Hamielec Et Al._1997_Mathematical Modeling of Multicomponet Chain-Growth Polymerization in...

Mathematical Modeling of Multicomponent Chain-GrowthPolymerizations in Batch, Semibatch, and Continuous Reactors: AReview

Marc A. Dube†

Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5

Joao B. P. Soares and Alexander Penlidis*

Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Archie E. Hamielec

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

A practical methodology for the computer modeling of multicomponent chain-growth polymeriza-tions, namely, free-radical and ionic systems, has been developed. This is an extension of apaper by Hamielec, MacGregor, and Penlidis (Multicomponent free-radical polymerization inbatch, semi-batch and continuous reactors. Makromol. Chem., Macromol. Symp. 1987, 10/11,521). The approach is general, providing a common model framework which is applicable tomany multicomponent systems. Model calculations include conversion of the monomers,multivariable distributions of concentrations of monomers bound in the polymer chains andmolecular weights, long- and short-chain branching frequencies, chain microstructure, and cross-linked gel content when applicable. Diffusion-controlled termination, propagation, and initiationreactions are accounted for using the free-volume theory. When necessary, chain-length-dependent diffusion-controlled termination may be employed. Various comonomer systems areused to illustrate the development of practical semibatch and continuous reactor operationalpolicies for the manufacture of copolymers with high quality and productivity. Thesecomprehensive polymerization models may be used by scientists and engineers to reduce thetime required to develop new polymer products and advanced production processes for theirmanufacture as well as to optimize existing processes.

1. Introduction

Mathematical models and their role in science/engineering are points of constant debate, especiallywhen models are employed in an industrial environ-ment. The role of a mathematical model is oftenmisinterpreted; as a result, we frequently blame themodel, instead of blaming our own lack of understand-ing about a process as well as our reluctance to experi-ment with a process in a meaningful and systematicway.Why, then, are models useful?1. Models enhance our process understanding since

they direct further experimentation. They act as thereservoir of one’s knowledge about a process, and hencethey may reveal interactions in a process that may bedifficult, if not impossible, to visualize/predict solelyfrom memory or experience, especially when manyfactors vary simultaneously. Since a model is a concise,compact form of process knowledge, models enhancetransferability of knowledge; they may act eventuallyas an “inference engine”, closely resembling the trainof thought of an experienced human. In a sense,mathematical modeling is the best way to find out whatone does not know about a process!2. Models are useful for process design, parameter

estimation, sensitivity analysis, and process simulation.

The significance of these is quite obvious. A valid modelallows one to test deviations from process trajectoriesusing a simulator in lieu of running experiments. Costeffectiveness implications are also obvious.3. Models are useful for process optimization, espe-

cially when dealing with highly nonlinear problems suchas grade changes/switchovers in batch, semibatch, andcontinuous reactors. Extensions to recipe modificationsand design are another application.4. Models are useful for safety/venting consider-

ations. It is very useful to be able to extrapolate todifferent operating conditions and anticipate “worst-casescenarios” or investigate the possible effects of processfactors. In this case one may be better prepared totackle situations that might not always be apparentfrom the outset.5. Models are useful for optimal sensor selection and

testing, sensor location, filtering and inference of un-measured properties, and process control. The trendsnowadays in process control are toward “model-based”control, and as the term signifies, application of ad-vanced control techniques may not be possible withouta model.6. Finally, since a model contains process knowledge

and is transferable, interactive models are extremelyuseful for the education and training of new (and old)personnel.In this paper, a practical methodology for the com-

puter modeling of multicomponent chain-growth poly-merizations, namely, free-radical and ionic systems, isdeveloped. This is an extension of the paper by Hamielec

* To whom correspondence should be addressed. Phone:(519) 888-4567. Fax: (519) 746-4979. E-mail: [email protected].

† E-mail: [email protected].

966 Ind. Eng. Chem. Res. 1997, 36, 966-1015

S0888-5885(96)00481-2 CCC: $14.00 © 1997 American Chemical Society

et al. (1987). The approach is general, providing acommon model framework which is applicable to manymulticomponent systems. Various comonomer systemsare used to illustrate the development of practicalreactor operational policies for the manufacture ofpolymers with high quality and productivity.Thorough general reviews of polymerization reactor

modeling have recently been published (Penlidis et al.,1985b; Hamielec et al., 1987; Rawlings and Ray, 1988).The model developed in this paper employs the “pseudo-kinetic rate constant method” (Hamielec et al., 1987;Tobita and Hamielec, 1991; Xie and Hamielec, 1993a,b).Our development uses many techniques and ideassimilar to those contained in Hoffman (1981), Broad-head et al. (1985), Penlidis et al. (1986), Hamielec et al.(1987), Mead and Poehlein (1988, 1989a), Rawlings andRay (1988), Maxwell et al. (1992b), Fontenot and Schork(1992-93a,b), Xie and Hamielec (1993a,b), Casey et al.(1994), and Urretabizkaia and Asua (1994).

2. Model Development: Free-RadicalPolymerizations

The objective of this section is to define the equationswhich form a mechanistic model to simulate bulk (orsuspension), solution, and emulsion free-radical homo-polymerization, copolymerization, and multicomponentpolymerization (three or more monomer types) in well-stirred batch, semibatch, and continuous modes. Theequations presented in this section are valid for thegeneral case of an unsteady-state CSTR. For a CSTRoperating at steady state, the accumulation derivativeterms can be set equal to zero to give a set of algebraicequations. For a semibatch reactor, the outflow termsshould be eliminated, and for a strictly batch reactor,all inflow and outflow terms should be eliminated.However, it is usually advantageous to consider com-plete equations since one has the flexibility of handlingall these reactor situations with a single model.The model is comprised of a set of mathematical

expressions which describe the physical and chemicalphenomena of polymerization. It consists of a set ofdifferential equations that describe material and energybalances on the reaction mixture. For computationalpurposes, the model is split into two categories. Thefirst category consists of bulk, suspension, and solutionpolymerization, while the other describes the emulsioncase. In the model development, we shall examine bothcategories in parallel.Bulk, suspension, and solution polymerizations are

characterized by the fact that all of the reaction stepsproceed in a single phase. A model for a reactorcarrying out such polymerizations would consist of a setof material balances describing the rates of accumula-tion, inflow, outflow, and disappearance by reaction ofthe various monomers, initiators, polymers, and otheringredients in the reactor. These polymerizations con-sist of initiation, propagation, termination, and transferreactions occurring simultaneously through the fullconversion range. Conventional emulsion polymeriza-tions usually occur in three stages and are comprisedof more than one phase (reactor head-space; the mono-mer droplets, which act as a monomer reservoir; the(continuous) aqueous phase, which can act as a locus ofpolymerization as well as a species transport medium;and the polymer particle phase, the main locus ofpolymerization). The first of the three common polym-erization stages involves the nucleation (birth) of poly-mer particles. This can occur by either micellar or

homogeneous (coagulative) nucleation. The secondstage involves the growth of the particles until themonomer droplets disappear. The third stage beginswith the disappearance of the monomer droplets andcontinues until the end of the reaction.The emulsion polymerization model can be briefly

described as follows. First, the initiation can be ac-complished via a redox mechanism or via thermaldecomposition of an initiator. The fate of radicals(initiator, monomeric, and oligomeric) in the waterphase is propagation with dissolved monomers in thewater phase, reaction with water-soluble impurities(WSI), termination in the water phase, possible recom-bination of initiator fragments, reaction with monomerdroplets, desorption from polymer particles, reabsorp-tion of desorbed radicals into polymer particles, captureby emulsifier micelles, and capture by polymer particles.The birth of particles can be accomplished by homoge-neous (aqueous phase) nucleation, micellar nucleation,and particle coalescence. Once captured by particles (ormicelles or droplets), the radicals may propagate, mutu-ally terminate, react with monomer-soluble impurities(MSI), react with chain-transfer agent (CTA), undergochain transfer to monomer, undergo chain transfer topolymer, and participate in internal and terminaldouble-bond polymerizations. The average number ofradicals per particle is followed by accounting for entry/absorption of radicals from the water phase, radical-radical termination, radical-MSI termination, anddesorption of radicals into the water phase. The par-titioning of monomer, monomer-soluble impurities, andCTA into the various phases is another important factor.For both the bulk/suspension/solution and emulsion

models, material balances on the various componentsof the polymerization are used to calculate the conver-sion, composition, molecular weight, and, in the case ofemulsion polymerizations, particle size and number.Other, not necessarily measurable, polymer propertiescan also be calculated or inferred. Finally, molecularweight averages dependent on termination, branching,and transfer reactions are estimated.The various symbols, subscripts and superscripts, and

variables are shown in the Nomenclature section. Theunits of the variables are also shown therein.2.1. Initiation. The first step in a polymerization

involves the creation of highly reactive free radicals.This is accomplished in the initiation stage. Bulk,suspension, and solution polymerizations involve organic-soluble initiators such as 2,2′-azobis(isobutyronitrile)(AIBN). The initiator is decomposed into free radicalsby thermal or photochemical (ultraviolet light) means.In emulsion polymerizations, there are two commonlyused initiation methods. The first, redox initiation, isused for low-temperature polymerizations, while initia-tion by thermal decomposition is used for the highertemperature range.Andersen and Proctor (1965) suggested the following

mechanism for the redox system persulfate (PS)/sodiumformaldehyde sulfoxylate (SFS)/iron (Fe).

(Symbols not explained in the text are given in the

S2O82- + Fe2+ 98

k1SO4

•- + Fe3+ + SO42- (1)

Fe3+ + RA98k2Fe2+ + X (2)

SO4•- + Mj 98

kpIjR1,j

• (3)

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 967

detailed Nomenclature section.) In the above mecha-nism, Mj refers to monomer of type j; S2O8

2- representsthe persulfate initiator, I; RA is the reducing agent (inthis case SFS), often referred to as activator; SO4

•- isthe initiator fragment that reacts with monomer, Mj,to produce primary radicals R1,j

• , i.e., radicals of chainlength 1 ending in monomer j. k1 is the rate constantfor the oxidation reaction, k2 is the rate constant for thereduction reaction, and kpIj is the propagation rateconstant for the addition of monomer j to an initiatorfragment. Radicals are generated in the water phaseby the reaction between the initiator (PS) and a complexof Fe2+ and ethylenediamine tetrasodium acetate (EDTA)(see eq [REF:eqn:ini1]). The complexation of iron andEDTA reduces the effective concentration of Fe2+ andprevents undesirable side reactions. Thus, the initiatoris reduced to a negatively charged free radical, whilethe Fe2+ is oxidized to Fe3+ (see eq 1). The SFS thenreduces Fe3+ to Fe2+ (see eq 2). The free radical, SO4

•-,reacts with monomer j to form radicals of chain-lengthunity (see eq 3). Broadhead et al. (1985) used a similarscheme to describe redox initiation for styrene/butadiene(SBR) polymerization.Performing material balances for the initiator (I or

PS) and the reducing agent (RA) gives

where Ni is the number of moles of component i, Fi,in isthe inflow of component i into the reactor in mol min-1,Vw is the volume of the water phase, VT is the totalvolume of the reaction mixture, and vout is the volumet-ric flow out of the reactor. The time dependence of allterms involved in the equations is not shown for thesake of brevity. One should note that k1 and k2 are“effective rate constants”, since other (unknown) el-ementary reaction steps may be occurring but have beenassumed to have a negligible contribution to the overallinitiation rate. This will simplify our set of equations.Further simplification occurs when we apply the reactorstationary-state hypothesis to ferrous and ferric ions toprovide the ion concentrations in terms of total ironconcentration as follows:

where

and

For the case of thermal decomposition of a persulfateinitiator, the commonly accepted mechanism is (Sarkaret al., 1988)

where kd is the initiator decomposition rate constant.This mechanism results in the following material bal-ance for the moles of initiator:

Finally, the overall rate of initiation, RI, is

where [I] is the initiator concentration and f is theinitiator efficiency factor. Equation 12 is implementedinto the computer model as shown above and includesthe rate of initiation by redox means and thermalmeans. However, when one of the methods is activated,the term representing the other option becomes negli-gible.The acceleration of the decomposition of potassium

persulfate (KPS) by “free” sodium dodecyl sulfate emul-sifier has been reported by Okubo et al. (1991). Sarkaret al. (1990) reported the acceleration of KPS decompo-sition due to the addition of vinyl acetate (VAc) mono-mer, but no emulsifier effect was detected. Consider-ations such as these may be responsible for somediscrepancies between model predictions and experi-mental data.Often, the initiator efficiency is considered to be

constant. However, in a high-viscosity regime theinitiator efficiency may decrease significantly (Garcia-Rubio and Mehta, 1986; Russell et al., 1988b; Zhu etal., 1990a). Since initiator decomposition occurs in thewater phase for conventional emulsion polymerizations,it is likely that initiator fragments are not subject to ahigh-viscosity environment and the initiator efficiencyis thus held constant (Adams et al., 1990). One can thenargue that, if the initiator efficiency changes, this“represents” water-soluble impurity effects. For bulkand solution polymerizations, however, the followingsemiempirical equation is used to describe the changinginitiator efficiency when the free volume of the reactionmixture (VF) becomes less than a critical free volume,VFcrif:

where fo is the initial initiator efficiency and C is aparameter which modifies the rate of change of theefficiency. The critical free volume, VFcrif, is dependenton temperature and initiator type. The initiator ef-ficiency typically becomes diffusion-controlled at veryhigh conversions (>80 wt %). The calculation of the freevolume, VF, will be discussed later.In the model, therefore, eqs 4-7 and 11-13 are used

directly to describe the initiation step.2.2. Water Phase Reactions. The discussion in this

section is restricted to emulsion polymerizations. Oncethe radicals are generated in the water phase, they canthen go on to propagate with monomer, react withvarious other species in the reaction mixture, andnucleate particles. Two approaches for particle nucle-ation are commonly employed nowadays: homogeneous(coagulative or aqueous phase) nucleation and hetero-geneous (micellar) nucleation. Several representativearticles dealing with these mechanisms follow: Fitch

dNI

dt) FI,in -

NI

VTvout -

k1NINFe2+

Vw(4)

dNRA

dt) FRA,in -

NRA

VTvout -

k2NRANFe3+

Vw(5)

NFe2+ )k2NFeNRA

k1NI + k2NRA(6)

NFe ) NFe2+ + NFe3+ (7)

dNFe

dt) FFe,in -

NFe

VTvout (8)

S2O82- 98

kd2SO4

•- (9)

SO4•- + Mj 98

kpIjR1,j

• (10)

dNI

dt) FI,in -

NI

VTvout - kdNI (11)

RI ) k1NFe2+

Vw[I] + 2fkd[I] (12)

f ) fo exp(-C(1/VF - 1/VFcrif)) (13)

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

and Tsai (1970, 1971), Hansen and Ugelstad (1978,1982), Poehlein et al. (1986), Maxwell et al. (1991,1992b), and Casey et al. (1994). Another interesting,slightly different nucleation case has recently beendescribed by Lepizzera and Hamielec (1994).We now examine the fate of the radical species in the

water phase:1. The radicals may react with monomers in the

water phase:

where the rate constants of propagation are used todefine reactivity ratios as follows:

Rn,i• represents a radical of chain length n ending in

monomer i. Equation 14 represents the reaction be-tween initiator fragments and monomers to form pri-mary radicals. Equation 15 represents propagation.Equation 15 describes the addition of monomer j to agrowing radical chain of length n ending in monomer i.The reaction, proceeding with a rate constant of kpij,results in a radical chain of length n + 1 ending inmonomer j. For the case of a terpolymerization, thereare nine different propagation reactions and six sepa-rate reactivity ratios. In principle, kpii in eq 16 may beobtained from homopolymerization data for each mono-mer type, while the six reactivity ratios may be calcu-lated from the three binary copolymerizations involvedin the terpolymerization. Copolymerization is describedusing four propagation reactions and two reactivityratios.The propagation reactions shown above represent

terminal model kinetics. That is, the reactivity of aradical center is assumed to depend only upon themonomer unit bound in the polymer chain on which itis located. Other alternative models could also beconsidered and are discussed later.2. The radicals may react with water-soluble impuri-

ties:

where P(WSI) is a dead molecule and kzj is the rateconstant for reaction of water-soluble impurity, j, witha monomer i-ended radical. The use of kzj as an overallrate constant, regardless of which monomer radical theimpurity is reacting with, is a simplification to ourmodel. It is assumed that the rate of reaction of WSI’sdoes not depend on the radical type. Water-solubleimpurities may cause an induction period by consuminglarge amounts of free radicals. Typical examples of suchimpurities are oxygen and other commonly used mono-mer inhibitors with a considerable water solubility atthe conditions of the polymerization (e.g., hydroqui-nones). The effects of impurities on polymerization rateand quality are poorly understood, yet they are one ofthe most important sources of variation in an industrialsetting (Huo et al., 1988; Penlidis et al., 1988; Chien andPenlidis, 1994a,b; Dube and Penlidis, 1997).3. The radicals may terminate upon encountering

another radical:

where Pm+n is a dead polymer molecule of chain lengthm + n. Note that in eq 18 termination may occur eitherby combination of the radical chains or by dispropor-tionation. ktw is often considered to be negligible(Urretabizkaia et al., 1992; Urretabizkaia and Asua,1994); however, in systems containing highly water-soluble monomers (Sarkar et al., 1988), or dependingon the emulsifier concentration (Song and Poehlein,1988b), water-phase termination may be significant.4. The initiator fragments may recombine. This

phenomenon is taken into account by use of the ef-ficiency factor, f, for the initiation step or by the use ofan “effective” initiator decomposition rate constant.5. The radicals may be captured by monomer drop-

lets. Most modeling efforts ignore this phenomenon dueto the fact that the surface area of the polymer particlesand micelles is far greater than that of the monomerdroplets. Thus, the likelihood of a radical speciesentering the monomer droplets is minimal. This as-sumption can be tested by the use of electron microscopyto determine if any abnormally large particles exist and,if so, how many. Droplet nucleation may occur shouldextremely high shear rates be used during mixing alongwith the appropriate emulsifier concentration. Also,this phenomenon usually occurs in the presence ofalcohol groups with ionic emulsifiers. The rate ofradical capture by monomer droplets can be defined as

where [RTOT• ]wdrop is the total concentration of radicals

able to enter a micelle, a particle, and/or a droplet (thecalculation of this quantity would be similar to that for[RTOT

• ]wmic and [RTOT• ]wpar in eqs 30 and 31, respec-

tively), and [drops] is the concentration of droplets. Inthe examples cited later in this paper, the rate constantfor the capture of radicals by monomer droplets, kcmd,is set to zero. For the interested reader, the case ofmonomer droplet polymerization has been described byUgelstad et al. (1973, 1974), Hansen and Ugelstad(1979), Song and Poehlein (1988a,b), and Fontenot andSchork (1992-93a,b).6. Radicals may desorb from polymer particles at the

following rate:

where Np is the number of polymer particles per literof water, nj is the average number of radicals perparticle, NA is Avogadro’s number, and Vw is the totalvolume of water. Several authors have reported expres-sions for the desorption rate constant, kdes, all of whichare based on the same principle. Information may befound in Nomura et al. (1971a), Ugelstad and Hansen(1976), Nomura and Harada (1981), Rawlings and Ray(1988), Mead and Poehlein (1989b), Asua et al. (1989),and Casey et al. (1994). Desorption is a phenomenonrestricted to small molecules, usually as a result of chaintransfer to monomer or monomer-soluble impurity orchain-transfer agent. However, the use of relativelylarge CTAmolecules (e.g., n-dodecylmercaptan) in manypolymerizations would obviate the need to include thechain transfer to CTA in the desorption equation. Thedesorption rate constant is discussed in detail later.

Rn,i• + Rm,j

• 98ktw

Pm+n or Pm + Pn (18)

Rcmd ) kcmd[RTOT• ]wdrop[drops] (19)

Fdes )kdesNpnjNAVw

(20)

RI• + Mj 98

kpIjR1,j

• (14)

Rn,i• + Mj 98

kpijRn+1,j

• (15)

rij )kpiikpij

(16)

Rn,i• + WSI98

kzjP(WSI) (17)


7. Desorbed radicals may reabsorb into the polymerparticles. Expressions for the reabsorption of previouslydesorbed radicals back into the particles have been putforth by Poehlein et al. (1986). This phenomenon, whilenot included in the desorption equation itself, is includedin the water-phase radical balances shown later.8. Radicals may be captured by micelles according

to

where kcm is the rate constant of capture by micellesand Am is the total free micellar area. The radicalconcentration [RTOT

• ]wmic is the concentration of radi-cals in the water phase which can enter micelles; it doesnot include those radicals which, due to their size,electric charge, and hydrophilicity, cannot be capturedby micelles. The derivation of [RTOT

• ]wmic is shownlater; it is defined in eq 30. Equation 38 gives the rateof particle nucleation by the micellar mechanism.9. Radicals may be captured by particles at the

following rate:

where kcp is the rate constant for capture by polymerparticles and Ap is the total surface area of the polymerparticles. The radical concentration [RTOT

• ]wpar is theconcentration of radicals in the water phase which canenter particles. The same arguments explained formicellar capture of radicals in the water phase applyhere as well. [RTOT

• ]wpar is defined later in eq 31.2.3. Particle Nucleation. The aspect of emulsion

polymerization that generates the most discussion andconflict is the debate regarding the nature of particlenucleation (Richards et al., 1989; Dunn, 1992; Hansen,1992, 1993). In this model, both micellar and homoge-neous particle nucleation mechanisms are accounted for.The assumption that particle sizes are monodisperse isalso employed in the following discussion; accountingfor a distribution of particle sizes is discussed in the nextsection.Reviews of the polymer particle formation mecha-

nisms may be found in Ugelstad and Hansen (1976) andHansen and Ugelstad (1982). Micellar and homoge-neous nucleation can both play a significant role inparticle formation (Fitch and Tsai, 1971; Fitch, 1981).The relative importance of homogeneous nucleationincreases with the solubility of monomer in the waterphase, while at high emulsifier levels, micellar nucle-ation dominates due to the high surface areas and rapidradical absorption rates from the water phase. Particle-particle coalescence has been neglected in modeling bymost workers. Notable exceptions include Min and Ray(1974), Hansen and Ugelstad (1978), Morbidelli et al.(1983), and Song and Poehlein (1988a,b). It has beendifficult to develop a general model for radical absorp-tion into micelles and polymer particles and desorptionfrom polymer particles (Hansen and Ugelstad, 1982;Nomura, 1982).We now proceed to form the expression for homoge-

neous particle nucleation. In this paper, a combinationof two methods, along with modifications to allow formulticomponent polymerizations and inhibitors, wasimplemented into the model. The first method, origi-nally proposed by Hansen and Ugelstad (1978), is morerigorous but requires knowledge of parameters that arenot readily known for all polymer systems. The second

method, that of Fitch and Tsai (1970, 1971), involvesthe use of grouped parameters, thus simplifying thestructure of the equations but not necessarily rewardingus with less uncertainty.According to the first method:

whereN is the total number of monomers in the systemand Nz is the total number of WSI’s in the system. Theabove expression equates the rate of radical generationto the rate of radical disappearance. In other words,we are invoking a steady-state hypothesis. On the left-hand side of eq 23 we have the overall rate of initiation(see RI of eq 12). The right-hand side of eq 23 showsthe disappearance of the initiator radicals by propaga-tion with monomer j in the water phase (see kpIj[Mj]w)and by reaction with water-soluble impurities (see kzi-[WSI]i). [RI

•] is the concentration of initiator radicalsin the water phase. [Mj]w is the concentration ofmonomer j in the water phase. We are making theassumption that initiator radicals will not terminate(this is accounted for by initiator efficiency) and thevisualization that radicals of this size will not enterparticles nor micelles nor monomer droplets due toelectrostatic forces and the hydrophilicity of such radi-cals.Next, we perform balances on radicals of chain length

1, ending in monomer i:

The above equation now includes termination with otherradicals in the water phase (see ktw[RTOT

• ]w in eq 24).The creation of radicals of chain length 1 by desorptionis not included in the above balance. The desorbedradicals are of a different nature compared to primaryradicals or oligomers in that they are a result of transferto monomer and transfer to chain-transfer agent reac-tions inside the polymer particles. Hence, desorbedradicals have an electric charge different from the otherradicals in the water phase and can be recaptured bythe particles. In fact, the desorbed radicals will notmove far beyond the particle from which they desorbed.That is, it can be argued that the desorbed radicals willnot necessarily enter the bulk of the water phase butwill stay in the vicinity of the particle surface. However,if there is a large amount of desorption, a significantamount of desorbed radicals may stray from the par-ticles.We now perform balances on radicals of chain length

k (2 e k e (jcr/2)), ending in monomer i. jcr/2 is thecritical chain length at which oligomers may be capturedby micelles, particles, and/or droplets.

