a simplified comprehensive kinetic scheme for modeling of ethylene/1-butene copolymerization using...

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A Simplified Comprehensive Kinetic Schemefor Modeling of Ethylene/1-buteneCopolymerization Using Ziegler-NattaCatalysts

Mostafa Ahmadi,* Mehdi Nekoomanesh, Hassan Arabi

A simplified kinetic scheme of eythylene/a-olefin copolymerization has been developed byadding reactions responsible for the unusual kinetic behavior to a general mechanism. Theestimation of rate constants has been simplified by making physically meaningful initialguesses. Rate constants affecting yield, MWD andcomonomer content have been estimated separ-ately. Experiments were designed to investigatethe effects of each rate constant independently.The obtained rate constants show that the siteswhich are responsible for formation of shortchains with higher 1-butene content are moreactive at the beginning of polymerization, whilethe sites which are responsible for formation oflonger chains with lower 1-butene units are moreactive at the final stages of polymerization.

Introduction

With the discovery of Ziegler-Natta catalysts in the middle

of the fifties, the polymer world experienced a great

revolution. Nowadays polyolefins, especially polyethylene

and poly(propylene), represent the highest fraction of

productionamongallpolymer types,withaproduction rate

of millions of tons per year. This high rate of production is

not only due to the cheapness of the raw materials but is

also the consequence of the ease of processability and the

broadness of mechanical and rheological properties that

M. Ahmadi, M. Nekoomanesh, H. ArabiDepartment of Polymerization Engineering, Iran Polymer andPetrochemical Institute, P.O. Box: 14965/115, Tehran, IranFax: þ982 1 4458 0021-3; E-mail: [email protected]

Macromol. React. Eng. 2010, 4, 135–144

� 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

can be obtained.[1] Quantification of the relationship

between these properties and the structural characteristics

of polymers has been a target since the first appearance of

polymer science.[2–4] Structural characteristics of polymers

can be predicted by polymerization process modeling. A

proper prediction of polymer microstructure through

mathematical models combined with quantitative struc-

ture properties relationships can construct a valuable

control loop that can be used for optimization of operating

conditions, development of new polymer grades and

improvement of a product’s properties for its end-use

application.

Polyethylene is produced in almost all types of reactor

configurations. While the macro-scale modeling differs in

different processes, the micro-scale modeling, which

explains the behavior of the catalyst, is not process

DOI: 10.1002/mren.200900039 135

M. Ahmadi, M. Nekoomanesh, H. Arabi

Table 1. Experimental conditions of the selected polymerization runs and the corresponding characterization results. Temperature¼ 82 8C;[catalyst]¼ 4.8� 10�5 M; stirring rate¼600 rpm; in 500mL hexane.

Run Time PC2 PH2 TEAL C4a) Yield Mw PDI C4

b)

min kPa kPa mol � L�1 g g Da wt.-%

1 15 350 350 1.09� 10�2 0 35.05 58 000 4.02 –

2 30 350 350 1.09� 10�2 0 45.68 65 000 3.38 –

3 60 350 350 1.09� 10�2 0 61.88 72 000 4.30 –

4 120 350 350 1.09� 10�2 0 70.34 70 000 4.88 –

5 60 300 350 1.09� 10�2 0 49.44 61 000 3.92 –

6 60 400 350 1.09� 10�2 0 62.64 69 000 4.23 –

7 60 350 300 1.09� 10�2 0 60.79 80 000 6.39 –

8 60 350 400 1.09� 10�2 0 58.86 66 000 3.88 –

9 60 350 350 5.43� 10�3 0 70.05 73 000 4.20 –

10 60 350 350 1.63� 10�2 0 48.00 – – –

11 60 350 350 1.09� 10�2 5 61.83 55 000 3.91 1.94

12 60 350 350 1.09� 10�2 10 63.51 45 000 3.84 3.24

13 30 350 350 1.09� 10�2 5 55.72 53 000 4.23 1.98

14 120 350 350 1.09� 10�2 5 91.32 60 000 3.83 1.75

a)Initially added amount; b)Final measurement.

136

dependent. Quantitative interpretation of a Ziegler-Natta

catalyst behavior in the copolymerization of ethylene with

a-olefins is the most complicated part of simulation of an

industrial process. This complexity arises from the special

kinetic features of ethylene/a-olefin copolymerization and

the mathematical description of such kinetics.

