, in order for the bimolecular terminations to overtake the monomolecu-


lar ones, the average number of active chains per particle should be higher than 0.5, so as to have enough particles containing at least two radicals. Accordingly, the decrease from the value Pd ) 2 occurs immediately for n j > 0 in a noncompartmentalized system, while, in a compartmentalized one, it occurs only for n j > 0.5. When examining the curves at q > 0, it can be seen that the effect of chain transfer to polymer is an expected broadening of the instantaneous MWD. The same behavior described above for bulk systems is observed, i.e., bimolecular termination in the presence of branching does not lead to larger polydispersity values than monomolecular termination unless branching is strong enough, with compartmentalization having again the only role of shifting the Pd values toward larger n j. Let us now analyze the case shown in Figure 13b, where termination by disproportionation is dominant. As far as linear chains are concerned (q ) 0), the polydispersity values for the pseudobulk system are constant and equal to 2 for increasing values of n j , since only monomolecular events are present in this system. On the other hand, the curve for the compartmentalized system exhibits a rather peculiar maximum at n j close to 0.5, which requires some physical explanation. As discussed above, for n j values close to zero the desorption of active chains is dominant and this implies Pd ) 2. As n j approaches 0.5, the bimolecular terminations are rather fast and they involve a long chain (growing inside the particle) and a very short one just entered from the aqueous phase. Hence, when comparing the case of termination by disproportionation with that of termination by combination discussed earlier, both µ1 and µ2 have the same value, since the contribution of the short chain of the colliding pair is negligible. However, the number of terminated chains µ0 is different: we have twice as many chains in the case of disproportionation as in the case of combination. Therefore, if the Pd value for a compartmentalized system dominated by combination (cf. Figure 13a at q ) 0) is 2, then Pd ) 4 is obtained, since Pd ) µ2µ0/µ12. Actually, the polydispersity values shown in Figure 13b for n j values around 0.5 are not really the double of those in Figure 13a, where combination is dominant. This is because in the systems under examination some transfer to monomer is always included and this tends to decrease polydispersity toward 2. As was first found by Lichti et al. (1980), in a zero-one system with linear chains and no radical desorption disproportionation leads to polydispersity equal to 4, which is exactly the double of that obtained in the case of combination. When including chain transfer to polymer, i.e., q > 0, then the polydispersity ratio increases, although we can observe that also in emulsion the effect of branching in widening the MWD is stronger in the presence of combination than in the presence of disproportionation only when branching is effective enough. On the other hand, the quite significant polydispersity increase due to compartmentalization (i.e., for n j between 0.5 and 1) discussed above is still present and actually strengthened by the presence of chain transfer to polymer. The results in Figure 13b indicate, in fact, the rather peculiar effect of compartmentalization on Pd in the presence of termination by disproportionation. On one side, with respect to a pseudobulk system, we have the usual effect, observed in the case of termination by combination, of shifting to n j values larger than 0.5, the point where bimolecular events become significant. On

Figure 14. Instantaneous polydispersity values as a function of the average number of active chains per particle at various q* values and q ) 0: (a) bimolecular termination by combination only; (b) bimolecular termination by disproportionation only (model parameter values as in Table 2).

the other side, compartmentalization, when coupled to diproportionation, leads to significant polydispersity increases with respect to a pseudobulk system. This justifies the typical shape of the curves shown in Figure 13b and their substantial difference with respect to noncompartmentalized systems. Finally, note that the above discussion holds true only for termination by disproportionation and not for chain transfer to monomer or to chain transfer agent. In other words, the grouping of the effect of these three different mechanisms in one single parameter (i.e., τ given by eq 5) is not possible in this case. The independent role of disproportionation and chain transfer to monomer or chain transfer agent is another peculiarity of emulsion polymerization and must be attributed to the active chain compartmentalization. 4.1.2. Cross-Linking. The calculated polydispersity ratio values are shown as a function of n j in Figure 14a in the case where the polymerization is dominated by cross-linking and termination by combination. As discussed earlier, combination has the effect of narrowing the MWD for linear chains (q* ) 0) and of widening the MWD for sufficiently cross-linked chains (i.e., q* ) 0.02). The competition between these two effects leads to maxima in the Pd values for intermediate degrees of cross-linking. This is evident in Figure 14a for a noncompartmentalized system (dotted curve) where, as n j increases, the bimolecular events become dominant. The same behavior is exhibited by the compartmentalized system (continuous curve), except for the shift toward larger n j values discussed above. The case where termination by disproportionation is dominant is shown in Figure 14b. Similarly to the case of chain transfer to polymer, compartmentalization has here a rather significant effect around n j ) 0.5 besides the usual shifting. 4.2. Criteria for Gel Formation. Using the kinetic approach, i.e., appropriate population balances for active


