Makromol. Chem., Macromol. Symp. 63,135-182 (1992) 135

KINETICS OF POLYMERIC NETWORK SYNTHESIS VIA FREE-RADICAL MECHANISMS - POLYMERIZATION AND POLYMER MODIFICATION

Shiping Zhu and Archie HamielecJ:

Institute for Polymer Production Technology (MIPPT) Department of Chemical Engineering McMaster University, Hamilton, Ontario, Canada L8S 4L7

Abstract: The kinetics of polymeric network formation via free radical mechanisms is an attractive research area because there are many phenomena which are not well understood and in addition, the commercial potential for crosslinked systems is great. Recently, a large research/development program was initiated at the McMaster Institute for Polymer Production Technology (MIPPT) to investigate the fundamentals and applications of polymeric network, in particular, the kinetics of synthesis via free-radical mechanisms and network characterization. The research on crosslinking involved both theoretical developments and experimentation. Herein is provided a comprehensive summary of this work. In the experimental polymerization, two comonomers, methyl methacrylate (MMA) / ethylene glycol dimethacrylate (EGDMA) and acrylamide (AAm) / N,N-methylene bisacrylamide (Bis), as model systems were studied in considerable detail. Measurements included: monomer conversions, radical concentrations sol/gel fractions, crosslink densities (equilibrium swelling and swollen-state "C-NMR) over the entire range of divinyl monomer levels as a function of polymerization time. In the polymer modification, high density polyethylenes were crosslinked using peroxides and y-radiation. For this system, crosslinking and chain scission occur simultaneously. In the theoretical studies, it was shown that in general, network formation by free-radical mechanisms is highly irreversible requiring that the classical equilibrium gelation theories after Flory/Stockmayer be generalized. The general model which was developed using the pseudo-kinetic rate constant method predicts the existence of a crosslink density distribution (crosslink density of a primary polymer chain depehds on its birth time) with a variance which can vary widely depending on network synthesis conditions.

INTRODUCTION

Polymer as a material plays an important role in almost every aspect of human life. By the year of 2000, the annual consumption of material per person will reach 286.9 liters, of which eighty percent will be polymeric materials, ie. plastics, rubbers and fibers, both synthetic and natural (Ref.1). Today, nearly half of the polymer products are produced by free-radical mechanisms.

Polymer molecules have various chain structures. They can be linear, branched (long and short branches) and crosslinked. According to Andreis and Koenig (Ref.2), large percentage of the polymer products in use today are crosslinked, eg. rubbers, thermoset plastics and superabsorbents. Free-radical polymerization with crosslinking is receiving more attention, due to its potential in the development of new polymer products. The potential applications of

0 1992 Huthig & Wepf Verlag, Base1 CCC 0258-0322/92/$04.00

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crosslinked polymer networks produced by free-radical polymerizations cover a wide range. However, most research work today has been directed at network characterization of final polymer products. Little attention has been paid to the polymerization kinetics. A knowledge of the kinetics is critical for the optimal control of polymer network structures during synthesis.

RADICAL

CROSS-LINKAGE PENDANT DOUBLE BOND

Fig.1 Schematic representation of lhe free-radical crosslinking mechanism via: (a) divinyl monomers, and @) chain transfer to polymer.

Free-radical polymerization with crosslinking can be accomplished using divinyl monomers and/or via chain transfer to polymer. In the case of copolymerization of vinyl/divinyl monomers as shown in Fig.1 (A), when a polymer radical propagates with a double bond on a divinyl monomer, the unreacted double bond on the same monomer molecule becomes pendant on the polymer chain. If a second polymer radical adds to the pendant double bond, a crosslinkage will form. Tetra-branched polymer chains are thus produced. Further branching leads eventually to crosslinking, ie. network formation or gel formation. In the case of network formation via chain transfer to polymer as shown in Fig.1 (B), the polymer chains must have abstractable atoms on their backbones. When a polymer radical abstract an atom from the backbone of another polymer chain, the radical center is then transferred to the backbone. When the backbone radical propagates with vinyl monomer, a tri-branched polymer chain forms. Two tri-branches form one cross-linkage by radical recombination. The continuous production of cross-linkages also leads to gelation. A gel molecule is a three-dimensional network. It cannot be dissolved in any solvent. Theoretically, it has an infinitely large molecular weight. Fig.2 schematically represents such a network. In contrast, those molecules which have finite molecular weight and which are soluble in their good

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solvents are defined as sol. Sol polymers can be linear and branched. It should be noted that, in the literature, words such as crosslinking, gelation, curing, network formation are often used interchangeably.

MNCUNC W N

CYCLE

CROSSUNKINC

PWDM DOUBLE BOND

Fig.2 Schematicrepresentation of thegelgrowlh. 'he four reactions are (1) propagation with monomers; (2) sol-gel radical recombination; (3) reaction of gel radicals with sol pendant double bonds; and (4) reaction of sol radicals with gel pendant double bonds.

Kloosterboer (Ref.3) has pointed out some special problems arising during free-radical polymerization with crosslinking: (1) early onset of the Trommsdorff effect, (2) incomplete conversion of pendant double bonds due to vitrification, (3) reactivity ratios change with conversion, (4) rates of polymerization are very sensitive to chain transfer to polymer, (5) trapped radicals induce post-copolymerization with oxygen, and (6) no available theory can fully describe crosslinking polymerization with extensive ring formation.

Free-radical polymerizations are often diffusion-controlled, even with linear chains, due to the overlap and entanglement of polymer chains. The branching and/or crosslinking certainly confound the diffusion-controlled reactions. The reasons for this are two-fold. Firstly, the diffusing polymer species may have much more complicated chain structures than merely linear. On the other side, the diffusion environment is also more complex, especially with the existence of network structures. The diffusion of polymer chains is critical to the reactions involving macromolecules, either dead or live polymer chains. Examples are not only the bimolecular radical termination, but also the propagation with pendant double bonds, and the chain transfer to polymer. However, there is no existing diffusion theory which is generally applicable to this type of system.

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A direct result of these diffusion-controlled reactions in crosslinking systems is the enhanced auto-acceleration in polymerization rate. The "gel" effect (Trommsdorff effect) due to physical entanglement of polymer chains is coupled with the gelation effect due to chemical crosslinking. With a high level of divinyl monomer, auto-acceleration can start at the very beginning of polymerization in almost pure monomer. When a 3-dimensional network forms, those reacting species chemically bound to network structures such as pendant double bonds and free radical centers have extremely small diffusion coefficients. The so-called "shielding" effect can significantly suppress their reactivities to an extent that they may be considered actually trapped. The trapping phenomenon strongly affects polymerization kinetics, and the latter is mainly responsible for the build-up of polymer chain structures affecting, in turn, polymer properties. It is well accepted that, because of diffusion control, most of the kinetic rate constants for elementary reactions in free-radical polymerization with crosslinking are not really constant and may change in orders by magnitude during the course of polymerization. These rate constants are the basis for the prediction of reaction behaviors and polymer chain properties. Experimental determination of these parameters, although difficult, is worth the effort required. In addition, micro-structure of the polymer networks formed in free-radical polymerization with crosslinking are non-ideal and often inhomogeneous. Realistic quantitative modelling of the network formation has been proven very challenging since the pioneering work of Flory (Ref.4) and Stockmayer (Ref.5).

Recently, the McMaster Institute for Polymer Production Technology (MIPPT) has launched a comprehensive study on the mechanisms and kinetics of free-radical polymerization with crosslinking in order to provide a better understanding of the molecular processes involved in polymer network formation for better control of network structures. The following aspects are emphasized: elucidation of the mechanisms of auto-acceleration in polymerization rates due to physical entanglements and chemical crosslinking of polymer chains, determination of kinetic parameters up to high monomer conversions and, development of comprehensive kinetical models for free-radical crosslinking.

The model systems chosen for experimental studies are methyl methacrylate (MMA) / ethylene glycol dimethacrylate (EGDMA), and acrylamide (AAm) / N,N-methylene bisacrylamide (Bis). However, there is no attempt to introduce such constraints on the kinetic modelling work. The principles should be generally applicable wherever possible.

EXPERIMENTAL WORK

Literature Review

The early experimental observations made focused mainly on the gelation point. Walling (Ref.6) was the first who reported the gelation points for the systems methyl methacrylate / ethylene glycol dimethacrylate and vinyl acetate / divinyl adipate by measuring the bubble rise rates in reaction solutions, and found that the observed extents of reaction at the gelation point were much higher than those predicted by the Flory gelation theory (Ref.4). He

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attributed this disagreement to the diffusion-controlled reactions in crosslinking. This work motivated many additional investigations. Gordon et al. (Ref.7) found that gelation always comes earlier than that of autoacceleration in polymerization rate and concluded that the supposed diffusion-controlled delay of gelation is illusory and that cyclization only is responsible for the observed discrepancy. Simpson et al. (Ref.8) quantitatively measured the amount of cyclization in the polymerization of many diallylesters, and found that cyclization alone is inadequate to explain the discrepancy. Loshaek et al. (Ref.9), Minnema and Staverman (Ref.10) measured the reactivity of pendant double bonds for different systems and concluded that there exists a shielding effect due to steric hindrance imposed by polymer segments which reduces the reactivity of pendant double bonds and delays gelation. Storey (Ref.ll), based on measurements of the initial rates and gel points for styrene / divinylbenzene, suggested that the gelation processes are in two stages: microgels form first, reactions between microgels lead to macrogelation, and that it is the diffusion-controlled reactions involved in macrogelation that delay gelation point.