Despite the catalyst being used, the most generally

accepted and used mechanism for ethylene/a-olefin

copolymerization reaction is based on the terminal group

model.[5] In the terminal group model, polymerization

kinetic constants depend on the type of monomer

participating in the reaction and the type of monomer at

the polymer chain end. This comprehensive engineering

approach has been recently reviewed by Soares and

traditionally used by different groups.[6–11] This type of

reaction scheme takes about 20 reactions into account that,

by considering 5 active center types, needs at least 100 rate

constants foran isothermalmodelingof theprocess.[12] This

highdegreeof freedomraises suchflexibility that themodel

can draw the experimental curves without any physically

meaningful kinetic parameters, if no attention paid to

proper selectionof rate constants. Inaddition, acquisitionof

the required experimental data and mathematical adjust-

ment of this large number of parameters is an over-

whelming task.

Recently, ThompsonandMcAuleydevelopedasimplified

kinetic scheme for olefincopolymerization that isnot based

on the terminal group model.[12] This approach decreases

the number of reactions and consequently the number of



unknown parameters. In this view, rate constants depend

only on the type of monomer taking place in the reaction

and there is no difference in chains ended by ethylene or

a-olefins. Since they all end with a �CH2 group, it sounds

plausible to disregard any selectivity.

In a series of papers, Kissin has emphasized the unusual

behavior of ethylene/a-olefin copolymerization reac-

tions.[13–18] The main special feature is the so-called

‘‘comonomer effect’’: ethylene is the most reactive

compound among all olefins in homopolymerization, at

least 3 times more reactive than propylene or 50 times

more reactive than 1-hexene, but ethylene never exhibits

the level of reactivity expected from its relative reactivity

in copolymerization reactions. Even the introduction of an

a-olefin into the ethylene polymerization noticeably

increases the reaction rate. This behavior, which never

appears in the copolymerization of propylene with

a-olefins, is believed to be due to the stability of the

Ti–C2H5 bond. This stability is the result of an unusually

strong b-agostic interaction between the hydrogen atom

of the methyl group and the Ti atom. The kinetic

schemes described by Soares and Thompson, missing this

important point, fail to capture all specific characteristics

that appear only in copolymerization of ethylene with

a-olefins.

In thiswork, a simplified comprehensive kinetic scheme,

not including terminal-group effects, that considers the

special reactions illustrated by Kissin, is developed in order

to overcome two aspects of difficulties encountered

DOI: 10.1002/mren.200900039

A Simplified Comprehensive Kinetic Scheme for Modeling . . .

in modeling of ethylene/a-olefin copolymerization:

multiplicity and the estimation of rate constants, and

prediction of the special reaction behavior that appears in

the copolymerization of ethylene with longer a-olefins.

A mechanistic algorithm has been devised for estimation

of different rate constants and the experimental part has

been designed according to the model requirements.

Table 2. Simplified comprehensive kinetic scheme for ethylene/1-butene copolymerization.

Process Reaction

catalyst activationTiþ Al �!

kiaTi� Et

chain initiationTi� Etþ C2 �!

kiiP1

Ti� Etþ C4 �!kiiP1

8<:

chain propagationPn þ C2 �!

kipaPnþ1

Pn þ C4 �!kipb

Pnþ1

8><>:

chain transfer to ethylenePn þ C2 �!

kita

Dn þ Ti� Et

chain transfer to 1-butenePn þ C4 �!

kitb

Dn þ P1

chain transfer to hydrogenPn þH2 �!

kitH

Dn þ Ti�H

chain transfer to TEALPn þ Al �!

kitAl

Dn þ Ti� Et

spontaneous deactivationPn �!

kid

Dn þ Cd

poisoning by TEALPn þ Al �!

kipAl

Dn þ Cd

re-initiationTi�Hþ C2 �!

kiria

Ti� Et

Ti�Hþ C4 �!kirib

P1

8<:

Experimental Part

Triethylaluminum (TEAL), with a purity of 93%, was purchased

fromMerck. Polymerization-grade ethylene and nitrogen gas with

a purity of 99.99% were supplied by Iran Petrochemical Co. and

were further purified by passing through columns of activated 4 A

molecularsieves. Industrialhexane, suppliedby IranPetrochemical

Co., was further purified by distillations on calcium hydride.

Polymerization-grade 1-butenewas usedwithout further purifica-

tion.

Slurry polymerization reactions were carried out in a 1 L

stainless steel Buchi reactor. In all reactions, 500mL hexane was

used as the polymerization environment. Reaction components

weremixedbyapaddlemechanicalmixerat600 rpm.Temperature

wascontrolledusingaHuberwater circulator,model PolysatCC1.A

slurry of Ziegler-Natta catalyst in hexanewas prepared in a round-

bottom flask and homogenized by a magnetic stirrer. Before each

reaction the reactor was purged with nitrogen for 1 h at 95 8C. Thereactor was filled with hexane and saturated with ethylene after

purging several times with nitrogen and ethylene. TEAL was

transferred to the reactor by a syringe through an injection nozzle

and after a fewminutes of mixing, the catalyst was introduced in

the same manner. The temperature was increased to the preset

value, after which hydrogen was introduced up to the desired

partial pressure. Finally, the reactor pressure was raised to the

desiredvalueby insertingethyleneandpolymerizationwasstarted

by switching themixer on. Thepressure dropwas compensated for

byaddingethyleneto the reactorusingamassflowcontroller.After

the desired polymerization time, the reaction was terminated by

switching the mixer off and evacuating the remained unreacted

gases. In copolymerization reactions, a weighed amount of

1-butene was directly introduced to the reactor before raising

the temperature to the preset value.