Figure 15. Gel fraction as a function of conversion (% wt) for different values of the cross-linking rate coefficient (various k*p values). Dotted line: Flory’s model. Solid line: kinetic model.

and dead polymer chains, it is possible to predict the occurrence of gelation also in emulsion polymerization (Mazzotti et al., 1996). This implies again the use of the numerical fractionation technique to compute the MWD evolution during the polymerization process, as we discussed earlier for a noncompartmentalized system. We do not report here the details of this analysis, but we limit ourselves to discuss the comparison between the prediction of the gelation point in the presence of cross-linking provided by the kinetic approach and Flory’s theory. Note that, due to the statistical nature of Flory’s treatment, this can be applied directly without any modification to the case of emulsion polymerization. In Figure 15, the gel fraction evolution predicted by the compartmentalized kinetic model (solid line) is compared to the one predicted by Flory’s theory (dotted line). It appears that, when polymer gelation occurs at large conversion values, a close agreement between the two models is observed. However, some discrepancies arise when the gelation occurs at low conversion values, i.e., at large k*p values. Once more, this is the result of the interaction between radical compartmentalization and larger polymer to monomer ratios in the reaction locus, which characterize emulsion polymerization. 5. Concluding Remarks The molecular weight distribution is probably the most important characteristic of polymer microstructure and has to be properly controlled in order to produce materials with specified applicative properties. A modeling technique, based on population balances for the polymer chains, has been used successfully to relate the characteristics of the MWD in free-radical polymerizations to the operating conditions and to the involved elementary reactions. These include propagation, chain transfer to monomer and to chain-transfer agent, termination by combination, and disproportionation in addition to the reactions leading to nonlinear chains, such as chain transfer to polymer, cross-linking, and terminal double-bond propagation. In the literature, several works have been devoted to this problem, and Hamielec, to whom this paper is dedicated, has given a fundamental contribution to its solution. In particular, explicit expressions for the instantaneous characteristics of the MWD are available which include each elementary reaction with an appropriate kinetic parameter related to the corresponding characteristic time. It is

worth mentioning that the termination reactions are grouped in two distinct parameters: one including disproportionation and chain transfer to monomer and chain transfer agent, referred to as monomolecular termination parameter, and the other including only combination and referred to as bimolecular termination parameter. In this work we have summarized and revisited some of these results by focusing in particular on the role and interactions of the various termination processes on one side and the reactions leading to nonlinear chains on the other. This provided the opportunity to discover and discuss on physical grounds several interesting features of the mechanisms of formation of MWDs, some of which are summarized in the sequel. With reference to instantaneous properties, it has been found that long-chain branching increases polydispersity to a larger extent when coupled with bimolecular terminations than when coupled with monomolecular terminations. While for linear chains bimolecular terminations lead to sharper MWDs than monomolecular ones, in the presence of branching this behavior is reversed and bimolecular terminations produce broader MWDs than the monomolecular ones do. It is interesting to note that, when considering another generation mechanism for nonlinear chains, such as cross-linking or terminal double-bond polymerization, the above behaviors are all opposite. In particular, the effect of these mechanisms in increasing the MWD polydispersity is more significant with mono- than with bimolecular terminations. The effect of compartmentalization, typical of emulsion polymerization, introduces some new features, but it does not affect the observations reported above. The most obvious one is a shift in the active chain concentration value where bimolecular terminations prevail over monomolecular ones. This requires an average number of active chains per particle larger than at least 0.5. More subtle is the interaction between compartmentalization and termination by disproportionation, which leads to instantaneous polydispersity values as large as 4 for linear chains and even larger in the presence of reactions generating nonlinear chains. Another aspect refers to the so-called monomolecular termination mechanisms. As mentioned above, disproportionation and chain transfer to monomer and to chain-transfer agent in the case of noncompartmentalized systems do not affect independently the MWD, but rather together through a unique dimensionless parameter. They are instead independent in compartmentalized systems, and, in fact, the above discussed effect of disproportionation on polydispersity does not apply to the two chain-transfer reactions. The cumulative properties of MWDs have also been discussed and, in particular, the occurrence of polymer gelation. The method of moments has been used to derive explicit criteria for the occurrence of gelation and the estimation of the corresponding time, while the numerical fractionation technique (Teymour and Campbell, 1994) has been used to solve the population balance equations beyond the gel point. As discussed in section 2.1, the study in the case of terminal double-bond propagation has been limited to reacting systems where no more than one terminal double bond can be present per polymer chain. This analysis allows one to conclude that in order for gelation to occur two necessary conditions must be satisfied: (i) a mechanism connecting the macromol-