It is generally agreed after a decade of exploration that these various effects coexist and are related to each other in free-radical polymerization with crosslinking. Cyclization was further studied by Soper et al. (Ref.12), Ishizu et al. (Ref.l3), Landin and Macosko (Ref.l4), Tobita and Hamielec (Ref.15). The shielding effect was studied by Hild et al. (Ref.l6), Whitney and Burchard (Ref.l7), Landin and Macosko (Ref.14)). The formation of microgels was studied by Shah et al. (Ref.l8), Dusek and Spevacek (Ref.l9), Such microgels were also prepared using emulsion techniques (Ref.20), dispersion (Ref.21) and bulk with chain transfer agents (Ref.22), with critical amounts of divinyl monomers (Ref.23).

The other aspects of the experimental studies on crosslinking include the measurements of reactivity (Ref.24) and Malinsky et al. (Ref.25), rate constants of propagation and termination (Ref.26), glass transition temperature (Refs.27-28), molecular weight of sol polymers (Refs.29-30), crosslink density (Refs.31-34), radical trapping (Ref.35), and effect of different crosslinkers (Ref.36).

Clearly, the studies of the free-radical polymerization with crosslinking provided data that were very scattered. Few comprehensive experimental investigations have been made on the kinetic aspects of crosslinking, particularly at high monomer conversions.

Polymerization of MMA/EGDMA

The reason for the choice of MMA as a model system is that the polymerization of MMA has the greatest auto-acceleration among vinyl monomers. As for EGDMA, it is chosen for its chemical similarity in molecular structure to that of MMA and the symmetry of the two vinyl groups. MMA and EGDMA were purified as follows: washed with a 10 wt% aqueous KOH solution to remove inhibitor, washed with deionized water, dried successively with anhydrous sodium sulfate and 4A molecular sieves, then distilled under reduced pressure collecting the middle fraction. The initiator, azobisisobutyronitrile (AIBN), was recrystallized from

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methanol, followed by drying in vacuum. Pyrex ampoules with various diameters (3 to 5 mm O.D. for different purposes) were used. The monomer solutions were prepared shortly before use by weighing the required amounts of MMA, EGDMA, and ABN. The samples were always degassed three or four successive freeze-thaw cycles under a reduced pressure using liquid nitrogen. The ampoules were filled with VHP N, to reduce gas bubbles produced during reaction.

Conversions

The monomer conversions to sol and gel for the copolymerization initiated with 0.3 wt% AIBN at 70 'C with and without carbon tetrabromide as a chain transfer agent were measured using gravimetric techniques (Ref.37). The polymerization was initiated by immersing the ampoule in an oil bath, and stopped by thrusting the ampoules into liquid nitrogen at required time intervals. The ampoules were broken and the contents were put into acetone with ppm level hydroquinone to inhibit possible further polymerization. The swollen gel, if present, was concentrated by centrifugation, extracted by acetone in a Soxhlet extractor, and then dried in a vacuum for days to constant weight. The sol polymer was precipitated from solution using methanol. The conversions to gel and sol were determined gravimetrically.

1

0 9

0 8

0 7

g 0 6

2 0 5

0 4

0 3

0 2

0 1

C

0: 0 20 4 0 60 80 100 1.20 140 I60

Time ( min. ) Fig3 Conversion versus time histories for the polymerization of MMA/EGDMA at 70 'C with 0.3 wt% AIBN and EGDMA levels: (circle) 0, (up triangle) 0.3, (square) 1, (diamond) 5, (down triangle) 15, (star) 25 wt%.

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Fig.3 shows the conversion data with EGDMA level up to 25 wt%. It can be clearly seen that crosslinking enhances the "Trommsdorf effect" significantly. The monomer conversion at which the autoacceleration in polymerization rate starts decreases with increase in divinyl monomer level. It is well known that the process involved in a free-radical polymerization basically consists of four types of reactions: initiation, propagation, radical termination and chain transfer. As temperature increases, the initiator molecules decompose into primary radicals. Some primary radicals experience geminate recombination, ie. the so-called cage effect. Others diffuse out of the cage to react with monomer double bonds to form polymer radicals. The polymer radicals grow rapidly by further propagating with double bonds on monomer and/or pendant on polymer chain. When one radical center meets another, they are terminated either by disproportionation or recombination, and two radicals are annihilated.

I . ( , , , , , I I . I ' I ' I ' 1 ' 1

Fig.4 Conversion versus time histories for the polymerization of MMA/EGDMA at 70 'C with 0.3 wt% AIBN, 1.0 wt% EGDMA and CBr, levels: (circle) 0, (up triangle) 1.0, (square) 2.0, (diamond) 4.0 wt%.

The bimolecular termination consists of three definable steps. First, two polymer chains far away from each other must come together by translational diffusion. Then, the radical centers reorient by segmental diffusion. Finally, they overcome the chemical barrier to react. Free radicals are often highly reactive. The third step is then very rapid. Therefore the termination is always diffusion-controlled. At low monomer conversions, it is segmental diffusion-controlled. When sufficient polymer chains are produced, they become physically entangled and/or chemically crosslinked, and the self-diffusion coefficients of chains fall rapidly with monomer conversion. The termination reaction then becomes translational diffusion-controlled. It is this diffusion-controlled termination reaction that causes the auto-acceleration in

142 ."/;I 0 8

0.8 1 I -1

Time ( rnin. )

Fig.5 Conversion versus time histories of the conversions of monomer to sol and gel during the polymerization of W G D h 4 . 4 at 70 'C with 0.3 wt% AIBN and 0.3 wt% EGDMA: (circle) total, (up triangle) sol, (square) gel, and (Xc) gelation point.

polymerization rate. Higher divinyl monomer level means more branching and crosslinking and larger chain molecules, and therefore results in more pronounced auto-acceleration. However, in many cases, certain chain transfer agent can be used to reduce the polymer chain size, and in turn the auto-acceleration. Fig.4 shows the effect of chain transfer agent on monomer conversion. The sudden levelling-off at high conversions in both Figs.3 and 4 is due to the fact that the reactions involving small molecules such as monomer and primary radical also become diffusion-controlled when the system approaches its glassy state.

Fig.5 gives a typical example of the sol/gel development during the polymerization. At low conversions, polymers produced are all soluble. They are either linear or slightly branched chains. When the conversion reaches to a certain point termed gelation point, the first insoluble molecule, ie. gel molecule, forms. Once a gel molecule is nucleated, it grows rapidly analogous to sponging by the following four reactions (refer to Fig.2). They are (1) propagation of gel radicals with monomers, (2) sol-gel radical recombination, (3) reaction of gel radicals with sol pendant double bonds, and (4) reaction of sol radicals with gel pendant double bonds. The last three reactions all consume sol polymers. This explains why the sol conversion decreases rapidly after the gelation point, although new sol polymer chains are continuously formed. Fig.6 summarizes how the gelation point varies with the crosslinker level. The higher the level of divinyl monomer, the earlier gelation occurs. The gelation point

143

0.2

0. I

I . . I . , . , . , ' , ,

0 . 9 } I

Coa.ccnlr.tioo of D i r i n j l Monomer ( rL.2 E C D W )

Fig6 Conversion at gelation point (Xc) versus EGDMA level for the polymerization of MMA/EGDMA at 70 'C with 0.3 wt% AIBN (circle) without CBr,, (up triangle) with 1.0 wt% CBr,.

is inversely proportional to the polymer size (discussed later). A chain transfer agent therefore delays gelation as shown in Fig.7.

The copolymerization of MMA/EGDMA under adiabatic conditions was also investigated using a DIERS VSP reactor (Ref.38). The monomer solutions were bubbled with nitrogen for more than half a hour in order to reduce oxygen content. The reactor cell was flushed with nitrogen three times. After injection of the monomer solution, the cell was heated rapidly to the set initial temperature. The temperature inside the cell was recorded continuously. The experiments were terminated when the temperature rise rate approached zero. Fig.8 shows the temperature profiles (equivalent to the conversion of double bonds) for five EGDh4A levels. These profiles have a similar trend to those in Fig.3. Fig.9 shows the sol/gel content of the final products. These polymerizations were dead ended due to the high decomposition rate of the initiator used at these elevated temperatures. At the final stage, there are significant amounts of monomer remaining unreacted. However, sol polymers hardly survive due to sponging by gel.

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Fig.7 Dependence of conversion at gelation pint (Xc) on CBr, concentration for the poly- merization of MMAEGDMA at 70 'C with 0.3 wt% AIBN and 1.0 wt% EGDMA.

Fig.8 Temperature histories (equivalent to the conversion of double bonds) for the adiabatic copolymerization of W G D M A with 0.5 wt% AIBN at six EGDMA levels: 0,0.2,1,5, 10, u) wt%. The initial temperature is 60 'C.

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1

Gel Polymer A A - A

I A , - 10 15 20

EGDMA WT% 5

Fig9 SoVgel fraction of the final produds of the adiabatic copolymerization of MMA/EGDMAwilh0.5wt%'oNandsixEGDMAlevels:O,0.2,1,5,10,20wt%measured wilh gravimelric method.

Radical Concentrations

Information on radical concentration development during polymerization is most desired for a kinetic study. The advent of modem electron spin resonance PSR) techniques has provided a powerful method for polymerization studies (Ref.39). Applications of ESR in direct measurement of radical type and concentration during polymerization are very promising (Refs.35,40-44). However, many questions are still without answers, particularly for a crosslinking system. A short summary of our work for the MMA/EGDMA copolymerization (Refs.45-48) is provided in the following. The reaction conditions are designed to couple with the corresponding conversion measurements.