Now a balance is performed on radicals of chainlength k (jcr/2 + 1 e k e (jcr - 1)), ending in monomer

Rcm ) kcm[RTOT• ]wmicAm/Vw (21)

Rcp ) kcp[RTOT• ]wparAp/Vw (22)

RI ) [RI•](∑

j)1

N

kpIj[Mj]w + ∑i)1

Nz

kzi[WSI]i) (23)

kpIi[Mi]w[RI•] ) [R1,i

• ](∑j)1

N

kpij[Mj]w + ∑i)1

Nz

kzi[WSI]i +

ktw[RTOT• ]w) (24)

(∑j)1

N

kpji[Rk-1,j• ])[Mi]w ) [Rk,i

• ](∑j)1

N

kpij[Mj]w +

∑i)1

Nz

kzi[WSI]i + ktw[RTOT• ]w) (25)


i. jcr is the critical chain length at which oligomers willprecipitate and form a polymer particle (homogeneousnucleation) given that there is remaining free emulsi-fier. We also see that all capture mechanisms (micellar,particle, droplet) have now been included (see the termskcmAm/Vw, kcpAp/Vw, and kcmd[drops], respectively, in eq26). The inclusion of the capture mechanisms reflectsthe visualization that the oligomeric radicals have nowgrown enough so that there are no repulsion barriers(i.e., the charge at the end of the oligomer is no longerstrong enough to create a repulsion from micelles,particles, or droplets).

Equations 23-26 represent a reasonable and practicalway to handle differing monomer solubilities in water.Our visualization that the capture of radicals by micellesand particles begins at about a length of jcr/2 units hasbeen independently supported by Poehlein (1990), Max-well et al. (1991, 1992b), and Kshirsagar and Poehlein(1994).jcr is calculated as a weighted function of the instan-

taneous composition of the polymer formed in the waterphase (Fjw):

Fjw is described later in eqs 132-134 as Fj.The concentration of radicals of chain length jcr

ending in monomer i is

A balance on the total amount of radicals in the waterphase (excluding initiator radicals) gives

Recall that radicals of chain length jcr/2 + 1 to jcrcan be captured by micelles, particles, and droplets.Also, radicals of chain length jcr can undergo homoge-neous nucleation. Droplet nucleation is neglected as asimplification to our model. Thus, a balance on theconcentration of radicals in the water phase able to becaptured (radicals of chain length > jcr/2) yields thefollowing equations. First, the concentration of radicalsin the water phase that may be captured by micelles isgiven by

Next, the concentration of radicals in the water phasethat may be captured by particles is given by

Finally, the concentration of radicals in the water phasethat may undergo homogeneous nucleation is definedby

Am represents the total free micellar surface area, thatis, the surface area created by the emulsifier remainingafter the coverage of droplets and particles. Am is givenby

[S]t is the total concentration of emulsifier in the reactor,[S]CMC is the critical micelle concentration, Sa is the areaoccupied by an emulsifier molecule, Ad is the area ofmonomer droplets, and Ap is the total surface area ofpolymer particles. As the reaction proceeds, the totalsurface area of polymer particles quickly becomes verymuch greater than the total surface area of monomerdroplets, so Ad is neglected as a simplification to ourmodel. In eq 34, Vp is the total volume of polymerparticles. Equation 34 is based on the assumption thatthe polymer particles are spherical. Sa (see eq 33) isaffected by the polarity of the adsorbing surface and istherefore affected by persulfate initiators (Ali andZollars, 1985).kh is the homogeneous nucleation rate constant

defined by Fitch and Tsai (1971) as

L in the above expression represents the critical radicaldiffusion length and is given by Einstein’s diffusion law:

Mwsat is the saturation concentration of monomer in thewater phase.The rate of change of the number of particles is

described by the following equation:

FNp,in is the inflow of particles (i.e., in the case of aseeded emulsion polymerization), while the second termon the right-hand side of eq 37 (Np(Vw/VT)νout) representsthe outflow of particles from the reactor. The rates ofhomogeneous and micellar particle generation are de-scribed by the third and fourth terms on the right-hand

(∑j)1

N

kpji[Rk-1,j• ])[Mi]w ) [Rk,i

• ](∑j)1

N

kpij[Mj]w +

∑i)1

Nz

kzi[WSI]i + ktw[RTOT• ]w + kcmAm/Vw + kcpAp/Vw +

kcmd[drops]) (26)

jcr ) ∑j)1

N

jcrjFjw (27)

[Rjcr,i• ] )

(∑j)1

N

kpji[Rjcr-1,j• ])[Mi]w

(kcmAm/Vw + kcpAp/Vw + kcmd[drops])(28)

[RTOT• ]w ) ∑

i)1

N

∑k)1

jcr-1

[Rk,i• ] (29)

[RTOT• ]wmic ) ∑

i)1

N

∑k)(jcr/2)+1

jcr-1

[Rk,i• ]

kcmAm/Vw

kcmAm/Vw + kcpAp/Vw

+

∑i)1

N

[Rjcr,i• ]

kcmAm/Vw

kcmAm/Vw + kcpAp/Vw + kh(30)

[RTOT• ]wpar ) ∑

i)1

N

∑k)(jcr/2)+1

jcr-1

[Rk,i• ]

kcpAp/Vw

kcmAm/Vw + kcpAp/Vw

+

∑i)1

N

[Rjcr,i• ]

kcpAp/Vw


[RTOT• ]whom ) ∑

i)1

N

[Rjcr,i• ]

kh


Am ) ([S]t - [S]CMC)VwSaNA - Ap - Ad (33)

Ap ) (πNp)1/3(6Vp)

2/3 (34)

kh ) kho(1 -LAp

4Vw) (35)

L ) (2DwjcrkpMwsat

)1/2 (36)

dNpVw

dt) FNp,in

- Np

Vw

VTvout +

dNhom

dtVw +

dNmic

dtVw -

kFNp2Vw (37)


side of eq 37, respectively. kFNp2Vw represents a crude

attempt to take particle coalescence into account (Ugel-stad and Hansen, 1976; Hansen and Ugelstad, 1978,1979). Song and Poehlein (1988a,b), Fontenot andSchork (1992-93a), and Chern and Kuo (1996) discussparticle coagulation in more detail. This is a difficultsubject, still under investigation, and hence, especiallydue to lack of specific data, we decided to neglect theterm in further simulations with the model.The rate of micellar nucleation, dNmic/dt, is given by

As can be seen from eq 30, when the surface area offree micelles, Am, goes to zero, micellar nucleation ishalted.The expression for the rate of generation of particles

by homogeneous nucleation becomes

The homogeneous nucleation rate constant, kh, tendstoward zero as the area of polymer particles, Ap,increases (see eq 35). This is because there is a higherprobability for an oligomer to be captured by a preexist-ing particle rather than form a new particle by homo-geneous nucleation.2.3.1. Emulsion Particle Size Distribution (PSD)

Calculation. If one wishes to relax the assumption ofa monodisperse particle size distribution, it is necessaryto account for different classes or ages of particles asshown below. In the first instance, it is assumed thatstatistical broadening can be neglected and, thus,polymer particles born at time τ with volume V0 willall have the same volume (V(t,τ)) at some later time t(at least for those particles which have not left thereactor in the case of a CSTR). Calculation of PSD withexact correction for statistical broadening requires thesolution of a large number of partial differential equa-tions (Behnken et al., 1963; Sundberg, 1979; Kiparis-sides and Ponnuswamy, 1981; Rawlings and Ray, 1988;Storti et al., 1989), and this appears to be impracticalat the present time except for special cases (the numberof polymer particles containing three or more radicalsis zero).The consumption rate of monomer in the reactor Rp-

(t) is given by

where Rp(t,τ) is the consumption rate of monomer attime t in polymer particles born at time τ. Np(t,τ) dτ isthe number of polymer particles in the reactor at timet which were born at time τ.The total polymerization rate,Rp(t,τ) in these particles

may be expressed as

where kp is defined in eqs 47 and 84, [M]p is theconcentration of monomer in the polymer particles, andnj is the average number of radicals per particle.The rates of polymerization for the individual species

Rpj(t,τ) may be found using Rp(t,τ) and reactivity ratios(see eq 126).

The volumetric growth rate of polymer in a polymerparticle is given by

where Vjmp(t,τ) is the volume of monomer j in theparticle born at time τ and φi is the mole fraction ofradicals in the particles ending in monomer i (see eq48). Equation 44 is defined by eqs 126 and 86 but withthe dependence on t and τ shown in the variables.The following algebraic relationship may be used to

calculate the volumetric growth rate for polymer par-ticles born at times τ > 0.

In this manner, one can calculate the full particle sizedistribution.2.4. Organic Phase Reactions. This section de-

scribes reactions in bulk and solution polymerizationas well as those reactions occurring in the bulk phasein emulsion polymerization once the radicals have beencaptured by particles (or micelles or monomer droplets).1. The radicals may propagate (as shown in eq 15):

For multicomponent polymerizations (e.g., in this case,terpolymerization), the overall propagation pseudo-kinetic rate constant can be defined as (Hamielec et al.,1987; Tobita and Hamielec, 1991; Xie and Hamielec,1993a,b)

where φi is the mole fraction of radicals in the particlesending in monomer i and fj is the mole fraction ofmonomer j in the particle or bulk phase. The φi’s arecalculated as above by invoking the quasi-steady-stateassumption (QSSA) for the radicals. The QSSA is notvalid at high conversion levels, but the conversion andMh n (number-average molecular weight) results will notbe significantly affected and the Mh w (weight-average

dNmic

dt) NAkcm[RTOT

• ]wmic/rmic (38)

dNhom

dt) NAkh[RTOT

• ]whomVw (39)

Rp(t) ) ∫0tRp(t,τ) V(t,τ) Np(t,τ) dτ (40)

Rp(t,τ) )kp[M]p(t) nj(t,τ)

NAV(t,τ)(41)

dVp(t,τ)

dt) (∑

j)1

N

MWjRpj(t,τ))V(t,τ)

Fp(t)(42)

V(t,τ) ) Vp(t,τ) + ∑j)1

N

Vjmp(t,τ) (43)

Rpj(t,τ) )

fj(t) (∑i)1

N

kpijφi(t))[M]p(t) nj(t,τ)

NAV(t,τ)(44)

dVp(t,τ)dt

)nj(t,τ)nj(t,0)

dVp(t,0)dt

(45)

Rn,i• + Mj 98

kpijRn+1,j

• (46)

kpo ) ∑i)1

N

∑j)1

N

kpijφifj (47)

φ1 ) (kp21kp31f12 + kp21kp32f1f2 + kp23kp31f1f3)/ψ (48)

φ2 ) (kp12kp31f1f2 + kp12kp32f22 + kp13kp32f2f3)/ψ (49)

φ3 ) 1 - φ1 - φ2 (50)

ψ ) (kp12kp31f12 + kp21kp32f1f2 + kp23kp31f1f3 +

kp12kp31f1f2 + kp12kp32f22 + kp13kp32f2f3 +

kp12kp23f2f3 + kp13kp21f1f3 + kp13kp23f32) (51)


molecular weight) predictions will be affected onlyslightly (Achilias and Kiparissides, 1994). The pseudo-kinetic rate constants for multicomponent polymeriza-tion described throughout this paper are in the contextof terminal kinetics. Tobita and Hamielec (1991) havederived equivalent expressions for pseudo-kinetic rateconstants in the context of the penultimate copolymer-ization model. The pseudo-kinetic rate constant methodis perhaps the only practical way of handling complexmulticomponent polymerization modeling.Equations 48-51, as shown above, have been devel-

oped for the terpolymerization case. The reduction ofeqs 48-51 to copolymerization and homopolymerizationand their extensions to higher multicomponent systemsshould be clear.2. The radicals may mutually terminate:

Equation 52 represents termination by combination,whereas eq 53 represents termination by disproportion-ation. The two termination rate constants are definedby the overall termination rate constant, kt, and γ, theratio of termination by disproportionation to the overalltermination:

Describing an overall termination pseudo-kinetic rateconstant in multicomponent polymerizations has beena source of constant debate. In this study, severaldifferent methods were attempted, but each method hadlimitations. The overall termination pseudo-kinetic rateconstant was calculated as

For the bulk/solution case, the cross-termination rateconstants were defined as

where Fi, the instantaneous polymer composition, isdefined later in eqs 132-134. The cross-terminationrate constants in the emulsion case were defined as

There has been little agreement as to how the cross-termination rate constants should be defined. There-fore, both eqs 57 and 58 were implemented for futurecomparison.An overall γ was calculated using a weighted average

of the homopolymerization γ’s based on the instanta-neous polymer composition.3. The radicals may react with monomer-soluble

impurities such as hydroquinone and tert-butylcatechol(TBC), which are commonly added to the fresh monomerby the suppliers due to their radical scavenging proper-ties:

where P(MSI) is a dead polymer molecule. kfmsi is therate constant for the reaction of monomer-solubleimpurities with radicals ending in monomer j. The useof kfmsi as an overall rate constant, regardless of whichmonomer the impurity is reacting with, is a simplifica-tion to our model. It is assumed that the rate of reactionof MSI’s does not depend on the monomer type. Inemulsion polymerizations, MSI’s are transferred into theparticles with monomer(s) during monomer diffusion.In the bulk/solution case, the MSI’s are in the samephase as the initiator, thus causing an induction time.The effect of monomer-soluble impurities, usually ig-nored or at least poorly understood, can be quitepronounced on the overall reaction rate, eventuallyaffecting particle growth (in emulsions), particle nucle-ation (in emulsions), and molecular weight (Huo et al.,1988; Penlidis et al., 1988; Chien and Penlidis, 1994a,b;Dube and Penlidis, 1997).The impurity effect on particle growth can be de-

scribed as follows. If a MSI partitions into the growingpolymer particles, it scavenges the free radicals. Thus,the particle growth rate is slowed down considerably.This leads to a prolonged particle nucleation stagebecause the micellar emulsifier, which is used to sta-bilize the particles, is consumed more slowly. Therefore,the average lifetime of a micelle is extended and moreparticles are nucleated. Since the rate of polymerizationis proportional to the number of particles, a significantincrease in the rate may result (Dube and Penlidis,1997).4. The radicals may react with a chain-transfer agent

such as a mercaptan:

where RSH represents the chain-transfer agent whichloses a labile hydrogen to the growing radical chain. kfctajis the rate constant for transfer to a CTA molecule forthe monomer j-ended radical type. The product of theabove reaction, RS•, continues propagating with mono-mer(s), thus lowering the average molecular weight. Theimportance of these reactions is stressed by the findingsof Broadhead et al. (1985), who showed that transfer toCTA dominated the SBR polymerization.In the case of multicomponent polymerization, an

overall chain transfer to CTA pseudo-kinetic rate con-stant (kfcta) is defined as

5. The radicals may undergo chain transfer tomonomer:

In this case, the active radical center is transferred toa monomer molecule which may propagate further, thuslowering the overall molecular weight.Chain transfer to monomer is an important reaction.

It greatly affects the molecular weight and is a precursorto terminal double-bond reactions and desorption. Forexample, the transfer reaction to VAc results in a stableradical which reinitiates slowly; thus, desorption frompolymer particles is possible (Litt, 1993).As previously with kfcta, an overall pseudo-kinetic rate

constant for chain transfer to monomer (kfm) is defined

Rn,i• + Rm,j

• 98ktc

Pm+n (52)

Rn,i• + Rm,j

• 98ktd

Pm + Pn (53)

γ ) ktd/kt (54)

kt ) ktc + ktd (55)

kto ) ∑i)1

N

∑j)1

N

ktoijφiφj (56)

ktoij ) ktoiFi + ktojFj (57)

ktoij ) (ktoiktoj)1/2 (58)

Rn,j• + MSI98

kfmsiP(MSI) (59)

Rn,j• + RSH98

kfctajHPn,j + RS• (60)

kfcta ) ∑j)1

N

kfctajφj (61)

Rn,i• + Mj98

kfmijPn,i + Mj

• (62)


as

6. The radicals may undergo chain transfer topolymer:

Transfer to polymer does not necessarily occur at theend of a dead polymer molecule. Thus, the species typeat the end of the radical chain is not important. Thepresence of chain transfer to polymer reactions has beenreported for BA and VAc polymerizations (Scott andSenogles, 1970, 1974; Friis et al., 1974; Hamielec, 1981;El-Aasser et al., 1981, 1983; Penlidis et al., 1985a; Dubeet al., 1991b; Lovell et al., 1991, 1992). Lovell et al.(1991, 1992) reported that the transfer to polymerreaction for BA occurred by the abstraction of a tertiaryhydrogen from a BA unit in the polymer chain. Friis etal. (1974) reported that transfer to polymer and transferto monomer reactions dominate the VAc polymerization.The chain transfer to polymer occurs at the methylhydrogen of VAc.The overall pseudo-kinetic rate constant for transfer

to polymer (kfp) is

The cumulative polymer composition, Fh j, is defined laterin eq 135.In a batch reactor when there is significant compo-

sitional drift, the use of the above equation is not strictlyvalid and one would have to resort to the cumbersomemethod of moments for the complete set of chemicalequations for polymerization. In many practical cir-cumstances compositional drift is either small or main-tained small (batch reactors and reactivity ratio pairswhich give small compositional drift, semibatch reactorswhere compositional drift is controlled to low levels andwell-mixed continuous stirred-tank reactors with mi-cromixing where only statistical spreading of composi-tion occurs).7. The radicals may undergo reactions with internal

and terminal double bonds. Once again, an effectiverate constant will be used. The rate constant forterminal double-bond reactions is k*p and that forinternal double-bond reactions is k*p*.

Terminal double-bond reactions yield trifunctional branchpoints, whereas internal double-bond reactions yieldtetrafunctional branch points. Hence, an overall pseudo-kinetic rate constant for terminal (k*p) and internaldouble-bond (k*p*) reactions are defined as

The importance of the terminal double-bond reactionhas been documented for the BA and VAc polymeriza-tions (Scott and Senogles, 1970, 1974; Friis et al., 1974;Hamielec, 1981; El-Aasser et al., 1981, 1983; Penlidiset al., 1985a; Dube et al., 1991b; Lovell et al., 1991,1992).As in the case for transfer to polymer, the validity of

the above pseudo-kinetic rate constant equations comesinto question in the presence of strong compositionaldrift.2.5. Diffusion-Controlled Rate Constants. When

discussing the various reactions taking place in apolymerization, many of the parameters were referredto as rate constants. This is somewhat of a misnomeras these so-called rate constants (e.g., kp, kt) vary withthe viscosity of the reaction medium. Thus, in a bulksystem, the viscosity increases due to the increase inpolymer concentration and, hence, affects the rate ofpropagation and termination. In the emulsion case, theviscosity in the main locus of polymerization (i.e., theparticles) is high from the onset of the reaction due tothe high polymer concentration. Thus, the rates oftermination and propagation may be diffusion-controlledeven at low conversion levels.The general chemical equations for diffusion-con-

trolled termination were expressed earlier in eqs 52 and53. The termination constants ktc and ktd can beredefined as ktc(n,m) and ktd(n,m), respectively. Thisillustrates their dependence, in general, on the chainlengths n and m of the polymer radicals, Rn,i

• and Rm,j•

(see eqs 52 and 53) but not on the polymeric radical type(i and j denote the monomer type on which the radicalcenter is located on the end of the polymer chain). Onemight expect that two backbone radical centers wouldmutually terminate at a significantly lower rate. Thedependence of this rate on the position of the radicalcenters in the polymer backbone is not clear and themodeling of this effect is beyond the scope of thisinvestigation. In other words, we limit the treatmentof polymer radical/polymer radical termination to linearchain systems where termination rates are controlledby radical centers on chain ends. The termination rateconstants for linear chain systems also depend on themass concentration, molecular weight distribution of theaccumulated dead polymer, and polymerization tem-perature. When most of the dead polymer chains(chains without a radical center) are made via chaintransfer to a small molecule (such as chain-transferagent, monomer, etc.) chain-length dependence of thetermination rate constant may not be an importantissue (accurate MWD’s of the dead polymer may becalculated without a knowledge of this chain-lengthdependence).The calculation of the total polymer radical concen-

tration and then the rate of polymerization wouldrequire at most a single number-average terminationconstant (khtN). However, when a significant number ofdead polymer chains are made via polymer radical/polymer radical termination, then to calculate the fullMWD or higher molecular weight averages (Mh w, Mh z,etc.), one must account for chain-length dependencewhen it is operative. When calculating MWD and itsaverages, chain-length dependence can be accounted forin different ways. One can use an approximate expres-sion to account for the dependence of kt(n,m) on chain

k*p* ) ∑i)1

N

∑j)1

N

k*p*ijφiFh j (69)

kfm ) ∑i)1

N

∑j)1

N

kfmijφifj (63)

Rn,i• + Rm,j98

kfpijPn,i + Pm,j

• (64)

kfp ) ∑i)1

N

∑j)1

N

kfpijφiFh j (65)

Rn,i• + Pm 98

k*pRm+n,i

• (66)

Rn,i• + Pm 98

k*p*

Rm+n,i• (67)

k*p ) ∑i)1

N

∑j)1

N

k*pijφiFh j (68)


lengths n and m (kt(n,m) ) ktc(n,m) + ktd(n,m)). Someof the earlier studies to use this approach includeBenson and North (1962), Duerksen and Hamielec(1967), Duerksen (1968), and others and more recentstudies by Cardenas and O’Driscoll (1976), Ito (1980,1981), Soh and Sundberg (1982a), Coyle et al. (1985),Russell et al. (1992), and O’Shaughnessy and Yu(1994a,b). These attempts to find a general and effec-tive expression for kt(n,m) based on theoretical andexperimental considerations are to be lauded eventhough to date an expression for kt(n,m) which can beconsidered effective and useful for commercial polymerreactor modeling is not available. In the meantime, itis recommended that a less general approach usingaverages of kt(n,m) be used (Boots, 1982; Olaj et al.,1987; Zhu and Hamielec, 1989) for reactor modeling.The use of kt(n,m) averages permits one to calculate therate of polymerization,Mh n,Mh w, and possiblyMh z of bothaccumulated dead polymer and dead polymer produced“instantaneously”. Individual empirical correlations forkhtN, khtW, and khtZ versus polymer concentration, temper-ature, and other significant variables as shown by Zhuand Hamielec (1989) are required. Methods based onaverages have yet to be comprehensively evaluatedalthough Vivald-Lima et al. (1994a) have recentlycompared the effectiveness of the CCS (Chiu et al., 1983)and MH (Marten and Hamielec, 1979) models.The number-average termination constant is given by

and the termination rate of polymeric radicals, Rt, by

where φn• is the mole fraction of polymeric radicals of

chain length n. The use of khtN to calculateMh w,Mh z, andhigher molecular weight averages will give estimatesthat are smaller than the true values when chain-lengthdependence is significant. For the correct calculationof higher molecular weight averages one should use khtw,khtz, etc. Examples are given in Zhu and Hamielec(1989).Anseth et al. (1994a) have made a comprehensive

experimental investigation of the effect of volumerelaxation on free-radical cross-linking kinetics. Thisphenomenon may explain observations made by Stickler(1983) of the effect of initiator level on limiting conver-sions for polymerization of methyl methacrylate in theabsence of cross-linking monomers. At higher polym-erization rates, the actual volume is greater than theequilibrium volume, permitting higher limiting conver-sions to be reached. Anseth et al. (1994b) have alsorecently shown dominance of termination by reactiondiffusion in highly cross-linked systems. Recent papersby Vivaldo-Lima et al. (1994b) and Hutchinson (1992-3) have accounted for diffusion-controlled terminationas affected by cross-linking.The validity of the stationary-state hypothesis (SSH)

was tested by direct experimentation for the homopo-lymerization of methyl methacrylate and for the copo-lymerization of methyl methacrylate and cross-linkerethylene glycol dimethacrylate using ESR for the firsttime (Zhu et al., 1990b). The SSH was shown to be validfor homopolymerization of methyl methacrylate and forlow levels of cross-linking. For high levels of cross-linking, however, the SSH is clearly not valid.

