Mathematical Modeling of the Long-Chain Branch Structure of Polyolefins Made with Two Metallocene Catalysts: An Algebraic Solution

184

Mathematical Modeling of the Long-Chain BranchStructure of Polyolefins Made with Two MetalloceneCatalysts: An Algebraic Solution

Jo¼o B. P. Soares

Institute for Polymer Research, Department of Chemical Engineering, University of Waterloo,200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1Fax: 1-519-746-4979; E-mail: [email protected]

Keywords: branched; metallocenes; modeling; polyethylene (PE); polyolefins;

IntroductionMetallocene catalysts that permit the formation of long-chain branches (LCBs) have been the subjects of consid-erable academic and industrial interest in the past fewyears. These catalysts produce polyolefins with narrowmolecular-weight distributions containing a limitedamount of LCB, typically from 1 to 5 LCB per 10000carbon atoms (C). Remarkably, the presence of even afew LCB is capable of significantly affecting the rheolo-gical and physical properties of these polymers, leadingto products with enhanced performance.

Monocyclopentadienyl complexes (sometimes calledconstrained geometry catalysts) were the first to producepolyethylene with LCB.[1–3] However, LCB formation isnot limited to this class of catalyst; several other metallo-

cene types have been used to synthesize branched poly-olefin chains,[4, 5] including the incorporation of brancheswith different chemical structures from the backbone.[6–8]

LCB formation with different metallocenes has beenobserved in solution, slurry, and gas-phase processes.

Regardless of the type of catalyst and reactor config-uration, the mechanism of branch-formation with metal-locenes is terminal branching: dead chains containing aterminal unsaturation (vinyl is the most reactive one)copolymerize with the growing chain to form a branchwith the length of the dead chain (minus two carbonatoms). These dead chains with terminal unsaturationsare usually called macromers or macromonomers. There-fore, LCB formation with metallocenes is simply copoly-merization with a long comonomer chain and thus it

Full Paper: A mathematical model was developed todescribe the populations of polymer chains containing dif-ferent numbers of long-chain branches (LCBs) made witha combination of two single-site catalysts. One of the cat-alysts produces only linear chains (linear-catalyst) and theother produces linear and long-branched chains (LCB-cat-alyst). The model shows that when the selectivity formacromer formation of the linear-catalyst is the same asthat of the LCB-catalyst, it is not possible to maximize thenumber of LCB per chain, even though the number ofLCB per 1000 carbon atoms (C) can be maximized. Onthe other hand, if the selectivity for macromer formationof the linear-catalyst is higher than that of the LCB-cata-lyst, both LCB/1000 C and LCB/chain pass through max-ima when varying the fraction of the linear-catalyst in thereactor. More importantly, polymer populations with dif-ferent numbers of LCB per chain will reach their maxi-mum values at different ratios of linear-catalyst to LCB-catalyst, thus permitting the maximization of individualpolymer populations in the mixture.

Macromol. Theory Simul. 2002, 11, No. 2 i WILEY-VCH Verlag GmbH, 69469 Weinheim 2002 1022-1344/2002/0202–0184$17.50+.50/0

The effect of the parameter v on the branching structure of apolymer made with an LCB-catalyst and a linear-catalystwith the same rates of macromer generation (c = cL). Simula-tion parameters: c = cL = 0, e = eL = 0, rn = rn,L = 4000, kp,L /kp = 1, s/(kLCBYT) = 1. v = 4.

Macromol. Theory Simul. 2002, 11, 184–198

Mathematical Modeling of the Long-Chain Branch Structure of Polyolefins ... 185

should be expected that catalysts that have high reactivityratios towards long a-olefins are better suited to producebranched polyolefins. This is indeed the case, as shownby several experimental investigations.[1–8]

Preformed macromers can be added to the polymeriza-tion reactor, in which case these polymerizations can betreated as common copolymerizations, or the macromercan be formed in situ by the very catalyst making thebranched chains (or by a second catalyst added for thisspecific purpose). In situ formation of polyethylenemacromers happens when living polyethylene chains areterminated via b-hydride elimination, or transfer to ethyl-ene. (A few other mechanisms can lead to macromerpolypropylene formation.[6]) If the latter mechanism ofLCB formation is predominant, the polymers made withthese catalysts have a complex LCB architecture com-posed of linear and branched (comb and tree) chains, asdescribed in detail in the literature.[9] We will be con-cerned with LCB formation by incorporation of in-situ-formed macromers in this publication.

Soares and Hamielec[10, 11] derived an analytical solu-tion for the distribution of molecular weight, chemicalcomposition, and long-chain branching for polymersmade with these catalysts. Different modelingapproaches[12, 13] led to the same results previouslyobtained by Soares and Hamielec. An even more in-depthpicture of the branching structure of these chains can beobtained via Monte Carlo models, such as the one devel-oped by Beigzadeh et al.[14] They found out that most ofthe high-molecular-weight chains had branches onbranches. This is a very important observation, sincethese highly branched structures can enhance polymerprocessability considerably.

For a given catalyst type, the LCB frequency of theresulting polymers can be enhanced by increasing theconcentration of macromer present in the reactor. Forsolution polymerization there are essentially three waysof achieving this objective: (1) operate the reactor at highconversions, (2) feed pre-formed macromer to the reactor,and (3) add a second catalyst with a higher rate of macro-mer formation to the reactor. Gas-phase polymerizationhas also been used to increase the concentration ofmacromers around the active sites,[15] but in this case bulkchain mobility is evidently severely decreased. Thismight also lead to a different branching structure, withunknown consequences to polymer properties.

The use of a dual metallocene catalyst system wasinvestigated by Beigzadeh et al.,[16–18] both from theoreti-cal and experimental approaches, confirming that LCBcan indeed be enhanced if two adequate metallocenes arecombined in the same reactor. However, as noticed by thesame authors,[9] this LCB-enhancement process wouldaffect the branching structure of the polymer since the pro-portion of chains with comb structures would be higherthan that present when a single constrained geometry cata-

lyst was used. It is well known that tree- and comb-branched polymers have quite different rheological behav-iors. Therefore, the LCB enhancement obtained by com-bining two metallocene catalysts might affect the finalrheological (and mechanical) properties of the polymer ina rather complex and so far unpredictable way.

