modeling of slurry polymerization of ethylene using a soluble cp2zrcl2/mao catalytic system

Full Paper

Modeling of Slurry Polymerization of EthyleneUsing a Soluble Cp2ZrCl2/MAO Catalytic System

Mostafa Ahmadi, Mehdi Nekoomanesh,* Roghieh Jamjah,Gholamhossein Zohuri, Hassan Arabi

The slurry homopolymerization of ethylene catalyzed by a Cp2ZrCl2/MAO catalytic systemwasstudied. A simple kinetic model including initiation, propagation, transfer to monomer andcocatalyst, spontaneous transfer and spontaneous deactivation was developed to predictdynamic yield of polymerization and molecularweight of final products. Kinetic constants wereestimated by numerical solution of polymeriza-tion kinetic model, combined with Nelder-Meadsimplex method. The model predicts that thepropagation reaction has the lower activationenergy in relation to chain transfer reactionswhich leads to decrease of molecular weight atelevated temperatures. The initiation reactionhas however, the highest activation energy thatresults in raising the peak of reaction rate athigher temperatures.

Introduction

The development of metallocene/aluminoxane catalytic

systems had a revolutionary effect on the production of

polyolefins. Themain advantages ofmetallocene catalysts,

such as extremely high activity and the ability to produce

polyolefins with controlledmicrostructure, are responsible

M. Ahmadi, M. Nekoomanesh, R. Jamjah, H. ArabiDepartment of Catalyst, Faculty of Polymerization Engineering,Iran Polymer and Petrochemical Institute, Pazhoohesh Blvd.,km 17, Tehran-Karaj Hwy, Tehran, P. O. Box: 14965/115, 14185/458,Postal code: 1497713115, I. R. IranFax: 00-98-2144580021-3; E-mail: [email protected]. ZohuriChemistry Group, Department of Science, Ferdowsi University,91775- Mashhad, I. R. Iran

Macromol. Theory Simul. 2007, 16, 557–565

� 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

for the growing interest of academic and industrial

researchers toward them. The structure of these catalysts

can be modified to give a number of active center types by

variation of: transition metal, ligand, bridge and sub-

stitutes on ligands and bridges.[1] This controllable catalyst

structure allows the formation of unique active centers

compared to traditional Ziegler-Natta catalysts, which

have several types of active centers with a very low level of

control on their characteristics. This variety of active

centers leads to complexity of kinetics and difficulty in

controlling the properties of the final product.

Because each active center has its own rate constants

and gives polymer chains with special structures, the bulk

polymer made with heterogeneous Ziegler-Natta catalysts

is in fact a mixture at molecular level. Therefore

investigation of ethylene polymerization with soluble

metallocene catalysts has many advantages such as:

DOI: 10.1002/mats.200700010 557

M. Ahmadi, M. Nekoomanesh, R. Jamjah, G. Zohuri, H. Arabi

558

(1) considering only one type of active center being

sufficient for quantitative study of polymerization

kinetics, while even in polymerization using supported

metallocene catalysts more than one type is involved;[2–4]

(2) non-encapsulation of catalyst particles by polymer

molecules which usually occurs in supported catalysts,

making mass and heat transfers difficult;[5,6] (3) simple

kinetic mechanism, that becomes complex in polymeriza-

tion of asymmetrical monomers;[7,8] and, (4) formation of

unique polymer structure that facilitates its characteriza-

tion and interpretation.

Olefin polymerization kinetics has been investigated for

a wide variety of polymerization systems, but in most

cases the kinetic constants are not estimated.[9–11] In some

works, estimates of kinetic constants are based on time

averaged values of reaction rates and final molecular

weights.[12–16] In more recent models, the kinetic para-

meters have been estimated using instantaneous reaction

rates and end of batch molecular weights.[4,7,8,17–32]

Mathematical modeling techniques for olefin polymer-

ization are well established, but the system’s predictabil-

ities are often weak because of uncertainty in model

quantitative parameters. Most mathematical models for

chemical processes are non-linear and complex, involving

different unknown parameters. No general methodology

has been developed that can guarantee good parameter

estimation in such non-linear models. There are vast

numbers of optimization methods that can be applied to

parameter estimation, like Sobol’s systematic search

method, Levenberg-Marquardt algorithm,[33] and ortho-

gonal collocation on finite elements.[7,8] These methods

make use of algebraic objective functions or are coupled

with non-linear differential equations.