The degassed ampoule (3 mm O.D.) filled with monomer solution was inserted into a TEllO cavity of a Bruker ERlOOD ESR spectrometer to measure the radical concentration on-line. The polymerization was implemented on-line to avoid possible changes of radical concentration during quenching. The temperature was controlled with a gas bath at 70 'C. An initiator level of 0.3 wt% AIBN and EGDMA level covering the full range were used. Absolute radical concentrations were found by calibrating with 2,2-diphenyl-l-picrylhydrazylhydrate (DPPK) dissolved in MMA.

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Fig.10 Radical concentrations (moV1) of the polymerizations with 0 (plus), 0.3 (cross), 1 (circle) wt % EGDMA as a function of time. T=70 'C; AIBN: 0.3wt%.

.. .. .. TIME IMINI

Fig.11 Radical concentrations of the polymerizations with 3 (circle), 5 (cross), 10 (square), 15 (plus) wt% EGDMA as a function of time. T=70 'c AIBN: 0.3wt%.

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Figs.10, 11 and 12 show the radical concentration histories for the polymerizations with the different EGDMA weight fractions. The characteristic features of these curves include: At low EGDMA levels (Figs.10 and l l ) , the polymerizations clearly show four phases in terms of the radical concentration behavior versus reaction time. Consider the 1 wt% EGDMA as an example. Coupled with corresponding conversion data (Ref.37, the radical concentration measurements are plotted in Fig.13. As is shown, the radical concentration first remains relatively constant. Then, both the radical concentration and the conversion rate show a synchronous rise. This verifies the hypothesis that the rapid autoacceleration in polymerization rate is due to a dramatic increase in radical concentration. This radical concentration rise is in turn due to a reduction in bimolecular termination rate of polymer radicals. Since the termination process involves two macroradicals diffusing together, such reduction in rate is caused by physical entanglement of linear polymer chains in an MMA polymerization. In the case of copolymerization with a divinyl monomer, this 'gel' effect is coupled with the gelation effect, an additional hindrance to radical termination due to the chemically crosslinked structure (chemical entanglement point or crosslink). The diffusion-controlled termination in a free radical polymerization with crosslinking is very complex. In fact, the macroradicals become highly branched prior to gelation. Such branched macroradicals likely experience limited translational diffusion in the reacting mass. Therefore, diffusion-controlled termination exists not only in the post-gelation period but also in the pre-gelation period. An experimental investigation of gelation with a 0.3 wt% EGDMA, for instance, gave a conversion at the gelation point of 0.24, while the autoacceleration starts at 0.15.

I0.I

10.'

lo"

"-'O 10 10 30 40 $0 60 10 80 PO 100

TIME [ M I N I

Fig.12 Radical concentrations of fhe polyrnerizafions with 25 (circle), 100 (triangle) wlW EGDMA BS a function of time. T=70 'C; MBN: 0.3W.

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1.0

0.9

0 1 0.0f-01

0.7

O.6OE-01 0.6

x 0.1 5 0.4 0.1OL.01

0.3

0.2 0.7f-01

0.1

0 10 10 10 10 SO 60 70 10 90 I00 110 0.0

TIME ( M I N I

Fig.13 The radial concentration (diamond) and monomer conversion (triangle) of the polymerization with 1 wt% EGDMA as a function of reaction time. T=70 "2; AIBN: 0.3wtl.

Fig.14 The radicalconcentration risentesduring theautoaccelerationsof thepolymerizations with (he different EGDMA weight fndions. T=70 'C; AIBN 0.3wtl .

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At high conversions when the system approaches its glassy state, the radical concentration falls slightly after reaching a peak value. Such a peak corresponds closely to the maximum conversion rate. Thereafter the conversion rate starts to fall. Finally, the radical concentration increases again, and gradually levels off. With high crosslinker levels (Fig.l2), the radical concentration increases monotonously from the onset of polymerization. Even in the glassy state, the increase is quite significant. These complex phenomena may be attributed to the competition of radical initiation and termination.

The effect of crosslinking on radical concentration is dramatic. In the case of pure EGDMA, the concentration reaches 10.’ molar. This clearly shows the severe constraints on molecular diffusion imposed by network structures. Those radicals dangled on the network (refer to Fig.2) are hardly mobile. The rapid rise of radical concentration during auto-acceleration may invalidate the steady-state hypothesis (SSH) which is widely used in kinetic modelling. During the autoacceleration in crosslinking polymerization, the radical Concentration rise rate can have a significant value compared to the initiation rate. We estimated the maximum rise rate using the experimental radical concentration versus time measurements. The initiation rate was estimated at the initial stage. An evaluation of the validity of SSH is shown in Fig.14 for different EGDMA weight fractions used. For high EGDMA levels, it reaches as high as

0.6. For EGDMA concentrations above ca. 10 wt%, the SSH introduces significant error.

It is clear that radical concentration information is essential for the estimation of kinetic parameters. In general, the radical concentration data shown in Figs.10, 11 and 12 coupled with the conversion data in Fig.3 permits one to estimate the propagation rate constant. The number-average termination rate constant (Ref.49) can be estimated using the post-effect measurements of radical concentration, ie. measuring the radical decay rate when initiation is suspended (Refs.44-45). Then the initiator efficiency can be estimated using the radical concentration profiles. However, all these constants are apparent kinetic parameters, particularly in the post-gelation period where there exist two macroradical populations. For example, the propagation rate constant estimated from the measurements of total double bond conversion and total radical concentration is an average rate constant for all the reactions which consume double bonds. It is also clear that the number-average termination rate constant defined in terms of the number of monomeric units on crosslinked polymers does not have a physical interpretation as it does for linear polymerization. A more realistic definition of the termination rate constant should include the distribution of mobility of the radical centers chemically bound to the crosslinked gel.

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Another important phenomenon observed in the copolymerization of M W G D M A is that, during the auto-acceleration, the ESR spectra changed from 13-line to 9-line. Fig.15 gives an example of this transition. The low conversion one (a) is a typical 13-line spectrum which is attributed to the radicals in the liquid state, while the last (e) is a 9-line spectrum attributed to those trapped in the glassy state. The model simulations (Ref.48) reveal that the intermediate spectra @-d) are due to an overlap of 13-line and 9-line spectra having different intensities. This strongly suggests that there exist two populations of radicals, radicals in the liquid and solid states, in the reacting mass. In other words, the reacting mass is heterogeneous in terms

of radical environment. The transition in ESR spectra indicates that the reacting system is approaching its glassy state. The amount of trapped radicals can be estimated either by measuring the residue ESR signals during the post-effect period, or by extracting the 9-line component from an overlapped spectrum.

Fig.15 The Iransition of 13-line spectra 10 9-line spectra during the polymerization with 15 wt% EGDMA. Microwave frequency: 9.45 GHz; modulation frequency: 100 KHZ; modu- lation amplitude: 3.2 Gpp; gain: (a)-@) 1 x 106, (c)-(e) 5 x Id; time: (a) 9'15", @) 10'30", (c) 12'20", (d) 13'15", (e) 20'30". T=70 'C; AIBN: 0.3wtk.

Fig.16 shows a typical example of the radical trapping data. It can be seen that the value of the trapped radical fraction increases dramatically from 0 to 1 during the rapid growth in radical concentration. This suggests that a strong relationship exists between the auto-acceleration and the radical trapping. The physical picture of this is as follows: During the auto-acceleration, radicals attached to long chains and network structure are entangled in the environmental polymer matrix and therefore bimolecular termination of these radicals are seriously constrained. However, the propagation reactions may still be unaffected. Certain radical centers exhaust monomers in their vicinities, and then experience a solid environment which consists of polymer chain segments under glassy temperature. The glass domains are

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micro-scaled (at molecular level) and may not have a clear phase boundary with their surrounding. Release of these trapped radical centers depends on the rate of monomer diffusion into glass domain and polymer chain relaxation time. However, in the glass state, neither of them is fast enough to compete with the auto-acceleration which takes only a few minutes. On the other hand, at the same time, radical centers which are attached to short chains and which have not experienced a glassy-state transition continue to both propagate and terminate.

Fig.16 Trapped fraction of propagating radicals fi the MMA/EGDMA matrix versus poly- merization time for copolymerization of 15 w t l EGDMA at 70 'C.

The polymer radicals obtained at low EGDMA levels and at final conversions are not stable at high temperatures, and their ESR spectra disappear in minutes after the temperature is raised to > T,, On the other hand, the radicals bound to highly crosslinked polymers are quite stable. These radicals hardly terminate even when the reaction temperature is raised to 150°C (Tg = 141°C for pure EGDMA polymer, Ref.28). Fig.17 gives an example of this radical stability (Ref.47). During the radical decay at elevated temperatures, ESR signals experience some structural changes. Fig.18 shows an example. It can be seen that the height ratio of the central line over the neighbor inner line falls significantly in the post-effect period. This change should reflect some changes in radical conformation and likely reactivity.

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These highly concentrated radical centers bound to gel may further react with other monomers added later in the polymerization. We used a sample of 50 wt% EGDMA polymerized at 70'C for 100 min, then swelled the gel with styrene monomer. After degassing, the sample was polymerized again at 80°C for some time. As shown in Fig.19, the styrene ESR signal was clearly detected. These so-called reactive gels may find some commercial applications.

Fig.17 Trapped radical concentration as function of temperature for MMA/EGDMA with 2.0 wt% AIBN and the EGDMA levels are: 100 (circle), 75 (triangle) and 50 (square).