For chemically-controlled termination, kt equals kht,and these are given by equations such as eq 56. Forisothermal polymerization and small compositionaldrift, kt can be treated as a constant. The instantaneousnumber-average molecular weight is given by

where Rpi is the rate of polymerization of monomer iand Rtc and Rtd are rates of termination by combinationand disproportionation, respectively. As mentionedearlier, the modeling to be discussed herein will use khtexclusively, with chain-length dependence neglected incalculating Mh w, Mh z, etc.Under diffusion-controlled termination a single ter-

mination rate constant can be employed to model therate of multicomponent polymerization and molecularweight development. In other words, the monomer typeon which the radical center is located is unimportant.However, when dealing with polymer radicals with longbranches, the self-diffusion coefficient of the polymermolecule will depend not only upon chain length or thenumber of monomer units in the chain but also uponthe number of long branches, branch lengths, and theirlocation on the polymer molecule. The position of theradical center (on the chain end or somewhere on thebackbone) should also affect the termination rate.Termination under these conditions is clearly verycomplex, and a good deal of empiricism is required tomodel termination reactions for polymer radicals withlong branches. The self-diffusion coefficient should alsodepend upon the distribution of monomer types in thepolymer chains. This is another complicating factorwhen compositional drift is important.Diffusion-controlled rate constants are modeled in this

paper using the free-volume approach (Marten andHamielec, 1979; Soh and Sundberg, 1982a-c; Hamielecet al., 1987). In order for the termination reaction tooccur, two macroradical chains must approach eachother via translational diffusion. Next, the diffusion ofthe chain segments containing the active centers towardeach other occurs. This is termed segmental diffusion.Finally, the termination reaction takes place. Thus,three diffusion-control intervals can be described.In the early stages of an isothermal, batch multicom-

ponent polymerization (typically at conversions lessthan about 10% in bulk polymerization), where termi-nation might be chemically controlled, compositionaldrift is usually small and hence a single constanttermination constant, kt (kt ) ktc + ktd), may be used tomodel the rate of polymerization and molecular weightdevelopment to the conversion where termination be-comes diffusion-controlled. However, when relativelyhigh molecular weight polymers are being produced atlow monomer conversions, the termination rate may becontrolled by segmental diffusion. Bhattacharya andHamielec (1986), Jones et al. (1986), and Yaraskavitchet al. (1987) have applied a model for segmental diffu-sion-controlled termination after Mahabadi and O’Driscoll(1977). The rate constant for termination controlled bysegmental diffusion is given by

where kto is given by eq 56 shown earlier, δ is aparameter dependent on the molecular weight of thepolymer radicals as well as solvent quality (monomers

khtN ) ∑n)1

∞

∑m)1

∞

kht(n,m) φn•φm

• (70)

Rt ) khtNYo2 (71)

Mh n ) ∑i)1

N

MWi

Rpi

(Rtc/2 + Rtd)(72)

ktseg ) kto(1 + δc) (73)


plus solvent if present), and c is the mass concentrationof accumulated polymer in the reaction mixture. As thepolymer concentration increases in a good solvent, thecoil size of polymer chains decreases, increasing thegradient of concentration of the radical center acrossthe coil and thus the diffusion rate of the radical center.The consequence of this is that the termination rateconstant increases with monomer conversion to thepoint where translational diffusion of the center of massof the polymer radical coils controls the termination rateand then kt begins to fall very rapidly. The contributionof δc is rather small in many cases (Bhattacharya andHamielec, 1986; Jones et al., 1986; Hamielec et al., 1987;Yaraskavitch et al., 1987) and could be neglected in mostreactor calculations.The next interval, translational diffusion control,

occurs up to high polymer concentrations (∼85-90%conversion in bulk). It is in this stage that the gel effector autoacceleration occurs. Using the free-volume ap-proach, the point at which the reaction becomes trans-lational diffusion-controlled is identified by a temper-ature-dependent term, K3, defined by Marten andHamielec (1979, 1982) as

where A andm are adjustable parameters but hopefullyindependent of temperature, radical initiation rate, andmonomer and polymer concentrations over wide rangesof these polymerization conditions. This has beenproven so for both homo- and copolymerization byStickler et al. (1984) and others (Bhattacharya andHamielec, 1986; Jones et al., 1986; Yaraskavitch et al.,1987). Mh wcrit is the accumulated weight-average mo-lecular weight at the monomer conversion at which thetermination rate is translational diffusion-controlled.Mh wcrit = Mwcrit, the instantaneous weight-average mo-lecular weight, and thus there is a direct connectionwith the size of the polymer radicals. VFcrit is the criticalfree volume corresponding to that conversion. K3 wasfound to have an Arrhenius temperature dependencefor both homo- and copolymerization (Stickler et al.,1984; Bhattacharya and Hamielec, 1986; Jones et al.,1986; Yaraskavitch et al., 1987). The frequency factorhad a small dependence on chain composition for thep-methylstyrene/acrylonitrile system (Yaraskavitch etal., 1987).In multicomponent polymerizations, the free volume

is given by

where i represents the monomer, polymer, and solvent.Ri is the difference in the thermal expansion coefficientsfor species i above and below its glass transitiontemperature, Tgi. T is the polymerization temperature,Vi is the volume of each species, and VT is the totalvolume of the reaction mixture.The translational diffusion-controlled termination

rate parameter, kT, is given by

where ktcrit is the value of kt when eq 74 is satisfied. nis an adjustable parameter.

For the modeling of homopolymerizations and copo-lymerizations, the adjustable parameters have beenarbitrarily set equal to 0.5 form and 1.75 for n (Martenand Hamielec, 1979, 1982; Garcia-Rubio et al., 1985;Bhattacharya and Hamielec, 1986; Jones et al., 1986;Yaraskavitch et al., 1987). Panke (1986) has shown thatsetting m ) n ) 0.5 gives an equally good fit to rateand molecular weight data for the homopolymerizationof methyl methacrylate. These values form and n alsogive the correct response when methyl methacrylate ispolymerized in a batch reactor with a high molecularweight heel.Finally, in extremely viscous environments, the reac-

tion diffusion-control termination rate constant, ktrd,becomes significant. Two methods of describing thisregime were employed in the model. The first (Sticklermethod) uses an expression by Stickler et al. (1984) andis given by

where

and NA is Avogadro’s number, δ is the reaction radius,D is the reaction diffusion coefficient, Vm is the molarvolume, ns is the number of monomer units in onepolymer chain segment, and lo is the length of themonomer unit.The second method (RNG method) involves what is

termed “residual termination”. Russell et al. (1988a)defined upper and lower bounds for the residual termi-nation rate parameter as

and

where a is the root-mean-square end-to-end distance persquare root of the number of monomer units, σ is theLennard-Jones diameter, and jc is the entanglementspacing of pure polymer, measured in monomer units.When attempting to model multicomponent polymeriza-tions, options to use residual monomer mole fractionweighted averages or overall values for a, σ, and jc weremade available. The lower limits of the RNG methodand the Stickler method give similar results. The upperand lower bounds are handled in the model with thefollowing expression:

where x is the conversion.As will be discussed later, the reaction diffusion (or

residual termination) concept may not be perfectlycorrect. Recent ideas regarding trapped radicals mayhave to be incorporated into future modeling efforts (Zhuet al., 1990a).

K3 ) Mh wcritm exp(A/VFcrit) (74)

VF ) ∑i)1

N

(0.025 + Ri(T - Tgi))Vi

VT

(75)

kT ) ktcrit(Mh wcrit

Mh w)n exp(-A( 1VF

- 1VFcrit

)) (76)

ktrd )8πNAδD1000

(77)

δ ) (6Vm

πNA)1/3 (78)

D )nslo

2

6kp[M] (79)

ktres,min ) 43

πkp[M]a2σ (80)

ktres,max ) 83

πkp[M]a3jc1/2 (81)

ktrd ) ktres,minx + ktres,max(1 - x) (82)


The point at which the reaction changes from trans-lational diffusion control to reaction diffusion control canbe somewhat ambiguous. Thus, the following overallkt is defined to circumvent this problem:

Equation 83 has been previously employed to handlethe various diffusion-control equations for kt by Martenand Hamielec (1979) and Soh and Sundberg (1982a-c). Vivaldo-Lima et al. (1994a) showed that this ap-proach (which they termed as a “serial” approach)resulted in as good, if not better, prediction of variouspolymerization data compared to the “parallel” approachsuch as that used by Russell (1994). The parallelapproach involves summing the reciprocals of each ktexpression from each diffusion-control interval. This isdiscussed further by Gao and Penlidis (1996).As will be discussed later, the prediction of the

termination rate is still under investigation. Recentpublications by Zhu et al. (1990a), Tobita and Hamielec(1991), Achilias and Kiparissides (1992), Maxwell andRussell (1993), Buback et al. (1994), Russell (1994), andTobita (1994a,b) have shown some insight into solvingthis problem while at the same time demonstrating awide variety of approaches.For free-radical polymerizations below the glass tran-

sition temperature of the polymer being synthesized,reaction mechanisms at very high monomer conversionscan be very complex due to the decrease in the diffusioncoefficients of small molecules: primary radicals andmonomer. These decreases in mobility will have oneor more of the following consequences: (1) decrease ofthe initiation efficiency; (2) the propagation reactionbecomes diffusion-controlled; (3) radical pair formation(Zhu and Hamielec, 1989). An experimental observationfor such polymerizations is the reduction in conversionrate to almost zero, even though appreciable concentra-tions of monomer and initiator exist in the polymeriza-tion mixture. Most polymerization models permit thepropagation constant to fall with monomer conversionwhile fixing the initiator efficiency at a constant value.This is in effect equivalent to letting the product f1/2kpfall with monomer conversion to fit the rate of polym-erization. There are practical reasons for doing thisbecause it is very difficult to measure f and kp separatelywithout the use of electron spin resonance spectroscopy(ESR). The use of data for polymerization rate and formolecular weights of accumulated polymer can, inprinciple, permit one to estimate f and kp separately ashas been done before (Hui and Hamielec, 1968; Duerk-sen and Hamielec, 1968). However, for the bulk po-lymerization of methyl methacrylate with the verypronounced diffusion-controlled termination followed bya glassy effect and the associated broad MWDwith bothlow and high molecular weight tails on the distribution,accurate molecular weight measurements are extremelydifficult if not impossible to obtain. ESR is the preferredmethod of measuring both f and kp at high monomerconversions for bulk polymerization of monomers suchas methyl methacrylate. Zhu and Hamielec (1989) didESR/rate of polymerization measurements and foundthat f and kp fall significantly with conversion at aboutthesame monomer conversion, while Ballard et al.(1986) observed that for the same monomer (methylmethacrylate) but with emulsion polymerization, kpbecame diffusion-controlled at higher monomer conver-sions. This interesting result may be due to the swellingof surface layers of polymer particles by water in

emulsion polymerization. Sulfate groups dragged intothe interior of polymer particles may be responsible forthis. In the event that f falls sooner than kp, the deadpolymer produced in the time interval between the fallin f and kp would have very high molecular weights. Thisshould affect high-order molecular weight averages.This, however, has not been confirmed experimentallyfor MMA bulk polymerization. Molecular weights aredifficult to measure as already mentioned, and there isthe possibility that chain transfer to monomer mightintervene to put a cap on molecular weights generated.For a discussion of phenomena such as radical pair

formation and heterogeneous effects during glassy-statetransition, one should refer to the original paper by Zhuand Hamielec (1989).In order to model diffusion-controlled propagation, i.e.,

when the polymerization temperature is less than theglass transition temperature of the polymer beingsynthesized, the propagation rate constants are thengiven by

where i is the radical type where the active center islocated, j is the monomer type being added to thepolymer chain, B is an adjustable parameter whichshould depend on monomer molecule type, and VFpcritis the critical free volume where the propagation reac-tion of monomer j adding to a polymer chain ending inmonomer i becomes diffusion-controlled. One mightexpect these critical free volumes to depend on polymerradical type as well as monomer type with propagationrates which are greater, becoming diffusion-controlledat lower monomer conversions. It is also reasonable toexpect that diffusion-controlled propagation rate con-stants for the same monomer are equal (kpij ) kpji). Thiscan be used to reduce the number of adjustable param-eters.When dealing with multicomponent polymerizations,

the diffusion-control expressions shown above weresimplified somewhat by taking weighted averages ofmany of the parameters. The parameter δ from thesegmental diffusion-controlled termination rate expres-sion (see eq 73) was averaged by the cumulative polymercomposition. A weighted average of the parameter Afrom eq 74 by the instantaneous polymer composition(Fi) was included as an option in the simulation package.As well, the sum of the K3 parameters of each monomerwas included as an option. The option for the VFpcritparameter in the diffusion-controlled propagation rateexpression (see eq 84) was a weighted average basedon the residual monomer mole fraction. Alternatively,overall values for A,K3, and VFpcrit were input. Althoughseveral parameters were mentioned above as beingadjustable, only the latter three (A, K3, VFpcrit) wereadjusted to fit the data. As will be discussed later, theywere not adjusted for each experiment.2.6. Free-Radical Concentrations. In order to

calculate the reaction rates, one must also know theconcentration of free radicals in the reactor. For thebulk, suspension, and solution case, one may use theoverall concentration of free radicals in the reactor, butfor the emulsion case, one requires an average numberof radicals per particle.2.6.1. Bulk/Suspension/Solution Case. The con-

centration of free radicals in the bulk, suspension, andsolution cases is given by

kt ) ktseg + kT + ktrd (83)

kpij ) kpijo exp(-B( 1VF- 1VFpcrit

)) (84)


Equation 85 describes the initiation of radicals byinitiator decomposition and the consumption of freeradicals by reaction with impurities.2.6.2. Emulsion Case. In the emulsion case the

overall concentration of free radicals in the particlephase is

where nj is the average number of radicals per polymerparticle. There have been several methods proposed toestimate nj (Stockmayer, 1957; O’Toole, 1965; Ugelstadet al., 1967; Ballard et al., 1981a; Huo et al., 1988; Liand Brooks, 1993). One can, in principle, write abalance around a polymer particle assuming a steadystate, where the disappearance of radicals from theparticle is equal to the entry of radicals into the particle.The phenomena to be accounted for in the balance areradical entry (absorption, capture) from the aqueousphase, radical-radical termination in the particle, reac-tion of the radical with monomer-soluble impurities inthe particle, and desorption of radicals into the waterphase. The balance is performed for Nn, the number ofparticles with n radicals:

where [RTOT• ]des is the concentration of desorbed radi-

cals in the water phase given by

vp in eq 87 is the average volume of a particle or Vp/Np.(Also note thatNn andNp in eqs 87, 90, and 91 representthe number of particles, not the number of particles perliter of water.) The first term (on the left-hand side) ineq 87 represents the capture of free radicals (includingdesorbed radicals) from the water phase (see also eq 31).The second term describes termination in the particles,while the third term represents radical desorption andreaction with monomer-soluble impurities. The solutionof the above recursive equation was originally proposedas a modified Bessel function; Ugelstad et al. (1967)proposed the following partial fraction expansion as anapproximation:

where

and

Equation 89 requires approximately 10 levels of frac-tions to obtain a converged value for the averagenumber of radicals per particle.More recently, Li and Brooks (1993) presented a more

universal model (does not assume a steady state) forestimating nj . An investigation of the Li and Brooks(1993) approach is recommended in future modelingefforts. In the computer implementation, we decidedto adopt the method of Ugelstad et al. (1967) due to itssimple form and reduced amount of computationaleffort.2.7. Radical Desorption. We will now look at the

rate of free-radical desorption from the polymer particles(restricted to the emulsion polymerization case). In themost general case, desorption occurs when a smallradical traverses from the particle to the aqueous phase.The desorbed radical is usually assumed to be no morethan 1 monomer unit in length. The formation of thesemonomeric radicals in the particles is a result of chaintransfer to monomer, to impurity, or to chain-transferagent. In our case, radicals formed as a result of chaintransfer to chain-transfer agent will be assumed not todesorb since typical CTAs are relatively large andinsoluble molecules (e.g., n-dodecylmercaptan) (Nomuraet al., 1982).It is commonly accepted that desorbed radicals will

not move far beyond the particle from which theydesorbed. That is, it can be argued that the desorbedradicals will not necessarily enter the bulk of the waterphase but will rather stay in the vicinity of the particlesurface. However, if there is a large amount of desorp-tion, a significant amount of desorbed radicals may strayfrom the particles. Thus, while the following modelequations are rigorous, there are certain underlyingassumptions that should always be checked with specificprocess data, in order to explain possible discrepancies.Expressions describing desorption rate constants

continue to be developed and reported in the literature.For large particles, desorption may be negligible; how-ever, in the early stages of a reaction (i.e., when theparticles are small) or when the rate of absorption ofradicals by the particles is slow, desorption may beimportant. In the following equations, any reactionswith monomer soluble impurities were assumed to yieldnonreactive products.Several groups have developed expressions for the

desorption rate constant. Their expressions attemptedto give the desorption rate constant as a function ofparticle diameter. Mead and Poehlein (1989b) derivedparticle-size-independent expressions for kdes. They alsorederived the equations put forth by other researchgroups while incorporating particle-size independence.The notion of a nonuniform distribution of radicals inthe particles is applicable to large particles and in caseswhere chain-transfer reactions do not dominate (Chernand Poehlein, 1990). It is due to the hydrophilicity ofend groups (initiator fragments) on the oligomers beingcaptured. de la Cal et al. (1990) also discussed thenonuniform distribution of radicals. As a simplification

Yo )

(∑i)1

Nz

kzi2[MSI]i

2 + 4ktRI)1/2 - ∑

i)1

Nz

kzi[MSI]i

2kt(85)

Yo )njNp

VpNA(86)

kcp([RTOT• ]wpar + [RTOT

• ]des)[Nn] +kt(n) (n - 1)[Nn]/vp + (kdes + kfmsi[MSI]p)n[Nn] )

kcp([RTOT• ]wpar + [RTOT

• ]des)[Nn-1] +kt(n + 2)(n + 1)[Nn+2]/vp + (kdes +

kfmsi[MSI]p)(n + 1)[Nn+1] (87)

[RTOT• ]des )

FdesVw

kcpAp(88)

nj ) R

m + 2R

m + 1 + 2R

m + 2 + 2Rm + 3 + ...

(89)

R )kcpAp([RTOT

• ]wpar + [RTOT• ]des)VpNA

2

Np2kt

(90)

m )(kdes + kfmsi[MSI]p)VpNA

Npkt(91)


to our model, particle-size-independent expressions werenot used. Recently, Fontenot and Schork (1992-93a)used the model developed by Nomura et al. (1976) andincorporated ideas by Mead and Poehlein (1989a,b) toaccount for nonuniform monomeric radical concentra-tions in particles. Fontenot and Schork (1992-93a)were dealing with large particles and droplet nucleation.The first desorption expression that we will examine

is that of Ugelstad and Hansen (1976):

In eq 92, kfm is the transfer to monomer rate constant,kp′ is the rate constant for reinitiation of oligomericradicals from desorbed monomeric radicals, Dw is thediffusivity of monomer radicals in the aqueous phase,a is the partition coefficient for monomer radicalsbetween the aqueous and particle phases, Dp is thediffusivity of monomeric radicals in the particles, anddp is the particle diameter.Harada et al. (1971) and Nomura et al. (1971a) have

theoretically derived an expression for the desorptionrate coefficient, kdes, using a stochastic approach andhave successfully applied it to vinyl acetate emulsionpolymerization. Their result is the following:

In eq 93, md is the partition coefficient for monomericradicals (same as Ugelstad and Hansen’s “a” andequivalent to 1/Kjwp in eq 110), dp is the average particlediameter, Dw is the diffusivity of monomer radicals inthe aqueous phase, kfm is the transfer to monomer rateconstant, and

where Dp is the diffusivity of monomeric radicals in theparticles. In eq 93, the term RI(1 - nj)/Npkp[M]pnjrepresents desorption of initiator radicals. In our modeldevelopment, it was assumed that initiator radicalswould not be captured by the particles due to theirhydrophilic nature and electric charge; hence, this termmay be safely neglected. As discussed above, radicalsformed as a result of chain transfer to CTA will notdesorb due to the size of the CTA assumed in this case.Thus, kfcta[CTA]p/kp[M]p in eq 93 can be neglected.Equation 93 can then be written as

Nomura and Harada (1981) derived exactly the sameexpression as in eq 95 using a deterministic approach.If nj is much less than 0.1 during the whole range ofconversions (which is not always the case), then thefollowing expression may be used:

The parameter δ (see eq 94) can be described as alumped coefficient or, alternatively, as the ratio of filmmass-transfer resistance to overall mass-transfer resis-tance (Nomura and Harada, 1981). At the start of thepolymerization, Dw and Dp are of the same order ofmagnitude because the particles are saturated withmonomer. Thus, δ assumes a value close to unity. Asthe conversion and, hence, the viscosity inside of theparticles increases, Dp decreases markedly and, hence,kdes decreases. An empirical model of the decrease inDp was used in the model (Friis and Hamielec, 1977):

Previous efforts in the literature to simulate emulsionpolymerizations (Harada et al., 1971; Nomura et al.,1971b; Friis and Nyhagen, 1973; Nomura et al., 1976)made use of eq 96 and assumed an average constantvalue for δ throughout the conversion range. Thus, inour case, eqs 94 and 95 were used. This approach isidentical to the one used by Penlidis (1986) to modelvinyl acetate and vinyl chloride emulsion polymeriza-tions. Furthermore, in order to accommodate the mul-ticomponent aspect of the polymerization, the param-eters Dw and md were calculated as weighted functionsof the mole fraction of monomer in the particles.Rawlings and Ray (1988) presented an expression

that included chain transfer to CTA. However, theiruse of an overall chain-transfer rate constant wouldenable one to modify other groups’ equations similarly.

where

and

ktrt is the chain transfer to CTA rate constant, [CTA]pis the concentration of CTA in the particle, ktrm is thechain transfer to monomer rate constant, dm is themonomer density, φ is the monomer volume fraction inthe particle,Mwt is the monomer molecular weight, andr is the particle radius.Asua et al. (1989) offered the following expression:

where

and

Dp ) Dpo(( 1 - x1 - 0.19x)

2+ 0.0017x) (97)

kdes )3Dm

ktrkp

3DmMwt

dmkpφ+ r2

(98)

Dm )DpDw

mdDp + Dw(99)

ktr ) ktrm(1 +ktrt[CTA]pktrm[M]p ) (100)

kdes ) kfm[M]pKo

âKo + kp[M]p(101)

Ko )12Dw/mddp

2

1 + 2Dw/mdDp(102)

kdes )kfmkp′[ 12Dw

(a + Dw/Dp)dp2] (92)

kdes ) (nj +kp[M]pmddp

2

12Dwδ )-1(kfmkp +kfcta[CTA]pkp[M]p

+

RI(1 - nj)Npkp[M]pnj)kp[M]p (93)

δ ) (1 +6Dw

mdDp)-1

(94)

kdes ) (nj +kp[M]pmddp

2

12Dwδ )-1

kfm[M]p (95)

kdes ) (12Dwδ

mddp2 )kfmkp (96)


ktw is the rate constant for termination in the waterphase, [M]w is the concentration of monomer in thewater phase, ka is the radical absorption (by particles)rate coefficient, NT is the total number of particles perunit volume of aqueous phase, φw

w is the volume frac-tion of water in the continuous phase, and NA isAvogadro’s number.Finally, the particle-size-independent expression de-

rived by Mead and Poehlein (1989b) is shown below:

where

R, in the above equations, is the radius of a monomer-swollen particle and r is the radial position within themonomer-swollen particle.Both the Nomura group and the Poehlein group

present expressions for the desorption rate constant forcopolymerization. The extension to the copolymer casefrom the homopolymerization equations shown aboveis quite straightforward. If one so desires, one may usea weighted average of two homopolymer equations.Also, one should check whether one of the comonomersis insoluble. This could lead to further simplificationsand modifications to the copolymer equations.After careful evaluation of the methods presented

above, we performed some tests with representativenumbers and found very little difference between thevarious methods. Hence, we have employed the expres-sion for kdes of Nomura and Harada (1981) (see eq 96).As an example, consider the monomers in the BA/