ecules must be active and (ii) the rate of generation of nonlinear chains (i.e., terminated polymer rebirth) must be proportional to chain length. Both these conditions apply in the case of cross-linking, while in the case of long-chain branching due to chain transfer to polymer, condition i requires the presence of termination by combination, and in the case of terminal double-bond propagation, condition ii is never satisfied. The developed criteria indicate that gelation does not occur in the presence of terminal double-bond propagation alone or in the presence of long-chain branching with monomolecular terminations only. Instead they provide an estimate of the gelation time, when this occurs, i.e., for chain transfer to polymer with bimolecular combination and for cross-linking in the presence of any type of termination. The reliability of these criteria, based on the population balance kinetic model, has been tested in the case of cross-linking by comparing their predictions with those obtained through the classical Flory’s theory, based on “topological” considerations only. The comparison has been extended to the case of compartmentalized systems. Appendix A.1. Population Balance Equations. With reference to the selected kinetic scheme, the following population balance equations can be written for the active and terminated polymer, respectively. When considering active chains, two different population balances have to be accounted for: one for monomeric radicals and the other for polymeric active chains with degree of polymerization larger than 1.

a cross-linking reaction can produce a polymeric active chain of length n but not a monomeric radical.

terminated chains (n g 1): dPn dt

∞

) kfmMRn + kfTTRn + kfpRn ∞

kfpnPn

∑

∞

Rr - kd p γPn

r)1

∞

) RI - kpMR1 - kfmMR1 + kfmM

dt

∑ Rr -

r)1 ∞

kfTTR1 + kfTT

∑

∞

Rr + kfpP1

r)1 ∞

∑

∞

Rr - kfpR1

r)1

∑rPr -

r)1 ∞

∞

∑rPr - kdp γR1r)1 ∑Pr - (ktc + ktd)R1r)1 ∑ Rr r)1

k*pRR1

(38)

where RI ) 2ηkI[I]is the initiation rate, η being the initiator efficiency.

polymeric active chains (n > 1): dRn dt

∞

∑

r)1 n-1

k* pR

∞

Rr - kfpRn

∑

∞

rPr - k* pRRn

r)1 ∞

∑rPr +

r)1 n-1

d Rn-rrPr - kd ∑ p γRn ∑ Pr + kp γ ∑ Rn-rPr r)1 r)1 r)1 ∞

(ktc + ktd)Rn

∑Rr

1

∑ Rr +

r)1 n-1

∑Rr + 2ktc r)1 ∑ Rn-rRr r)1

ktdRn

(40)

Accordingly, terminated chains are produced by chain transfer to monomer or to chain-transfer agent, chain transfer to polymer, and termination both by disproportionation and by combination. The negative terms in the balance are due to chain rebirth, which can occur by chain transfer to polymer, by cross-linking or by propagation to terminal double bond. Moment Equations. In order to obtain the moments of the chain length distribution, it is convenient to introduce the associated generating functions both for the active and for the terminated polymer chain distributions. They are defined as follows: ∞

H(u) )

unRn ∑ n)0

(41)

∞

G(u) )

unPn ∑ n)0

(42)

From these functions, the moments of any order of the unknown distributions (λj for active chains and µj for terminated polymer) are readily obtained. For example, the first three moments of the distribution Pn are obtained by evaluating G(u) and its first and second derivatives when u ) 1, by means of the following formulas:

G(u ) 1) ) µ0

| |

dG du

u)1

) µ1

d2G du2

u)1

) µ2 - µ1

(43)