Network Properties

The microstructures of polymer networks are mainly developed in the post-gel period. The characterization of network properties, such as crosslinking density, content of pendant double bonds, cyclizations, dangling chain ends and radicals (refer to Fig.2), is very important, but difficult. However, an effort was made to correlate the line broadening in I3C nuclear magnetic resonance (Nh4R) spectra of crosslinked polymers to the crosslinking density (Refs.32-33). The earlier studies in this area involved aqueous gels of poly(sodium acrylate) crosslinked with N,N'-methy he-bis(acry1amide). It was observed that both the methine and methy lene peaks showed substantial broadening with increasing crosslinking. Fig.20 shows some spectra for poly(sodium acrylate) dissolved or swollen with D,O. The line widths of the methine and methylene peaks increase rapidly up to ca. 0.02 mole fraction of Bis and then more slowly reach a maximum at about 0.08 mole fraction. A maximum width of approximately 20 times the original width was observed. Fig31 shows the variation of the

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Fig.18 ESR spectra recorded in the post-effect period at 180 'C for MMAEGDMA with 75 wt% EGDMA. The operation conditions are: microwave frequency 9.45 GHz, modulation frequency 100 KHz, modulation amplitude 3.2 Gpp, Power 20 dB, gain 2x10' and reaction time (a) O'OO", @) 1'30", (c) 3'30", (d) 6'00".

line widths with mole fraction of crosslinker. The origin of the broadening was examined by a number of different experiments, such as to investigate the line width as a function of field strength, relaxation measurements and hole burning experiments, and was attributed to the chemical shift dispersion caused by crosslinking. Very recently, a further effort is being made to measure the solvent penetration rate into MMA/EGDMA gels swollen with chloroform using the 13C NMR technique to elucidate structural properties of the gels (Tkf.50).

An alternative technique to estimate the crosslinking density is the equilibrium swelling experiment. The crosslinked polymer samples of MMA/EGDMA were swollen with either acetone or chloroform at room temperature (Ref.51). The swelling ratios were then measured. Fig.22 shows an example of the typical results. The swelling behavior of a polymer network can be described by a balance between mixing and elastic free energy. Many models have been developed to interpret the relationship between crosslinking density and swelling ratio. The measured swelling ratios were then used to estimate the crosslinking density. Fig.23 shows some data obtained using this technique.

The content of pendant double bonds were also measured (Ref.52) using a Raman spectrometer (conventional Spex, cat. 14018, er. 5420, double-monochromator, 0.85 m, 1800 grooves/mm). Fig.24 shows a spectrum of the h4MA/EGDMA polymer sample. The 1641 cm" band, due to the C=C stretching mode (Refs.53-54), is characteristic of the unreacted double bonds. The carbonyl group, GO, at 1729 cm", does not change during polymerization and therefore was used as an internal standard. Fig.25 shows an example of the ratio of the

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Fig.19 The change in the ESR spectra of (a) 50 wt% EGDhfA, polymerized for 100 min; lo

@) reacted with styrene monomer for 10 hrs thereafter. gain: (a) 5 x lo4; @) 5 x Id, the other ESR conditions Same as in Figure 5.3.7. T = 70 'C; AIBN: 0.3wt%.

peak areas at 1641 and 1729 cm-'. It was found that the pendant double bonds once trapped in polymer network likely remain unreacted due to the severe diffusion constraint.

2 0 X x l i n k .... -Y 4 % d i n k

li".., I, 100 110 1.1 I t 0 111 ,," 800 I 0 so I 0 10

Fig20 "C NMR spectra of linear poly(scdium acrylate) in D,O solution and of crosslinked poly(sodium acrylate) swollen with 40.

155

-CROSSLINKIN G MONOMER (MF I

Fig.21 Variation of CH (circle) and CH, (triangle) line widths with mole fraction of cross- linking monomer.

0- 0 0 5 1.0

CONVERSION

Fig.22 Swelling experiments for gel molecules in chloroform and acetone at mom temper- ature. Each gel w z sepanted from sol completely.

156

s m o x i m u a

4 0.01 7

0 05 1.0 X

Fig.23 Elastic crosslinking density estimated using the swelling data based on the Flory swelling theory. The solid line jS the prediction result of the Tobita-Hamielec model.

Fig.24 An example of the Raman spectra of MU'VEGDMA polymer. The 1641 an.' band is charaderistic of the unreacled double bonds. The carbonyl group at 1729 an.' does not change during polymerization and therefore was used as an internal standard.

157

Polymerization of AAmBis

Cyclizatwn

The copolymerization of AAm/Bis was implemented in ion-exchanged water at 25 OC (Refs.15, 51). The redox system potassium persulfate (PS) / triethanolamine (TEA) was used as initiator. Isopropyl alcohol (IPA) was also used as a chain transfer agent. During the polymerization, nitrogen was continuously bubbled to provide good mixing. The reaction was stopped by adding acetone containing 4-methoxyphenol at a desired time. The polymer was successively separated by centrifugation, washed with acetone, and dried. The conversion was determined gravimetrically. Number of pendant double bonds was measured by bromometry (bromate/bromide titration). The end point was determined when the solution remained colorless for over 15 minutes. The concentrations used were 56.6 g/l for comonomers, 2.48~10' moln for PS and 4 . 2 3 ~ 1 0 ~ molb for TEA. Fig.26 shows the monomer conversion histories. The point with an abrupt increase in polymerization rate was assigned as the gelation point. It was found that the gelation point came much later than predicted by existing theories (discussed later). This delay was adequately explained by the large amount of primary cyclization existing in this system. Fig.27 shows the measured, pendant double bond conversion plotted as a function of monomer conversion. The intercept at zero monomer conversion gives the degree of primary cyclization. Amazingly, it was found that at least 80% of the pendant double bonds were consumed by primary cyclization at low conversions. The effect of initial mole fraction of divinyl monomer on primary cyclization for various systems are summarized in Fig.28. The experiments of AAmBis and acrylic acid (AA) with the full range of Bis using inverse microsuspension polymerization techniques are underway to investigate the effect of electrostatic interactions among charged species (Ref.55).

THEORETICAL WORK

Literature Review

Carothers (Ref.56) was the first to define gel as a three-dimensional molecule with infinitely large molecular weight. Many theories have been developed since then to describe the molecular processes involved in gel formation during various polymerizations. The creative genius of Flory (Ref.4) and Stockmayer (RefS), led to the now called classical theory, established the framework for further developments. Most statistical theories derived in the following decades are fully equivalent, differing only in mathematical language, eg. the cascade theory by Gordon (Ref.57), the various probabilistic theories such as Macosko and Miller (Ref.58); Pearson and Graessley (Ref.59); Durand and Bruneau (Ref.60).

However, it is also well known that the derivation of the classical gelation theory employed three basic assumptions which are rather limiting. They are: (1) all functional groups (ie. double bonds in the free-radical copolymerization of vinyl/divinyl monomers) have equal reactivity; (2) there is no cyclization; (3) all functional groups react independently of one another. Although much effort has been made.to account for unequal reactivity (Refs.61-62), substitution effect (Refs.63-64), and cyclization (Refs.65-67) the mathematics involved are often complex. And above all, these statistical models require the random formation of crosslink points on accumulated polymers and thus inherently treat gelation as a process in thermodynamical equilibrium.

1

EGDMA \NT%

0.8

" 1 a h 50 I 46

I I I I

0 0.2 0.4 0.6 0.8 1

Monomer Conversion

Fig.2.5 Variation of the ratio of peak areas at 1641 and 1729 an.' with monomer conversion during the ooplymerization of MMA/EGDMA.

Reactions involved in free-radical polymerization with crosslinking are however kinetically controlled. The development of chain properties depends strongly on the reaction path. In terms of this point, the statistical models are not realistic. Although some attempts have been made to modify statistical models to accommodate this point (Refs.14, 68-69), the simpler remedy for this is to derive kinetic models accounting for the dependence of polymer properties on reaction path. This approach has been tried by Dusek (Ref.70), Mikos et al. (Ref.71), Batch and Macosko (Ref.72). But, only very recently, Tobita and Hamielec (Refs.51, 73-74) made a rather systematic attempt. Their model has been successfully applied to some aspects of the systems MMA/EGDMA and AAm/Bis.

159

0 10 20 30 4C 50 60 70 TlMF Emin.]

Fig.26 Conversion history of the copolymerization of AAm/Bis for f,=0.07 with 0.094 moVl of P A at 25 'C. ?he point with an abrupt increase in polymerization rate was assigned as the gelation point.

It should be pointed out that the above gelation models are all based on mean-field theory, but, that there is a non-mean-field theory available, ie. percolation theory (Refs.75-78). Although the mean-field theory for gelation is not exact (Ref.79), the non-mean-field theory such as percolation on the other hand is still too premature for practical use.

Crosslinking Density Distribution

Tobita-Hamielec model was developed based on the pseudo-kinetic rate constant method (Refs.80-81). The key point of the method is the birth conversion dependence of crosslink density. Let us make a brief review of this model. The crosslink density is defined as a ratio of the number of crosslinked units (two such units form per crosslinkage) over the total number of monomer units chemically bound in primary chains. The primary chain is a rather imaginary linear polymer molecule which would exist if all crosslinks connected to it were severed. At present conversion @, the total crosslink density p(0,@) of primary polymer chains born at birth conversion 0 (0 s @) is the sum of the instantaneous crosslink density pi(@) and the additional crosslink density pe(O,Q),

160

z 0.8

0 0.8 m

f,, =0.07 with IPA

2 0.8

a 0 1.0

MONOMER CONVERSION

Fig.27 Pendant double bond conversion versus monomer conversion measured by broma- tebromide titration. This measurement permits one to estimate primary cyclization during network formation.