MMA/VAc system. VAc is known to exhibit significantradical desorption (Nomura et al., 1971a; Friis andNyhagen, 1973; Penlidis, 1986), while MMA desorbs toa lesser extent (Ballard et al., 1981b) and the desorptionfor BA is considered negligible (Mallya and Plamthot-tam, 1989). Urretabizkaia et al. (1992) and Urretabi-zkaia and Asua (1994) reported that desorption wasnegligible for the seeded semibatch emulsion polymer-ization of BA/MMA/VAc. Considering the dominanceof the acrylic monomers at lower conversions (<60 wt%, see Dube and Penlidis, 1995a-c), it is likely thatdesorption will not play a role in the BA/MMA/VAcemulsion polymerizations.2.8. Species Concentrations in the Different

Phases. We shall now look at defining the concentra-tion of monomers and of various other components thatmay partition between different phases (i.e., waterphase, monomer droplet phase, particle phase). Underequilibrium conditions, the concentrations of monomersin the particles are determined by the balance betweenthe gain in interfacial free energy caused by an increasein surface area upon swelling and the loss in free energycaused by dissolving the monomer in the polymer. Aknowledge of interfacial tension and the Flory-Huggins

interaction parameters would permit one to makereasonable estimates of these concentrations.There are several methods available to define parti-

tioning of species between several phases: a rigorousthermodynamic approach, an experimental approach(based on xc, the conversion at which the monomerdroplets disappear), an empirical approach, and the useof partition coefficients.Several groups in the literature employ the following

equation (Dougherty, 1986):

where a2 is an adjustable parameter that has a valueless than 1 for a water-soluble monomer and is equalto 1 otherwise. Fm and Fp are the densities of monomerand polymer, respectively. xc is the critical conversionat which the monomer droplets disappear. After theconversion has reached xc, one can replace xc in theabove equation with the conversion level x.The thermodynamic approach presented by Guillot

(1985) is as follows:

Equation 107 represents the chemical potential ofspecies i in the monomer droplets. φi is the volumefraction of component i, m is a coefficient related tovolume characteristics, and øij is the Flory-Hugginsinteraction parameter. In the water phase:

and for the particle phase a similar expression can bewritten. Asua et al. (1989), Dimitratos et al. (1989),Mead and Poehlein (1989b), Urquiola et al. (1991), Noelet al. (1994), and Schoonbrood et al. (1994) have alsoused this thermodynamic approach successfully. It isevident from eqs 107 and 108 that several parameters(e.g., the Flory-Huggins interaction parameter, øij; thevolume coefficients, mi and mw) must be known. Theuncertainties around these parameters make use of thethermodynamic approach less practical compared to theone described below, which is based on defining parti-tion coefficients. Noel et al. (1994) recently publisheda simplification to the thermodynamic approach whichwas extended for the case of multiple monomers bySchoonbrood et al. (1994). While the thermodynamicapproach is rigorous, partitioning may also be a functionof emulsifier concentration and monomer feed (fi) (Guil-lot, 1985; Emelie et al., 1991).According to the empirical approach, one tries to

correlate the points where the monomer droplets disap-pear with the weight fraction of monomer in the initialcharge. This approach involves a good deal of experi-mentation and will, of course, be practical only whennarrow ranges of reactor operation are of interest.Finally, the method of partition coefficients based on

that proposed by Omi et al. (1987) was adopted in thecurrent model with ideas from Maxwell et al. (1992a)and Urretabizkaia and Asua (1994). Recent updates ofthese ideas are also available (Armitage et al., 1994;Gugliotta et al., 1995a). This general and valid ap-proach for modeling uses partition coefficients betweenthe monomer/water phases and the water/polymerphases combined with the overall material balances on

â )kp[M]w + ktw[R

•]wkp[M]w + ktw[R

•]w + kaNTφww/NA

(103)

kdes′ )

kfmkp′× 12Dw(π/6)

2/3

(a +Dw[sinh(φ)(1 + 3/φ2) - 3 cosh(φ)/φ]

kp′[M]pR2[cosh(φ)/φ - sinh(φ)/φ2] )

(104)

φ ) r[kp′[M]pDp

]1/2 (105)

[M]psat )a2FmMwt

1 - xc1 - xc(1 - Fm/Fp)

(106)

µi ) µo,i + RT(ln φi + (1 - m)φj + øijφj2) (107)

µiw ) µo,i

w + RT(ln φi + (1 - mi/mw)φw + miøiφw2)(108)


the monomers, the polymer, and the water. Partitioncoefficients for monomer j between the monomer dropletphase and the water phase (Kjmw) and for monomer jbetween the water phase and the particle phase (Kjwp)are defined as

where Vjmd, Vjmw, and Vjmp are the volumes of monomerj in the droplets, aqueous phase, and particles, respec-tively. Vmd is the volume of the monomer droplets, Vaqis the volume of the aqueous phase, and Vp is the volumeof particles.The total volume of monomer j, Vmj, is given by

where Vjmd and Vjmw are derived from eqs 109 and 110,respectively, as follows:

Combining eqs 111-113 gives

The algorithm used to calculate the partitioning ofthe monomers into the different phases is as follows.Initially, when there are no particles present, thevolumes of monomer j in the droplet and water phasesare calculated by guessing values for Vaq (using mono-mer solubilities) and Vmd and iterating over the follow-ing equations:

When particles are generated, the volume of particles,Vp, is calculated using the differential equations shownlater in eqs 141 and 142. The volume of the aqueousphase, Vaq, is calculated from the moles of monomer inthe aqueous phase (see eq 125). The volume of themonomer droplet phase, Vmd, is calculated by thedifference between the total monomer in the reactor andthe sum of the monomer in the particle and waterphases. Then, eq 114 is calculated followed by eqs 112and 113. The concentrations of the monomers in the

various phases can then be calculated. The abovemethod has a rigorous thermodynamic basis as shownin Mead and Poehlein (1988); at the same time, theapproach is quite practical and amenable to experimen-tal confirmations.In order to obtain the concentration of the monomer-

soluble impurities and the chain-transfer agent in theparticles, partition coefficients were also used as shownbelow:

where

Equations 119 and 120 both assume that monomer-soluble impurities and chain-transfer agents are notconsiderably soluble in water. Nmsi and Ncta representthe total number of moles of MSI and CTA in thereactor, respectively.2.9. Material and Population Balances: Pro-

ductivity and Quality. We now present the variousmaterial balances for all of the species involved (savethe initiating system which was presented earlier) andthe rate equations to which they are related. For theemulsion case, the terms with subscripts p and wrepresent the particle and water phases, respectively.In the bulk, suspension and solution cases, recallingthat the reaction steps proceed in only one phase, theterms with the subscript w should be ignored, whilethose with subscript p represent the bulk reactionphase. Note that terms for semibatch and continuousreactors which allow for the inflow and outflow of thevarious species have been included here. In this section,the inflow of species i is designated by Fi,in and voutrepresents the volumetric flow out of the reactor.The balances for the monomers comprise the inflow

and outflow of monomers and their depletion by polym-erization in the organic (or particle) phase as well as inthe aqueous phase (for the emulsion case):

The inflow of monomer increases the rate of accumula-tion of the number of moles of monomer j (dNmj/dt) inthe reactor. The outflow (second term on the right-handside of eq 123) and the reaction of monomer (third andfourth terms on the right-hand side of eq 123) result ina decrease in dNmj/dt. The reaction takes place in theparticles, hence the term RpjpVp in eq 123, and in thewater phase, hence the term RpjwVw in the sameequation.Similarly, the balances for the monomers bound in

the polymer involve accumulation of bound monomerby polymerization in both the particles and the waterphase. In a batch reactor, where there is no inflow oroutflow of a polymer from the reactor, the total amountof polymer formed and its composition can be obtainedfrom the monomer material balances directly by con-sidering the net change in moles of each monomer j.

Kjmw )Vjmd/Vmd

Vjmw/Vaq(109)

Kjwp )Vjmw/Vaq

Vjmp/Vp(110)

Vmj ) Vjmd + Vjmw + Vjmp (111)

Vjmd )VmdVjmpKjmwKjwp

Vp(112)

Vjmw )VaqVjmpKjwp

Vp(113)

Vjmp )Vmj

1 +VmdKjmwKjwp

Vp+VaqKjwp

Vp

(114)

Vjmd ) Vmj(1 +Vaq

VmdKjmw) (115)

Vjmw ) Vmj - Vjmd (116)

Vmd ) ∑j)1

N

Vjmd (117)

Vaq ) Vw∑j)1

N

Vjmw (118)

[MSI]p )Nmsi

Vp + VmKmsi(119)

[CTA]p )Ncta

Vp + VmKcta(120)

Kmsi ) KMSIwpKMSImw (121)

Kcta ) KCTAwpKCTAmw (122)

dNmj

dt) Fmj,in -

Nj

VTvout - RpjpVp - RpjwVw (123)


However, in a semibatch or continuous-flow reactor withinflow and outflow of polymer and monomer, additionalbalances are needed for the moles of each monomer j inthe reactor that are bound as polymer, i.e.

The moles of monomer j bound in the polymer (Npolj)are a result of polymerization in the particles (RpjpVp)and in the water phase (RpjwVw) as well as inflow (Fpj,in)and outflow ((Npolj/VT)vout). Polymerization in the mono-mer droplets is neglected for the reasons presentedearlier.A separate balance on the number of moles of mono-

mer j in the water phase is used to calculate the volumeof monomers in the water phase. During the intervalwhen monomer droplets exist, the number of moles ofmonomers in the water phase is held constant from theinitial values based on the partitioning equations shownearlier (see eq 116). When the droplets are depleted,the monomer in the water phase is calculated by:

The polymerization rates are defined as

where [RTOT• ]w is the total concentration of radicals in

the water phase (see eq 29).From the above balances, a number of variables of

interest can be calculated directly. For example, thetotal mass conversion of monomers to polymer can becalculated by

The instantaneous polymer composition (Fj) in theparticles can now be calculated as:

Expressions for the instantaneous polymer compositionbased solely on the reactivity ratios (defined earlier ineq 16) and the monomer feed mole fractions werederived from the Alfrey-Goldfinger equations for ter-polymers (Alfrey and Goldfinger, 1944). The Alfrey-Goldfinger equations are shown below:

After some algebraic manipulation, one may obtain

where num(Fi/Fj) refers to the numerator of the equa-tions for Fi/Fj (see eqs 130 and 131) and den refers totheir denominator. Equations 132-134 may be reducedto the homo- or copolymerization case by setting thereactivity ratios, which include the missing monomer,equal to 1.The cumulative polymer composition (Fh j) is calculated

as

The instantaneous polymer composition is calculatedmainly for its use as a weighting function for the variousparameters in the model and is calculated for twoseparate cases. One is based on the polymer beingformed in the particles and the other on that beingformed in the water phase. On the other hand, thecumulative polymer composition takes into account thepolymer formed in both phases, i.e., the total polymerformed in the reactor. The cumulative polymer com-position includes both water phase and particle phasepolymerizations in order to account for the contributionto the composition due to the capture of oligomericradicals from the water phase by the particles.A balance is now performed on the solvent, which is

in the emulsion case, water, comprising of inflow andoutflow:

Similarly, a balance for the emulsifiers results in

Note that the emulsifier can be located in four areas:

dNpolj

dt) Fpj,in -

Npolj

VTvout + RpjpVp + RpjwVw (124)

dNmjw

dt) -RpjwVw (125)

Rpjp ) Yo[M]pfj∑i)1

N

kpijφi (126)

Rpjw ) [RTOT• ]w[M]wfjw∑

i)1

N

kpijφiw (127)

x )

∑j)1

N

(NpoljMWj)

∑j)1

N

((Nmj + Npolj)MWj)

(128)

Fj )Rpjp

∑j)1

N

Rpjp

(129)

F1

F3)

f1(f1r23r32 + f2r23r31 + f3r21r32)(f1r12r13 + f2r13 + f3r12)

f3(f1r12r23 + f2r13r21 + f3r12r21)(f1r32 + f2r31 + f3r31r32)(130)

F2

F3)

f2(f1r13r32 + f2r13r31 + f3r12r31)(f1r23 + f2r21r23 + f3r21)

f3(f1r12r23 + f2r13r21 + f3r12r21)(f1r32 + f2r31 + f3r31r32)(131)

F1 )num(F1/F3) - 2num(F2/F3) - 2den

num(F1/F3) - num(F2/F3) - den(132)

F2 )num(F2/F3)

num(F1/F3) - num(F2/F3) - den(133)

F3 ) dennum(F1/F3) - num(F2/F3) - den

(134)

Fh j )Npolj

∑j)1

N

Npolj

(135)

dNs

dt) Fs,in -

NS

VTvout (136)

dNem

dt) Fem,in -

Nem

VTvout (137)


as micelles, stabilizing monomer droplets, and stabiliz-ing particles and dissolved in the aqueous phase.Now, balances for the water-soluble (z1, z2) and

monomer-soluble impurities (MSI) are performed:

Water-soluble impurities such as oxygen and hydro-quinones can enter the reaction via various feed streamsand consume radicals in the water phase. Monomer-soluble impurities can arise from the feed streams andrecycle streams.A balance for the consumption of chain-transfer agent

(CTA) is

The volume of polymer particles is defined in twoseparate regimes: when monomer droplets exist andwhen there are no longer any droplets. The balancesrespectively are

where φp is the volume fraction of polymer in theparticles, Fmj is the density of monomer j, and Fp is thedensity of the polymer. Fp is calculated using theindividual polymer densities weighted by the molefractions of each polymer in the particles. For the caseof seeded emulsion polymerizations or a series ofconstant stirred-tank reactors (CSTRs), a term describ-ing the inflow of polymer particles can be added to theequations. Note that one may be required to includean outflow term to describe the flow of polymer out ofthe reactor. Since the outflow term, vout, includes boththe organic (monomer and polymer) and aqueous phases,it should be multiplied by the percentage of polymer inthe reactor. In any case, one must satisfy the massbalances. The first term in eq 141 describes theincreasing volume of the particles due to the saturationof the particles by monomer. In other words, as polym-erization within the particles proceeds, monomer isconsumed and replaced by monomer from the dropletsin amounts required to maintain a constant monomerconcentration in the particles. RpjwVw takes into accountthe growth of the particles by capture of oligomericradicals formed in the aqueous phase. Equation 141 isactive while the monomer droplets exist. Equation 142represents the stage of the emulsion polymerizationwhen the monomer droplets disappear as a separatephase. At this stage, the volume of the particles canincrease only by capture of radicals from the waterphase. This growth is counteracted by shrinkage due

to the density differences between the monomers andthe polymer. What results is a net decrease in the totalvolume of particles.The particle diameter of the “swollen” particles, under

the “monodispersed” approximation, is defined as

The particle diameter of the “unswollen” particles (thequantity measured using a disc centrifuge), under themonodispersed approximation, is defined as

The volume of the monomer droplets can be calculatedby the difference between the volume of the organicphase and the sum of the volume of polymer particlesand the volume of monomer in the aqueous phase. Thebalance for the volume of the organic phase is

The volume of the organic phase changes due to theinflow of monomer and polymer, the outflow of monomerand polymer, and volume shrinkage during the polym-erization. Vo corresponds to the total volume in a bulkpolymerization. Recall that vout can be included in theequation to describe the outflow of monomer andpolymer.2.10. Molecular Weight, Long-Chain Branching,

and Cross-Linking Development. 2.10.1. LinearCopolymer Chains. With linear copolymer chains,polymer molecules once formed are inert and do not takepart in reactions such as transfer to polymer andreaction with internal double bonds bound in polymerchains, and, therefore, the method of instantaneousmolecular weight distribution can profitably be em-ployed for homo-, co-, and, in general, multicomponentpolymerizations. This approach, shown below, was firstintroduced for binary copolymerization by Stockmayer(1945), who derived the instantaneous bivariate distri-bution for linear copolymer chains, assuming that bothmonomer types have the same molecular weight. Acorrection factor to account for differing monomermolecular weights was recently derived by Tacx et al.(1988) and is given by eq 210 shown later.

where W(r,y) dr dy is the weight fraction of copolymerof chain length in the range r to r + dr and compositiondeviation in the range y to y + dy for copolymerproduced instantaneously (in the time interval t to t +dt in a batch reactor).

dNzi

dt) Fzi,in -

Nzi

VTvout - kziNzi[RTOT

• ]w (138)

dNmsi

dt) Fmsi,in -

Nmsi

VTvout - kfmsi[MSI]pYoVp (139)

dNcta

dt) Fcta,in -

Ncta

VTvout - kfcta[CTA]pYoVp (140)

dVp

dt)

∑j)1

N

MWmj(RpjpVp + RpjwVw)

φpFp(141)

dVp

dt)

∑j)1

N

MWmjRpjwVw

Fp- ∑

j)1

N

MWmjRpjpVp(1/Fmj - 1/Fp)

(142)

dp ) ( Ap

πNp)1/2 (143)

dp ) (6Vpφp

πNp)1/3 (144)

dVo

dt) ∑

j)1

N Fmj,inMWmj

Fmj-

∑j)1

N

MWmj(RpjpVp + RpjwVw)( 1Fmj -1

Fp) (145)

W(r,y) ) (τ + â)(τ + â/2(τ + â)r)r exp(-(τ +â)r(1/2πσ2)1/2 exp(-y2/2σ2) (146)

y ) y1 - F1 (147)

σ2 ) F1(1 - F1)K/r (148)


where y1 is the mole fraction of monomer 1 in thecopolymer and F1 is the overall mole fraction of mono-mer 1 in the copolymer and was given earlier in eq 129and

The pseudo-kinetic rate constants for multicomponentpolymerization were given earlier by eqs 61 (for transferto CTA), 63 (for transfer to monomer), 83 (for diffusion-controlled termination), and 84 (for diffusion-controlledpropagation).The instantaneous molecular weight distribution of

the copolymer is given by

For long polymer chains, one can assume with smallerror that all chains produced instantaneously have thesame composition F1, F2, .... These overall compositionscan be found using eq 129.The bivariate distribution for polymer accumulated

in a batch or semibatch reactor may be found byintegration of eq 146 as follows:

Instantaneous number- and weight-average chain lengthsare given by

Number- and weight-average chain lengths of theaccumulated polymer in batch or semibatch reactors aregiven by

Tobita and Hamielec (1991) generalized Stockmayer’sbivariate distribution to account for chain-length de-pendence of polymer radical/polymer radical termina-tion. Expressions equivalent to eqs 155 and 156 when

chain-length-dependent termination is significant maybe found in Zhu and Hamielec (1989).2.10.2. Polymers with Long Branches. The

“method of instantaneous molecular weight distribution”which is such a powerful method for the calculation ofmolecular weight distribution of linear polymer chainsmust be modified when long-chain branching reactionssuch as transfer to polymer and reaction with internalor terminal double bonds are significant. In this case,dead polymer chains are no longer inert and can reactwith polymeric radicals (and other radical types) andnow the instantaneous molecular weight distribution isno longer a permanent quantity but changes with timeas its chains react. One must keep track of these chainsin order to predict the MWD of the accumulated linearchains. In principle, this can be done for chains having1, 2, or more long branches (Soares and Hamielec,1996c). Alternatively, one can use Monte Carlo simula-tions to calculate the full MWD for chains with longbranches as illustrated by Tobita (1995a,b).Teymour and Campbell (1994) developed a novel

method coined “numerical fractionation” to calculate thefull MWD of sol before and after the gelation pointwithout the normal discontinuity that plagued earliermethods using the method of moments. This methodis presently being comprehensively evaluated by variousresearch groups, and we await the results of thesestudies. Of course, when the full MWD is not essentialand average molecular weights will do, it is recom-mended that the method of moments be used. The useof the pseudo-kinetic rate constant method for copolym-erization with long-chain branching has been illustratedby Xie and Hamielec (1993b,c).Using the method of moments, the leading moments

of the distribution (Q0, Q1, Q2) can be calculated asfollows. Invoking the stationary-state hypothesis forradicals, one can readily derive the following momentequations (Ray, 1972; Broadhead et al., 1985; Hamielecet al., 1987; Xie and Hamielec, 1993a,b).

where

K ) (1 - 4F1(1 - F1)(1 - r1r2)) (149)

τ )ktdYo

kp[M]p+ Cfm +

Cfcta[CTA]p[M]p

+Cfmsi[MSI]p

[M]p(150)

â )ktcYo

kp[M]p(151)

W(r) ) ∫-∞

∞W(r,y) dy ) (τ + â)(τ + â/2(τ + â)r)r

exp(-(τ + â)r) (152)

Wh (r,y) )∫0tW(r,y) RpW(t) V(t) dt

∫0tRpW(t) V(t) dt(153)

RpW(t) ) ∑i)1

N

RpiMWi (154)

rN ) 1τ + â/2

(155)

rW ) 2τ + 3â(τ + â)2

(156)

rjN )∫0tRpW(t) V(t) dt

∫0t(1/rN)RpW(t) V(t) dt(157)

rjW )∫0trWRpW(t) V(t) dt

∫0tRpW(t) V(t) dt(158)

dVpQ0

dt) (τ + â/2 -

CkQ1

[M]p-KQ0

[M]p)kp[M]pYoVp (159)

dVpQ1

dt)

(1 + Cfm +Cfcta[CTA]p

[M]p-Cfmsi[MSI]p

[M]p )kp[M]pYoVp

(160)

dVpQ2

dt) (factor + 2(1 +

KQ1 + CkQ2

[M]p )bracketdenom+

â(bracketdenom )2)kp[M]pYoVp (161)

τ )ktdYo

kp[M]p+ Cfm +

Cfcta[CTA]p[M]p

+Cfmsi[MSI]p

[M]p(162)

â )ktcYo

kp[M]p(163)


Cfm, Cfcta, and Cfmsi are the constants for transfer tomonomer, to chain-transfer agent, and to monomer-soluble impurities, respectively. They are defined as

Cfp, Ck, and K are the constants for transfer to polymer,internal double-bond, and terminal double-bond reac-tions, respectively. They are defined as

The accumulated number- and weight-average mo-lecular weights are then defined as

The effective molecular weight, Mweff, is calculated bysumming the individual monomer molecular weightsand weighting them by the cumulative polymer com-position.To calculate instantaneous number- and weight-

average chain lengths, one can simply divide eq 160 byeq 159 and then eq 161 by eq 160, respectively. SettingCfp, Ck, and K to zero should then give the average chainlengths for linear polymer chains (see eqs 155 and 156).The branching frequencies for both trifunctional and

tetrafunctional branching are calculated as follows:

2.10.3. Cross-Linking and Gelation. Reactionwith internal double bonds may rapidly lead to a three-dimensional network and the onset of gelation. In thecontext of the present kinetic models, the onset ofgelation occurs when Q2 (or the weight-average molec-ular weight) goes to infinity. In other words, the firstcross-linked gel particle of infinite molecular weightforms in the polymerizing mixture. To illustrate the useof eq 161 to calculate the onset of gel formation, alimiting form of this equation will be used. It is

assumed that transfer to monomer and reaction withinternal double bonds control molecular weight develop-ment, with an insignificant number of polymer chainsproduced by termination reactions. In addition, it isassumed that, in the batch reactor, the volume of thereacting mixture is constant. Equation 161 thus re-duces to

Introducing monomer conversion, eq 177 can berearranged to give

where

and

Equation 178 can be solved analytically to give

where xc is the monomer conversion at the point ofgelation.The solution (eq 181) applies for a homopolymeriza-

tion of a divinyl monomer whose double bonds haveequal reactivity. For the copolymerization of a divinylmonomer with a vinyl monomer, one must redefine ψas follows:

where F1 is the mole fraction of divinyl monomer in thecopolymer chains.The physical interpretation of these results is straight-

forward. The more active the double bonds bound inthe copolymer chains (larger k*p*) and the greater thenumber of these double bonds (larger F1), onset ofgelation occurs at lower monomer conversions. Themore transfer to a small molecule such as monomer orchain transfer agent (larger kfm), the further in conver-sion is the onset of gelation delayed.Models which can be used to calculate the quantity

of gel and details of gel structure are under development(Tobita and Hamielec, 1989, 1992; Tobita, 1992, 1993a,b;Charmot and Guillot, 1992; Teymour and Campbell,1994).2.11. Energy Balance Equations. Two energy

balances, one around the reactor contents and the otheraround the reactor cooling jacket, will enable one toinvestigate nonisothermal operation, variable heat-transfer coefficients, reactor runaways, and temperaturecontroller design). For heat transfer through the reactorwalls (reactor jacket) and with the jacket full of perfectlymixed heat-transfer fluid, the two heat balance equa-tions may be written as follows for a well-mixed vessel:

factor ) 1 + Cfm +Cfcta[CTA]p

[M]p+Cfmsi[MSI]p

[M]p(164)

denom ) τ + â +CfpQ1

[M]p(165)

bracket ) 1 + Cfm +(Cfp + Ck)Q2

[M]p+KQ1

[M]p+

Cfcta[CTA]p[M]p

+Cfmsi[MSI]p

[M]p(166)

Cfm ) kfm/kp (167)

Cfcta ) kfcta/kp (168)

Cfmsi ) kfmsi/kp (169)

Cfp ) kfp/kp (170)

Ck ) k*p*/kp (171)

K ) k*p/kp (172)

Mh w ) Q1/Q0Mweff (173)

Mh w ) Q2/Q1Mweff (174)

dVpQ0BN3

dt) kpYoVp(CfpQ1 + KQ0) (175)

dVpQ0BN4

dt) kpYoVpCkQ1 (176)

dQ2

dt) ( 2

Cfm(1 + Q2Ck/[M]p)

2)kp[M]pYo (177)

dQh 2

dx) 2ψ(1 +

Qh 2

1 - x)2

(178)

Qh 2 ) k*p*Q2/kp[M]p (179)

ψ ) k*p*/kfm (180)

Qh 2 )4ψ(1 - x)(1 - (1 - x)x1+8ψ)

(1 + 4ψ - x1 + 8ψ)[(1 - x1 - xc)

x1+8ψ- 1]

(181)

ψ: 0 0.1 0.2 0.3 0.4 0.5 1.0 5.0xc: 1 0.943 0.835 0.778 0.647 0.577 0.370 0.094

ψ ) F1k*p*/kfm (182)


Controller Equation. Reactor temperature can becontrolled by manipulating the temperature of thecooling jacket water feed. The temperature adjustmentof the cooling jacket water feed is assumed to beinstantaneous. A proportional-integral controller canbe simulated as follows:

where the error e is the difference between the desiredand actual reactor temperature.