In this paper we developed a simple mathematicalmodel to predict the branching structure of polyolefinsmade with two metallocene catalysts, in which one metal-locene makes only linear chains (linear-catalyst) and theother produces linear and branched chains (LCB-cata-lyst). This can be achieved in practice by combining twometallocenes with very different reactivity ratios towardsa-olefins. The most attractive feature of the model is thatit is computationally simple and uses just a few dimen-sionless parameters that can be obtained via 13C NMRanalysis of chain ends and the branch structure of poly-mers made with each metallocene catalysts separately. Inorder to keep the model simple enough to allow foranalytical solution, molecular weight was not described.More complete models (but also more complex) havebeen published elsewhere. This model can be used toanswer some questions posed by Beigzadeh et al.[9] inregard to the branch structure of polymers made withcombined metallocene catalysts.

Model DevelopmentThe model that will be developed below is applicable to adual catalyst system where one of the catalysts is able toproduce LCB via terminal branching (LCB-catalyst) andthe other catalyst can only produce linear polymer chains(linear-catalyst). The linear catalyst is added to the sys-tem to produce macromers and therefore affect thebranching structure of the polymer made with the LCBcatalyst.

Polymerization Kinetics

Since molecular weight is not monitored in this model,only the first insertion step, transforming a monomer-freesite into a growing polymer chain, will be considered:

C þM ggggskp P0 (1)

CL þM ggggskp;L p (2)

Equation (1) describes the polymerization with theLCB-catalyst, where C represents monomer-free activesites, P0 stands for linear living polymer chains of anylength, and kp is the propagation rate constant. Equation(2) describes the polymerization with the linear-catalyst,where CL, p, and kp,L have equivalent meanings to C, P0

and kp. (The subscript L will always refer to the linear-catalyst throughout this paper.) Monomer is representedby M. Subsequent monomer insertions do not affect theconcentration of linear polymer chains in the reactor, i.e.,

186 J. B. P. Soares

P0 þM ggggskp P0 (3)

pþM ggggskp;L p (4)

The formation of branched chains is described by thegeneral equation:

Pi þ Dj ggggskLCB Piþjþ1 (5)

In Equation (5), Pi and Dj represent living chains andmacromers (dead chains with vinyl unsaturations) with iand j LCB, and kLCB is the kinetic rate constant for LCBincorporation, assumed to be independent of the size ofthe macromer. For instance, the propagation of a livinglinear chain (i = 0) with a linear macromer (j = 0) willlead to the formation of a living polymer chain with oneLCB (i + j + 1 = 1). The incorporation of D0 macromersto linear chains leads to the formation of comb-branchedchains, whereas the incorporation of Di macromers,where i A 1, leads to the formation of dendritic (branches-on-branches) chains.

Four types of transfer reactions are considered:

b-Hydride elimination: Pn ggggskb Dn þ C (6)

p ggggskb;L D0 þ CL (7)

Transfer to monomer: Pn þM ggggskM Dn þ C (8)

pþM ggggskM;L D0 þ CL (9)

Transfer to hydrogen: Pn þ H2 ggggskH Ds

n þ C (10)

pþ H2 ggggskH;L Ds

0 þ CL (11)

Transfer to aluminum: Pn þ Al ggggskAl Ds

n þ C (12)

pþ Al ggggskAl;L Ds

0 þ DL (13)

where Dsn represents a saturated dead chain with n LCB,

H2 is hydrogen, Al is the cocatalyst, and kb, kM, kH and kAl

are the rate constants for the several transfer mechanismsconsidered in Equation (6) to (13).

Population Balances for Dead Chains

The molar balance for linear macromers made by bothcatalysts is given by:

dD0

dt¼ðkbþkMMÞP0þðkb;LþkM;LMÞpÿðkLCBYþ sÞD0

¼ K¼P0 þ K¼L pÿ ðkLCBY þ sÞD0 ð14Þ

where s is the space-velocity or the reciprocal of the aver-age residence time in the reactor and Y is the total con-centration of active sites of the LCB-catalyst:

Y ¼ C þXvi¼0

Pi ð15Þ

Similarly, for the saturated linear dead chains:

dDs0

dt¼ ðkHH2þkAl AlÞP0þðkH;LH2þkAl;L AlÞpÿ sDs

0

¼ KsP0 þ KsLpÿ sDs

0 ð16Þ

The parameters K= and Ks are defined below:

K¼ ¼ kb þ kM M ð17Þ

K¼L ¼ kb;L þ kM;L M ð18Þ

Ks ¼ kH H2 þ kAl Al ð19Þ

KsL ¼ kH;LH2 þ kAl;L Al ð20Þ

The molar balance for macromers with n LCB is givenby

dDn

dt¼ K¼Pn ÿ ðkLCBY þ sÞDn ; for n F 1 ð21Þ

The equivalent balance for saturated dead chains isgiven by:

dDsn

dt¼ KsPn ÿ sDs

n ; for n F 1 ð22Þ

Population Balances for Living Chains

The molar balance for living chains made with the linear-catalyst is:

dpdt¼ kp;L MCLÿðkb;LþkM;LMþkH;LH2þkAl;L Alþ sÞp

¼ kp;L MCL ÿ ðKL þ sÞp ð23Þ

Similarly, for the linear living chains made with theLCB-catalyst:

dP0

dt¼ kp MCÿðkbþkMMþkHH2þkAl AlþkLCBQþ sÞP0

¼ kp MC ÿ ðK þ kLCBQþ sÞP0 ð24Þ

where Q is the total concentration of macromers in thesystem,

Q ¼Xvi¼0

Di ð25Þ

and the parameters KL and K are defined as follows:

KL¼ kb;LþkM;L MþkH;LH2þkAl;L Al¼K¼L þKsL ð26Þ


K ¼ kbþkM MþkHH2þkAl Al¼K¼þKs ð27Þ

According to the LCB-forming reaction, Equation (5),the population balances for living branched chains isgiven by:

dP1

dt¼ kLCB P0 D0 ÿ ðK þ kLCBQþ sÞP1 ð28Þ

dP2

dt¼ kLCBðP0 D1þP1 D0ÞÿðKþkLCBQþ sÞP2 ð29Þ

dP3

dt¼ kLCBðP0 D1þP1 D1þP2 D0Þ

ÿðKþkLCBQþ sÞP3 ð30Þ

dPn

dt¼ kLCB

Xnÿ1

i¼0

Pi Dnÿiÿ1 ÿ ðKþkLCBQþ sÞPn ;

for n F 1 ð31Þ

Population Balances for Other Species

The total concentration of macromers in the reactor is cal-culated using the equation:

dQdt¼ K¼

Xvi¼0

Pi þ K¼L pÿ ðkLCBY þ sÞQ ð32Þ

Since,

Y ¼ C þXvi¼0

Pi ð15Þ

and,

YL ¼ CL þ p ð33Þ

then,

dQdt¼K¼ðYÿCÞþK¼L ðYLÿCLÞÿðkLCBYþ sÞQ ð34Þ

Similarly, the total concentration of saturated deadchains is given by:

dQs

dt¼KsðYÿCÞþKs

LðYLÿCLÞÿ sQs ð35Þ

Balances for the monomer-free active sites of both cat-alysts give:

dCdt¼ _YYin þ KðY ÿ CÞ ÿ kp MC ÿ Cs ð36Þ

dCL

dt¼ _YYL;in þ KLðYL ÿ CLÞ ÿ kp;L MCL ÿ CLs ð37Þ

where _YYin and _YYL;in are the molar flow rates of the LCB-catalyst and of the linear-catalyst fed to the reactor,respectively.

Active site deactivation will be neglected in this model.Equation (36) and (37) can be easily modified to accountfor active-site deactivation if needed.

The system defined by Equation (23) to (37) can besolved numerically if a dynamic solution of the popula-tion balance is required. However, a steady-state solutionis more adequate for our present objectives and will bedeveloped in the next section.

Steady-State Solution

The steady-state solution for concentration of macromersand saturated dead polymer chains is easily derived fromEquation (14), (16), (21) and (22):

D0 ¼K¼P0 þ K¼L p

kLCBY þ sð38Þ

Dn ¼K¼Pn

kLCBY þ s; for n F 1 ð39Þ

Ds0 ¼

KsP0 þ KsLp

sð40Þ

Dsn ¼

KsPn

s; for n F 1 ð41Þ

Similarly, Equation (23), (24), and (31) are used toobtain the steady-state solution for the living polymerchains:

p ¼ kp;L MCL

KL þ sð42Þ

P0 ¼kp MC

K þ kLCBQþ sð43Þ

Pn ¼kLCB

Xn¼1

i¼0

PiDnÿiÿ1

K þ kLCBQþ s; for n F 1 ð44Þ

The concentration of monomer-free sites is obtainedfrom Equation (36) and (37):

C ¼_YYin þ KY

K þ kp M þ sð45Þ

CL ¼_YYL;in þ KLYL

KL þ kp;L M þ sð46Þ

Assuming that there is no accumulation of active sitesin the reactor, and _YYin = sY and _YYL;in = sYL, therefore:

188 J. B. P. Soares

C ¼ ðsþ KÞYK þ kp M þ s

ð47Þ

CL ¼ðsþ KLÞYL

KL þ kp;L M þ sð48Þ

The total concentration of macromers is obtained fromEquation (34), assuming that C s Y and CL s YL:

Q ¼ K¼ðY ÿ CÞ þ K¼L ðYL ÿ CLÞkLCBY þ s

XK¼Y þ K¼L YL

kLCBY þ sð49Þ

A similar expression is derived from Equation (35) forthe concentration of saturated dead polymer chains:

Qs ¼ KsðY ÿ CÞ þ KsLðYL ÿ CLÞ

sX

KsY þ KsLYL

sð50Þ

Equation (38) to (50) can be solved for the concentra-tion of the several species present in the reactor. How-ever, it is more useful to solve for the molar fractions ofthese species. The advantages of this approach willbecome apparent in the next section.

Steady-State Solution for Molar Fractions of Livingand Dead Chains

The molar fraction of all living linear chains, y0,T, is givenby:

y0;T ¼ y0 þ y0;L ¼P0 þ p

Y þ YL

¼ P0 þ p

YT

ð51Þ

where YT is the total concentration of active sites in thereactor, y0 is the molar fraction of linear polymer madewith the LCB-catalyst, and y0,L is the molar fraction ofpolymer made with the linear-catalyst. Substituting Equa-tion (42) and (43) into Equation (51):

y0;T ¼kp MC

K þ kLCBQþ sþ kp;L MCL

KL þ s

� �1YT

ð52Þ

or,

y0;T ¼C

1rn

þ kLCBQkp M

þ e

þ CL

1rn;L

þ eL

0BB@1CCA 1

YT

ð53Þ

where,

e ¼ skp M

X 0 ð54Þ

eL ¼s

kp;L MX 0 ð55Þ

1rn

¼ Kkp M

ð56Þ

1rn;L

¼ KL

kp;L Mð57Þ

The parameter rn,L is the number-average chain lengthof polymer made with the linear-catalyst, while theparameter rn is the number-average chain length of poly-mer made with the LCB-catalyst in the absence of LCBformation. Therefore, rn,L has a clear physical meaning,while rn is only a reference value: the actual number-aver-age chain length of polymers made with the LCB-catalystwill be always higher than rn.

The dimensionless parameters e and eL are ratios of thereciprocal of the average residence time in the reactor (orspace-velocity) and the turnover frequency for the cata-lysts, both given in units of reciprocal time. The space-velocity is the number of reactor volumes of feed that canbe treated in unit time and the turnover frequency is thenumber of monomer insertions per unit time. Therefore,the parameters e and eL are ratios of reactor and chaingrowth dynamics. Since chain growth dynamics are muchfaster than reactor dynamics for most olefin polymeriza-tions, e and eL are negligible for most polymerization con-ditions. However, they will be kept in the model for thesake of completeness.

Equation (47) and (48) can be rearranged using thedefinitions shown in Equation (54) to (57):

C ¼eþ 1

rn

1þ eþ 1rn

Y ¼ sY ð58Þ

CL ¼eL þ

1rn;L

1þ eL þ1

rn;L

YL ¼ sLYL ð59Þ

Substituting Equation (58) and (59) into Equation (53):

y0;T ¼s

1rn

þ kLCBQkp M

þ e

n

þ sL

1rn;L

þ eL

ð1ÿ nÞ ¼ y0 þ y0;L ð60Þ

where n = YYT

is the molar fraction of the LCB-catalyst inthe reactor and consequently 1 – n = YL

YT.