Estimation of kinetic parameters from experimental

data is a key point to achieve quantitative predictions and

explain experimental results. The goal of this work is to

investigate slurry polymerization of ethylene using

soluble Cp2ZrCl2/MAO catalytic system and find the best

estimate of kinetic parameters using numerical solution of

kinetic model combined with the Nelder-Mead simplex

method as an optimization algorithm.[34,35]

Experimental Part

Materials

The metallocene catalyst of Cp2ZrCl2 and methylaluminoxane

(10 wt.-% in toluene) was purchased from Sigma-Aldrich Chemie

GmbH, (Steinheim, Germany). Polymerization grade ethylene

with purity of 99.99% supplied by Iran Petrochemical Co. and

nitrogen gas with purity of 99.99% purchased from Roham Co.

(Iran, Tehran) was further purified by passing through columns of

activated 13X and 4 Amolecular sieves and columns of P2O5, KOH,

activated silica gel and 4 Amolecular sieve, respectively. Industrial



toluene, supplied by Iran Petrochemical Co. was further purified

by repeating distillations on sodium wire and benzophenone.

Decaline with purity of 97% was supplied by Merck Schuchardt

OHG (Hohenburn, Germany) and was used with 0.1% antioxidant

(2,6-di-tert-butyl-p-cresol).

Polymerization

Polymerization runs were carried out in a 1 L stainless steel Buchi

reactor model bmd 300. Monomer consumption wasmeasured by

a pressflow gas controller. Reactor temperature was controlled by

water circulation using a Huber circulator model Polysat CC3. The

reactor was purged with nitrogen gas at 90 8C for about 30 min

to ensure the absence of moisture and oxygen. The reactor was

filled with 500 mL toluene and was subsequently evacuated five

times and nitrogen was used to refill it. This procedure was

repeated using ethylene at least five further times. Agitation was

started by a paddle mixer at 800 rpm and the reactor was filled

with ethylene to saturate solvent. MAO was added to the reactor

by means of a syringe. After a few minutes stirring, the catalyst

was introduced in a same manner. After filling the reactor with

ethylene to desired monomer pressure, the reaction was begun by

mixing. At the end of the polymerization time, reaction was

terminated by degassing of the reactor. The viscosity average

molecular weights of polymers were determined by Ubbelohde

viscometer and decaline at 135 8Cwas used as a solvent, where the

constants of the Mark-Houwink equation were k¼6.77�10�2

(mol � g�1) and a¼0.67.[36]

Results and Discussion

Experimental Results

Results of polymerization yields and viscosity average

molecular weights are listed in Table 1. Equilibrium

concentrations of ethylene in toluene, estimated from

the Peng-Robinson equation of state based on an algorithm

presented by Atiqullah et al. are also listed in Table 1.[38]

Figure 1 shows the effects of temperature and pressure

on activity. The results are consistent with previously

reported metallocene catalyzed ethylene polymeriza-

tions.[2,3,12,15,16,39] Figure 1 indicates that activity decreases

at elevated temperatures and increases with pressure. It

also clarifies that pressure is the parameter that most

affects activity. Figure 2 shows the effect of temperature

and pressure on viscosity average molecular weight. As it

was expected Mv decreased with increasing temperature

being the most effective parameter. Details of exper-

imental results and design of experiments can be found

elsewhere.[40] Explanation of these observations requires a

detailed knowledge of polymerization kinetics.