P(@, @I - Pi (@) + PA@, @I (1)

The instantaneous crosslinks are formed during the growth of the primary chains. The additional ones are formed in the conversion-interval 0 to @ by consuming unreacted pendant double bonds in the primary chains born at 0. The fundamental equation for pa(@,@) is

(2) dP,(@, @I KJ@) Fz(0) - P,(@) - P,(@ @I - PA@, @I --- a@ K,(@) 1 - @

Please note that a correction to this equation as presented in the original work has been made to account for the fact that the instantaneous secondary cyclization does not consume pendant double bonds on primary chains born at the conversion 0, and hence p,,(0) should be absent from p,(0,@). It is also worth mentioning here that the Tobita-Hamielec crosslink density distribution is with respect to the birth conversion (time) of primary polymer chains. p(0,@) is an average value for all the chains born at the conversion 0. Actually, there is also a crosslink density distribution among the chains born at the same time, eg. The instantaneous crosslink points are clearly formed in a manner analogous to the Stockmayer composition distribution.

Ki(@) and K,(@) are the pseudo-kinetic rate constants for propagation with pendant double

bonds and the monomers. Fz(0 ) is the mole fraction of divinyl monomer bound in the primary chains, and p,(0) is an intramolecular (primary) cyclization. The intramolecular cyclization

161

1.0 I I I I I I 1

-0 on X

acrylamide/N,N'-Rethylene-bis -acrylamide 5 6 . 6 9 6 i n water [ S e c t i o n 3 . 2 1

s t y r e o e / p - D M IOvol% in cyclohexanc [soper et a l . (197211

0.5

styrene/p-DVE bulk p l y n e r l z a t l o n [Hallnsky et a l . (1971); IkraeK (198211

Mu/EGWA bulk p o l m r i z a t i o n

0 0 0.1 0.2 0.3

f20

Fig.28 Effect of initial mole fradion of divinyl monomer on primary cycliition for some vinyVdiviny1 comonomer systems.

occurs within a primary polymer chain. It should be pointed out that there actually are two types of intramolecular cyclization. Those, formed by connecting the two double bonds within a single divinyl monomer next to each other through chain propagation, have not been explained in the Tobita-Hamielec work. Divinyl monomers, like ethylene glycol dimethacrylate and N,N'-methylene-bis-acrylamide but not divinyl benzene, with more than five units between the two vinyl groups are likely to form this type of cycle, if not chemically prohibited. The cycle formed does not make contributions to either gel formation or elastic properties of the polymer network. According to the "random flight model" (Ref.65),

p,,(@, @) is an additional intermolecular cyclization. The intermolecular cyclization occurs

between two primary polymer chains (note that in the Tobita-Hamielec work multi-chain cyclization is excluded). Although this type of cycle does not contribute to gel formation either, they may be elastically effective.

162

where q(O,@) is the average number of intermolecular cycles per crosslink. It is equal to a constant K,, using the "zeroth approximation". Fig.29 gives the pictorial representation for the definitions of the discussed cyclization type.

INTERMOLECULAR CYCLIZATION

-n

BETWEEN TWO CHAlNS MULTI-CHAIN

INTRAMOLECULAR CYCUZATlON

SINGLE DNINYL MONOMER UNlT

Fig.29 Schematic representation of the various cyclizations which may occur during the free-radical polymerization with mosslinking.

Since additional crosslinks need their partners, pi(@) is therefore given by

In general, the total crosslink density p(0, @) can be solved numerically using Eqs.(l) to (5). The average crosslink density ;(@) is

The major contribution of the Tobita-Hamielec work is that it has theoretically demonstrated the inherent heterogeneity of polymer networks formed in free-radical polymerization with crosslinking. As it can be seen from Fig.30, the crosslink densities for the primary chains born at different conversions can be very different. Such information may be important for the design of microstructure for network polymer products. An application of this model can be found in Fig.23.

163

0.021, , , , , , , , , ,

0.000 0.0 0.1 0 . ) 0.3 0.4 0 s 0.6 0.7 0.8 0.8 l.0

BIRTH CONVERSION

Fig.30 Crosslink density distribution of the primary polymer chains with respect lo the birth conversion at the present conversions: 0.1,0.3,0.5,0.7,0.9 predicted by the Tobita-Hamielec model.

Molecular Weight Development

The Stockmayer instantaneous bivariate distribution of chain length and composition for linear polymers (Refs.82-83) provides a good starting point for copolymerization studies using the terminal model. Although the derivation of the distribution is quite straightforward, the algebra involved is lengthy and rather laborious. The following is a brief summary of the distribution (note: their equations did not account for the recombination reaction, ie. f3 - 0).

where

F1F$ d-- r

K - 41 + 4FlF2(r,r2 - 1)

and

164

w(r,y,x) is the weight fraction of the polymers born at monomer conversion x, with a chain length r and a composition y (deviated from the average valueFl). K’s are the reaction rate constants with the subscripts p, h, ff, td and tc standing for propagation, transfer to monomer, transfer to agent or solvent, termination by disproportionation and recombination, respectively. M, T and R are the concentrations of monomer, agent and radical.

When the molecular weights of the comonomers, m, and m,, are not equal, a correction factor,

m, - m.

is needed (Refs.51, 84).

The bivariate distribution is of course the most informative chain property. However, the equivalent general treatment for copolymerization with crosslinking is rather complicated due to the different double bond reactivities. The system is therefore a multi-component polymerization with at least three different types of double bonds (on comonomers and pendant). The extension of the Stockmayer distribution to multi-component system is a prerequisite for the distribution modelling of the copolymerization with crosslinking, but unfortunately is not available. However, the composition drift of long polymer chains is not significant, and in practice, it is often enough to know some averages instead of tracing the full detail information. A practical alternative is the use of the method of moments based on the pseudo-kinetic rate constant method. It should be clarified that the pseudo-kinetic rate constant method does not change the nature of copolymerization. The advantage of the method is to permit one to treat copolymerization as homopolymerization. The latter is, of course, much simpler. However, it does not carry detailed information on copolymer composition and chain sequence distribution.

In the following, we take the reaction of recombination termination in copolymerization as an example to illustrate the use of the pseudo-kinetic rate constant method. The reactions are

KUO Rs,i +Rr-s*j -* P r

i, j = 1, 2. The rate of polymer formation by this reaction is

K, is a pseudo-kinetic rate constant. Clearly, the last expression is the same as for the

homopolyerization case. It is assumed that the radical fractions, I$,,i, are chain-length independent, ie.

165

$i - - Ori - ... - $,,i - ... Therefore, the pseudo-kinetic rate constant is

K i e - z zKioij@i$j $ 1

and all the other rate constants in copolymerization can be treated in the same way. It is clear that the pseudo-kinetic rate constants are expressed as functions of the radical fractions, 3. Such defined rate constants are not really “constant” because they often change during the course of polymerization. Therefore, the calculation of these radical fractions is the most fundamental step for further modelling using the method.

The moments of a chain distribution for radical and polymer are defined as - - X riR, i - 1

and

ei - ,i rip, 1.1

respectively. The relative moments are yi - q/Yo and qi - Qi/Mo with M, the initial monomer

concentration. In general, the method of moments cannot provide the full molecular weight

distribution, but only the average properties, eg. The i-average chain length yi - 5 (i = 1, 2,

3, ... represent number, weight, z-, ...). Theoretically, distribution and its moments are interchangeable, and it is noted that Bamford and Tompa (Ref.85) have developed a method to calculate of the full molecular weight distribution from the corresponding moments. However, their method is effective only for simple unimodal distributions which are not often met in practice. The reason is that the moments are integer moments (i = 1, 2, 3, etc.). Within the dominant range of a distribution, only three integer moments of lowest order (zeroth, first and second) exist, and these do not carry sufficient information for the full distribution. It is obvious that the molecular weight distribution is the most desired property in a kinetic study. However, it is not always accessible, particularly in nonlinear free-radical polymerization. In contrast, the moment method is simple and tractable in most cases.

The moment equations governing free-radical polymerization with crosslinking including both sol and gel populations were derived (Ref.86). For radical,

C,p. r + B + r

r+B+*+- Y: - cpw (Cb +C;).w# (16)

166

and for polymer,

with

KfPh C, -- KP

where %’ and K, are the rate constants for propagation with pendant double bonds and transfer to polymer. f and &, are the ratios of pendant double bonds and abstractable hydrogens (or other atoms) over the total monomeric units of polymer chains. It should be noted that y: + yo8 = 1, w, + wa - 1 and q; = xw,. Therefore,

where w’s are weight fractions and the subscripts s and g denote sol and gel, respectively.

Pre-Gelation Period

In the pre-gelation period, wg=O. The above equations therefore can be simplified as follows:

Y o - 1 (16a)

167

The above simplified equations are the same as those of Tobita and Hamielec (Tkfs.51, 73-74). The latter did not take into account gel participation in the sol molecular weight development. Therefore they are applicable only in the pre-gelation period. Fig.31 shows the application to the system of styrene / m-divinyl benzene (DVB) in benzene. It should be pointed out that the modelling of conversion data (refer to Fig.3) did not involve any detail of the polymer chain structures. The data could always be fitted by adjusting the apparent kinetic parameters of initiation, propagation and termination.

In general, Eqs.(l6a) to (20a) can be solved numerically. However, under limiting conditions, analytic solutions can be obtained. These analytical solutions may provide greater insight into the formation mechanisms of the polymer chain structure. The following special cases are discussed using constant t, p, Cfp and Ci.

Case 1:

Cfp - 0; C i - 0, ie. onLy linear polymer chains are produced.

where i = 1, 2, 3 represent number-, weight-, z-average chain length, etc. In the polymerization with crosslinking, this equation is readily applicable to the primary polymer chains.

Case 2:

C, - 0, ie. gelation is only by propagation with pendant double bonds, with an additional

condition, xgc, << 1 -

168

where the subscript p of r denotes primary. At the gelation point, where C,’xFwp -+ 1, the

infinitely large molecule starts to form. The weight- and z-average chain lengths approach infinity simultaneously.

It should be pointed out that the limiting condition xgcl << 1 can be easily removed in this case. But, the equations are much more complicated, eg.