Equation (60) can be further simplified using Equation(49) for Q:

kLCBQkp M

¼ kLCB

kp MK¼Y þ K¼L YL

kLCBY þ s

¼ K¼

kp MkLCBY

kLCBY þ s1þ K¼L

K¼YL

Y

� �ð61Þ


Defining three new dimensionless parameters,

v ¼ K¼LK¼

ð62Þ

c ¼ Ks

K¼ð63Þ

l ¼ kLCBYkLCBY þ s

¼ n

nþ skLCBYT

ð64Þ

Equation (61) becomes:

kLCBQkp M

¼ l

rnðcþ 1Þ 1þ v1ÿ n

n

� �� ð65Þ

Substituting Equation (65) into Equation (60), onefinally obtains:

y0;T ¼ y0 þ y0;L ¼sn

1rn

1þ l

cþ 11þ v

1ÿ nn

� �� þ e

þ sLð1ÿ nÞ1

rn;L

þ eL

ð66Þ

The three dimensionless parameters defined by Equa-tion (62) to (64) have important physical meanings: v isthe ratio of the rate of macromer formation by the linear-catalyst to the rate of macromer formation by the LCB-catalyst; c is the ratio of the rate of saturated dead chainformation to the rate of macromer formation by the LCB-catalyst; l is similar to e, but with respect to the turnoverfrequency for macromer insertion. When v = 1 the linear-catalyst makes as many macromers as the LCB-catalysts.When c = 1, the LCB-catalyst makes 50% of saturateddead chains and 50% of macromers. It will be shown laterthat in the limiting case when l = 1 (i.e., kLCBY S s), thedegree of LCB reaches a maximum value for these poly-mers.

These dimensionless parameters are very useful forsimulation studies, since very often the values of the indi-vidual polymerization kinetics parameters are unknown.On the other hand, their ratios can be more easilyobtained. For instance, both v and c can be obtained fromthe 13C NMR analysis of chain ends and reactor produc-tivity data.

The molar fraction of living chains with one LCB perchain is calculated from Equation (44) setting n = 1:

P1 ¼kLCBP0 D0

K þ kLCB Qþ s¼ kLCBðy0YTÞðx0 QÞ

K þ kLCB Qþ s

¼ kLCBQy0x0

K þ kLCB Qþ sYT ð67Þ

The variable x0 is the molar fraction of linear macro-mer, with respect to the total concentration of macromerin the reactor. Expressions for the calculation of x0, x1, x2,..., xn will be derived later.

Therefore,

y1 ¼P1

YT

¼

kLCBQkp M

Kkp M

þ kLCBQkp M

þ skp M

y0x0

¼

l


n

� �� 1rn

1þ l

cþ 11þ v

1ÿ nn

� �� þ e

y0x0

¼ Uy0x0 ð68Þ

Evidently a clear pattern emerges for the other molarfractions:

y2 ¼ Uðy0x1 þ y1x0Þ ð69Þ

y3 ¼ Uðy0x2 þ y1x1 þ y2x0Þ ð70Þ

yn ¼ UXnÿ1

i¼0

yixnÿiÿ1 ; n F 1 ð71Þ

where,

U ¼

l


n

� �� 1rn

1þ l

cþ 11þ v

1ÿ nn

� �� þ e

ð72Þ

The molar fraction of linear macromer with respect tothe total concentration of macromer in the reactor is cal-culated from Equation (38) and (49):

x0 ¼D0

Q¼ K¼P0 þ K¼L p

K¼Y þ K¼L YL

¼

P0

YT

þ K¼LK¼

p

YT

YYT

þ K¼LK¼

YL

YT

¼ y0 þ vy0;L

nþ vð1ÿ nÞ ð73Þ

Similarly, for branched macromers,

xn ¼Dn

Q¼ K¼Pn

K¼Y þ K¼L YL

¼ yn

nþ vð1ÿ nÞ ; n F 1 ð74Þ

Equation (73) and (74) are used with Equation (67) to(71) to solve for the molar fraction of living chains in thereactor.

190 J. B. P. Soares

The molar fractions of dead polymer chains with differ-ent degrees of LCB are the most important results in thissimulation, since they account for the majority of thechains present in the reactor. The fraction of linear deadpolymer chains is defined as:

X0 ¼D0 þ Ds

0

Qþ Qsð75Þ

Substituting Equation (38), (40), (49), and (50) intoEquation (75) and after some algebraic manipulationsone obtains:

X0 ¼y0ð1þ cÿ lÞ þ y0;Lvð1þ cL ÿ lÞ

nð1þ cÿ lÞ þ vð1ÿ nÞð1þ cL ÿ lÞ ð76Þ

where the new dimensionless parameter, cL, is the ratio ofthe rate of saturated dead chain formation to the rate ofmacromer formation by the linear-catalyst:

cL ¼Ks

L

K¼Lð77Þ

A similar equation can be derived for the molar frac-tion of branched dead chains:

Xn ¼Dn

Qþ Qs

¼ ynð1þ cÿ lÞnð1þ cÿ lÞ þ vð1ÿ nÞð1þ cL ÿ lÞ ð78Þ

The equations developed in this section can be solvedto obtain the molar fractions of all species present in thereactor. Table 1 summarizes these equations and Table 2lists the definitions of all dimensionless parameters usedin the model.

Model Simplifications

Table 1 lists the equations for the general model devel-oped above, together with some model simplificationsthat might be applicable to specific cases.

The first logical simplification is to set e = eL = 0. Asmentioned before, the parameters e and eL are ratios of the

Table 1. Model equations for the general solution and some particular cases.