Kinetic Scheme

A simple kinetic model was developed to explain the

available experimental results. The kinetic mechanism is

DOI: 10.1002/mats.200700010

dC�

dt

Modeling of Slurry Polymerization of Ethylene Using a Soluble Cp2ZrCl2/MAO Catalytic System

Table 1. Design of experiments and polymerization results. Polymerization conditions: [Zr]¼ 1.125� 10�6 mol/L Toluene: 500 mL, Stirrerspeed: 800 RPM, reaction time: 1 hr.

Run Standard order T P MAO Ma) Yield Mv

-C bar mol � LS1T 103 mol � LS1T 10 g T10S5

1 9 60 4 5.432 3.51 37.00 2.99

2 3 50 6 5.432 6.13 46.78 3.65

3 7 70 2 5.432 1.42 23.38 1.47

4 4 70 4 3.621 3.08 37.39 2.51

5 14 60 2 3.621 1.66 33.00 2.84

6 5 50 4 7.243 4.02 39.21 3.38

7 12 70 6 5.432 4.74 36.86 2.13

8 11 50 2 5.432 1.94 26.88 3.97

9 1 60 6 7.243 5.38 39.36 3.47

10 8 60 6 3.621 5.38 44.89 3.27

11 10 50 4 3.621 4.02 41.02 4.46

12 13 70 4 7.243 3.08 32.93 2.55

13 6 60 4 5.432 3.51 37.90 2.80

14 2 60 2 7.243 1.66 26.27 2.13

15 15 60 2 5.432 1.66 44.59 2.39

a)M¼monomer concentration.

shown in Table 2. Chain growth is initiated by insertion of

the first monomer to the active center. Chain transfer

reactions include transfer to monomer, transfer to

cocatalyst and spontaneous chain transfer. Each active

center or growing chain can undergo a deactivation

reaction as well.[14]

Figure 1. Effects of temperature and pressure on catalyst activity.



Material balances for active centers, cocatalyst, live

chains and dead chains are given below:

dAl

dt

dP1dt

dPndt

dDn

dt

¼ �ðkiMþ kdÞC� þ ðktcAlþ kbÞl0 (1)

¼ �ktcAll0 (2)

¼ kiC�M � P1ðkpM þ ktc½Al� þ kb þ kdÞ (3)

¼ kpMðPn�1 � PnÞ � ðktmM þ ktcAlþ kb þ kdÞPn (4)

¼ ðktmM þ ktcAlþ kb þ kdÞPn (5)

The method of moments was used to calculate

dynamic production yield and molecular weight, where

ln ¼P1

i¼1 inPi is the nth moment of live chains and

www.mts-journal.de 559


Figure 2. Effects of temperature and pressure on viscosity averagemolecular weight.

560

mn ¼P1

i¼1 inDi is the nth moment of dead chains:

Tabof

Kin

Cha

Cha

Cha

Cha

Spo

Spo

Macrom

� 2007

dl0dt

¼ kiMC� � ðktcAlþ kb þ kdÞl0 (6)

dl1
dt
¼ kiMC� � ðktcAlþ kb þ kdÞl1 þ kpMl0

þ ktmMðl0 � l1Þ (7)

dl
2dt
¼ kiMC� � ðktcAlþ kb þ kdÞl2

þ kpMð2l1 þ l0Þ þ ktmMðl0 � l2Þ (8)

dm
i
dt¼ ðktmMþ ktcAlþ kb þ kdÞli (9)

It is reported thatMv is usuallywithin 20% of theweight

average molecular weight.[41] Number andweight average

molecular weights and polymerization yield were calcu-

le 2. Kinetic model for metallocene catalyzed polymerizationethylene.

etic Model

in initiation C� þM �!ki P1

in propagationPn þM �!kp Pnþ1

in transfer to monomer Pn þM �!ktm Dn þ P1

in transfer to cocatalyst Pn þ Al �!ktc Dn þ P0

ntaneous chain transfer Pn �!kb Dn þ P0

ntaneous deactivation Pn �!kd Dn

ol. Theory Simul. 2007, 16, 557–565

WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

lated from the following equations, where M0 is the

molecular weight of monomer:

Mn ¼ M0l1 þ m1

l0 þ m0

(10)

l2 þ m2
Mw ¼ M0 l1 þ m1
(11)

Yield ¼ M ðl þ m Þ (12)
0 1 1
Parameter Estimation

Kinetic rate constants were estimated from experimental

and calculated instantaneous yields and end of batch

molecular weights. Reaction rate constants were assumed

to have Arrhenius dependence on temperature:

k ¼ k0 exp � Ea

RT

� �(13)

Where Ea is the activation energy and k0 is a

pre-exponential factor. For estimation of each kinetic

constant, its activation energy and pre-exponential factor

should be determined.

Application of the kinetic model requires the initial

concentration of active centers. There is no analytical

method to determine this, because it depends strongly on

cocatalyst concentration, polymerization temperature and

amount of impurity showing the purity of monomer and

other reaction components.[42]

Direct measurement methods include chemical label-

ing[43] and quenched flow.[44] Chien et al. have reported that

two-third of the catalyst rac-Et-(4,5,6,7-H4-1-Ind)2ZrCl2/

MAO became active at 30 8C for MAO/Zr ratios larger than

3 500.[45] Indirect estimation methods are based on kinetic

models, which correlate the initial concentration of active

centers to the number of generated polymer chains at early

stages of polymerization.[8] This assumption gives a big

error especially for supported catalytic systems. Quevedo-

Sanches et al. have reported that 60% of zirconocenes

became active for rac-Et-(Ind)2ZrCl2/MAO at 40 8C and a

MAO/Zr ratio of 3 000, and 30% of zirconocenes became

active for rac-Et-(4,7-Me2-1-Ind)2ZrCl2/MAO at 40 8C for a

MAO/Zr ratio of 1 000.[8]

In our operational procedure, adequate premixing of

catalyst and cocatalyst was not applied, before the starting

of polymerization which prevented production of the

ultimate probable amount of active centers. Considering

the variation of polymerization conditions at each run

affecting concentration of active centers, wewere forced to

estimate active center concentration at each run. This

DOI: 10.1002/mats.200700010


estimation was carried out in the first step of parameters

estimation algorithm to reach the best agreement between

predicted and measured instantaneous polymerization

yields.

Estimation of kinetic parameters was performed in

three steps. The polymerization yield was assumed to be

mainly affected by initiation, propagation, and deactiva-

tion reactions. So first isothermal optimizations were

performed to estimate ki, kp and kd at each temperature

level. Results of runs 2, 6, 8, 11 were used for optimization

at 50 8C, runs 1, 5, 9, 10, 13, 14, 15 for optimization at 60 8Cand runs 3, 4, 7, 12 for optimization at 70 8C. Then pre-

exponential factors and activation energies were calcu-

lated by a simple regression according to equation given

below:

Macrom

� 2007

LnðkÞ ¼ Lnðk0Þ �Ea

RT(14)

The results from the above step were used as the set of

initial guesses for determination of pre-exponential

factors and activation energies of initiation, propagation,

and deactivation reactions at all temperatures, using

instantaneous yields of all runs. An average error was

defined and minimized by variation of six parameters:

v1¼ {ki0, Eai, kp0, Eap, kd0, Ead}. The error was not

normalized to minimize errors corresponding to higher

yields similar to the lower yields:

x1 ¼ 1

Ni

XNi

i¼1

1

Nj

XNj

j¼1

yiðtjÞ � yiðtjÞ�� (15)

Where yiðtjÞis the measured yield at time tj for run i and

yiðtjÞ is the corresponding calculated yield. Ni is the

number of runs and Nj is the number of reported yields at

each run.