From the population balance equations the expressions for the time derivatives of H(u) and G(u) can be obtained:

) kpMRn-1 - kpMRn - kfmMRn - kfTTRn + kfpnPn

∑

∞

Rr - k* pRnPn

r)1 ∞

monomeric radicals (n ) 1): dR1

∑rPr -

r)1

(39)

r)1

The differences between the two equations are due to the different way the monomeric and polymeric radicals are born. In fact, both the initiator decomposition and the transfer to monomer or to transfer agent reactions produce a monomeric radical, but not a polymeric radical. On the other side, a propagation or

dH ) uRI + kpMuH - kpMH - kfmMH + kfmMuλ0 dt kfTTH + kfTTuλ0 + kfpλ0uG′u - kfpµ1H + k* pRuG′uH d d k* pRµ1H + kp γGH - kp γµ0H - (ktd + ktc)λ0H (44)

dG ) kfmMH + kfTTH + kfpµ1H - kfpλ0uG′u dt 1 2 k*pRλ0uG′u - kd p γλ0G + ktdλ0H + ktcH (45) 2 where G′u indicates the first-order derivative of G(u). From eqs 43 and 44, using the quasi-steady-state assumption (QSSA) and the dimensionless parameters


defined by eqs 5-9, the following expressions for the first three moments of the MWD of the active chains, λi, are obtained:

λ0 )

x

{

(46)

µ3 λ0 λ1 1 + τ + β + 2 + cp + τ + β + cp λ0 µ1 µ3 λ µ λ1 2 D 1 c*p + 2MD + cd + 2MD n Pd p n µ1 λ0 µ0 λ0

[

]}

] [

(48)

where MD n is the cumulative number-average molecular weight of the terminated polymer at a given time t and PD d its polydispersity ratio. Through the same procedure, from eqs 43 and 45 the following moment equations are obtained for the terminated polymer:

dµ0 β d ) Rp τ + - c* p - cp dt 2

(

)

(49)

dµ1 ) Rp(1 + τ + β) dt

[

(50)

λ dµ2 D D d D 1 ) Rp 1 + τ + β + 2(1 + c* + pMn Pd + cp Mn ) dt λ0 λ1 2 β (51) λ0

( )]

where Rp ) kpMλ0 indicates the propagation rate. Note that in eq 48 the third-order moment of the terminated polymer appears. Thus, the moment equations 46-51 are not in closed form and moments of order larger than the maximum calculated one have to be evaluated. This is usually performed through a suitable closure formula. Among different possibilities (cf. Bamford and Tompa, 1953, 1954; Frenklach and Harris, 1987), we selected the following one, based on the approximation of the unknown distribution as a truncated series of Laguerre polynomials using a Γ distribution weighting function (Hulburt and Katz, 1964):

µ3 )

µ2(2µ2µ0 - µ12) µ1µ0

dt dnC

RI ktc + ktd

D D d D 1 + τ + β + (cp + c* p)Mn Pd + cp Mn λ0 (47) λ1 ) τ + β + cp

λ2 )

dnB

(52)

The accuracy of such a formula is essentially determined by the resemblance between the unknown distribution and the selected model distribution, the Γ one in this case. The adequacy of this closure formula in the case of nonlinear chains has been recently confirmed (Baltsas et al., 1996). Besides the MWD moment equations, other properties of branched or cross-linked polymers are of interest when dealing with the polymer microstructure. In particular, the equations for evaluating the number of trifunctional branches produced by chain transfer to polymer, nB, the number of cross-linkages, nC, and the number of trifunctional points produced by terminal double-bond propagation, nd, are here reported:

dt dnd dt

∞

) kfp

∑

∞

nPn

n)1

∑ Rn ) Rpcp

∞

) k*pR

∑

∞

nPn

n)1

∑ Rn ) Rpc*p

(54)

n)1

∞

) kd pγ

(53)

n)1

∞

Rn ) Rpcd ∑ ∑ p n)1 n)1 Pn

(55)