Case 3:

C,‘ - 0, ie. gelation by transfer to polymer, also with xgel c< 1

This equation clearly gives the necessary condition for polymer network formation, ie. both reaction types, chain transfer to polymer and radical recombination are required. The essential condition is

It can be seen that if the termination reaction is recombination exclusively, the reaction rate of chain transfer to polymer should be higher than that of radical recombination in order to gel the system. It should be pointed out that gelation by this mechanism does not necessarily mean copolymerization. As long as the polymer chains bear abstractable atoms, gel formation can also occur in homopolymerization. However, it is easier to gel a system by propagation with pendant double bonds than by chain transfer. The former forms tetra-functional branch points while the latter tri-functional.

Post-Gelation Period

In the post-gelation period, the gel fraction, wg, is not zero. The governing equations, Eqs.(l6) to (20) are not closed. Therefore, closure conditions, which express qf as a function of the lower order moments (i-1, i-2, ... 0), are needed. It should be pointed out that, in practice, one closure condition is actually sufficient to solve the whole set of governing

169

" 0 0.1 0.2 0.3 0.4 0.5 MOLAR CONVERSION

Fig31 Average molecular weight development in the pre-gelation period for the copolym. erization of styrene/m-DVB in benzene at 60 'C (Ref.16).

equations. A closure condition can be proposed for any specific i. However, the one that expresses the sol/gel fraction as a function of monomer conversion is the most desired. The reasons for this are two-fold. First, the sol/gel fraction has a much clearer physical meaning than any of the higher order moments. Second, there has been some related work in the literature which may provide the basis for the derivation of this function. Once this function is known, the other moments, q:, can be easily calculated from the governing equations. Take q; as an example, instead of using the equation for dq'ddx, one can use dq'ddx which itself is known, and solve qS2 using the right side of the equation.

The most well known formulation of the sol/gel fraction is the Flory equation (Ref.87):

where wp(r) is the molecular weight distribution of primary polymer chains. p is the crosslink

density, which is defined as the number of crosslink points over the total number of monomer units in polymer chains. Please note that in most of the literature, the crosslink point is actually defined as a branch point. Therefore, one crosslinkage consists of two crosslink points. The Flory equation is derived for the network formation with tetra-functional crosslinkage units. We therefore focus on the mechanism of gelation by the propagation with pendant/intemal double bonds in the following analysis.

170

The derivation of the Flory equation requires that the accumulated crosslink points are randomly distributed over the accumulated primary polymer chains over the entire conversion range. This actually indicates that the polymerization is in thermodynamic equilibrium at aoy monomer conversion, although this was not explicitly stated in the original derivation. However, free-radical polymerization with crosslinking is kinetically-controlled. The extension of the Flory equation for the kinetically-controlled gelation has been attempted by Tobita and Hamielec (Refs.51, 73-74). Based on their successful formulation of the birth-conversion dependent crosslink density distribution, p(0, Q), Tobita and Hamielec proposed the following equation for the sol/gel fraction of the primary polymer chains born at 0:

Cp is the present monomer conversion (using their nomenclature). The overall sol/gel fraction

is then

Unfortunately, this extension is not generally correct. The reason for this is the following. Eq.(30) is a direct application of Eq.(29) to the primary polymer population born at conversion 0. In reality, primary chains born at 0 are not isolated from the chains born at the conversions other than 0. Chains born at any conversion are supposed to be distributed over the whole reaction system, ie. there is no spatial heterogeneity (this is the basis for the expression of reaction rates in deriving a kinetic model). Furthermore, the crosslink points in p(0, Cp) according to the original derivation are formed between the primary chains born at different conversions, but not between the chains born at the same conversion 0. This contradicts the definition of the crosslink density in Eq.(30). Actually, the crosslink densities between chains born at the same conversion are assumed to be zero in the derivation of the Tobita-Hamielec model (this assumption is valid because statistically the chains born over an infinitely small conversion range are infinitely small in amount, and hence seldom meet one another to implement crosslinkings among themselves). It should be pointed out that the calculated results using Eq.(29) are not much different from those of Eqs.(30) and (31).

Tobita and Hamielec (Ref.51) have also solved Eq.(23) as an alternative approach for the sol/gel fraction calculation. They used the Flory-Stockmayer expression for the weight-average chain length of sol polymers as a closure condition to eliminate the q; terms.

171

where <p is the weight-average chain length of the primary polymers which belong to the sol

population, and p' is the crosslink density of sol polymers. Both <, and p' can be calculated from the Flory-Stockmayer theory (Refs.4-5). In fact, Eq.(32) is equivalent to Eq.(29).

Here let us recall the mechanism of gel formation in the free-radical copolymerization of vinyl/divinyl monomers by referring to Fig.2. With statistical theories, the nucleation and growth of gel molecule are ingeniously described using the recursive feature of the gelation process coupling techniques of probability and statistics. Eq.(29) is an excellent example. However, the kinetic formulation, which accounts for polymerization path and is thus more realistic, is not as successful in this regard. We have tried many kinetic approaches without a breakthrough. In summary, the kinetic formulation of the sol/gel fraction as a function of the monomer conversion has been proven very challenging. However, it deserves much effort because this function is the essence of kinetic modelling for the whole post-gelation period. As an approximation (Ref.86), we have developed a statistical model by introducing the Tobita-Hamielec crosslink density distribution into the Macosko-Miller approach (Ref.58). This model can be used to investigate the effect of the crosslink density on the molecular weight development, the gelation point, the sol/gel fraction, the weight fraction of linear polymers, the dangling chains in gel. It is found that the crosslink density distribution is in favour of the gel formation. At gelation point, we have

where 0 and @ are the birth conversion and the present conversion, r is chain length, p is the

crosslink density, w,(r,O) is the instantaneous weight chain length distribution at conversion 0. When both the molecular weight distribution and the crosslink density distribution are independent of the birth conversion, Eq.(33) reduces to the result of the classical gelation theories. Fig.32 shows an example. According to our simulation, the differences between the predictions of gelation properties with and without taking the crosslink density distribution into account are often minor except for those under limiting conditions. It should be kept in mind that this model is still a statistical model, not a kinetically-controlled one.

CHEMICAL MODIFICATION

The commercial incentive for the chemical modification of commodity polymers by grafting, chain scission, long chain branching, crosslinking is the enhancement of physical and chemical properties of commodity polymers and polymer mixtures (alloy, blend, additive etc.) and/or the improvement of their processability. In many cases, the extruder has been found to be an effective chemical reactor in which chemical modification of polymer can be done economically. The extruder may be considered as a continuous reactor with a narrow

172

CONVERSION

Fig.32 Development of weight-average molecular weight rJrpw in the pre-gel period. The solid line are calculated by taking thecrosslinking density distribution into account, and the dotted assuming that the crosslink densities are all the Same for the primary polymer chains which have different birth conversions.

residence time distribution which is usually highly desireable from the points of view of productivity and product quality.

Fig.33 schematically presents the modification mechanism. The initiator molecules, often peroxides, decompose into primary radicals at elevated temperature. The active primary radicals abstract atoms, often protons, from prepolymer backbones and produce backbone radicals. Generally, for a backbone radical, there exist three possible fates: (1) It experiences chain scission, breaking the chain into two parts, one with a radical center on its chain end, the other a dead polymer; (2) It is terminated by another radical. If the termination is recombination between two macroradicals, a cross-linkage will be formed; or (3) It reacts with an additive molecule, if available, which is a case of grafting. Additives are often used either to improve the properties of polymer product or to orient the reaction direction, ie. The relative level of chain scission and crosslinking. For example, pure polypropylene undergoes preferentially chain scission. However the use of additives such as multi-functional monomers (Refs.88-89) and quinones (Refs.90-91) can result in quite high levels of crosslinking. Under limiting conditions, one modification reaction type may be dominant over the others. But, in general, these reactions occur simultaneously.

A general equation describing the molecular weight development during simultaneous chain scission and crosslinking in the presence of additives has been derived (Refs.86, 92).

173

~ N ~ T ~ ~ T O R PRIMARY RADICAL PEROXIDE .. DECOMPOSITION 0

ABSTRACTAOLE PROTON BACKBONE RADLCAL

J J

CHAIN SCISSION

9 --3 CROSSLINKINC

GRAFTING

- T& Fig.33 Schematic representation of h e free-radical mechanism of chemical modification of pre-polymers in reactive p m i n g .

l aw(r z ) -- - - w ( r , z ) + 2 a r a t

(34)

where z is a parameter associated with the degrees of random chain scission and crosslinking. In the absence of chain scission, it is exact given the degree of crosslinking (crosslinking density). Together with a and P’s, it can be calculated once the reaction rate constants are known. The five terms on the right hand side have clear physical interpretations. The first term represents the polymer chains consumed by hydrogen abstraction becoming the backbone radicals (note that when the backbone radicals are terminated by the primary radicals or by the polymer radicals by disproportionation, the polymer chains are actually not consumed); The second term represents the chains produced by chain scission (one scission

174

makes two chains with one stable polymer and the other chain-end polymer radical which also becomes a stable polymer chain when terminated by the primary radicals or the polymer radicals by disproportionation). This term is derived based on an assumption that there are only two ways to produce a chain with length r from a longer chain with length s (x). This is of course true for linear chains; The third term.represents the chains produced by recombination of backbone radicals with tetra-branching points; The forth term represents the chains produced by recombination of backbone radicals and chain-end radicals with tri-branching points; And the last term represents the chains produced by recombination of chain-end radicals forming linear chains. Eq.(34) is a very complicated integro-differential equation. However it can be simplified under limiting conditions.