In the second step, the viscosity average molecular

weight was considered to be mainly affected by transfer

reactions. Then, similar to the previous step, first transfer

reactions rate constants were found at each temperature.

The initial guesses for activation energies and pre-

exponential factors were calculated by regression. A new

average error was defined and minimized by variation of a

new set of parameters: v2¼ {ktm0, Eatm, ktc0, Eatc, kb0, Eab}.

The error was normalized to reach the same fitness for lower

molecular weights similar to the higher ones:

x2 ¼XNi

i¼1

XNj

j¼1

ðMiv � Mi

vÞ2

Mi2v

(16)

Finally, because the first set of estimation parameters v1has some effects onmolecularweight and the second set of

ol. Theory Simul. 2007, 16, 557–565

WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

estimation parameters v2 has some effects on production

yield, the final optimization problem was solved by a

combined set of variables. The results of the two previous

steps were used as the set of initial guesses for the third

step. An average error was defined by combining two

previous errors usingweighting factors to roughly equalize

their values.

The multivariable non-linear optimization problemwas

solved by the Nelder-Mead simplex method.[34,35] This is a

direct search method which attempts to minimize a

scalar-valued non-linear function of n real variables using

only function values with no information about its

derivatives. At each iteration, the method updates a

non-vanishing volume, which is the convex hull of nþ 1

vertex. One or more test points are computed using simple

geometric transformations (like contraction or expansion)

that often consist of moving the current vertex where the

function value is largest. The function is evaluated at each

of these candidate vertices and new simplex is generated

such that its vertex values satisfy a suitable descent

condition compared to the previous simplex. The iteration

terminates when simplex diameter becomes less than a

specified value.[34,35]

Modeling Results

As explained above, parameter estimation was carried out

in three steps including optimization of kinetic constants

responsible for polymerization yield, optimization of

kinetic constants responsible for molecular weight, and

combined optimization. The results of steps 1 and 2 are

reported in Table 3. It is obvious that each step includes

two levels: isothermal optimization at three temperatures

and non-isothermal optimization for all runs. These results

were used as the initial guesses for step 3 which included

twelve parameters. Results of step 3 and rate constants

at three temperature levels are given in Table 4. Comparing

Table 3 and 4 one can say that initiation and chain transfer

to cocatalyst reactions are strongly interdependent. This is

because we assumed that the produced active center

from chain transfer to cocatalyst is the same as the initial

active centers. The results of pre-exponential factors and

activation energies are in a logical agreement with

previously reported values in different polymerization

systems.[7,8,12,16,21,26–29] The propagation reaction has the

lowest activation energy (7.10� 101 kJ �mol�1) while the

chain transfer and deactivation reactions have higher ones

(Eatm¼ 9.18� 101 kJ �mol�1, Eatc¼ 9.87� 101 kJ �mol�1,

Eab¼ 1.23� 102 kJ �mol�1, Ead¼ 1.08� 102 kJ �mol�1). This

difference results in decreasing molecular weight while

temperature increases. The simulation results for poly-

merization yield are compared with experimental ones in

Figure (3)–(5). The figures show that themodel makes good



Table 4. Parameter estimation results at step 3.

ki kp kd ktm ktc kb

molS1 � sS1 molS1 � sS1 sS1 molS1 � sS1 molS1 � sS1 sS1

ln(k0) 91.96 33.30 31.81 31.60 35.86 40.48

Ea (kJ �molS1) 2.49T 102 7.10T 101 1.08T 102 9.18T 101 9.87T 101 1.23T 102

T (-C)

50 4.93T 10S1 9.82T 102 2.71T 10S4 7.67T 10S2 4.14T 10S1 5.42T 10S3

60 7.96T 100 2.17T 103 9.02T 10S4 2.14T 10S1 1.25T 100 2.14T 10S2

70 1.09T 102 4.58T 103 2.80T 10S3 5.62T 10S1 3.52T 100 7.78T 10S2

Table 3. Parameter estimation results at step 1 and step 2.