Note that these balances allow us to evaluate average properties only, such as the number-average trifunctional branch points, F3 ) (nB + nd)/(λ0 + µ0), or the corresponding quantity for the tetrafunctional branch points, F4 ) nC/(λ0 + µ0). It is useful to remark here that the rates of these reactions depend upon the characteristics of the terminated polymer through the d parameters cp, c* p, cp , but while the branching and cross-linking reactions are proportional to the first-order moment of the terminated polymer, the terminal doublebond propagation is proportional to the zeroth-order moment. Relevant instantaneous properties of the polymer, such as the polydispersity ratio and the average molecular weights, are readily obtained from the moment equations. Assuming the propagation rate to be considerably larger than any termination rate, i.e., the longchain assumption (LCA), the parameters τ and β are negligible with respect to unity, i.e., τ, β , 1. This is a quite common assumption when dealing with polymeric materials of practical interest, where typical chainlength values are of thousands of units. By substituting the expressions for λ0 and λ1 from eqs 46 and 47 into the moment equations for the terminated polymer, eqs 49-51, the expressions for the average molecular weights and polydispersity ratio reported in eqs 2-4 are obtained after some rearrangement. Nomenclature cp ) chain transfer to polymer dimensionless parameter c* p ) cross-linking dimensionless parameter cd p ) terminal double-bond propagation dimensionless parameter cjp ) chain transfer to polymer dimensional parameter [cm3/ mol] cjp* ) cross-linking dimensional parameter [cm3/mol] cjd p ) terminal double-bond propagation dimensional parameter [cm3/mol] c ) termination frequency in a latex particle [1/s] G(u) ) moment generating function for the terminated polymer distribution H(u) ) moment generating function for the active chain distribution I ) initiator concentration [mol/cm3] I0 ) initial initiator concentration [mol/cm3] k ) desorption frequency [1/s] kfm ) chain transfer to monomer rate constant [cm3/ (mol s)] kfT ) chain transfer to chain transfer agent rate constant [cm3/(mol s)] kI ) initiator decomposition rate constant [1/s] kp ) propagation rate constant [cm3/(mol s)] 3 k* p ) cross-linking rate constant [cm /(mol s)] d kp ) terminal double-bond propagation rate constant [cm3/(mol s)] ktd ) termination by disproportionation rate constant [cm3/ (mol s)]

1300 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 ktc ) termination by combination rate constant [cm3/(mol s)] M ) monomer concentration [mol/cm3] M0 ) initial monomer concentration [mol/cm3] Mn ) instantaneous number-average degree of polymerization Mp ) monomer concentration in the latex particle [mol/ cm3] Mw ) instantaneous weight-average degree of polymerization MD n ) number-average degree of polymerization of the preexisting polymer MD w ) weight-average degree of polymerization of the preexisting polymer MwP ) weight-average degree of polymerization of the primary chains NA ) Avogadro’s number nB ) concentration of trifunctional points produced by branching [mol/cm3] nC ) concentration of cross-linkages [mol/cm3] nd ) concentration of trifunctional points produced by terminal double-bond propagation [mol/cm3] n j ) average number of active chains per particle Pd ) instantaneous polydispersity ratio PD d ) polydispersity ratio of the preexisting polymer P∞d ) asymptotic polydispersity ratio Pn ) concentration of terminated polymer with length n [mol/cm3] q ) ratio between chain transfer to polymer and termination frequencies q* ) ratio between cross-linking and termination frequencies RI ) initiation rate [mol/(cm3 s)] Rp ) propagation rate [mol/(cm3 s)] Rn ) concentration of active chains with length n [mol/ cm3] T ) chain-transfer agent concentration [mol/cm3] tc ) critical gelation time [s] Vp ) particle volume [cm3] wg ) gel polymer weight fraction wp(r) ) weight chain-length distribution of the primary chains Greek Letters R ) double-bond density in a polymer chain β ) bimolecular termination parameter γ ) fraction of terminated chains with a terminal double bond λi ) ith-order moment of the active chain distribution [mol/ cm3] µi ) ith-order moment of the terminated polymer distribution [mol/cm3] ν ) concentration of cross-linked units [mol/cm3] τ ) monomolecular termination parameter F ) cross-linking density F3 ) number-average number of trifunctional branch points F4 ) number-average number of tetrafunctional branch points

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Received for review August 6, 1996 Revised manuscript received December 17, 1996 Accepted December 18, 1996X IE9604841 X Abstract published in Advance ACS Abstracts, February 15, 1997.

Nonlinear Chain-Length Distributions in Free-Radical Polymerization

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