The integral term of the controller can be simulatedwith the following differential equation

where

The control action is implemented through the followingequation:

where ∆Tj,in represents the control action and Tc asteady-state or bias term.The controller gain, Kc, and integral time, τI, can be

tuned using a linearized process model and stabilityanalysis. U and Aj are the overall heat-transfer coef-ficient and jacket heat-transfer area, respectively. QRand Qj are uncontrolled heat losses for reactor andjacket.

3. Model Development: Olefin Polymerizationwith Homogeneous and HeterogeneousZiegler-Natta Catalysts

Polyolefins are very important commodity polymers.Polyethylene and polypropylene are today the majortonnage plastic materials worldwide, accounting for 44%of all U.S. plastic sales in 1988 and reaching a capacityof about 45 million tons in 1990 (Elias, 1992; Whiteleyet al., 1992).Most industrial processes for the production of poly-

olefins utilize heterogeneous Ziegler-Natta catalysts.Conventional soluble Ziegler-Natta catalysts have notfound widespread industrial applications, mainly be-cause of insufficient catalytic stability and stereochem-ical control. This picture, however, will likely change

in the near future with the advent of metallocene-basedcatalyst systems. Metallocene catalysts with aluminox-ane and other cocatalysts are able to produce polyolefinsat a very high productivity with a degree of microstruc-tural control not possible to achieve using conventionalZiegler-Natta catalysts (Hamielec and Soares, 1996;Huang and Rempel, 1995; Soares and Hamielec, 1995a).Ziegler-Natta catalysts are formed by a transition

metal salt of metals of groups IV-VIII and a metal alkylof a base metal of groups I-III (known as cocatalyst oractivator). However, not all combinations are equallyefficient and can be used for all monomers. From theindustrial point of view, most heterogeneous catalystsare based on titanium salts and aluminum alkyls.Most types of heterogeneous Ziegler-Natta catalysts

have a common intriguing characteristic: they yieldpolymer with broad molecular weight distribution (MWD)and, in the case of copolymerization, broad chemicalcomposition distribution (CCD). There is now a generalagreement that heterogeneous Ziegler-Natta catalystspossess more than one type of active site, each one withdistinct ratios of chain transfer to propagation rates,comonomer reactivity ratios, and stereoselectivities.Each site type makes polymer chains that have differentaverage chain lengths, comonomer compositions, comono-mer sequence lengths, and, in the case of asymmetricmonomers, different degrees of stereoregularity. Con-sequently, heterogeneous Ziegler-Natta catalysts pro-duce a mixture, at the molecular level, of polymer chainshaving dissimilar average properties. These dissimilaraverage properties are reflected in broad MWDs andCCDs. Additionally, intraparticle heat- and mass-transfer resistances during the polymerization maysomewhat broaden these distributions even further.On the other hand, polyolefins made with most

soluble Ziegler-Natta catalysts, including several met-allocene systems, have narrow MWD, and copolymersalso have narrow CCD. This behavior supports themultiple-site-type hypothesis for heterogeneous cata-lysts. Soluble Ziegler-Natta catalysts consist of rea-sonably well-defined, single catalytic species, probablynot subject to significant heat- and mass-transfer re-sistances during polymerization.The complexity of MWDs and CCDs of polyolefins

made with Ziegler-Natta catalysts constitutes a chal-lenging problem for polymer quality control. Mostproperties of polyolefins are routinely measured only asaverage values. Measurements of melt flow index (asan estimator for a molecular weight average), melt flowindex ratio (as an estimator of polydispersity), and bulkdensity (as an estimator of copolymer composition ordegree of short-chain branching) are common practicein industry. However, macroscopic properties of poly-mers, in general, and polyolefins, in particular, cannotbe uniquely determined by average values, since poly-mers that have some average properties in common canpossess other properties that differ markedly. Even aknowledge of the full molecular weight distribution forpolypropylene may not be sufficient for many practicalapplications, due to differences in stereoregularity of thechains. In the same way, determining average composi-tions of copolymers or average degrees of branching willnot entirely define the polymer in question. The wholedistribution of composition in addition to chain lengthis necessary to accomplish this task. This issue becomesvery complex with polyolefins made with heterogeneous

Reactor Balance

d(∑j

MiCpi(T - Tref))

dt) ∑

i

Fi,inCpi(Tin - Tref) -

∑i

Fi,outCpi(T - Tref) + ∑i

Rpi(-∆Hpi)Vp - UAj(T -

Tj) - QR (183)

Cooling Jacket Balance

dTj

dt)(FjwCpw(Tj,in - Tj) + UAj(T - Tj) - Qj)

FwVjCpw(184)

∆Tj,in ) Kce (Kc

τI∫0te dt (185)

e ) T - Tset (186)

dI/dt ) T - Tset (187)

I ) ∫0te dt (188)

Tj,in ) Tc + ∆Tj,in (189)


Ziegler-Natta catalysts because polymers with broadand sometimes multimodal MWDs and CCDs can beproduced.The concern about the breadth of MWD and CCD of

polyolefins is far from academic. Those distributionsaffect the final mechanical and rheological propertiesof polyolefins and ultimately determine their applica-tions. Polyethylenes with broad MWD are easier toprocess because of greater flowability in the molten stateat high shear rate, while polyethylenes with narrowMWD have greater dimensional stability, higher impactresistance, greater toughness at low temperatures, andhigher resistance to environmental stress cracking. Forpolypropylene, narrow MWD is required for rapid mold-ing of products with good mechanical properties. Nar-row MWD and especially narrow CCD are required forthe production of elastomers. Polymers with broad CCDmight have some chains containing long sequences ofone monomer type, increasing the crystallinity of thepolymer, which is highly undesirable for elastomers.3.1. Mathematical Models. It is convenient to

classify mathematical models for polymerization pro-cesses in three levels: microscale, mesoscale, and mac-roscale (Ray, 1988). Microscale models define thekinetics of polymerization and the types of active siteson the catalyst. Mesoscale models define interparticleand intraparticle mass- and heat-transfer resistancesin the polymer particle. Macroscale models describe themacroscopic behavior of the polymerization reactor, suchas imperfect mixing, residence time distribution, gas-liquid mass transfer, and removal of heat of polymeri-zation. The final application of the model determinesthe degree of complexity required in each modeling level.Figure 1 shows schematically these three levels ofmathematical modeling.Knowledge of the kinetics of polymerization is evi-

dently fundamental for the mathematical modeling ofZiegler-Natta polymerization reactors. The kinetics ofpolymerization with heterogeneous and homogeneousZiegler-Natta catalysts has been the subject of detailed

study and will not be covered in this review. Goodreviews on polymerization kinetics are available in theliterature for both heterogeneous (Keii, 1972; Cooper,1976; Bohm, 1978; Boor, 1979; Tait and Watkins, 1989)and homogeneous (Hamielec and Soares, 1996; Huangand Rempel, 1995; Soares and Hamielec, 1995a) Zie-gler-Natta catalysts.Mathematical models that can predict MWD and CCD

of polymer chains will be reviewed in the next sections.3.1.1. Heterogeneous Catalysts. Heterogeneous

Ziegler-Natta catalysts are porous secondary particles,formed by loosely aggregated primary particles (Tait,1989, Noristi et al., 1994). During polymerization, thegrowing polymer chains fragment these secondaryparticles, forming an expanding particle containingprimary particles and living and dead polymer chains(Figure 2). This catalyst fragmentation mechanism hasbeen documented for several types of heterogeneousZiegler-Natta catalysts (Hock, 1966; Buls and Higgins,1970; Wilchinsky et al., 1973; Boor, 1979; Kakugo et al.,1989; Noristi et al., 1994). One of its consequences isthe well-known replication phenomenon: the particle-size distribution of the polymer particles at the end ofbatch or semibatch polymerization closely approximatesthe particle-size distribution of the catalyst at thebeginning of polymerization (Simonazzi et al., 1991).Good replication is supposed to occur when there is anadequate balance between the mechanical strength ofthe particle and catalytic activity. Replication factorsof 40-50 (ratio of average polymer particle diameter toaverage catalyst particle diameter) are obtained withthird generation Ziegler-Natta catalysts (Galli andHaylock, 1992). For the case of continuous reactors,residence time distribution can significantly affect thisparticle size distribution (Soares and Hamielec, 1995d;Choi et al., 1994).Based on this well-known experimental evidence,

some researchers advocate that, due to diffusion resis-tances, catalyst fragments in different radial positionsare exposed to different concentrations of monomer and

Figure 1. Levels of mathematical modeling.


chain-transfer agent (generally hydrogen) and conse-quently produce polymer with chain-length averagesthat differ spatially inside the polymeric particle. Forcopolymerization, monomers with different effectivediffusivities and reactivities may be responsible forspatial compositional heterogeneity in the polymericparticle. In addition, if there is appreciable heat-transfer resistance, hot spots can occur inside thepolymer particle, altering reaction rates and furtherbroadening MWD and CCD (Figure 3). Models basedon this approach are generally called physical models(Schmeal and Street, 1971, 1972; Singh and Merrill,1971; Crabtree et al., 1973; Nagel et al., 1980; Taylor etal., 1983; Floyd et al., 1986a-c; Galvan and Tirrell,1986a; Hutchinson and Ray, 1988; Skomorokhov et al.,1989; Sarkar and Gupta, 1991, Soares and Hamielec,1995b).

A second category of mathematical models neglectsheat- and mass-transfer resistances in the polymerparticle. In these models, multiple types of catalyticsites are responsible for the production of polymer withbroad MWDs and CCDs ( chemical models). Each sitetype has its own kinetic constants and produces polymerwith often very different MWDs, CCDs, and stereoregu-larities, as shown in Figure 4 for a three-site-typecatalyst (Tait and Wang, 1988; de Carvalho et al., 1989;Rincon-Rubio et al., 1990; McAuley et al., 1990; Loren-zini et al., 1991; Vela-Estrada and Hamielec, 1994; Leeet al., 1994; Soares and Hamielec, 1995b). There is vastexperimental evidence supporting the existence of mul-tiple site types on heterogeneous Ziegler-Natta cata-lysts (Cozewith and VerStrate, 1971; Boor, 1979; Keii,1982; Zucchini and Cecchin, 1983; Keii et al., 1984;Chien et al., 1985; Usami et al., 1986; Spitz, 1987; Chengand Kakugo, 1991; Soares and Hamielec, 1996a,b).The effects of transfer resistances and of multiple site

types on MWD and CCD broadening are evidently notexclusive; the heterogeneity caused by the presence ofmultiple site types can be increased by transfer resis-tances during the polymerization. Some models com-bining these two effects have been proposed recently andseem to be highly successful in predicting the qualitativebehavior of polymerization with Ziegler-Natta cata-lysts. These models will be called hybrid models sincethey combine the two previous modeling strategies(Galvan and Tirrell, 1986b; Floyd et al., 1988; Ray, 1988;Hutchinson et al., 1992; Sau and Gupta, 1993; Soaresand Hamielec, 1995b).Physical Models. Different mathematical models

have been proposed to describe interparticle and intra-particle mass and heat transfer during polymerizationwith heterogeneous Ziegler-Natta catalysts. Some arevery simplified pictures of the polymerization processand are only useful as a reference point for comparisonwith more sophisticated models. Among those, the most

Figure 2. Fragmentation of heterogeneous Ziegler-Natta catalyst (secondary) particles during polymerization. The growing polymerchains break the secondary catalyst particles into primary particles, forming an expanding particle made of catalyst fragments surroundedby dead and living polymer chains. This behavior is responsible for the replication phenomenon observed in heterogeneous Ziegler-Natta catalysis.

Figure 3. Generic representation of physical models. T )temperature; [M] ) monomer concentration; %M1 ) percent ofcomonomer type 1 in the polymer;Mn ) number-average molecularweight. In physical models, intraparticle mass- and heat-transferresistances are responsible for MWD and CCD broadening.


representative for this review are the solid core modeland the polymeric core model (Figure 5).The solid core model does not model the breakup of

the catalyst particle. The polymer is considered to growaround a solid catalyst core containing all active siteson its surface. This model using a single type of catalystsite cannot predict broad MWDs (Schmeal and Street,1971, 1972; Nagel et al., 1980). This result should notcome as a surprise since the polymerization takes placeonly at the surface of the catalyst. For a given polym-erization time, the monomer concentration is constantat the surface of the catalyst and therefore all polymerchains are produced with the same average properties.Only if the concentration of monomer at the surface ofthe catalysts changed significantly in time (because ofincreasing transfer resistance caused by the growth ofpolymer around the catalytic particle) would the solidcore model predict significant broadening of MWD.Additionally, the solid core model is clearly in contradic-tion with experimental data on catalyst breakup. De-spite of these limitations, Brockmeier and Rogan (1976)adjusted the solid core model to data collected in a

semibatch reactor and used this model to predict thebehavior of a continuous reactor. The predictions wereaccurate for polymer yield, but no molecular weightresults were shown.With the polymeric core model, polymer grows around

a nonexpanding polymeric core formed by polymer andcatalyst particles (Figure 5). Although this model is animprovement over the first one, it still is not able toaccount for broad MWDs (Schmeal and Street, 1971,1972; Singh and Merrill, 1971). It shows, however, theimportance of a core of polymer-catalyst fragments inthe explanation of broad MWD.The models that best represent the polymerization in

this category of single-site-type, transfer-controlledmodels are the so-called expansion models. Expansionmodels consider the fragmentation of the catalystparticle and the formation of an expanding particle ofpolymer and catalyst fragments.Growing polymer chains and catalyst fragments form

a continuum in the polymeric flow model (Schmeal andStreet, 1971, 1972; Singh and Merrill, 1971; Galvan andTirrell, 1986a). Diffusion of reagents as well as heattransfer occurs in this polymeric particle. If the reactionis diffusion-controlled, the radial profiles of monomerand chain-transfer agent in the particle may causeMWD and CCD broadening. The effective mass- andheat-transfer parameters in the polymeric particle haveto be estimated to use this model (Figure 6). Thepolymeric flow model can predict significant broadeningof MWD using reasonable physical parameters.The multigrain model (Crabtree et al., 1973; Nagel et

al., 1980) considers two levels of mass- and heat-transferresistances. The polymeric particle ( macroparticle orsecondary particle) is formed by an agglomerate ofmicroparticles or primary particles. Each microparticleconsists of a fragment of the original catalyst particle,with all active sites on its surface, surrounded by dead

Figure 4. Generic representation of chemical models. 1, 2, and 3 indicate sites of different types; open and shaded circles indicate differenttypes of monomers. In chemical models, multiplicity of active site types is responsible for MWD and CCD broadening.

Figure 5. Schematic representation of the solid core model andof the polymeric core model.


and living polymer (the solid core model describes thebehavior of the microparticles). Monomer diffusesthrough the pores of the macroparticle and through thelayer of polymer surrounding the catalyst fragment inthe microparticle. Polymerization occurs on the surfaceof the catalyst fragment. Electron microscopy studiesconfirm the formation of primary and secondary struc-tures in polymerizations with Ziegler-Natta catalysts(Hock, 1966; Buls and Higgins, 1970; Kakugo et al.,1989; Noristi et al., 1994). It is necessary to estimatethe number of microparticles and the transfer param-eters in the microparticle and macroparticle to use themultigrain model (Figure 6).The multigrain model is probably the most studied

expansion model for polymerization of olefins withheterogeneous Ziegler-Natta catalysts. It has beenextensively used to study heat- and mass-transferresistances for the homopolymerization of ethylene andpropylene in slurry and gas-phase reactors. Someimportant conclusions obtained with the multigrainmodel are as follows: (1) In most cases, intraparticletemperature gradients are negligible. (2) Concentrationgradients in the macroparticles are likely to be moreimportant in slurry reactors. For gas-phase reactors,these gradients may be significant in the microparticles.(3) Interparticle mass- and heat-transfer resistances arenegligible except for high activity catalysts. (4) Forslurry reactors, gas-liquid mass-transfer resistancesare generally negligible for sparged reactors but can beimportant for unsparged reactors (Nagel et al., 1980;Taylor et al., 1983; Floyd et al., 1986a-c). Most mass-and heat-transfer resistances are only significant atearly stages of polymerization as suggested by detailedmathematical modeling of catalyst particle fragmenta-tion (Laurence and Chiovetta, 1983; Ferrero and Chio-vetta, 1987a,b, 1991).Hutchinson and Ray (1988) used the multigrain model

to analyze the influence of monomer adsorption on thesurface of the microparticle. In the multigrain model,before diffusing into the microparticle, adsorption of themonomer on the outer surface of the microparticle hasto take place. Rate enhancement in copolymerizationand differences between slurry and vapor-phase reac-tions were explained based on differences of solubilitiesin each case, since more crystalline polymer layers havelower monomer solubility and effective diffusivity.Hutchinson et al. (1992) further expanded the multi-grain model to describe copolymerization and particlemorphology.Modeling equations for intraparticle radial profiles of

monomer concentration and temperature for all expan-sion models are very similar. The radial profile ofmonomer concentration in the secondary particle is

described by the well-known diffusion-reaction equationin spherical coordinates:

whereDs ) effective diffusivity of monomer in secondaryparticle (cm2 s-1), ks ) mass-transfer coefficient in anexternal film (cm s-1),Mb ) monomer concentration inthe reactor (mol cm-3), Ms ) monomer concentrationin the secondary particle (mol cm-3), Ms0 ) initialmonomer concentration in the secondary particle (molcm-3), rv ) volumetric rate of polymerization is thesecondary particle (mol cm-3 s-1), rs ) radial positionin the secondary particle (cm), Rs ) radius of thesecondary particle (cm), and t ) polymerization time (s).Notice that rv is the average rate of polymerization

at a given radial position in the secondary particle. Inthe multigrain model, the polymerization is supposedto only take place on the surface of the catalyst fragmentin the primary particle. Mathematically, it is intro-duced as a boundary condition for the mass balanceequation of the primary particles.The radial profile of monomer concentration in the

primary particles is given by the equations for the solidcore model:

where Dp ) effective diffusivity of monomer in theprimary particle (cm2 s-1),Meq ) equilibrium concentra-tion of monomer in the interface between the primaryand secondary particles (mol cm-3), Mp ) monomerconcentration in the primary particle (mol cm-3), Mp0) initial monomer concentration in the primary particle(mol cm-3), rc ) rate of polymerization on the surfaceof the catalyst fragment (mol cm-3 s-1), Rc ) radius ofthe catalyst fragment in the primary particle (cm), rp )radial position in the primary particle (cm), and Rp )radius of the primary particle (cm).The rate of polymerization on the surface of the

catalyst fragment is generally given by

where kp ) average rate constant for monomer propaga-tion (cm3 mol-1 s-1), C* ) concentration of active siteson the catalyst surface (mol cm-3), and Mc ) monomerconcentration on the catalyst surface (mol cm-3).

Figure 6. Schematic representation of the polymeric flow modeland of the multigrain model.