T Step 1 Step 2 Runs

ki kp kd ktm ktc kb

-C molS1 � sS1 molS1 � sS1 sS1 molS1 � sS1 molS1 � sS1 sS1

50 7.97T 103 8.63T 102 2.46T 10S4 7.67T 10S2 1.37T 10S5 3.57T 10S3 2,6,8,11

60 1.41T 105 2.77T 103 1.24T 10S3 2.86T 10S1 8.40T 10S4 3.77T 10S2 1,5,9,10

13, 14,15

70 1.01T 105 2.85T 103 1.67T 10S3 4.84T 10S1 6.46T 10S4 5.28T 10S2 3,4,7,12

Final results

ln(k0) 60.42 31.16 27.35 30.89 41.30 40.60 All Runs

Ea (kJ �molS1) 1.69T 102 6.51T 101 9.52T 101 9.00T 101 1.07T 102 1.23T 102

Figure 3. Predicted (solid lines) and measured (dashed lines) yields at 50 8C: run 8(. . .), run 11(- - -).

562

predictions of polymerization

behavior despite difficulties

associated with slow tem-

perature control, which is very

important especially at the

beginning of polymerization.

Both simulation and exper-

imental results show that

with increasing temperature,

peak of reaction rate or high-

est slope of polymerization

yield increases. This is because

the activation energy of the

initiation reaction is the

highest (2.49� 102 kJ �mol�1)

among others. In addition,

the final polymerization rate

decreases, because of the

higher activation energy of

the deactivation reaction


� 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim DOI: 10.1002/mats.200700010


Figure 4. Predicted (solid lines) and measured (dashed lines) yields at 60 8C: run 14(. . .), run 15(- - -).

compared to the propagation reaction. Predicted and

measured viscosity average molecular weights are com-

pared in Figure 6. It is clear that predicted values are very

close to measured molecular weights. Estimated percen-

tages of active zirconocenes, predicted and measured

polymerization yields and viscosity average molecular

weights are listed in Table 5. There was no meaningful

relation between estimated percentages of active zirco-

nocenes and polymerization conditions. This could be due

to variation of environmental errors such as component

impurities and operational mistakes. Modeling results

Figure 5. Predicted (solid lines) and measured (dashed lines) yields a



illustrate that estimated kinetic parameters are successful

in predicting polymerization behavior in selected opera-

tional window.

Conclusion

A kinetic model for semi-batch slurry polymerization of

ethylene using Cp2ZrCl2/MAO as the catalytic system has

been developed. A Box-Behnken design of experiment

with three different temperatures (50, 60 and 70 8C),

t 70 8C: run 3(. . .), run 7(- - -).



Figure 6. Predicted (~) and measured (&) viscosity average molecular weights.

564

pressures (2, 4 and 6 bar), and cocatalyst concentrations

(3.621� 10�3, 5.432� 10�3 and 7.243� 10�3 mol � L�1) was

used to collect themost experimental data from the lowest

number of experiments. A simple kinetic model was

developed according to the experimental operational

Table 5. Estimated percentages of active zirconocenes and predicted and measured polymerization yieweights.

Run Percentage of active

zirconocenes

Experimental

yield

Predicted

yield

Ex

mol-% g g

1 68.00 37.00 38.59

2 86.10 46.78 46.22

3 83.10 23.38 21.68

4 56.70 37.39 32.22

5 87.60 33.00 36.75

6 91.50 39.21 38.54

7 62.50 36.86 32.87

8 61.60 26.88 27.14

9 53.80 39.36 39.57

10 78.10 44.89 43.78

11 68.80 41.02 42.03

12 96.30 32.93 33.12

13 68.10 37.90 38.98

14 63.70 26.27 26.54

15 108.00a) 44.59 45.17

a)Could be due to weighing error.



window. Estimation of kinetic

parameters was performed

using online measurements

of polymerization yield and

end of batchmeasurements of

viscosity average molecular

weight. Multivariable non-

linear optimization problem

was solved using the Nelder-

Mead simplex method, which

is a direct search method with

no need for functional deriva-

tives. Parameter estimation

was performed by combining

numerical solution of kinetic

model with optimization

algorithm, without analytical

solutions of polymerization

yield or molecular weight.