General e = 0 e = c = cL = 0 e = c = cL = 0,l = 1

y0,LsLð1ÿ nÞ

1rn;L

þ eL

1 – n 1 – n 1 – n

y0

sn1rn

1þ l

cþ 11þ v

1ÿ nn

� �� þ e

n

1þ l

cþ 11þ v

1ÿ nn

� �� n

1þl 1þ v1ÿ n

n

� �� n

2þv1ÿ n

n

� �

yn UXnÿ1

i¼0

yixnÿiÿ1 UXnÿ1

i¼0

yixnÿiÿ1 UXnÿ1

i¼0

yixnÿiÿ1 UXnÿ1

i¼0

yixnÿiÿ1

U

l


n

� �� 1rn

1þ l

cþ 11þ v

1ÿ nn

� �� þ e

l

cþ 11þ v

1ÿ nn

� �� 1þ l

cþ 11þ v

1ÿ nn

� �� l 1þ v1ÿ n

n

� �� 1þl 1þ v

1ÿ nn

� �� 1þv1ÿ n

n

� �2þv

1ÿ nn

� �

x0y0 þ vyL;0

nþ vð1ÿ nÞy0 þ vy0;L



nþ vð1ÿ nÞ

xn

yn

nþ vð1ÿ nÞyn

nþ vð1ÿ nÞyn

nþ vð1ÿ nÞyn

nþ vð1ÿ nÞ

X0y0ð1ÿ lþ cÞ þ y0;Lvð1ÿ lþ cLÞ

nð1ÿ lþ cÞ þ vð1ÿ nÞð1ÿ lþ cLÞy0ð1ÿ lþ cÞ þ y0;Lvð1ÿ lþ cLÞ

nð1ÿ lþ cÞ þ vð1ÿ nÞð1ÿ lþ cLÞy0ð1ÿ lÞ þ y0;Lvð1ÿ lÞ

nð1ÿ lÞ þ vð1ÿ nÞð1ÿ lÞy0 þ y0;Lv

nþ vð1ÿ nÞ

Xnynð1ÿ lþ cÞ

nð1ÿ lþ cÞ þ vð1ÿ nÞð1ÿ lþ cLÞynð1ÿ lþ cÞ

nð1ÿ lþ cÞ þ vð1ÿ nÞð1ÿ lþ cLÞynð1ÿ lÞ

nð1ÿ lÞ þ vð1ÿ nÞð1ÿ lÞyn

nþ vð1ÿ nÞ


space-velocity to the turnover frequency of the catalyst.In the majority of cases kp M, kp,L M S s for olefin poly-merization reactors, therefore this is a very good assump-tion. Table 1 (2nd column) shows that, as a consequenceof this simplification, the chain-length term, rn, drops outof the model equations. This is also expected, since chainlength should not have any appreciable effect on themolar fractions of the different polymer populations inthe reactor.

In order to maximize LCB per chain, ideally all deadchains should be macromers, i.e., Ks = Ks

L = 0. Conse-quently, c = cL = 0. Table 1 (3rd column) also shows howthe model equations are reduced for the case whenc = cL = 0 and e = eL = 0. This is evidently a limiting solu-tion, since most catalysts will not have a 100% selectivitytowards vinyl unsaturations.

Finally, if besides all these simplifications one assumesthat kLCBY S s then l = 1, and the model equations are

reduced to the form shown in the last column of Table 1.This solution corresponds to the maximum possible LCBformation achievable with a combination of one LCB-catalyst and one –linear-catalyst. However, this conditionis unlikely to be reached in practice, since it implies thatthe turnover frequency for LCB insertion is much largerthan the space-velocity in the reactor. This can only beobtained at very large average residence times, i.e.,s e 0, or at very high catalyst concentration. Both condi-tions lead to very high macromer concentration inside thereactor, and thus to maximum LCB incorporation. Thus,this condition is only useful as a maximum limit condi-tion for LCB formation.

Equations to Calculate Average LCB

The average number of LCB per 1000 carbon atoms (C),k, can be calculated from the expression:

Table 2. Definition of model parameters.

Parameter Definition Equation Notes

es

kp M(54) Ratio of reactor to chain growth dynamics for LCB-catalyst.

(Generally e X 0)

eL

skp;L M

(55) Ratio of reactor to chain growth dynamics for linear-catalyst.(Generally eL X 0)

1rn

Kkp M

(56) Reciprocal of the number-average chain length of polymermade with the LCB-catalyst in absence of LCB formation.

1rn;L

KL

kp;L M(57) Reciprocal of the number-average chain length of polymer

made with the linear-catalyst.

seþ 1

rn

1þ eþ 1rn

(58)

sL

eL þ1

rn;L

1þ eL þ1

rn;L

(59)

vK¼LK¼

(62) Ratio of the rate of macromer formation by the linear catalystto that of the LCB-catalyst.

cKs

K¼(63) Ratio of the rate of saturated dead chain formation to the rate

of macromer formation by the LCB-catalyst.

cLKs

L

K¼L(77) Ratio of the rate of saturated dead chain formation to the rate

of macromer formation by the linear-catalyst.

ln

nþ skLCBYT

(64) The degree of LCB reaches a maximum when l = 1.

192 J. B. P. Soares

k ¼ 500kLCBQY

kp MY þ kp;L MYL

¼ 500kLCB Qkp M

1

1þ kp;L MYL

kp MY

¼ 500l

rnðcþ1Þ 1þv1ÿ n

n

� �� 1

1þ kp;L MYL

kp MY

ð79Þ

However,

1

1þ kp;L MYL

kp MY

¼ 1

1þ Rp;L

Rp

¼ Rp

RpþRp;L

¼ m ð80Þ

where m is the molar fraction of polymer made by theLCB-catalyst, and Rp and Rp,L are the rates of polymeriza-tion of the LCB-catalyst and linear-catalyst, respectively.Therefore:

k ¼ 500l


n

� �� m ð81Þ

The relation between m and n is easily derived if oneknows the ratio of monomer propagation rate constantsfor both catalysts:

m ¼ 1

1þ kp;L YL

kp Y

¼ 1

1þ kp;L

kp

1ÿ nn

� � ð82Þ

The average number of LCB per polymer chains, B, iscalculated from the equation:

B ¼ kLCBQYKY þ KLYL ÿ kLCBQY

¼ l½nþ ð1ÿ nÞv�nð1þ cÿ lÞ þ vð1ÿ nÞðcL þ 1ÿ lÞ ð83Þ

Results and Discussion

Limiting Solution for LBC-Catalyst Only

It is interesting to examine what would be the limitingsolution (maximum LCB) for the case of a single LCB-catalyst in a CSTR. In this case, e = c = 0 and l = 1, i.e.,the LCB-catalyst produces only vinyl-terminated deadchains (macromers), and the residence time in the reactoris very long (s e 0); conditions that lead to maximumconcentration of macromers in the reactor. Equations forthis model are obtained from the last columns of Table 1,setting n = 1. Therefore:

y0 ¼12

ð84Þ

y1 ¼12

y0x0 ¼12ðy0Þ2 ¼

18

ð85Þ

Notice that x0 = y0 when only one catalyst is present inthe reactor.