∂Ms

∂t) 1rs2∂

∂rs (Dsrs2 ∂Ms

∂rs ) - rv (190)

∂Ms

∂rs(rs)0,t) ) 0 (191)

Ds

∂Ms

∂rs(rs)Rs,t) ) ks(Mb - Ms) (192)

Ms(rs,t)0) ) Ms0 (193)

∂Mp

∂t) 1rp

2∂

∂rp (Dprp2 ∂Mp

∂rp ) (194)

4πRc2Dp

∂Mp

∂rp(rp)Rc,t) ) 4

3πRc

3rc (195)

Mp(rp)Rp,t) ) Meq e Ms (196)

Mp(rp,t)0) ) Mp0 (197)

rc ) kpC*Mc (198)


The effective diffusivity in the secondary particle canbe estimated as usually done for heterogeneous catalyticreactions (Froment and Bischoff, 1990), using the valueof monomer diffusivity in the reaction medium, Db:

where ε and τ are the porosity and tortuosity of thesecondary particle, respectively.For the primary particles, one has to correct the

diffusivity of monomer in amorphous polymer, Da,according to the equation:

where R and â are correction factors to account forpolymer crystallinity and chain immobilization of theamorphous fraction (Floyd et al., 1986b).Analogous equations can be used for the radial

temperature profiles. For the secondary particle:

where Cp ) heat capacity of the secondary particle (calmol-1 K-1), h ) film heat-transfer coefficient (cal cm-2

s-1 K-1), ∆Hp ) heat of polymerization (cal mol-1), ke) effective heat conductivity in the secondary particle(cal cm-1 s-1 K-1), Tb ) reactor temperature (K), Ts )temperature in the secondary particle (K), Ts0 ) initialtemperature in the secondary particle (K), and Fp )density of the polymer particle (mol cm-3).Similarly, for the primary particles the temperature

profile is given by

where Tp0 is the initial temperature in the primaryparticle.Equations for intraparticle monomer concentration

and temperature radial profiles for the polymeric flowand related models are similar to the ones for thesecondary particles of the multigrain model. The reac-tion term in eqs 190 and 201 are calculated using theaverage concentrations of monomer, Mh , and catalyticsites, Ch*, in the secondary particle:

Equations for the moments of the molecular weightdistribution and copolymer composition have then to be

solved simultaneously with the intraparticle mass- andheat-transfer equations for each radial position in orderto calculate the molecular weight and compositionaverages in the polymeric particle.The existence of yet another level of mass transfer is

suggested by Skomorokhov et al. (1989). In their model,microparticles (described by the solid core model) ag-glomerate to form subparticles. The polymer particle(macroparticle) results from the agglomeration of thesesubparticles. Growing polymer rapidly fills the poresof the subparticles, and therefore the monomer diffusionin the subparticle is in the order of magnitude ofdiffusion in pure polymer. This model evidently predictshigher diffusion resistances than that predicted by themultigrain model, since the effective diffusion in thelarger subparticles is of the same magnitude as that inthe microparticles. These transfer resistances can, insome polymerization conditions, decrease polymer yield.Unfortunately, no predictions of MWD or CCD werereported. Some experimental results on particle mor-phology seem to support this model (Bukatov et al.,1982; Skomorokhov et al., 1987, Kakugo et al., 1989).Some mathematical models combine the approaches

of the polymeric flow model and the multigrain model.In the polymeric multigrain model (Sarkar and Gupta,1991), catalyst fragments are assumed to be in acontinuum of polymer with only one level of diffusionalresistance for the monomer. This model can predicthigher values of polydispersity than the multigrainmodel.In the polymeric multilayer model (Soares and

Hamielec, 1995b), the polymeric particle is divided intoconcentric spherical layers as in the multigrain modelbut microparticles are not considered. Therefore, theintraparticle radial profiles of monomer concentrationand temperature for the multilayer model are the sameas those of the polymeric flow model; i.e., mass- andheat-transfer processes that take place in the primaryparticles are not explicitly considered in the model. Priorto polymerization all layers have the same concentrationof active sites as the whole catalyst particle; i.e., thereis no radial profile of active sites. This assumption issupported by some recent electron microscopy studiesof heterogeneous Ziegler-Natta catalysts (Noristi et al.,1994). Population balances for each active species arecalculated inside each layer. The volume of eachconcentric layer is updated using average monomerconcentrations and temperature after a given integra-tion interval. The population balances derived for eachlayer are the same as those for a model with no transferresistances. If transfer resistances are found to be oflittle importance, the same equations can still be usedwith the bulk monomer concentration and polymeriza-tion temperature.Additionally, the polymeric multilayer model esti-

mates the distributions of molecular weight and chemi-cal composition in each model layer and whole polymerusing a theoretically sound equation, Stockmayer’sbivariate distribution (Stockmayer, 1945), as shown inFigure 7. The former expansion models could onlyestimate averages of molecular weight and composition.The instantaneous chain length and composition

distribution proposed by Stockmayer and corrected fordiffering monomer molecular weights is given by

Ds ) Dbε/τ (199)

Dp ) Da/Râ (200)

FpCp

∂Ts

∂t) 1rs2∂

∂rs (kers2 ∂Ts

∂rs ) + (-∆Hp)rv (201)

∂Ts

∂rs(rs)0,t) ) 0 (202)

ke∂Ts

∂rs(ts)Rs,t) ) h(Tb - Ts) (203)

Ts(rs,t)0) ) Ts0 (204)

FpCp

∂Tp

∂t) 1rp

2∂

∂rp (kerp2 ∂Tp

∂rp ) (205)

-4πRc2ke

∂Tp

∂rp(rp)Rc,t) ) (-∆Hp)

43

πRc3rc (206)

Tp(rp)Rp,t) ) Ts (207)

Tp(rp,t)0) ) Tp0 (208)

rv(polymeric flow) ) kpCh*Mh (209)w(r,y) ) (1 + yδ)τ2r exp(-τr) dr 1

x2πâ/rexp(-y2r/2â) dy (210)


where

and Fh1 ) average mole fraction of monomer type 1 oncopolymer, M1, M2 ) molecular weights of monomertypes 1 and 2, respectively, r ) chain length, r1, r2 )reactivity ratios, and y ) deviation from averagecopolymer composition.For single-site catalysts, the principal conclusions of

the polymeric multilayer model confirm and comple-ment the conclusions obtained with the polymeric flowmodel and the multigrain model: (1) Mass-transferresistances can reduce the polymerization rate, decreasemolecular weight averages, and affect copolymer com-position for large, highly active catalysts. (2) Particu-larly important for supported catalysts, the concentra-tion of highly active sites can increase mass-transferresistances and have undesirable results in catalystperformance and product quality. (3) Mass-transferresistances may also be a source of copolymer composi-tion heterogeneity for highly active and large catalystparticles, if the comonomers have reactivities that differsignificantly. (4) Temperature gradients in the poly-meric particle are not expected to be a significant factorfor reactions carried out in slurry reactors.The conclusions obtained with single-site-type expan-

sion models are especially important for the technologyof supported metallocene catalysts, where single-site,highly active catalytic species may be subjected tosignificant mass- and heat-transfer resistances (Soaresand Hamielec, 1995a,b).Chemical Models. It is generally accepted that,

under most polymerization conditions, the effect ofmultiple-site types on polymer properties is far moreimportant than mass- and heat-transfer resistances(Ray, 1988; Soares and Hamielec, 1995b). Under theseconditions, each site type instantaneously produces

polymer that has Flory’s most probable MWD. There-fore, the instantaneous MWD of accumulated polymermade with heterogeneous Ziegler-Natta catalysts canbe considered an average of that produced by theindividual site types, weighted by the weight fractionof polymer produced by each site type, as shown inFigure 8 (Soares and Hamielec, 1995c).The same treatment can be extended to copolymers

by using Stockmayer’s bivariate distribution to describethe instantaneous bivariate distribution of chain lengthand composition. Therefore, if one assumes that eachsite type produces copolymer that obeys distinct bivari-ate distributions of chain length and copolymer compo-sition, the bivariate distribution of the accumulatedcopolymer can be considered an average of that pro-duced by the individual site types, as shown in Figure9 (Soares and Hamielec, 1995e).If these hypotheses are valid, broad MWDs and CCDs

result from the superposition of narrower distributionsproduced on each site type. Additionally, from theknowledge of the global MWDs and CCDs, one can use

Figure 7. Prediction of the bivariate distribution of chemicalcomposition and chain length with the polymeric multilayer model.r ) chain length;W(r) dr ) chain-length distribution; y ) deviationfrom copolymer average composition; W(y) dy ) compositiondistribution; s1, s2, s3 site types; l1, l2, ..., l9 ) model layers; l1 isthe innermost layer; l9 is the outermost layer.

â ) Fh1(1 - Fh1)K (211)

K ) [1 + 4Fh1(1 - Fh1)(r1r2 - 1)]0.5 (212)

δ )1 - M2/M1

M2/M1 + Fh1(1 - M2/M1)(213)

Figure 8. Instantaneous chain-length distribution (CLD) of apolyolefin made with a multiple-site-type catalyst as a superposi-tion of four individual Flory’s most probable weight chain-lengthdistributions (solid line indicates CLD of whole polymer, anddotted lines represent CLD of polymer made on distinct active sitetypes).

Figure 9. Instantaneous chemical composition distribution (CCD)of a binary olefin copolymer made with a multiple-site-typecatalyst as a superposition of five individual Stockmayer’s CCDsconsidering all chain lengths (solid line indicates CCD of wholepolymer, and dotted lines represent CCDs of polymer made ondistinct active sites).


mathematical models to infer the number of active sitetypes present on the catalyst (Vickroy et al., 1993;Soares and Hamielec, 1995c,e). Multiple-site-type mod-els without heat- and mass-transfer resistances haveachieved considerable success in simulating polymeriza-tions using heterogeneous catalysts in slurry reactors(Rincon-Rubio et al., 1990; Lee et al., 1994), series ofslurry reactors (de Carvalho et al., 1989, 1990; Soares,1994), gas-phase reactors (McAuley et al., 1990; Xie etal., 1994), and high-pressure-high-temperature tubularreactors (Lorenzini et al., 1991).Hybrid Models. Hybrid models provide the most

complete description of polymerization with heteroge-neous Ziegler-Natta catalysts since they account formultiple-site types and transfer resistances simulta-neously. Although multiple-site types are probably themain explanation for the characteristic behavior ofheterogeneous Ziegler-Natta catalysts, transfer resis-tances may have a secondary effect on these propertiesin certain polymerization conditions (Ray, 1988; Soaresand Hamielec, 1995b).Several of the expansion models described above have

been modified to include the presence of more than onetype of active site: the polymeric flow model (Galvanand Tirrell, 1986b), the multigrain model (Ray, 1988),the polymeric multilayer model (Soares and Hamielec,1995b), and the polymeric multigrain model (Sau andGupta, 1993). All these models indicate that the pres-ence of more than one type of active site is mainlyresponsible for the broad MWDs and CCDs observed inpolymer produced with heterogeneous Ziegler-Nattacatalysts. In general, mass- and heat-transfer resis-tances are regarded as secondary effects. However,under certain polymerization conditions with large,highly active catalyst particles, mass-transfer resis-tances can have an important role in further broadeningthe MWD and CCD of polymers made with heteroge-neous Ziegler-Natta catalysts.The polymeric multilayer model uses Stockmayer’s

bivariate distribution to predict MWDs and CCDs ofpolymer chains per site type in each concentric layer ofthe model. These distributions are averaged over allparticles to obtain the MWD and CCD of the wholepolymer. Figure 10 shows the CCDs per site type of anethylene-propylene copolymer made with a three-site-type catalyst averaged over all model layers. Figure 11shows the CCDs of the same copolymer per layeraveraged over all site types. Finally, Figure 12 shows

the MWDs per site type for the same polymer averagedover all model layers (Soares and Hamielec, 1995b).Among the expansion models, only the polymeric mul-tilayer model can predict complete distributions ofmolecular weight and copolymer chemical composition.3.1.2. Homogeneous Catalysts. Soluble Ziegler-

Natta catalysts, including metallocene catalysts, canmake polymer with narrow MWD and CCD (Kaminsky,1986a, 1991). This behavior supports the multiple-site-type hypothesis for heterogeneous catalysts since thesecatalysts consist of reasonably well-defined, singlecatalytic species and should produce polymer that hasnarrow MWD and CCD.Some metallocenes, however, can synthesize polymer

with broad MWD and CCD. This behavior has beenlinked to the presence of different site types (Kaminsky,1986b). In fact, a combination of different types ofmetallocene catalysts can be used to produce polymerwith broad MWD and CCD (Welborn, 1987, 1991, 1992,1993), which is in good agreement with the chemicalmodels presented above.In most processes using homogeneous Ziegler-Natta

catalysts, in general, and metallocenes, in particular,the polymer is not soluble in the reaction medium andprecipitates after a critical chain length is achieved. Ifafter chain termination the active site returns to the

Figure 10. CCDs for each site type over all model layers forpropylene-ethylene copolymerization (Soares and Hamielec, 1995b).

Figure 11. CCDs considering all site types in different polymerlayers for propylene-ethylene copolymerization. Layer 1 is theinnermost layer, and layer 9 is the outermost layer (Soares andHamielec, 1995b).

Figure 12. Chain-length distribution for each site type over alllayers for propylene-ethylene copolymerization (Soares andHamielec, 1995b).


solution, then interparticle mass and heat transfershould not influence the polymerization. In this case,chemical models would be adequate to describe thepolymerization. However, if the active sites are trappedinside polymer particles, interparticle mass- and heat-transfer resistances could become significant. A math-ematical model for particle growth during polymeriza-tions catalyzed with metallocenes has been proposedrecently (Herrmann and Bohm, 1991). It is possiblethat, as the polymer particles grow by agglomeration,some active sites will remain inside the polymer par-ticles. This is especially more likely in processes wherethe active sites are exposed to high concentrations ofmonomer, as in liquid monomer polymerizations. Inthis case, due to the high activity of metallocenecatalysts, transfer resistances may play a role inbroadening MWD and CCD.Very few mathematical models have been proposed

to describe polymerization using homogeneous Ziegler-Natta catalysts. Soares and Hamielec (1995b) proposedthat the polymeric multilayer model could be used tosimulate polymerization with homogeneous Ziegler-Natta catalysts if one neglects mass- and heat-transferresistances in the polymeric particle. This would berigorously valid only if active sites are not trapped insidepolymer particles during polymerization. Vela-Estradaand Hamielec (1994) were the first to model the polym-erization of ethylene with bis(cyclopentadienyl)zirco-nium dichloride and methylaluminoxane. Experimentalpolymerization rates and MWD results could be wellrepresented by a two-site-type model. This was the firststudy to use size-exclusion chromatography to establishthe number of active site types quantitatively forpolymers synthesized with homogeneous Ziegler-Nattacatalysts. In fact, a bimodal distribution gave a poly-dispersity index of 2 for each mode. This result is inexcellent agreement with the model proposed in Figure8.Supported metallocenes have also been studied ex-

tensively. Among the several incentives for supportingmetallocenes are the use of already existing polymeri-zation reactors for heterogeneous catalysts, the enhance-ment of stereochemical and regiochemical control, de-crease in the methylaluminoxane/catalyst ratio, andproduction of polymer with better particle properties.The models described before for heterogeneous Ziegler-Natta catalysts can also be applied to this class ofcatalysts (Soares and Hamielec, 1995b). In this case,in the presence of only one active site type on thecatalyst, transfer resistances and statistical broadeningwould be the only factors causing broadening of MWDand CCD.Hoel et al. (1994) developed a simplified flow model,

with no MWD prediction capabilities, and used it tointerpret their experimental data on ethylene-pro-pylene polymerization using two silica-supported met-allocene catalysts in a liquid-propylene slurry reactor.Both catalysts produced polymer with narrow MWD ata constant rate of polymerization. However, one cata-lyst produced polymer with narrow CCD and the otherproduced polymer with broad CCD and decreasingamount of ethylene in the copolymer as a function ofpolymerization time (unfortunately, the catalyst typeswere not specified). The radial copolymer compositionwas measured by FTIR-microscope analysis for micro-tomed 250-750 µm particles, and it was found to bericher in propylene in the center than in the exteriorlayers of the particle, which is also in good agreement

with the multilayer model predictions for single-site-type catalyst (Soares and Hamielec, 1995b), as shownin Figure 13. Although their model could qualitativelypredict the radial profile of chemical composition, itcould not account for the decrease in ethylene contentduring the polymerization. In reality, their modelpredicted an increase in ethylene content with polym-erization time, and so would any other physical modelpublished in the literature, since the concentration ofactive sites in the polymeric particle decreases duringthe polymerization, therefore decreasing the effect ofintraparticle mass-transfer resistances. The observeddecrease in ethylene content in the copolymer wastentatively attributed to variations in monomer diffu-sivity and particle porosity during the polymerization.Recently, Bonini et al. (1995) applied a modified form

of the multigrain model to the polymerization of pro-pylene with silica-supported Me2Si(Ind)2ZrCl2/MAO.Their model takes into account the mechanism ofcatalyst fragmentation by the growing polymer chainsto explain the observed induction period when thesupported metallocene is used. In their model, thecatalyst particle is fragmented by the growing polymerin successive concentric shells, from the external surfaceto the center of the particle, as proposed by Ferrero andChiovetta (1991). Their model agreed well with experi-mental rates of polymerization, including the inductionperiod and number-average molecular weight, but couldnot predict the relatively broad MWD (polydispersitiesfrom 2.4 to 3.8). The simulated polydispersities werealways less than 2.1, surprisingly low values.3.1.3. Modeling of Long-Chain Branching. Con-

strained geometry metallocene catalysts can be used tosynthesize polyolefins with narrow molecular weightdistributions and significant degrees of long-chainbranching. These polymers have excellent mechanicalproperties combined with good processability (goodshear thinning, delayed melt fracture, and improvedmelt strength) (Hamielec and Soares, 1996).The most suitable catalyst types for long-chain branch

formation appear to be those with an “open” metal activecenter, such as the Dow Chemical constrained geometrycatalysts. The active center of these catalysts is basedon group IV transition metals that are covalentlybonded to a monocyclopentadienyl ring and bridged with

Figure 13. CCDs in different polymer layers for propylene-ethylene copolymerization made on a single-site-type catalyst.Layer 1 is the innermost layer, and layer 10 is the outermost layer(Soares and Hamielec, 1995b).


a heteroatom, forming a constrained cyclic structurewith the transition-metal center. Strong Lewis acidsare used to activate the catalyst to a highly effectivecationic form. This geometry allows the titanium centerto be more open to the addition of ethylene and higherR-olefins but also for the addition of vinyl-terminatedpolymer molecules (Lai et al., 1993a,b).The most likely long-chain branch formation mecha-

nism with metallocene catalyst systems is terminalbranching, a mechanism which has been known in thefree-radical polymerization literature for many years.In free-radical polymerization, macromonomers aregenerated via termination by disproportionation and viachain transfer to monomer. With metallocene catalystsystems, the facile â-hydride elimination reaction ap-pears to be responsible for in-situ macromonomerformation. Other reaction types, such as â-methylelimination and trans, may also generate dead polymerchains with terminal unsaturation (Resconi et al., 1992).â-Methyl elimination can actually be the most importanttransfer mechanism in propylene polymerization when(Me5Cp)2Ti, (Me5Cp)2Zr, and (Me5Cp)2Hf complexes areused (Huang and Rempel, 1995). Therefore, thesecatalytic systems have the potential of producing poly-propylene with long-chain branches.The chain-length distribution of polymer chains pro-

duced with metallocene catalysts that permit long-chainbranch formation via the terminal double-bond mech-anism can be described analytically for each populationcontaining a different number of long-chain branchesper polymer molecule (Soares and Hamielec, 1996c). Thepolymerization kinetic model involves the followingsteps:

where Pr,i is a living polymer molecule of chain lengthr containing i long-chain branches, Dq,j

d is a deadpolymer molecule of chain length q containing j long-chain branches and having terminal vinyl unsaturation,Dq,j is a dead polymer molecule of chain length qcontaining j long-chain branches and a saturated chainend, M is the monomer, CTA is a chain-transfer agent,kp is the propagation rate constant for monomer, kpLCBis the propagation rate constant for dead polymer withterminal vinyl unsaturation, kCTA is the rate constantfor transfer to chain-transfer agent, and kâ is the rateconstant for â-hydride elimination. Chain initiation isassumed to be instantaneous.Dead polymer chains having terminal vinyl unsat-

uration, Dq,jd , can coordinate to the catalytic active site

and insert in the growing chain, forming a trifunctionallong-chain branch. Dead polymer chains with saturatedchain ends, Dq,j, cannot polymerize again. Observe thatby examining the mechanism of chain formation, onecan conclude that the instantaneous molecular weightdistributions of live polymer, dead polymer with vinylchain-end unsaturation, and dead polymer with satu-rated chain ends will be the same. However, therelative amount of dead polymer with and without

terminal vinyl unsaturation is proportional to the ratesof â-hydride elimination (producing dead polymer chainswith terminal vinyl unsaturation) and transfer to chain-transfer agent (commonly to hydrogen, producing deadpolymer chains with saturated chain ends).The frequency distribution of chain length for polymer

populations with n long-chain branches per chain isgiven by (Soares and Hamielec, 1996c):

where r represents chain length, n represents thenumber of long-chain branches per polymer molecule,and τ is given by

where Râ is the rate of â-hydride elimination, Rp is therate of monomer propagation, RCTA is the rate oftransfer to chain-transfer agent, and RLCB is the rateof macromonomer propagation or long-chain branchformation.The chain-length averages of the polymer populations

with different numbers of long-chain branches per chainare related by the simple relationships:

where i indicates the number of long-chain branches perchain, and rjn,i, rjw,i, rjz,i, and pdii are the number, weight,and z-average chain lengths and polydispersity, respec-tively.Parts a and b of Figure 14 show the predicted weight

chain-length distributions for a polyolefin produced ina CSTR for a given value of τ. Figure 15 shows theeffect of varying the rate of long-chain branch incorpo-ration in the chain-length distribution. The chain-length distribution becomes broader as the long-chainbranching level increases, but the distribution is stillvery narrow as compared with the ones for polyolefinsmade with heterogeneous Ziegler-Natta catalysts.The analytical solution for the chain-length distribu-

tion of homopolymer with long-chain branches (eq 218)is also valid for copolymerization when appropriatepseudo-kinetic rate constants are used to calculate τ.Additionally, for the case of binary copolymerization,Stockmayer’s distribution can be used to obtain thechemical composition distribution of the copolymer. Thisis possible because the branched chains are formed bylinear copolymer chains which follow Stockmayer’sbivariate distribution (Soares and Hamielec, 1996c).

4. Determination of Binary Reactivity Ratios

The determination of reactivity ratios (terminal model)with small confidence intervals requires sensitive ana-lytical techniques, carefully planned experiments, andthe use of statistically valid methods of estimation.Unfortunately, most of the binary reactivity ratio datapublished to date were not found using statistically valid

Pr,i + M98kpPr+1,i (214)

Pr,i + Dq,jd 98

kpLCBPr+q,i+j (215)

Pr,i + CTA98kCTA

Dr,i + P0,0 (216)

Pr,i 98kâDr,i

d + P0,0 (217)

f(r,n) ) 1(2n)!

r2nτ2n+1 exp(-τr) (218)

τ )Râ

Rp+RCTA

Rp+RLCB

Rp(219)

rjn,i ) (1 + 2i)rjn,0 (220)

rjw,i ) (1 + i)rjw,0 (221)

rjz,i ) (1 + 2i/3)rjz,0 (222)

pdii ) ( 1 + i1 + 2i)pdi0 (223)


methods and confidence intervals were usually notgiven. The reactivity ratios are highly correlated withmajor disagreement in values measured in differentlaboratories (O’Driscoll and Reilly, 1987; Dube et al.,1991a; Burke et al., 1993; Rossignoli and Duever, 1995).4.1. Statistically Valid Methods for Estimation

of Reactivity Ratios. Traditional methods for esti-mating reactivity ratios (Mayo and Lewis, 1944; Fine-man and Ross, 1950; Braun et al., 1973; Kelen andTudos, 1975) are based on first transforming the in-

stantaneous copolymer composition equation into alinear form in the parameters r1 and r2, for example

and then estimating the reactivity ratios by graphicalplotting or by linear least squares. However, theseapproaches, aside from requiring that the instantaneouscopolymer composition equation be valid, are statisti-cally unsound because the “independent variable”

a

b

Figure 14. Weight chain-length distribution for polymer populations containing different numbers of long-chain branches per polymerchain: (a) weight chain-length distribution normalized with respect to the weight of individual populations; (b) weight chain-lengthdistribution normalized with respect to the total weight of the whole polymer. The global distribution (overall polymer populations) isindicated by the bold line. The table in the top right corner indicates number and weight fractions of each population.

(f1/f2)(F2/F1 - 1) ) -r2 + r1(f1/f2)2(F2/F1) (224)


(f1/f2)2(F2/F1) has error, and the dependent variable(f1/f2)(F2/F1 - 1) does not have constant variance. Bothof the latter assumptions are necessary for linear leastsquares to be a statistically valid estimation method.As a result, they have been shown often to lead to verypoor estimates with misleading confidence intervals(Tidwell and Mortimer, 1965, 1970). The common useof these statistically invalid estimation procedures inthe past is one of the reasons for the wide variation inreactivity ratios reported by different researchers.It should be mentioned here that the Kelen and Tudos

(1975) method, although statistically invalid, can beused to at least obtain “good initial” r1 and r2 estimates,provided the experiments have been suitably designed.More details are given in McFarlane et al. (1980) andLaurier et al. (1985).Reactivity ratios are nowadays most usually esti-

mated using procedures based on the statistically validerror-in-variables model (EVM) or on its modifications(Box, 1970; Britt and Luecke, 1973; Sutton and MacGre-gor, 1977; Garcia-Rubio et al., 1985; Yamada et al., 1978;Van der Meer et al., 1978; Patino-Leal et al., 1980;Patino-Leal and Reilly, 1981; Dube et al., 1991a; Burkeet al., 1993; Rossignoli and Duever, 1995). Extensionof these EVM methods to the estimation of reactivityratios in terpolymerizations has been discussed inDuever et al. (1983). These methods allow one to takeproperly into account all the sources of experimentalerror. No groups of variables are considered to be“independent” and free of error or “dependent” with“constant” error. All measured variables (e.g., [Mj]m, j) 1,2 comonomers) are considered on an equal basis andrepresented as coming from some true (but unknown)value [Mj]t which is contaminated with measurementerror, that is

The estimates obtained by these EVM approaches aresuperior to those obtained by arbitrary graphical orarbitrary least-squares procedures.Although EVM procedures require more information

than arbitrary least-squares procedures, namely, themeasurement error variances and covariances, the pointestimates of the reactivity ratios obtained by theseprocedures have been shown to be very insensitive tochanges in these covariance values. The great improve-ment provided by the EVM estimation methods comesfrom simply accounting correctly for the major sourcesof error.4.2. Design of Experiments. For any experimental

investigation, an optimal experimental design is of greatimportance since it enables one to perform the minimumnumber of experiments and obtain the most preciseparameter estimates. In copolymerizations, such adesign is that using the Tidwell-Mortimer (1965)criterion. The criterion employs the Mayo-Lewis model(e.g., see O’Driscoll and Reilly (1987) or Hamielec et al.(1989)) and is based on the sensitivity of the reactivityratio estimates to the errors encountered in the deter-mination of copolymer composition. Tidwell and Mor-timer (1965) recommended running several replicatesat two different monomer feed compositions, f ′1o and f ′′1o,defined by the following equations:

It is evident from eqs 226 and 227 that preliminaryestimates of the two reactivity ratios, r1 and r2, areneeded. These estimates can be obtained from theliterature (e.g., Polymer Handbook, 1989), a set of

Figure 15. Effect of varying macromonomer addition rate on weight chain-length distribution.