When there is no recombination of polymer radicals, the parameters are

z - u

a - 1

P’S - 0

u is the degree of chain scission. In this case of solely chain scission, Eq.(34) is reduced to the Saito equation (Refs.93-94). The Saito equation was derived for random chain scission of linear chains by radiation. The difference with the peroxide-initiated chain scission lies in the calculation of the degree of chain scission as a function of time and reaction conditions.

The other limiting case is when there is no chain scission, the parameters are

z-- v

a - 0

v is the degree of crosslinking. This is the case of solely random crosslinking.

The analytical solution to the general equation of simultaneous chain scission and crosslinking may be formidable and is yet not available. Numerical methods are therefore recommended. However, there is an approach (Refs.93-94) which has been often used, and which may provide a reasonable approximation in special cases. In the approach, it is assumed that (1) chain scission and crosslinking occur independently; (2) the reactions occur in two stages, the chain scission occurs first and the crosslinking is then based on the scissored polymer population (as primary chains). There is no doubt that errors of unknown significance will be introduced by this assumption. It is obviously contradictory to have recombination of polymer radicals in the second stage but not in the first. Since there are no

175

chain-end radicals existing in the second stage (because there is no chain scission), the formation of tri-branching polymers is therefore magically prevented. However, when the degrees of chain scission and crosslinking are low, the assumption may be acceptable.

Using the Saito assumption, polymer chain properties can be easily calculated using the respective equations for crosslinking with the initial values replaced by those of the scissored polymers from chain scission. Take the sol/gel fraction as an example. Substituting a molecular weight distribution (if it is a random distribution) for polymers, which have experienced random chain scission, as the primary chain distribution into Eq.(29), we can easily get

where F,,,' is the initial weight-average chain length. Eq.(35) is the widely used

Charlesby-Pinner equation (Ref.95).

In chain scission experiments (Refs.96-97), polypropylene PP), produced by a conventional process using Ziegler-Natta catalyst, was used. The peroxide, 2,5-dimethyl- 2,5-di(t-butylperoxy) hexane (Lupersol 101), was carefully mixed with the polymer by dissolving and then evaporating methanol. The molecular weight distributions were measured by a high temperature size exclusion chromatography (SEC) with trichlorobenzene as solvent at 140 'C. The chain degradation was implemented either in a differential scanning calorimeter cell (du Pont Instrument 910) or a Killion single screw extruder at 220 'C. Fig.34 shows an example of the change of molecular weight distribution. Fig.35 shows the result of weight-average molecular weight as a function of the degree of chain scission. It was found that PP underwent almost exclusively random chain scission.

The crosslinking experiments were carried out in an extrusion plastometer using LLDPE (0.924 g/cm3) and HDPE (0.951g/cm3) prepolymers (Refs.98-99). For LLDPE at low peroxide levels, the experiments were also conducted in the Killion single-screw extruder. The melt flow index of the produced polymers was measured at 190 or 230 OC with a weight of 10 kg using a Monsanto extrusion plastometer. The density of the crosslinked HDPE was measured using a flotation technique involving the addition of two miscible nonsolvents (isopropyl alcohol and water) in a manner similar to that employed in a titration. The end point is reached when the polymer becomes suspended in the mixture. The gel content was extracted by boiling decalin for 8 hrs, followed by methanol for 4 hrs. Fig.36 shows the MFI result for LLDPE. Low temperature was found to be in favour of crosslinking. As temperature increased, chain scission became more and more significant. Fig.37 shows the density change of HDPE extruded at 190 OC. The MFI and density measurements provides easy assessments for the modification direction. However, they are empirical methods since the general relationship between the melt flow index and polymer structures is not available. Fig.38 shows an example of the gel development during extrusion of HDPE. The crosslinking was

176

almost instantaneously occuring in only a few minutes. Continuous studies in this area are underway (Ref.lOO).

Perox: 0.1 %wt

Fig.34 Molecular weight distribution of PP after degration in a DSC cell with 0.1 wt%

peroxide.

300000

200000 = 5.

100000

0

= m

T " " : '

0 0.0005 0.001

Degree of chain scission

Fig.35 Weight-average molecular weight as a function of the degree of chain scission for PP degraded in a single-screw exuuder at 220 ''2.

177

x

- p 12[1

Extrusion Temperature (C)

n

0 280

.- Peroxide wl%

Fig.36 Melt flow index for LLDPE extruded in the presence of peroxides. The effect of

. Measured a1 230 C I lo kg \ yo

0.952 7

0.95 - 0948 - 0.946 - 0.944 -

8 0.942 - Extruded at 19OC . m 0.94 -

.$ 0.938 ~ Residence time 20 min. 2 0.936 -

0.934 - 0.932 - 0.93 -

0.928 - 0.926 - 0.924 1

0 0.2 0.4 0.6

56 Peroxide

Fig.37 Densily versus peroxide concentration of HDPE polymers extruded at 190 'C, for 20 min.

1

Fig.38 Effect of the residence lime on the gel fraction of HDPE at 190 'C, and peroxide 0.2, 0.5 wt%.

178

CONCLUSIONS

We have been involved in the kinetic study of free-radical polymerization and modification with crosslinking for the past five years. The following areas were emphasized: elucidation of mechanisms of the diffusion-controlled reactions due to both physical entanglement and chemical crosslinking of polymer chains; measurements of the kinetic parameters under various polymerization conditions; development of new techniques to characterize polymer network; the development of comprehensive kinetic models for polymer network formation by various free-radical mechanisms. The experimental work provides substantial kinetic data for the free-radical polymerization of MMA/EGDMA and AAm/Bis. In theoretical work, a gelation model taking the polymerization path into account was derived.

Our future research activities should be directed to the following. (1) The radical trapping effect: The polymerizing system of MMA/EGDMA is heterogeneous in terms of radical environments. The crosslinking seriously enhances the trapping effect. The formation of glass domains, where radicals are trapped, during auto-acceleration should be described quantitatively by taking into account the radical life-time and the relaxation of polymer network. It should be kept in mind that the process involved in domain formation cannot be in thermodynamic equilibrium because the short time interval of autoacceleration does not allow the chain entanglements to fully relax. (2) Diffusion-controlled reactions: In free-radical polymerization with crosslinking, due to the constraints exerted by the polymer network, most reactions become diffusion-controlled, particularly, those involving long polymer chains, such as bimolecular termination, propagation with internal/pendant double bonds, chain transfer to polymer. The modern diffusion theories should be introduced into the framework of gelation models to give proper description of the rate constants. (3) Gel nucleation and growth: Since the gel/sol fraction calculation provides the basis for modelling the whole post-gelation period, the more realistic mathematical formulation for kinetically-controlled polymerization-path dependent gelation is therefore very critical and deserves further effort. (4) Kinetics of hydrogel formation: Effort should be directed to study the interaction of the ionic centers on polymer chain backbones, and the effect of electric field on the elementary reactions involved in network formation, particularly those like cyclization which strongly depend on the chain conformation. (5 ) Modelling of extruders: A flow model considering the geometric parameters of extruder should be developed to couple with kinetic models.

ACKNOWLEDGEMENTS

We wish to express our appreciation to Ontario Center for Materials Research and Natural Sciences and Engineering Research Council of Canada for their financial supports.

179

Richardson, T.L., IndustrialPlastics: Theory and bnpl ication, 2nd ed., Delmar

Publishers Inc., NY, (1989) Andreis, M.; Koenig, J.L., Adv. PoIV- ' , Kloosterboer, J.G., Adv. Polvm. Sc i, @, (1988) Flory, P.J., -, a, 3083; 3091; 3096, (1941) Stockmayer, W.H., J. Chem. Phvs ., u , 4 5 , (1943); 12, 125, (1944) Walling, C., J, Am. Chem. s oc., 61,441, (1945) Gordon, M.; Roe, R.-J., ' a , 2 7 ; 39; 57; 75, (1956) Simpson, W.; Holt, T.; Zetie, R . J . , J J & G h . , ' u 4 8 9 , (1953) Loshaek, S., J. Am. Chem. Sc L, z. 3544, (1953)

69, (1989)

(10) Minnema, L.; Staverman, A.J., J. Polvm. S&, ' 2 , 2 8 1 , (1958) (11) Storey, B.T., J, Polvm. Sci .. Part A, 3,265, (1965) (12) Soper, B.; Haward, R.N.; White, E.F.T., J. Polvm. Sci.. Al, .lQ, 2545, (1972) (13) Ishizu, K.; Kuwabara, S.; Chen, H.; Mizuno, H.; Fukutomi, T., J, Polvm. Sci.. Polvm,

Chem. Ed., a, 1735, (1986) (14) Landin, D.T.; Macosko, C.W., Jvlacromolecule s, 2, 846, (1988) (15) Tobita, H.; Hamielec, A.E., Polvmq, in press, (1991). (16) Hild, G.; Okasha, R., Makromol. Chem, , J&, 93; 389; 407, (1985) (17) Whitney, R.S.; Burchard, W., Makromol. Chem., 181,869, (1980) (18) Shah, A.C.; Parsons, I.W.; Haward, R.N.,polvmer, 825, (1980) (19) Dusek, K.; Specvacek, J., Polvme~, 21,750, (1980) (20) Obrecht, W.; Seitz, U.; Funke, W., Makromol, Ch em,, m, 3587, (1974); flli, 2771,

(1975); m, 1877, (1976) (21) Okay, 0.; Funke, W., Macromolecules. 22 2623, (1990)

2123, (1984) (22) Chen, H.; Ishizu, K.; Fukutomi, T.; Kakurai, T., J. Polvm. Sci.: Polvm. &m. Ed, ,a, (23) Ambler, M.R.; McIntyre, D., Polvm. &L ' ,a, 3237, (1977) (24) Wiley, R.H.; Rao, S.P.; Jin, J.4.; Kim, K.S.,J. Macromol. Sci.. Chem, ,Be 1453,