Similarly:

y2 ¼12ðy0x1 þ y1x0Þ ¼

12ðy0y1 þ y1y0Þ

¼ 12

218

12

� �¼ 1

16ð86Þ

Molar fractions of chain with more LCB are easilyobtained with the general expression:

yn ¼12

Xnÿ1

i¼0

yiynÿiÿ1 ; for n F 1 ð87Þ

For polymerization with the LCB-catalyst, the weightfractions of the different polymer populations can also becalculated. Soares and Hamielec[10] showed that for poly-mers made with a LCB-catalyst, the number-averagelength of chains with i LCB, rn,i , is related to the number-average length of the linear chains, rn,0 , by the equation:

rn;i ¼ ð1þ 2iÞrn;0 ð88Þ

Therefore,

wj ¼yj rn;jXv

i¼0

yi rn;j

¼ yjð1þ 2iÞXvi¼0

yið1þ 2iÞð89Þ

where wj is weight fraction of polymer with n LCB perchain.

From Equation (81), LCB/1000 C becomes:

k ¼ 500rn

ð90Þ

Equation (90) sets a useful theoretical upper limit forthe achievable LCB/1000 C for a polymer made with asingle-site catalyst with LCB formation via terminalbranching.

Interestingly, the number of LCB per chain tends toinfinity for this limiting case, as described by Equation(83):

B ¼ 11ÿ 1

e v ð91Þ

This “anomaly” happens because the steady-stateassumption made when deriving the model equationsimplies that there is no accumulation of macromer in thereactor. Consequently, when s = 0 (no flow out of thereactor) most of the macromers produced must be incor-porated in the growing polymer chains, leading to chainswith an infinite number of LCB. It is clear that this limit-ing case cannot be achieved in practice, but it is illustra-


tive to have it set as a limiting condition for these cata-lysts.

It is also interesting to compute the molar, qn, and massratios, xn, of each polymer population with respect to thelinear chains, i.e.:

qn ¼yn

y0

¼Xnÿ1

i¼0

yiynÿiÿ1 ð92Þ

xn ¼wn

w0¼ ynð1þ 2iÞ

y0¼ ð1þ 2iÞqn ð93Þ

Equation (93) indicates that weight contribution of thebranched populations is greater than their molar contribu-tion, as expected. Table 3 shows values of yn, qn and xn

for chain populations of up to 10 LCB/chain for this lim-iting case. Thus, polymer populations made with anLCB-catalyst have a very rigid branching structure.Methodologies to affect this branching structure will beinvestigated next.

Simulation Results for a Single LCB-catalyst

Figure 1 and 2 show simulation results for polymeriza-tions with a single LCB-catalyst. These results agree withthe ones presented by Soares and Hamielec[10] and areshown here for comparison with the dual metallocenesystems discussed later and to introduce the effect of twoimportant model parameters, c and l.

Figure 1a shows how the molar fractions of chains con-taining 0 to 5 LCB vary as a function of the parameter l.For the case of a single catalyst, the expression for l

reduces to:

l ¼ 1

1þ skLCBY

ð94Þ

As discussed in the previous section, l = 1 indicatesinfinite residence time in the reactor and maximumbranching, as clearly seen in Figure 1a. For l = 1, themolar fractions for the polymer populations are thoseshown in Table 3. It is interesting to see how, as l

decreases (reactor residence time decreases for a givencatalyst), the fraction of chains with no LCB increases.Figure 1b shows a similar result for the weight fractions

Table 3. Fractions of linear and branched chains for the limit-ing case of one LCB-catalyst with e = c = 0 and l = 1.

LCB/chain yn qn xn

0 12

1 18

14

34

2 116

18

58

3 5128

564

3564

4 7256

7128

56128

5 211 024

21512

231512

6 332 048

331 024

4291 024

7 42932 768

42916 384

6 43516 384

8 71565 536

71532 768

12 15532 768

9 2 431262 144

2 431131 072

46 189131 072

10 4 199524 288

4 199262 144

88 179262 144

Figure 1. The effect of the parameter l on the branching struc-ture of a polymer made with a LCB-catalyst. Simulation param-eters: n = 0, c = 0, e = 0, rn = 4000. (a) molar fraction, (b) weightfraction.

194 J. B. P. Soares

of polymers with different numbers of LCB. As expected,branched chains have a more significant contribution tothe total mass of the polymer.

Figure 2a and 2b examine the effect of the parameter con the molar and mass fractions of chains containing 0 to5 LCB. As expected, the maximum LCB is obtainedwhen c = 0, i.e., when all dead polymer chains are macro-mers.

It is clear from the results presented in Figure 1 and 2that the LCB structure of polymers made with a singleLCB-catalyst is somewhat inflexible. In the limiting casestudied in the previous section, 1/2 of all chains are lin-ear, 1/8 contain only one LCB, 1/16 contain two LCB,etc. It will be shown in the next section that combining anLCB-catalyst with a linear-catalyst can provide an effi-cient way of affecting the LCB distribution of these poly-mers.

Simulation Results for a Mixture of a LCB-Catalystand a Linear-Catalyst

Table 4 lists the simulation parameters used to generatethe results shown in Figure 3 to 5. These figures investi-gate the effect of several reactor parameters on the frac-tion of dead polymer chains, X, containing 1 to 5 LCB.

Figure 2. The effect of the parameter c on the branching struc-ture of a polymer made with a LCB-catalyst. Simulation param-eters: n = 0, l = 0.8, e = 0, rn = 4000. (a) molar fraction, (b)weight fraction.

Table 4. Simulation parameters.

Parameters Figure 3 Figure 4 Figure 5

v 1, 2, 4 1, 2, 4, 10 4c 0 1 1cL 0 0 0

s/(kLCBYT) 1 1 0.1, 0.5, 2e 0 0 0eL 0 0 0rn 4000 4000 4000rn,L 4000 4000 4000

kp,L /kp 1 1 1

Figure 3. The effect of the parameter v on the branching struc-ture of a polymer made with an LCB-catalyst and a linear-cata-lyst with the same rates of macromer generation (c = cL). Simu-lation parameters: c = cL = 0, e = eL = 0, rn = rn,L = 4000, kp,L /kp = 1, s/(kLCBYT) = 1. (a) v = 1, (b) v = 2, (c) v = 4.