[Mj]im ) [Mj]i

t + εi (225)

f ′1o ) 22 + r1

(226)

f ′′1o )r2

2 + r2(227)


preliminary experiments, a calculated guess using theQ-E scheme, etc. (O’Driscoll and Reilly, 1987). Inaddition, the copolymerizations should be carried outto a conversion low enough that composition drift (i.e.,a change in the monomer feed composition, f10) isnegligible (e.g., conversion <5-8%).

5. Alternative Copolymerization Kinetic Models

Deviations from the terminal model have been notedfor various comonomer pairs and for various polymer-ization systems including free-radical, anionic, andheterogeneous Ziegler-Natta. In order to explain somedeviations from the terminal model in free-radicalpolymerization, several alternative models have beenproposed to describe copolymerization kinetics (Har-wood, 1987). A model first formulated by Merz et al.(1946), the penultimate unit effect model has receivedmuch attention over the last decade. This modeldescribes how the reactivity of a radical center maydepend on the type of monomer unit bound in the chainend adjacent to the radical center. Much of the recentinterest in the penultimate model is due to the work ofFukuda et al. (1985, 1987, 1989, 1991). Using thepseudo-kinetic rate constant modeling approach de-scribed earlier, extension of these alternative models tomulticomponent systems is straightforward.With the existence of several alternatives to describe

the polymerization mechanism, it is then left to distin-guish between the models to find which gives the “best”description of experimental data. Early model discrimi-nation attempts were geared toward fitting models tomeasurements of copolymer composition and diad andtriad fractions (Hill et al., 1982, 1984, 1989; Brown etal., 1993). Some researchers have proposed that co-polymer composition data can usually be adequatelyrepresented by terminal model kinetics since differencesbetween model predictions are typically small (Fukudaet al., 1987). On the other hand, results by Burke et al.(1994a) have indicated that, with the use of statisticalexperimental design techniques, these data may bemore useful in discriminating between copolymer mod-els than previously thought. However, sequence dis-tribution and copolymer composition data do not provideenough information to estimate all of the parameterscontained in some models (e.g., the penultimate model).Thus, in recent attempts to discriminate between com-peting copolymerization models, polymerization ratedata were used to gain additional information on modeladequacy (Fukuda et al., 1985; Davis et al., 1989).Typical attempts at identifying the best model haveinvolved spreading experiments over the entire rangeof feed compositions. This may not be optimal for eithermodel discrimination or parameter estimation (Burkeet al., 1994a). Rather, the use of statistical modeldiscrimination methods which include the design ofexperiments and the measurement of copolymer com-position, triad fractions, and copolymerization rate ispreferred (Burke et al., 1994a,b, 1995, 1996). Theresults of Burke et al., which were verified experimen-tally, showed that the use of model discriminationtechniques should improve our ability to discriminatebetween proposed copolymerization kinetic models.

6. Semibatch Monomer Feed Policies

In the absence of an azeotrope and when one mono-mer is more reactive than the other in a binary, batchcopolymerization (say r1 > 1 and r2 < 1), the instanta-

neous copolymer composition will decrease in monomer1 with an increase in monomer conversion. The extentof compositional drift which leads to a copolymerheterogeneous in composition depends on the ratio ofreactivity ratios r1/r2 (increase with an increase inr1/r2), the initial monomer composition (f10), and themonomer conversion (x). A copolymer which is hetero-geneous in composition usually has inferior properties,and therefore industrial semibatch processes have beendeveloped to reduce composition hetereogeneity. Thereare, however, certain semibatch emulsion polymeriza-tion processes where heterogeneous copolymers areproduced in the latex particles to achieve certainproperty improvements (Bassett and Hamielec, 1981).There are two basic monomer feed policies, originally

shown in Hamielec et al. (1987), which may be used insemibatch copolymerization to minimize compositionaldrift. Effective commercial processes are usually basedon one or a combination of these feed policies. Someadditional promising derivations of these policies havebeen presented (Arzamendi and Asua, 1989, 1990, 1991;Arzamendi et al., 1991, 1992; van Doremaele et al., 1992;Leiza et al., 1993; Schoonbrood et al., 1993, 1996;Canegallo et al., 1994; Canu et al., 1994; Urretabizkaiaet al., 1994a; de la Cal et al., 1995; Echevarria et al.,1995). These feed policies (Policy I and Policy II) aredescribed below using binary copolymerization as anexample.Policy I. All of the slower monomer and sufficient

enough of the faster monomer (to give the desiredcomposition F1) are added to the reactor at time zero.Thereafter, the faster monomer is fed to the reactor witha time-varying feed rate to maintain N1/N2 (the ratio ofthe number of moles of monomer 1 to that of monomer2 in the reactor) and F1 constant with time or grams ofcopolymer produced.Policy II. A heel of monomers 1 and 2 at the desired

concentration levels (to give the desired F1) is added tothe reactor at time zero. Thereafter, monomers 1 and2 are fed to the reactor with time-varying feed rates tomaintain [M1], [M2] and F1 constant with time or gramsof copolymer produced.The equations to be solved to determine the required

feed rates to produce a homogeneous binary copolymerin a semibatch process are given by eqs 228-233.

dN1

dt) -N1(k11φ1 + k21φ2)[P

•] + F1,in (228)

dN2

dt) -N2(k12φ1 + k22φ2)[P

•] + F2,in (229)

dVdt

)F1,inMW1

Fm1+F2,inMW2

Fm2- [Rp1MW1( 1

Fm1- 1

Fp) +

Rp2MW2( 1Fm2

- 1Fp)]V (230)

Initial Conditions

t ) 0, N1 ) N1o, N2 ) N2o, V ) Vo (231)

Conditions

Policy I:

F2,in ) 0 andd(N1/N2)

dt) 0 (232)


Policy II:

d[M1]dt

)d[M2]dt

) 0 (233)

Since f1 and F1 are constant, φ1 is also constant.Given the time variation of the total polymer radical

concentration, [P•], one can readily solve for F1,in andF2,in, the time-varying monomer feed rates, using theabove equations.We will now investigate Policies I and II with respect

to cross-linking using the criterion of onset of gelationalready discussed. Monomer 1 is the cross-linker (di-vinyl monomer) and a chain-transfer agent (CTA) isused to delay the onset of gelation.6.1. Policy I. For constant [P•] and F2,in ) 0, eq 229

may be solved analytically for N2 to give

where

For constant N2/N1, N1 is given by

One may now solve eq 228 for F1,in to get

Assuming that the densities of monomers and copolymerare the same (Fm1 ) Fm2 ) Fp), one can integrate eq 230analytically to get

The total monomer concentration in the reactor istherefore given by

where

We will now investigate gelation for a binary copo-lymerization of vinyl and divinyl monomers. The divin-yl monomer is considered the more reactive monomer,and therefore, with Policy I, it is fed into the reactorover time. Low levels of the cross-linking monomer areto be used and therefore to simplify the analysis, onecan assume that the volume of the reacting mixture, V,is constant and equal to the initial volume, V0. We nowapply eq 161 for constant V, â ) Cp1 ) Cp2 ) 0 and τ )kfCTA[CTA]/(kp[M]) with Policy I to give

with Qh 2 at t ) 0 where

and

Q2 goes to infinity at the gelation point t ) tc and tc isgiven approximately by

We employ the following set of kinetic parameters: kfCTA) 102 L mol-1 s-1, [CTA] ) 0.1 mol L-1, a constant k*p*) 102 L mol-1 s-1, F1 ) 0.01, kp ) 103 L mol-1 s-1, Y0 )10-7 mol L-1, [M]0 ) 10 mol L-1, and tc ) 5000 s (1.4h).Employing eqs 159, 160, and 176 for this example of

Policy I, it can readily be shown that, at the gelationpoint (t ) tc), BhN4 = 1/3. In other words, at the gelationpoint one polymer molecule in three on the average hasa tetrafunctional long-chain branch point.6.2. Policy II. To simplify the analysis, we assume

that the densities of monomers and copolymer are thesame and that the molecular weight of the monomersare the same, in addition to constant [P•]. Equation 229may be solved analytically to give the following:

where

Since the ratio F1,in/F2,in is independent of time, the twomonomers may be premixed and fed to the reactor inthe same stream.

We will now investigate gelation for a binary copo-lymerization of a vinyl and divinyl monomer. Tosimplify the analysis, it is assumed that the concentra-tion of chain-transfer agent, [CTA], is maintainedconstant. We now apply eq 161 under the followingconditions: â ) Cp1 ) Cp2 ) 0 and τ ) kfCTA[CTA]/(kp[M]) constant with time in the context of Policy II.

with Qh 2 ) 0 and t ) 0. The definitions of Qh 2 and ψ arethe same as those used with eq 241. Equation 251 canbe solved numerically (setting ψ ) 4.54 × 10-4 s-1 andR ) 1.515 × 10-4 s-1) to give

N2 ) N2oe-φ2t (234)

φ2 ) (k12φ1 + k22φ2)[P•] (235)

N1 ) N1oe-φ2t (236)

F1,in ) N1o(φ1 - φ2)e-φ2t (237)

V ) Vo + (N1oMW1

Fm1 (φ1 - φ2

φ2))(1 - e-φ2t) (238)

[M] )N1 + N2

V)

(N1o + N2o)e-φ2t

Vo + R(1 - e-φ2t)(239)

R ) N1oMW1

φ1 - φ2

φ2/Fm1(240)

dQh 2/dt ) (1 + Qh 2)2e-φ2t (241)

Qh 2 )k*p*Q2

kp[M](242)

ψ )2F1k*p*kp[M]0Yo

kfCTA[CTA](243)

tc ) 1/ψ )kfCTA[CTA]

2F1k*p*kp[M]0Yo(244)

N1 ) N1oeRt (245)

N2 ) N2oeRt (246)

F1,in ) (R + φ1)N1oeRt (247)

F2,in ) (R + φ2)N2oeRt (248)

R )f1oφ1 + f2oφ2

VoFm(MW(N1o + N2o) - 1) (249)

V ) VoeRt (250)

dQ2/dt ) ψ(1 + Qh 2)2 - RQh 2 (251)

t (s) 0 100 200 300 400 500 700 900Qh 2 0 0.0472 0.0983 0.154 0.215 0.282 0.438 0.636

t (s) 1100 1300 1500 1700 1900 2100 2150 2200Qh 2 0.894 1.249 1.769 2.61 4.22 8.57 11.13 15.60

t (s) 2250 2300 2310 2320 2330Qh 2 25.4 64.0 91.1 157.0 556.0


Neglecting RQh 2 in eq 251, one gets an analytical solutionfor Qh 2

and

Using eq 253, one finds tc ) 2203 s. RQh 2 in eq 251accounts for the volume increase due to monomer fedand has the effect of delaying the onset of gelation. Thefollowing set of kinetic parameters has been used tocalculate ψ and R and will be used to calculate thenumber- and weight-average molecular weights andbranching frequency (BhN4) with Policy II.

In the context of Policy II, one can solve eqs 159, 160,and 176 analytically to give

Employing the above kinetic parameters in eqs 254, 255,and 256 and eqs 155 and 156, one can calculate rjN, andrjW, and BhN4 versus time of polymerization to give

At this point of gelation, t ) tc ) 2350 s, rjN ) 1513, rjW) ∞, and BhN4 ) 0.333.6.3. Beyond the Point of GelationsGel Growth

and Structure. Beyond this critical point, the calcula-tion of Q2 and rjW is no longer possible as they are both

infinite; however, one can continue to calculate rjN andBhN4 beyond the point of gelation. Experimental dataclearly show that once gel is formed, it grows veryrapidly at the expense of the sol. The gel acts like agreat sponge, sucking in the polymer from the sol.Radical centers located on polymer chains in the gel arelonger lived because of their greatly reduced mobility.These radical centers are terminated by reaction diffu-sion (these radical centers move in the gel by the act ofmonomer addition or propagation). The greatly reducedtermination rates of radical centers located on chainsin the gel result in a very high concentration of theseradical centers, and this in part is the cause of the rapidgrowth of the gel at the expense of the sol. One canidentify the following reactions as responsible for thegrowth of the gel.1. k*p*Y1,b[Rs

•] consumption rate of polymer radicalsin the sol by addition to double bonds bound in polymerchains in the gel (Y1,b equals the concentration ofmonomer bound in the gel and [Rs

•] is the concentrationat radical centers on the gel).2. kp[M][Rb

•] consumption rate of monomer by radicalcenters located on polymer chains in the gel.3. k*p*Q1,s[Rb

•] addition rate of polymer chains fromthe sol to radical centers located on the chains in thegel (Q1,s is the concentration of monomer bound in thepolymer chains in the sol).4. ktsb[Rs

•][Rb•] termination rate of polymer radicals

in the sol with those on the gel.Termination of radical centers on the gel may result

from the following reactions in addition to number 4above:ktrd[Rb

•]2 termination rate of radical centers on chainsin the gel by reaction diffusion (this reaction typeproduces intramolecular cross-links and leads to adenser gel with lower absorbency and higher gelstrength).As soon as gel is formed, it grows rapidly at the

expense of the sol with the above reactions playing amajor role in its rapid growth.

7. Some Recent Case Studies

7.1. Introduction. In the previous review byHamielec et al. (1987), an evaluation of the free-volumetheory for the modeling of diffusion-controlled termina-tion and propagation in binary copolymerization as wellas the pseudo-kinetic rate constant method was pre-sented. Several case studies involving the implementa-tion of these ideas are summarized in Table 1. Earlyapplication of the pseudo-kinetic rate constant methodwas made for the batch, semibatch, and continuousemulsion polymerization of styrene/butadiene rubber byBroadhead et al. (1985). Four other comonomer systems(styrene/acrylonitrile, styrene/p-methylstyrene, p-me-thylstyrene/acrylonitrile, and p-methylstyrene/methylmethacrylate), modeled using the free-volume theory toaccount for diffusion-controlled termination and propa-gation in batch reactors (Garcia-Rubio et al., 1985;Bhattacharya and Hamielec, 1986; Jones et al., 1986;Yaraskavitch et al., 1987), were discussed in detail inHamielec et al. (1987). Since then, Storti et al. (1988,1989) employed the pseudo-kinetic rate constant methodto model the emulsion copolymerization of styrene/acrylonitrile and acrylonitrile/methyl methacrylate aswell as the emulsion terpolymerization of styrene/acrylonitrile/methyl methacrylate. Xie and Hamielec(1993c) modeled solution polymerizations with added

Qh 2 ) ψt1 - ψt

(252)

tc ) 1/ψ (253)

r1 ) 2, r2 ) 0.5(monomer 1 is the divinyl monomer), F1 ) 0.2

k11 ) k22 ) 2 × 103 L mol-1 s-1

φ1• ) 0.333, f1 ) 0.1111, φ1 ) 3.33 × 10-4,

φ2 ) 1.667 × 10-4 s-1

Fm ) Fm1 ) Fm2 ) Fp ) 1000 g L-1

MW ) MW1 ) MW2 ) 100

Vo ) 1 L, N1o ) 0.5 mol, N2o ) 4 mol

[M1] ) 0.5 mol L-1, [M2] ) 4 mol L-1

kp ) 2.52 × 103 L mol-1 s-1, k*p* ) 10 L mol-1 s-1

kfCTA ) 102 L mol-1 s-1, [CTA] ) 0.1 mol L-1,

Yo ) 10-7 mol L-1

R ) 1.515 × 10-4 s-1, ψ ) 4.54 × 10-4 s-1

VQ0 ) (F1k*p*[M]Yo2Vo/R)[t - (1/R)(eRt - 1)] +

(1/R)(kfCTA[CTA]YoVo)(eRt - 1) (254)

VQ1 ) (kp[M]YoVo/R)(eRt - 1) (255)

BhN4 )(F1k*p*[M]Yo

2Vo/R)[(1/R)(eRt - 1) - t]

VQ0(256)

t (s) 0 100 500 1000 2000 2500rjN 1134 1147 1201 1274 1445 1544rjW 2270 2380 2930 4070 16 650 ∞BhN4 0 0.0113 0.0592 0.1242 0.274 0.362


cross-linker (styrene/ethylene dimethacrylate, styrene/divinylbenzene) using similar techniques.Further applications of the free-volume theory were

demonstrated by Xie et al. (1991) for the suspensionpolymerization of vinyl chloride and by Penlidis et al.(1992) for the solution polymerization of MMA. Thelatter showed how to simplify the general model (similarto the one described herein) for a particular case. Ofmost significant note, however, is the work by Gao andPenlidis (1996) which demonstrated the general ap-plicability of the modeling techniques presented herein.They modeled bulk and solution polymerizations ofseveral monomers as well as some suspension polym-erization cases (see Table 1). The most convincingaspects of that work were the good model predictionsof data from a large number of research groups andwidely varying reaction conditions.7.2. Styrene/Ethyl Acrylate Model Prediction

Case Study. An experimental study of the bulk free-radical copolymerization of styrene (Sty)/ethyl acrylate(EA) was carried out by McManus and Penlidis (1996).This thorough kinetic study included conversion, com-position, and molecular weight data for reactions at avariety of feed compositions, initiator concentrations,and temperatures, with and without added chain-transfer agent. The model described herein was appliedto these data, and some representative results areshown below.In Figures 16 and 17 are shown model predictions

which correctly predict the conversion vs time data forthree different monomer feed compositions at 50 and

60 °C, respectively, for a given initiator concentration.The cumulative copolymer composition is plotted againstconversion in Figure 18 for three different monomer feedcompositions at 60 °C. These results are representativeof all of the composition plots. The good agreementbetween the composition data and the model predictions

Table 1. Some Examples of Practical Applications of Copolymerization Models

Broadhead et al., 1985 batch, semibatch, and continuous emulsion styrene (Sty)/butadieneGarcia-Rubio et al., 1985 bulk Sty/acrylonitrile (AN)Bhattacharya and Hamielec, 1986 bulk Sty/p-methylstyreneJones et al., 1986 bulk p-methylstyrene/MMAYaraskavitch et al., 1987 bulk p-methylstyrene/ANStorti et al., 1988 emulsion Sty/AN, AN/MMA, Sty/AN/MMAPenlidis et al., 1989 solution MMAStorti et al., 1989 emulsion AN/Sty/MMAXie et al., 1991 suspension VCXie and Hamielec, 1993c solution Sty/ethylene dimethacrylate, Sty/divinylbenzeneGao and Penlidis, 1996 bulk and solution MMA, Sty, AN, BA, ethyl acrylate, methyl acrylate,

VAc, p-methylstyrene, carboxylic acid

Figure 16. Bulk Sty/EA: model predictions of conversion vs timedata from McManus and Penlidis (1996) for [AIBN] 0.05 M at 50°C.

Figure 17. Bulk Sty/EA: model predictions of conversion vs timedata from McManus and Penlidis (1996) for [AIBN] 0.05 M at 60°C.

Figure 18. Bulk Sty/EA: model predictions of copolymer com-position vs conversion data from McManus and Penlidis (1996)for [AIBN] 0.05 M at 60 °C.


is not surprising since the reactivity ratios used in thisstudy were estimated using the statistically validmethods outlined earlier.Finally, representative plots of number- and weight-

average molecular weight vs conversion are shown inFigures 19 and 20 for two different initiator concentra-tions at 60 °C. The model correctly predicts the trendof increased molecular weight as a result of an increasein the initiator concentration. The important point toconsider in this case study is that only one set ofdatabase parameters was used to predict the data overa wide variety of reaction conditions.7.3. BA/MMA/VAcModel Prediction Case Study.

A comprehensive study of the butyl acrylate (BA)/methylmethacrylate (MMA)/vinyl acetate (VAc) terpolymersystem was recently reported in Dube (1994) and Dubeand Penlidis (1995a-c, 1997). While the eventualobjective of the work was to develop an understandingof the emulsion terpolymerization system, bulk, solu-

tion, and emulsion homo- and copolymerization datawere also collected. The model described herein wasapplied to these systems, and some representativeresults are discussed below. The parameter values usedin the model are found in Dube, 1994.As an example of a bulk homopolymerization, the

prediction of bulk BA data from Dube et al. (1991b), isshown in Figure 21. From these data, values for thepropagation and termination rate constants (kp and kt,respectively) were estimated using only one set of data,hence, more accurate values of these rate constants arerequired (see Dube, 1994). Updated values of these rateconstants are now available from pulsed-laser polym-erization studies (Buback et al., 1989, 1994; Buback andDegener, 1993; Lyons et al., 1996). Values for the free-volume parameters A and VFpcrit were fit to the data atone set of reaction conditions. For the BA system, rateconstants for transfer to polymer and terminal double-bond reactions are unknown, thus necessitating theacquisition of molecular weight data. This is particu-larly difficult due to the presence of significant amountsof gel in the samples (Dube et al., 1991b). Although nomolecular weight plots are shown for these data, it wasnoted that Mh n and Mh w are extremely sensitive to thevalue of kfm. As mentioned earlier, a thorough inves-tigation of bulk and solution modeling showing a largeset of examples (including BA, MMA, and VAc bulkhomopolymerization) is found in Gao and Penlidis(1996).Bulk copolymerization data reported in Dube and

Penlidis (1995a) (the reaction conditions can be foundtherein) are plotted along with model predictions inFigures 22-25. Conversion vs time data for bulk BA/MMA runs are plotted in Figure 22. The free-volumeparameters A, K3, and VFpcrit were fit to data at one setof reaction conditions. The reader is referred to Dubeand Penlidis, 1995a, for the model prediction of copoly-mer composition. Molecular weight data are shown inFigure 23. No parameter fitting to the molecular weightdata was attempted. As was the case for the bulkhomopolymerization modeling of BA, the molecularweight predictions were quite sensitive to the value forthe transfer to monomer constant, kfm, for BA. Resultsfrom a bulk BA/VAc and a bulk MMA/VAc run are

Figure 19. Bulk Sty/EA: model predictions of number- andweight-average molecular weight vs conversion data from Mc-Manus and Penlidis (1996) for [AIBN] 0.05 M at 60 °C.

Figure 20. Bulk Sty/EA: model predictions of number- andweight-average molecular weight vs conversion data from Mc-Manus and Penlidis (1996) for [AIBN] 0.10 M at 60 °C.

Figure 21. Bulk BA: model predictions of conversion vs time datafrom Dube et al. (1991b) at 60 °C.


shown with model predictions in Figures 24 and 25,respectively. As was the case for the BA/MMA runs,the free-volume parameters A, K3, and VFpcrit were fitto the data. The prediction of the composition data forboth BA/VAc and MMA/VAc are reported in Dube andPenlidis, 1995a.Values for the free-volume parameters A, K3, and

VFpcrit were fit to the BA/MMA/VAc conversion data fromDube and Penlidis, 1995b, for one set of reactionconditions. Conversion vs time data and model predic-tions for the bulk terpolymerizations are plotted inFigures 26 and 27. The prediction of the compositiondata is plotted in Figure 28. The predictions of theterpolymer composition data were achieved through theuse of copolymerization reactivity ratios estimated usingthe optimal methods described earlier. Molecular weightpredictions also followed similar trends (see Dube,1994).

In the prediction of emulsion polymerizations, a largernumber of physical constants and rate parameters wererequired. It was not possible to obtain all of theconstants from the open literature, nor was it possibleto obtain empirical estimates of the parameters. There-fore, many of the parameters used in the prediction ofthe emulsion polymerization data were arrived at byway of calculated guesses. In several cases, unknownparameters were given values previously reported forother monomers (see Dube, 1994). Prediction of conver-sion for the BA emulsion polymerization run describedin Dube and Penlidis (1995c) is plotted in Figure 29.Despite the uncertainty in the various rate parametersfor BA, the model prediction was quite good. Emulsionterpolymerization composition vs conversion data areshown in Figures 30 and 31. This is yet anotherconfirmation of the accuracy of the reactivity ratios aswell as the partition coefficients used in the model(Dube, 1994).

Figure 22. Bulk BA/MMA: model predictions of conversion vstime data from Dube and Penlidis (1995a) for f10 ) 0.439 at 60°C.

Figure 23. Bulk BA/MMA: model predictions of molecular weightvs conversion data from Dube and Penlidis (1995a) for f10 ) 0.439,[AIBN] ) 0.005 mol/L at 60 °C.

Figure 24. Bulk BA/VAc: model prediction of conversion vs timedata from Dube and Penlidis (1995a) for f10 ) 0.8, [AIBN] )0.00054 mol/L at 60 °C.