Simulation results were in

acceptable agreement with

experimental data. Simulation shows that the propagation

reaction has the lowest activation energy while chain

transfer reactions have higher activation energies, which

is responsible for drops in molecular weight at elevated

temperatures. The initiation reaction has the highest

lds and viscosity average molecular

perimental

Mv

Predicted

Mv

T105 T105

2.99 3.46

3.65 3.55

1.47 1.78

2.51 2.48

2.84 2.75

3.38 3.48

2.13 2.78

3.97 4.03

3.47 3.76

3.27 3.15

4.46 4.38

2.55 2.50

2.80 3.10

2.13 2.71

2.39 2.75

DOI: 10.1002/mats.200700010


activation energy, which leads to an increase of the

maximum reaction rate at higher temperatures. This

kinetic model and results are restricted to this special

polymerization system and in polymerization of other

monomers with different catalytic systems it could be

more complex.

Acknowledgements: The authors would like to thank Mirzaee forhis valuable comments and Mivehchi for her nice contribution toediting of the text.

Received: January 28, 2007; Revised: April 4, 2007; Accepted: April5, 2007; DOI: 10.1002/mats.200700010

Keywords: ethylene polymerization; kinetic constants; metallo-cene catalysts; parameter estimation; polymerization modeling

[1] J. B. P. Soares, A. E. Hamielec, Polym. React. Eng. 1995, 3, 131.[2] H. Frauenrath, H. Keul, H. Hocker, Macromol. Chem. Phys.

2001, 202, 3543.[3] H. Frauenrath, H. Keul, H. Hocker, Macromol. Chem. Phys.

2001, 202, 3551.[4] P. G. Belelli, M. L. Ferreira, M. H. Lacunza, D. E. Damiani, A.

Brandolin, Poly. Eng. Sci. 2001, 41, 2082.[5] R. A. Hutchinson, C. M. Chien, W. H. Ray, J. App. Poly. Sci.

1992, 44, 1389.[6] G. Fink, B. Tesche, F. Korber, S. Knoke,Macromol. Symp. 2001,

173, 77.[7] R. A. Gonzalez-Ruiz, B. Quevedo-Sanchez, R. L. Laurence,

M. A. Henson, E. B. Coughlin, AIChE J. 2006, 52, 1824.[8] B. Quevedo-Sanchez, J. F. Nimmons, E. B. Coughlin, M. A.

Henson, Macromolecules 2006, 39, 4306.[9] J. M. Vele Estrada, A. E. Hamielec, Polymer 1994, 35, 808.[10] H. Yiannoulahis, A. Yiagopoulos, P. Pladis, C. Kiparissides,

Macromolecules 2000, 33, 2757.[11] J. B. P. Soares, A. E. Hamielec, Polym. Reac. Eng. 1996, 4, 153.[12] P. A. Charpentier, Ph.D. thesis, McMaster University, Hamil-

ton Ontario, Canada 1997.[13] W. J. Wang, D. Yan, S. Zhu, A. E. Hamielec, Macromolecules

1998, 31, 8677.[14] N. P. Khare, Ph.D. thesis, Virginia Polytechnic Institute and

State University, Virginia, USA 2003.[15] M. F. Bergstra, G. Weickert,Macromol. Mater. Eng. 2005, 290,

610.[16] W. Kaminskey, F. Muller, O. Sperber,Macromol. Mater. Eng.

2005, 290, 347.[17] J. Huang, G. L. Rempel, Ind. Eng. Chem. Res. 1997, 36, 1151.



[18] M. Lahelin, E. Kokko, P. Lehmus, P. Pitkanen, B. Lofgren, J.Seppala, Macromol. Chem. Phys. 2003, 204, 1323.