(The molar fraction of linear chains is not shown to avoidovercrowding the figures. It is evident that X0 = 1 – RXi.)Notice that e = eL = 0 for all simulations, which is a rea-sonable assumption, as explained before. In this case, thevalues of rn and rn,L only affect the value of k. Addition-ally, kp,L /kp was kept at 1 for all simulations, since thisparameter only affects the value of k, as indicated inEquation (81) and (82).

Figure 3 studies the case when both the LCB-catalystand the linear-catalyst produce only macromers(c = cL = 0). Figure 3a shows that when both catalystsproduce the same amount of macromers (v = 1) the molarfraction of the branched species increases monotonicallyfrom n = 0 to 1. This is to be expected, since replacingthe LCB-catalyst with a linear-catalyst that makes thesame amount of macromer will not increase the overall

Figure 4. The effect of the parameter v on the branching structure of a polymer made with an LCB-catalyst and a linear-catalystwhen the linear-catalyst is more selective than the LCB-catalyst towards macromer generation (c A cL). Simulation parameters: c = 1,cL = 0, e = eL = 0, rn = rn,L = 4000, kp,L /kp = 1, s/(kLCBYT) = 1. (a) v = 1, (b) v = 2, (c) v = 4, (d) 1st derivatives for v = 4, (e) v = 10, (f)1st derivative for v = 10.

196 J. B. P. Soares

macromer concentration and thus lead to lower LCBaverages.

When the linear-catalyst produces twice as manymacromers as the LCB-catalyst (v = 2), LCB/1000 Cpasses through a maximum (Figure 3b, insert). Thisbehavior has been predicted theoretically and observedexperimentally by Beigzadeh et al.[16–18] Interestingly,LCB/chain still increases monotonically from n = 0 to 1.This has some important implications: rheological and

mechanical properties are probably more affected byLCB/chain than by LCB/1000 C. Therefore, this LCB“maximization” might not lead to improved polymerproperties. Figure 3c shows that the same behavior isrepeated when the linear-catalysts makes 4 times moremacromer than the LCB-catalyst (v = 4). In this case, thefraction of the branched chains decreases more rapidlywith decreasing molar fraction of the LCB-catalyst in thereactor. The same behavior is observed for any other cat-alyst combination in which the LCB-catalyst and the lin-ear-catalyst have the same ratio of macromer to saturateddead-chain production, i.e., for any case when c = cL, butare not shown here for brevity.

This picture changes when a linear-catalyst that ismore selective towards macromer formation than theLCB-catalyst is used, i.e., when cL a c. Figure 4a to 4fshow simulation results for the case when the LCB-cata-lyst produces the same amount of macromer and deadsaturated chains (c = 1), while the linear-catalyst pro-duces only macromers (cL = 0). For this case, when v = 1(Figure 4a), the fractions of branched polymer dependless strongly on the fraction of LCB-catalyst in the sys-tem. Since the linear-catalyst is more selective towardsthe production of macromers, the replacement of part ofthe LCB-catalyst with the linear-catalyst actuallyincreases the total macromer concentration in the reactor.When v is increased to 2 and to 4 (Figure 4b and 4c), themolar fractions of all polymer populations pass throughmaximum values.

This seems to be a very powerful strategy to maximizeLCB formation in these systems, since both LCB/1000 Cand LCB/chain can be maximized (see inserts in Fig-ure 4b and 4c). Additionally, since each polymer popula-tion has a maximum located at a distinct n value, it is pos-sible, at least in theory, to maximize a given polymerpopulation over the others. This is more clearly shown inFigure 4d, where the 1st derivatives of the curves shownin Figure 4c with respect to the molar fraction of theLCB-catalyst (dX/dn) are plotted as a function of the frac-tion of LCB-catalyst. For this particular combination,chains with 1 LCB are maximized when n L 0.86, chainswith 2 LCB when n L 0.75, chains with 3 LCB when n L

0.65, etc. Analogous results for the case when v = 10 areshown in Figure 4e and 4f. These findings are generallyapplicable to any catalyst combination when cL a c.

Finally, Figure 5a to 5c show how the ratio of space-velocity to macromer insertion affects the LCB structureof these polymers. Low space-velocities (high residencetimes) lead to increased macromer concentration and thusto higher LCB incorporation. It is interesting to see thatincreasing the residence time will also affect the positionof the LCB maximum for the several populations. Forinstance, when s/(kLCBYT) varies from 0.1 to 2, the maxi-mum fraction of chains with 1 LCB is achieved when nchanges from 0.78 to 0.85.

Figure 5. The effect of the ratio s/(kLCBYT) (see definition of l,Equation (64)) on the branching structure of a polymer madewith an LCB-catalyst and a linear-catalyst when the linear-cata-lyst is more selective than the LCB-catalyst towards macromergeneration (c A cL). Simulation parameters: v = 4, c = 1, cL = 0,e = eL = 0, rn = rn,L = 4000, kp,L /kp = 1. (a) s/(kLCBYT) = 0.1, (b) s/(kLCBYT) = 0.5, (c) s/(kLCBYT) = 2.


ConclusionsA simple mathematical model was developed to describethe populations of polymer chains containing differentnumber of LCB made with a combination of two single-site catalysts. One of the catalysts produces only linearchains (linear-catalyst) and the other produces linear andlong-branched chains (LCB-catalyst).

The mathematical model contains three dimensionlessparameters for the linear-catalyst (eL, rn,L, and cL), four forthe LCB-catalyst (e, rn, c and l) and a parameter thatquantifies the relative rate of macromer formation by thetwo catalysts (v). The parameters e and eL are negligiblefor most polymerization conditions, in which case rn andrn,L will influence only the value of k (LCB/1000 C) butnot the rest of the simulation. The remaining four param-eters can be obtained from 13C NMR analysis of polymermade with each catalyst individually.