Figure 25. Bulk MMA/VAc: model prediction of conversion vstime data from Dube and Penlidis (1995a) for f10 ) 0.3, [AIBN] )0.01 mol/L at 60 °C.


7.4. Semibatch Production of MulticomponentPolymers. The practical implementation of monomerfeed policies requires the use of on-line (or possibly off-line) measurements to permit one to adjust for uncon-trolled variations in recipe impurities such as O2 (whichis a radical scavenger at temperatures below about 100°C) and other impurities in the recipe components whichcan affect radical concentration. An extensive reviewof on-line sensors for polymerization reactors waspublished by Chien and Penlidis (1990). Examples ofon-line calorimetry, gas chromatography, and densito-metry are discussed below.7.4.1. Calorimetric Control of Monomer Feed in

a Copolymerization. The instantaneous heat genera-tion due to polymerization (VQ) is given by

where V is the volume of polymerizing mixture, Q isthe instantaneous heat generation rate due to polym-erization in cal L-1 s-1, -∆H1 is the heat of polymeri-zation when monomer 1 adds to either polymer radicaltype and similarly for -∆H2.The feed rate of M1, F1,in, is given by

for any monomer feed policy, where

Dividing eq 257 by eq 258 and thus eliminating the totalpolymer radical concentration, one obtains

Figure 26. Bulk BA/MMA/VAc (30/30/40 wt %): model predic-tions of conversion vs time data from Dube and Penlidis (1995b)at 50 °C.

Figure 27. Bulk BA/MMA/VAc (30/30/40 wt %): model predic-tions of conversion vs time data from Dube and Penlidis (1995b)at 70 °C.

Figure 28. Bulk BA/MMA/VAc (30/30/40 wt %): model predic-tions of terpolymer composition vs conversion data from Dube andPenlidis (1995b).

Figure 29. Emulsion BA: model prediction of conversion vs timedata from Dube et al. (1995c).

VQ ) [(k11φ1• + k21φ2

•)N1(-∆H1) + (k12φ1• +

k22φ2•)N2(-∆H2)] (257)

F1,in ) [ φ1 - φ2

1 - R(N1/N2)]N1 (258)

R ) F2,in/F1,in (259)


If the two monomers are premixed and fed to the reactorin one stream, R is a constant. It should be recalledthat φ1• and φ2• are both functions of N1/N2 and polym-erization temperature. φ1• may be expressed as

The temperature dependence of φ1• should be smallbecause the activation energies of the cross-propagationconstants are similar in magnitude. In fact, the right-

hand side of eq 260 should have a small temperaturedependence. In other words, to maintain constant F1for the copolymer being produced, one should controlthe monomer feed rate (F1,in or (1 + R)F1,in) to maintainVQ/F1,in constant with time. The on-line measurementof VQ is obviously a convenient method for the controlof copolymer composition with any monomer feed policy(N.B. when R ) 0, Policy I is being employed). On-linemeasurement of VQ can also be used to achieve adesired variation of N1/N2 and spread in copolymercomposition.To maintain the desired production level of copolymer,

one can increase polymerization rate by increasing thetotal polymer radical concentration, [P•]. This can bedone by increasing the reaction temperature whilemaintaining VQ/F1,in constant. Hendy (1975) usedcalorimetric control to produce high AN copolymers ofstyrene/acrylonitrile of uniform composition in emulsionpolymerization. Two monomer feeding methods wereemployed, and both provided composition control. Inone, a monomer mixture with the same composition asthat desired for the copolymer was fed to the reactor(Policy II with monomer-starved feeding), and with thesecond feeding method, only the more reactive monomerwas fed to the reactor (Policy I). For the high ANcopolymers synthesized by Hendy, it was found thatPolicy I (feeding in styrene alone) gave better copolymerproducts than Policy II (with monomer-starved feeding).The specific reasons for this were not given. In theproduction of AN/S copolymers by feeding styrene, aconstant feed rate of styrene was employed and thisappeared to give uniform composition copolymer at leastup to 80% conversion. Samples were withdrawn duringthe polymerization for analysis, and it was found thatat about 80% conversion of AN the copolymer wasslightly richer in styrene (2% higher in styrene) thanthe copolymer produced up to 50% conversion. This wasattributed to the partitioning of AN into the water phase(∼10% soluble in water) and the effect on the AN/S ratioin the polymer particles at higher conversions. Equa-tion 260 should only apply to stage III in emulsionpolymerization when both monomers have very lowsolubilities in the water phase. Hoffman (1984) foundsimilar results.Moritz (1989) reviewed the state of the art of isother-

mal bench-scale calorimeters and their application topolymerization reactions. He developed a microcom-puter bench-scale polymerization calorimeter with on-line determination of polymerization rate and monomerconversion and studied the batch and semibatch emul-sion polymerization of vinyl acetate. Lately, the use ofon-line reaction calorimeters in emulsion polymerizationhas been gaining favor because on-line gas chromato-graph setups allow only for discrete and delayed mea-surements.An automated reaction calorimeter was used to moni-

tor the rate of emulsion polymerization of styrene usingdifferent emulsifier and initiator concentrations inconjunction with off-line measurements of the evolutionof the particle-size distributions by de la Rosa et al.(1996). Their experimental results suggested that theend of nucleation and the disappearance of monomerdroplets take place at approximately the same conver-sion (x = 36-42 wt %). A method to determine theminimum time monomer addition policy for compositioncontrol in the semibatch unseeded emulsion copolym-erization of BA/VAc using calorimetric measurementswas developed by Gugliotta et al. (1995b). An iterative

Figure 30. Emulsion BA/MMA/VAc (30/30/40 wt %): modelpredictions of terpolymer composition vs conversion data fromDube (1994).

Figure 31. Emulsion BA/MMA/VAc (35/15/50 wt %): modelpredictions of terpolymer composition vs conversion data fromDube (1994).

VQF1,in

)

(k11φ1• + k21φ2

•)(-∆H1) + (k12φ1• + k22φ2

•)(-∆H2)(N2/N1)

(k11 - k12)φ1• + (k21 - k22)φ2

•

1 - R(N1/N2)(260)

φ1• ) 1

1 + k12N2/k21N1(261)


empirical approach using semistarved conditions wasused. This was accomplished using pure and technical-grade monomers. Saenz de Buruaga et al. (1996)presented further results for the same system. Casestudies involving the emulsion copolymerizations of BA/MMA, BA/VAc, and MMA/VAc were examined by Ur-retabizkaia et al. (1993) using a similar setup.7.4.2. On-Line Monomer Concentration Mea-

surements for Control of Monomer Feed. On-linemeasurements of the monomer ratio N1/N2 in thepolymerizing phase (polymer particles in emulsionpolymerization) with a feedback system to adjust mono-mer feed rates can be used to control copolymer com-position. Guyot et al. (1981) used an on-line GC tocontrol copolymer composition in the emulsion copolym-erization of styrene/acrylonitrile. They employed PolicyI but found in their subsequent analysis of the copoly-mers that copolymer composition drifted toward higherstyrene levels. They showed that this was due tosignificant partitioning of acrylonitrile into the waterphase. This was not corrected for in the monomer feedrates employed to control copolymer composition at auniform level.Askill and Gilding (1981) produced uniform composi-

tion copolymers of methyl methacrylate/acrylic acid(rMMA ∼ 2.3 and rAA ∼ 0.31) using semibatch solutionpolymerization and Policy II (with monomer-starvedfeed). The solvent used was butanone. On-line GCmeasurements were used to monitor the monomer ratio(NMMA/NAA) in the reactor and adjust monomer feedrates. For most of the semibatch runs under monomer-starved conditions, there was no need to adjust feedrates. Narrower molecular weight distributions werefound for the uniform composition copolymers producedby use of Policy II (with monomer-starved feed). Thisconfirms that reactions which form long trifunctionalbranches such as transfer to polymer and reaction withterminal double bonds are insignificant for the MMA/AA system.Guyot et al. (1984) monitored monomer concentrations

by GC and used semibatch copolymerization of buta-diene/acrylonitrile to obtain uniform composition poly-mers (Policy I). Constant composition copolymers showeda single glass transition temperature, and the produc-tion of cross-linked gel was delayed and sometimesavoided. The production of gel occurred due to polymerradicals adding to internal double bonds in l,2-butadieneunits bound in the copolymer chains. Reactivity ratioswere estimated to be rB ) 0.3 and rA ) 0.04. Semibatchcopolymerization not only produced copolymers uniformin composition but also polymer with lower levels oftetrafunctional long-chain branching and cross-linkedgel. The reactive internal double bonds have the vinyl1,2-structure, and these were responsible for the forma-tion of cross-linked gel. In butadiene homopolymeriza-tion about 20% of the internal double bonds have the1,2-structure. Apparently, during semibatch synthesisof uniform copolymer the incorporated acrylonitrileorientated the added butadiene unit to the less reactive1,4-structure because the sequence length of butadienewas kept small. In batch copolymerization, acrylonitrilewas consumed preferentially, and at high conversions,copolymer containing long butadiene sequences andhigher levels of vinyl 1,2-structure was formed. This,of course, was responsible for the formation of the highlevel of cross-linked gel.Snuparek and Kaspar (1981) investigated the semi-

batch emulsion copolymerization (Policy II with mono-

mer-starved feed) of ethyl acrylate/butyl acrylate. Thecopolymerization was followed by gas chromatography.There is evidence that the copolymerization may becontrolled by monomer diffusion in the polymer par-ticles, with both reactivity ratios approaching unity.El-Aasser et al. (1983) investigated the batch and

semibatch emulsion copolymerization of vinyl acetate/butyl acrylate. Policy II with monomer-starved feed wasemployed. Chain transfer to vinyl acetate bound in thecopolymer chains produced long trifunctional branchesand led to broader molecular weight distributions whichwere bimodal.Leiza et al. (1993) used an on-line GC in their

development of a semiempirical approach to determineminimum time optimal monomer addition policies toproduce homogeneous methyl methacrylate/ethyl acry-late copolymers. The method involves a series ofsemibatch emulsion copolymerizations carried out undersemistarved conditions.The composition control of a semibatch BA/MMA/VAc

emulsion terpolymerization was achieved with an on-line GC by Urretabizkaia et al. (1994b). Their strategyis based on comparing on-line measurements to modelpredictions and using the difference as input to anonlinear adaptive plus proportional-integral control-ler. The controller calculates the feed rates of the morereactive monomers that have to be fed into the reactorto make sure that, after a sampling time interval, themonomer ratios in the polymer particles lead to theformation of a terpolymer of the desired composition.Experimental verification was performed using a 55 wt% solids content and a molar terpolymer compositiontarget of BA/MMA/VAc ) 50/35/15.7.4.3. On-Line Density Measurements for Com-

position Control. The use of on-line density measure-ments has gained increasing importance as attested bysome earlier references in Chien and Penlidis (1990).More recently, Canegallo et al. (1993) used an on-linedensitometer coupled with a kinetic model to monitorconversion and copolymer composition in batch emulsionhomopolymerizations (Sty and MMA) as well as copo-lymerizations (Sty/MMA, AN/MMA, and MMA/VAc).Canu et al. (1994) then developed an a priori monomerfeed policy which requires implementation in conjunc-tion with the densitometer described above. Experi-mental verifications with emulsion copolymerizations(MMA/VAc, Sty/BA) and a terpolymerization (BA/MMA/VAc) were reported.7.4.4. Policies I and II in Semibatch Emulsion

Copolymerization. It should be mentioned that thestrict application of Policies I and II is not alwayspossible in emulsion copolymerization. In stages I andII, monomer droplets are also feeding the polymerparticles (the major site for polymerization), and thismust be properly accounted for when calculating therequired feed rates of the monomers, F1,in and F2,in. Instage III, however, if the solubilities of the monomersin the water phase are negligible, Policies I and II canbe strictly applied.

8. Concluding Remarks

There is no doubt that the multidisciplinary polymerreaction engineering horizons will be extended evenfurther in the future as more value-added specialtypolymers (especially those arising out of and applied tobiological systems) are developed, and the materialscience field develops in this area. Certainly more effortis required in relating the fundamental properties of


polymers, i.e., average molar mass and molar massdistribution, monomer sequence distribution, branching,etc., to particular requirements and attributes of specificapplications. Developing a polymerization model is notan isolated effort; rather, it requires a commitment tothe long-term development in parallel of a comprehen-sive polymerization database for characteristics of mono-mers, initiators, solvents, emulsifiers, chain-transferagents, inhibitors, etc. Property databases, with thepotential to provide an effective computer-based bridgebetween the simulation phase and detailed engineering,may have a lot to offer to these future directions.

Acknowledgment

M.A.D., A.P., and J.B.P.S. thank Professor Hamielecfor being a great mentor. Financial support for thisresearch supplied by the Natural Sciences and Engi-neering Research Council (NSERC) of Canada, theOntario Centre for Materials Research (OCMR), ICIWorldwide, and Uniroyal Chemical Co. over the yearsis gratefully acknowledged.

Nomenclature

a ) partition coefficient for monomer radicals betweenwater phase and particles

a ) root-mean-square end-to-end distance per square rootof the number of monomer units

A ) free-volume theory parametera2 ) adjustable parameterAd ) total surface area of monomer droplets (dm2)Aj ) jacket heat-transfer area (m2)Am ) total free micellar area (dm2)Ap ) total surface area of polymer particles (dm2)B ) free-volume theory parameterBN3 ) average number of trifunctional branch points (#)BN4 ) average number of tetrafunctional branch points (#)c ) mass concentration of accumulated polymer in reactionmixture (g L-1)

C ) parameter modifying rate of change of initiatorefficiency

Cfcta ) transfer to chain-transfer agent constantCfm ) transfer to monomer constantCfmsi ) transfer to monomer-soluble impurities constantCfp ) transfer to polymer constantCk ) internal double-bond reaction constantCpi ) heat capacity of species i (cal g-1 K-1)[CTA]p ) concentration of chain-transfer agent in theparticles (mol L-1)

D ) reaction diffusion coefficientdm ) density of monomer (kg L-1)dp ) particle diameter (dm)Dp ) diffusivity of monomer radicals in the particles (cm2

s-1)[drops] ) concentration of monomer droplets (mol L-1)Dw ) diffusivity of monomer radicals in the water phase(cm2 s-1)

f, fo ) overall and initial initiator efficiencyFi,in ) flow of species i into the reactor (mol min-1)Fj ) instantaneous polymer composition (mole fractionmonomer j bound in polymer)

Fh j ) cumulative polymer composition (mole fraction mono-mer j bound in copolymer)

fj ) mole fraction of monomer j in the particlesFjw ) mole fraction of monomer j bound in the polymerformed in the water phase

∆Hpi ) heat of polymerization of monomer i (cal g mol-1)[I] ) concentration of initiator (mol Lw

-1)jc ) entanglement spacing of pure polymer (# monomerunits)

jcr ) overall critical chain length of polymer radical formedin the water phase (#)

jcrj ) critical chain length of polymer radical formed in thewater phase ending in species j (#)

K ) terminal double-bond reaction constantk1 ) iron oxidation rate constant (L mol-1 min-1)k2 ) iron reduction rate constant (L mol-1 min-1)K3 ) free-volume theory parameterka ) radical absorption by particles rate coefficient (dmmin-1)

Kc ) controller gainkcm ) capture by micelles rate constant (dm min-1)kcmd ) capture by monomer droplets rate constant (dmmin-1)

kcp ) capture by particles rate constant (dm min-1)Kcta ) partition coefficient for chain-transfer agent betweenthe droplet and particle phases

kd ) initiator decomposition rate constant (min-1)kdes ) radical desorption rate constant (min-1)kdes′ ) particle-size-independent radical desorption rateconstant (min-1)

kF ) particle coagulation rate constant (Lw min-1)kfcta ) overall rate constant for transfer to chain-transferagent (L mol-1 min-1)

kfctai ) rate constant for transfer to chain-transfer agentfrom radical ending in monomer i (L mol-1 min-1)

kfidb, k*p* ) rate constant for reaction with internal doublebonds (L mol-1 min-1)

kfm ) overall rate constant for transfer to monomer (Lmol-1 min-1)

kfmij ) rate constant for transfer to monomer j from radicalending in monomer i (L mol-1 min-1)

kfmsi ) rate constant for transfer to monomer-solubleimpurity (L mol-1 min-1)

kfp ) overall rate constant for transfer to polymer (L mol-1min-1)

kfpij ) rate constant for transfer to monomer j on a deadpolymer from monomer i (L mol-1 min-1)

kftdb, k*p ) rate constant for reaction with terminal doublebonds (L mol-1 min-1)

kh, kho ) homogeneous nucleation rate constant (min-1)Kjmw ) partition coefficient for species j between themonomer droplet and water phases

Kjwp ) partition coefficient for species j between the waterand particle phases

Kmsi ) partition coefficient for monomer-soluble impuritybetween the droplet and particle phases

kp, kpo ) overall propagation rate constant (L mol-1 min-1)kp′ ) rate constant for reinitiation of oligomer radicals frommonomer radicals (L mol-1 min-1)

kpij ) propagation rate constant for radical ending inspecies i adding monomer j (L mol-1 min-1)

kpIj ) propagation rate constant for addition of monomer jto an initiator radical (L mol-1 min-1)

kt ) termination rate constant (L mol-1 min-1)kT ) translational diffusion-controlled termination rateconstant (L mol-1 min-1)

kto ) termination rate constant at zero polymer concentra-tion (L mol-1 min-1)

ktc ) termination by combination rate constant (L mol-1min-1)

ktd ) termination by disproportionation rate constant (Lmol-1 min-1)

khtN, khtW, khtZ ) number-, weight-, and Z-average terminationconstants (L mol-1 min-1)

ktr ) transfer rate constant (L mol-1 min-1)ktrd ) reaction diffusion-controlled termination rate con-stant (L mol-1 min-1)

ktres ) “residual” termination rate constant (L mol-1 min-1)ktrm ) transfer to monomer rate constant (L mol-1 min-1)ktrt ) transfer to CTA rate constant (L mol-1 min-1)


ktsb ) termination rate constant of polymer radicals in thesol with those on the gel (L mol-1 min-1)

ktseg ) segmental diffusion-controlled termination rateconstant (L mol-1 min-1)

ktw ) water phase termination rate constant (L mol-1min-1)

kzj ) rate constant for reaction with water-soluble impurityj (L mol-1 min-1)

L ) critical chain length for water phase polymerizationlo ) length of a monomer unitm ) grouping of terms representing radical exit from thepolymer particles

m ) adjustable parameter for free-volume theorym, mi ) coefficient relating to volume characteristics inpartition coefficient equations

md ) partition coefficient for monomer radicals betweenthe water phase and the particles

Mj ) monomer of type j[Mj]m, [Mj]p, [Mj]w ) concentration of monomer j in thedroplets, particles, and water phase, respectively (molL-1)

Mh n ) accumulated number-average molecular weight[M]p ) total concentration of monomer in the particles (molLp

-1)[M]psat ) total (saturation) concentration of monomer inthe particles (mol Lp

-1)[MSI]p ) concentration of monomer-soluble impurities inthe particles (mol Lp

-1)[M]w ) concentration of monomer in the water phase (molLw

-1)Mh w ) accumulated weight-average molecular weightMwc ) saturation concentration of monomer in the waterphase (mol Lw

-1)Mh wcrit ) critical accumulated weight-average molecularweight

Mweff ) effective molecular weight of copolymer (g mol-1)MWi ) molecular weight of species i (g mol-1)Mwsat ) saturation concentration of monomer in the waterphase (mol Lw

-1)Mwt ) molecular weight of monomer (g mol-1)Mh z ) accumulated z-average molecular weightn ) adjustable parameter for free-volume theorynj ) average number of radicals per particleNA ) Avogadro’s number (molecule mol-1)Nhom ) number of particles formed by homogeneousnucleation (# Lw

-1)Nj ) moles of species j (mol)Nmj ) moles of monomer j (mol)Nmic ) number of particles formed by micellar nucleation(# Lw

-1)Np, NT ) total number of particles in the reactor (# Lw

-1)Npolj ) moles of monomer j bound in the copolymer (mol)ns ) number of monomer units in one polymer chainsegment

Pm, Pn ) dead polymer molecule of chain length m or nQ0, Q1, Q2 ) zeroeth, first, and second moments of themolecular weight distribution (mol)

Qj, QR ) heat losses in jacket and reactor, respectively (calmin-1)

r ) particle radius (dm)r ) radial position in monomer-swollen particle (cm)R ) universal gas constant (cal mol-1 K-1)R ) radius of monomer-swollen particle (cm)rij ) reactivity ratioRcm ) rate of capture of radicals by micelles (mol Lw

-1

min-1)Rcmd ) rate of capture of radicals by monomer droplets (molLw

-1 min-1)Rcp ) rate of capture of radicals by particles (mol Lw

-1

min-1)

[RI•] ) concentration of initiator radicals (mol Lw

-1)RI ) rate of initiation (mol Lw

-1 min-1)[Rj,i

• ]w ) concentration of radicals of chain length j endingin monomer i in the water phase (mol Lw

-1)rmic ) radius of a micelle (dm)rn, rjn ) instantaneous and accumulated number-averagechain lengths, respectively

rw, rjw ) instantaneous and accumulated weight-averagechain lengths, respectively

Rn,i• ) radical of chain length n ending in monomer i

Rp ) overall rate of polymerization (mol Lp-1 min-1)

Rpj ) rate of polymerization of monomer j (mol Lp-1 min-1)

Rpjp ) rate of polymerization of monomer j in the particles(mol Lp

-1 min-1)Rpjw ) rate of polymerization of monomer j in the waterphase (mol Lw

-1 min-1)Rtc, Rtd ) rate of termination by combination and dispro-portionation, respectively (mol L-1 min-1)

[RTOT• ]des ) concentration of desorbed radicals in the waterphase (mol Lw

-1)[RTOT

• ]w, [R•]w ) total concentration of radicals in thewater phase (mol Lw

-1)[RTOT

• ]wdrop, [RTOT• ]wmic, [RTOT

• ]wpar ) concentration of radi-cals in the water phase able to be captured by droplets,micelles, and particles, respectively (mol Lw

-1)[RTOT

• ]wdrop ) concentration of radicals in the water phasethat may undergo homogeneous nucleation (mol Lw

-1)Sa ) surface area covered by one molecule of emulsifier i(dm2 molecule-1)

[S]CMC ) critical micelle concentration (mol Lw-1)

[S]t ) total concentration of emulsifier (mol Lw-1)

t ) time (min)T, Tj, Tref, Tset ) reaction, jacket, reference, and setpointtemperatures, respectively (K)

Tgi ) glass transition temperature of species i (K)Vaq ) volume of the aqueous phase (L)VF ) free volume (L)VFcrif, VFcrit, VFpcrit ) critical free volumes (L)Vi ) volume of species i (L)Vjmd, Vjmp, Vjmw ) volumes of monomer j in the droplets,aqueous phase, and particles, respectively (L)

Vm ) volume of the monomer droplet phase (L)Vm ) molar volume (L mol-1)Vmj ) total volume of monomer j (L)Vo ) total volume of the organic (monomer and polymer)phase (L)

vout ) total volumetric flow out of the reactor (L min-1)vp ) average volume of a polymer particle (Lp)Vp, V, Vo ) total volume of polymer particles (Lp)VT ) total volume of reaction mixture (L)Vw ) volume of water (L)W(r,y) ) weight fraction of polymer of chain length r andcomposition deviation y

[WSI]i ) concentration of water-soluble impurity i (molLw

-1)x ) monomer conversion on a mass basisxc ) critical monomer conversion where droplets disappearYo ) total concentration of radicals in the particles (molLp

-1)

Greek Letters

R ) grouping of terms representing radical entry into thepolymer particles

â ) molecular weight contribution due to termination bycombination

δ ) reaction radiusγ ) fraction of termination by disproportionationε ) measurement errorµi ) chemical potential of species i


Fdes ) rate of radical desorption from the particles (molLw

-1 min-1)Fm ) density of monomer (kg L-1)Fp ) density of polymer (kg L-1)σ ) Lennard-Jones diameterτ ) molecular weight contribution due to termination bydisproportionation and transfer to small molecules

τ ) time (min)φi ) mole fraction of radicals ending in monomer i in theparticles

φp ) volume fraction of polymer in the particlesøi, øij ) Flory-Huggins interaction parameter

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Received for review August 1, 1996Revised manuscript received January 10, 1997

Accepted January 10, 1997X

IE960481O

X Abstract published in Advance ACS Abstracts, March 1,1997.


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