[19] Y. V. Kissin, Makromol. Chem. Macromol. Symp. 1993, 66,83.

[20] Y. V. Kissin, R. I. Mink, T. E. Noelin, J. Polym. Sci., Part A:Polym. Chem. 1999, 37, 4255.

[21] G. B. Meier, G. Weickert, W. P. M. Van Swaaij, J. Polym. Sci.,Part A: Polym. Chem. 2001, 39, 500.

[22] V. Matos, A. G. Matos Neto, J. C. Pinto, J. App. Poly. Sci. 2001,79, 2076.

[23] V. Matos, A. G. Matos Neto, M. Nele, J. C. Pinto, J. App. Poly.Sci. 2002, 86, 3226.

[24] G. C. Han-Adebekun,M. Hamba,W. H. Ray, J. Polym. Sci., PartA: Polym. Chem. 1997, 35, 2063.

[25] M. Hamba, G. C. Han-Adebekun,W. H. Ray, J. Polym. Sci., PartA: Polym. Chem. 1997, 35, 2075.

[26] Z. Gene Xu, S. Chakravarti, W. H. Ray, J. App. Poly. Sci. 2001,80, 81.

[27] S. Chakravarti, W. H. Ray, J. App. Poly. Sci. 2001, 80, 1096.[28] S. Chakravarti, W. H. Ray, J. App. Poly. Sci. 2001, 81, 2901.[29] S. Chakravarti, W. H. Ray, S. X. Zhang, J. App. Poly. Sci. 2001,

81, 1451.[30] B. Kou, K. B. McAuley, C. C. J. Hsu, D. W. Bacon, Macromol.

Mater. Eng. 2005, 290, 537.[31] B. Kou, K. B. McAuley, C. C. J. Hsu, D. W. Bacon, K. Z. Yao, Ind.

Eng. Chem. Res. 2005, 44, 2428.[32] K. Z. Yao, B. M. Shaw, B. Kou, K. B. McAuley, D. W. Bacon,

Poly. Reac. Eng. 2003, 11, 563.[33] J. B. P. Soares, A. E. Hamielec, Polymer 1995, 36, 2257.[34] J. A. Nelder, R. Mead, Computer J. 1965, 7, 308.[35] J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, SIAM J.

Optim. 1998, 9, 112.[36] J. Brandrup, E. H. Immergut, ‘‘Polymer Handbook’’, 4th Ed.

Wiley-VCH Weinheim 1999, p. VII/8.[37] M. Myers, D. C. Montgomery, ‘‘Response Surface Method-

ology: Process and Product Optimization Using DesignedExperiments’’, 2nd Ed. Wiley-VCH, 2002.

[38] A. Atiqullah, H. Hammawa, H. Hamid, Eur. Poly. J. 1998, 34,1511.

[39] G. Zohuri, R. Jamjah, S. Ahmadjo, Iran. Polym. J. 2005, 14,111.

[40] M. Ahmadi, R. Jamjah, M. Nekoomanesh, G. Haghighi, H.Zohuri, Arabi, Iran. Polym. J. Submitted.

[41] G. Odien, ‘‘Principles of Polymerization’’, 4th Ed. Wiley-VCH,2004, p. 22.

[42] J. D. Kim, J. B. P. Soares, G. L. Rempel, J. Polym. Sci., Part A:Polym. Chem. 1999, 37, 331.

[43] Z. Liu, E. Somsook, C. R. Landis, J. Am. Chem. Soc. 2001, 123,2915.

[44] V. Busico, R. Cipullo, V. Esposito,Macromol. Rapid Commun.1999, 20, 116.

[45] J. C. W. Chien, R. Sugimoto, J. Polym. Sci., Part A: Polym. Chem.1991, 29, 459.


modeling of slurry polymerization of ethylene using a soluble cp2zrcl2/mao catalytic system

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