This model can be used to investigate polymerizationconditions and catalyst combinations that will maximizethe formation of a given polymer population. It wasshown that when the selectivity for macromer formationof the linear-catalyst is the same as that of the LCB-cata-lyst (c = cL), it is not possible to maximize LCB/chain,even though the number of LCB/1000 C can be maxi-mized. On the other hand, if the selectivity for macromerformation of the linear-catalyst is higher than that of theLCB-catalyst (c A cL), both LCB/1000 C and LCB/chainpass through maxima when varying the fraction of linear-catalyst in the reactor. More importantly, each polymerpopulation will reach its maximum value at a differentratio of linear-catalyst to LCB-catalyst, thus permittingthe maximization of individual polymer populations inthe mixture.

One of our main objectives in this development was toobtain a model that was simple enough to be implemen-ted easily in a spreadsheet and to be used in catalystscreening studies. In order to achieve this objective, thismodel does not calculate molecular weight that is, ofcourse, very important for determining the properties ofthese polymers. A complete model for these catalystscombinations, including molecular-weight predictions, isunder development in our group and will be publishedshortly.[19]

NomenclatureB average number of LCB per chainC LCB-catalyst monomer-free active sitesCL linear-catalyst monomer-free active sitesDi macromer with i LCBDs

i saturated dead polymer chain with i LCBK defined in Equation (26)KL defined in Equation (27)K = defined in Equation (17)

K¼L defined in Equation (18)Ks defined in Equation (19)Ks

L defined in Equation (20)kAl transfer to aluminum rate constant for LCB-catalystkAl,L transfer to aluminum rate constant for linear-cata-

lystkb b-hydride elimination constant for LCB-catalystkb,L b-hydride elimination constant for linear-catalystkH transfer to hydrogen rate constant for LCB-catalystkH,L transfer to hydrogen rate constant for linear-catalystkLCB LCB formation rate constantkM transfer to monomer rate constant for LCB-catalystkM,L transfer to monomer rate constant for linear-catalystkp propagation rate constant for LCB-catalystkp,L propagation rate constant for linear-catalystM monomerm molar fraction of polymer made with the LCB-cata-

lystn molar fraction of LCB-catalyst in the mixture, num-

ber of LCBP0 linear chain growing on LCB-catalystPi polymer chains with i LCB growing on the LCB-

catalystQ total concentration of macromer, Equation (25)Qs total concentration of saturated dead chainsrn number-average chain length of polymer made with

LCB-catalyst in absence of LCB formationrn,i number-average chain length of polymer with i

LCB (for single LCB-catalyst only)rn,L number-average chain length of polymer made with

linear-catalysts reactor space-velocityXi molar fraction of dead polymer chains (macromer

and saturated) with i LCBxi molar fraction of macromer with i LCB calculated

with respect to the total concentration of macromerwi weight fraction of polymer with i LCB (for single

LCB-catalyst only)Y total number of active sites for LCB-catalyst, Equa-

tion (15)YL total number of active sites for linear-catalyst,

Equation (33)_YYin molar flow rate of LCB-catalyst fed to the reactor_YYL;in molar flow rate of linear-catalyst fed to the reactory0 molar fraction of living chains growing on the LCB-

catalysty0,L molar fraction of living chains growing on the lin-

ear-catalysty0,T molar fraction of all living chainsyi molar fraction of living chains with i LCB

Greek Letters

v model parameter, Equation (62)e model parameter, Equation (54)

198 J. B. P. Soares

eL model parameter, Equation (55)H model parameter, Equation (72)c model parameter, Equation (63)cL model parameter, Equation (77)k average number of LCB per 1000 C atomsl model parameter, Equation (64)xn defined in Equation (93)p linear chain growing on linear-catalystqn defined in Equation (92)

Received: September 5, 2001Revised: November 15, 2001

Accepted: November 27, 2001

[1] US 5272236 (1993), Dow Chemical, invs: S. Y. Lai, J. R.Wilson, G. W. Knight, J. C. Stevens, P. W. S. Chum.

[2] US 5665800 (1997), Dow Chemical, invs: S. Y. Lai, J. R.Wilson, G. W. Knight, J. C. Stevens.

[3] P. Walter, S. Trinkle, J. Suhm, D. Mäder, C. Friedrich, R.Mülhaupt, Macromol. Chem. Phys. 2000, 201, 604.

[4] E. Kokko, A. Malmberg, P. Lehmus, B. Löfgren, J. Sep-pälä, J. Polym. Sci., Part A: Polym. Chem. 2000, 38, 376.

[5] A. Malmberg, J. Liimatta, A. Lehtinen, B. Löfgren, Macro-molecules 1999, 32, 6687.

[6] W. Weng, E. J. Markel, A. Dekmezian, Macromol. RapidCommun. 2000, 21, 1103.

[7] E. J. Markel, W. Weng, A. J. Peacock, A. Dekmezian,Macromolecules 2000, 33, 8541.

[8] T. Shiono, S. M. Azad, T. Ikeda, Macromolecules 1999,32, 5723.

[9] D. Beigzadeh, J. B. P. Soares, T. A. Duever, Macromol.Symp. 2001, 173, 179.

[10] J. B. P. Soares, A. E. Hamielec, Macromol. Theory Simul.1996, 5, 547.

[11] J. B. P. Soares, A. E. Hamielec, Macromol. Theory Simul.1997, 6, 591.

[12] H. Yannoulakis, A. Yiagopoulos, P. Pladis, C. Kiparissidis,Macromolecules 2000, 33, 2757.

[13] D. J. Read, T. C. B. McLeish, Macromolecules 2001, 34,1928.

[14] D. Beigzadeh, J. B. P. Soares, T. A. Duever, A. E. Hamie-lec, Polym. React. Eng. 1999, 7, 195.

[15] European Patent Application 94309546.3 (1994), UnionCarbide, invs: F. J. Karol, E. P. Wasserman, S. C. Kao, R.C. Brady.

[16] D. Beigzadeh, J. B. P. Soares, T. A. Duever, Macromol.Rapid Commun. 1999, 20, 541.

[17] D. Beigzadeh, J. B. P. Soares, A. E. Hamielec, J. Appl.Polym. Sci. 1999, 71, 1753.

[18] D. Beigzadeh, J. B. P. Soares, A. E. Hamielec, Polym.React. Eng. 1997, 5, 143.

[19] L. C. Simon, J. B. P. Soares, Macromol. Theory Simul.2002, 13, 222.

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