(1970) (25) Malinsky, J.; Klaban, J.; Dusek, K., J. MacromoLScL. Chem, &, 1071, (1971) (26) Hayden, P.; Melville, H., J. Polvm. Sc i, a, 201; 215, (1960) (27) Loshaek, S.; Fox, T.G., J. Polvm. SG + , fi. 391,(1955) (28) Horie, K.; Otagawa, A.; Muraoka, M.; Mita, I., J. Polvm. Sci.. Polvm. Chem. Ed, ,a

445, (1975) (29) Fink, J.K., J. Polvm. Sci.: P w . Chem. ErL, L& 195, (1981) (30) Elias, H.-G., U a c r omol. SCI.. C h e m B, 239, (1983) (31) Hwa, J.C.H., J. Polvm. Sc L, 3 , 7 1 5 , (1962) (32) Neppel, A.; Eaton, D.R.; Hunkeler, D.; Hamielec, A.E., polvma a 1338, (1988) (33) Bain, A.D.; Eaton, D.R.; Hamielec, A.E.; Mlekus, M.; Sayer, B.G., Macromolecules, 2 3561, (1989)

180

(34) Arndt, K.F.; Zander, P., w i d Polvm. Sc L, a 806, (1990) (35) Atherton, N.M.; Melville, H.; Whiffen, S.H., J. Polvm. Sc i., (36) Matsumoto, A.; Inaue, H.; Matsumoto, T.; Oiwa, M., J,, Sc i.. Chem, ,426,

1279; 1475, (1989)

(37) Li, W.-H.; Hamielec, A.E.; Crowe, C.M., Polvmer. a 1513, (1989) (38) Zhu, S.; Hamielec, A.E., Polym. Material Sci. and (39) Ranby, B.; Rabek, J.F., ESR Snectra in Polvm- Springer-Verlag, NY, (1977) (40) Bamford, C.H.; Jenkins, A.D.; Symons, C.R.; Townsend, M. G., J. Polvm. SG * ,&I, 181,

(1959)

(41) Kamachi, M.; Kuwae, Y.; Kohno, M.; Nozakura, S., m, (42) Ballard, M.J.; Gilbert, R.G.; Napper, R.G.; Pomery, P.J.; O'Donnell, J.H.,

Macromolecules, 12,504, (1984) (43) Shen, J.; Tian, Y.; Zeng, Y.; Qiu, Z., ,8,615,

(1987)

(44) Tian, Y. "Kinetics of Bulk Free-Radical Polymerization of MMA", Ph.D The sis, Jilin University, China, (1988)

(45) Zhu, S.; Tian, Y.; Hamielec, A.E.; Eaton, D.R., Macromolecules. a 1144, (1990) (46) Zhu, S.; Tian, Y.; Hamielec, A.E.; Eaton, D.R., m, z 154, (1990) (47) Zhu, S.; Tian, Y.; Hamielec, A.E.; Eaton, D.R., Polvmu, (48) Tian, Y.; Zhu, S.; Hamielec. A.E.; Fulton, B.; Eaton, D.R., Polvmer, in press, (1991) (49) Zhu, S.; Hamielec. A.E., Macromole- 22,3039, (1989) (50) Bain, A,; Mlekuz, M.; Hamielec, A.E., Macromolecules. submitted, (1991) (51) Tobita, H., "Crosslinking Kinetics in Free Radical Copolymerization", &J) Thesis,

McMaster University, Canada, (1990)

(52) Zhu, S.; Hamielec, A.E., unpublised results, (1990) (53) Schneider, B.; Stokr, J.; Schmidt, P.; Mihailov, M.; Dirlikov, S.; Peeva, N., polvmer, a 705, (1979)

(54) Chu, B.; Lee, D.C., b4acromolecules, u, 926, (1984) (55) Zhao, X., Ph.D. work, McMaster University, (1991) (56) Carothers, W.H., Trans. Faradav Soc,, 2, 39, (1936) (57) Gordon, M., Proc. Rov. Sac. f .London), -240, (1962) (58) Macosko, C.W.; Miller, D.R., Macromolecules, 2 199; 206, (1976) (59) Pearson, D.S.; Graessley, W.W., Macromolecules, 1l, 528, (1978) (60) Durand, D.; Bruneau, C.-M., J. Polvm. Sci.: Polvm. Phvs. Ed, , u, 273; 295, (1979) (61) Miller, D.R.; Macosko, C.W., Macromolecules. u, 656, (1978) (62) Durand, D.; Bruneau, C.-M., polvmec, 2,588, (1983) (63) Miller, D.R.; Macosko, C.W., Mg.romolecules , J.& 1063, (1980) (64) Durand, D.; Bruneau, C.-M., w, 24 592, (1983) (65) Jacobson, H.; Stockmayer, W.H., J. Chem. Phvs, 18,1600, (1950)

(66) Gordon, M.; Scantlebury, G.R., J. Polvm. Sci.. Part C , a, 3933, (1968)

199, (1959)

' ,a, 787, (1988)

541, (1985)

1726, (1990)

181

(67) Stepto, R.F.T., "Intra-Molecular Reaction and Gelation in Condensation or Random Polymerization", Develonment in Polv m, Haward, R.N., Ed., Applied Science Publishers Ltd., London, (1982)

(68) Dotson, N.A.; Galvan, R.; Macosko, C.W., -molecules, 2l, 2560, (1988) (69) Galina, H.; Szustalewicz, A., &gomolecules, 22,3124, (1989) (70) Dusek, K., a l l e c tion Czechhoslov. Chem. C o r n ,%, 1891, (1969). (71) Mikos, A.G.; Takoudis, C.G.; Peppas, N.A., Macrom&,wh, 19,2174, (1986) (72) Batch, G.; Macosko, C., m E C , 1039, (1988) (73) Tobita, H.; Hamielec, A.E., Macro- 22,3098, (1989) (74) Tobita, H.; Hamielec, A.E., "Crosslinking Kinetics in Free Radical Polymerization",

Polvmer Reaction, Reichert, K.-H.; Geiseler, W., Eds., VCH Verlagsgesellschaft, New York, a, (1989)

(75) Broadbent, S.R.; Hammersley, J.M., Proc. C smb. Ph il. SOC, a 642, (1957) (76) Kirkpatrick, S., Rev. Modern Phvs, , fi 574, (1973) (77) Herrmann, H.J.; Stauffer, D.; Landau, D.P., -h. G e L JJj, 1221, (1983) (78) Durand, D. Eplvmer Ye arbook, 3,229, (1986) (79) De Gennes, P.G., -ts in P o l e ' , Cornell Universty Press,

Ithaca, NY, (1979) (80) Hamielec, A.E.; MacGregor, J.F. "Modelling Copolymerizations - Control of

Composition, Chain Microstructure, Molecular Weight Distribution, Long Chain Branching and Crosslinking", Polvm er Reactionfie ineerh, Reichert, K.H.; Geiseler, W., Eds.; Hanser Publisher, New York, 2l, (1983)

(81) Broadhead, T.O.; Hamielec, A.E.; MacGregor, J.F. (1985), Makromol. C h u ,

(82) Simha, R.; Herman, B., J. Chem. Phvs,, 12,253, (1944) (83) Stockmayer, W.H., J.Chem., fi 199, (1945) (84) Tacx, J.C.J.F.; Lissen, H.N., German, A.L., J. Polvm. Sc i.. Polvm.

(1988) (85) Bamford, C.H.; Tompa, H., JJolvm. Sc L, fi 345, (1953) (86) Zhu, S., "Advances in Free-Radical Polymerization Kinetics", Ph.D. Thes is, McMaster

University, (1991) (87) Flory, P.J., Principle s of Polvmer Ch Q&&Y, Cornell University Press, Ithaca, NY,

(1953) (88) Chodak, I.; Fabianova, K.; Borsig, E.; Lazar, M., Die A w e wandte Makromol. Chemic,

@, 107, (1978) (89) Borsig, L.; Fiedlerova, A; Lazar, M., J. Macr omol. Sc i . - CherQ fi 513, (1981) (90) Chodak, I.; Lazar, M., J.vm. Sc L, a 5431, (1986) (91) Chodak, I.; Romanov, A.; Ratzsch, M.; Haudel, B., Acta Polv . meric& a 672, (1987) (92) Hamielec, A.E.; Gloor, P.E.; Zhu, S., (93) Saito, O., J, Polvm. Sci. Japan, 13, 198, (1958)

105, (1985)

61,

J. Che- a 611, (1991)

182

. . (94) Saito, 0. "Statistical Theory Of Crosslinking", The Radiation C h e w

ornolecules, 6, Dole, M., Ed., Academic Press, New York, (1972). (95) Charlesby, A.; Pinner, S.H., && 367, (1959) (96) Tzoganakis, C., "Peroxide Degradation of Polypropylene during Reactive Extrusion",

Thesis, McMaster University, Canada, (1988) (97) Triacca, V.J.; Gloor, P.E.; Zhu, S.; Hrymak, AN.; Hamielec, A.E., Polvm. enp i . and

&, submitted, (1991)

217, (1989) (98) Tang, Y.; Tzoganakis, C.; Hamielec, A.E.; Vlachopoulos, J., Adv. in Polvm. Tech, ,e

(99) Hamielec, A.E.; Gloor, P.E.; Zhu, S.; Tang, Y., Procee- of the h p r e s s on a m p a tib-d Re active Polvmer Allov

1 '

Schotland Bussiness Research Inc., NJ, (1990)

(100) Gloor, P., Ph.D. work, McMaster University, (1991)