The molecular weight distributions (MWDs) of the polymers

were determined by a PL-210 gel permeation chromatographusing

1,2,4-trichlorobenzene as solvent at 140 8C. The compositions of

ethylene/1-butene copolymers were measured by Fourier-trans-

form infrared (FTIR) spectroscopy.[19] This technique, which was

calibrated using 13C NMR evaluated standards, is designed for

levelsofcomonomerregularlyappear inhigh-densitypolyethylene

(HDPE) grades.

The experimentswere divided into two parts. First the effects of

polymerization time, ethylene, hydrogen and TEAL concentration

on homopolymerization of ethylene were studied and then the

effects of polymerization time and 1-butene concentration on

copolymerization behavior were investigated. All experiments

were carried out within a one week period in order to keep the

environmental conditions stable and increase the reproducibility

of the results. Table 1 shows the polymerization conditions and



characterization results of some selected polymerization runs.

Thesedataarepresented for comparisonwithmodeling results and

parametric studies of the effects of different variables on

polymerization behavior; polymer properties will be published

elsewhere.

Kinetic Scheme

A simplified comprehensive kinetic scheme for copolymer-

ization of ethylene with 1-butene was developed based on

the results derived from the mechanisms suggested by

Soares, Thompson and Kissin.[5,12,13] No differences were

taken into account for selectivity of a growing polymer

chain ended with either ethylene or 1-butene; in other

words, no terminal group effects were considered. The

kinetic mechanism is shown in Table 2, where C2 and C4

stand for ethyleneand1-butene, respectively, andPnandDn

represent, respectively, growing and dead polymer chains

of n units. Determination of the fraction of active centers

has been a controversial subject,[20–22] and since the

building up of a generalized model was being attempted,

a catalyst activation step was considered. Polymerization

initiates with addition of the first monomer unit to the

stable Ti–Et compound, and subsequently continues.While

in most previous mechanisms[6–11] no differences were

www.mre-journal.de 137


138

taken into accounts between active centers formed

from chain transfer to ethylene or 1-butene, in this work,

transfer to ethylene makes the stable Ti–Et compound, but

transfer to 1-butene avoids formation of Ti–Et, by-passing

this step by forming an initiated chain of P1. Chain transfer

to hydrogen forms a different type of active center, Ti–H,

while chain transfer to TEAL still forms the stable Ti–Et

compound. All active centers and growing polymer chains

can undergo spontaneous deactivation or be poisoned by

excess amount of cocatalyst, which competes with the

monomers in complexation with the active centers.[23]

The poisoning role of hydrogen was dismissed from the

kinetics due to the dominant poisoning effect of the excess

amount of TEAL, as shown by the yield results in Table 1.

Finally dormant Ti–H compound can be re-initiated by

insertion of ethylene or 1-butene. The insertion of ethylene

leads to formation of the stable Ti–Et compound, but

insertion of 1-butene avoids the rate-limiting step of Ti–Et

formation and results in creation of an initiated chain of P1.

These two reactions – transfer to and re-initiation with 1-

butene – are responsible for the synergistic comonomer

effect.

Modeling

Themass balances for different components defined in the

suggested kinetic scheme are given in Table 3. The method

of moments was used for calculation of polymerization

Table 3. Population balance equations for polymerization componen

d½Ti�dt

¼ �Pi

ðkia½Ti�½Al�Þ

d½Al�dt

¼ �Xi

kia½Ti� þ kitAlli0 þ kipAl ½Ti� Et�i þ ½Ti�H�i þ li0

�h

d½Ti� Et�i

dt¼ kia½Ti�½Al� � kii ½C2� þ ½C4�ð Þ þ kid þ kipAl½Al�

h i½Ti� E

þ kitma½C2� þ kitAl½Al��

li0 þ kiria½C2�½Ti�H�i

d½Ti�H�i

dt¼ kitHl

i0½H2� � kid þ kipAl½Al� þ kiria½C2� þ kirib½C4�

� �½Ti

d½C4�dt

¼ �Xi

kii½Ti� Et�i þ kipbli0 þ kitmbl

i0 þ kirib½Ti� H�i

� �½C4

d½H2�dt

¼ �Xi

kitHli0½H2�

dli0dt

¼ kii ½C2� þ ½C4�ð Þ½Ti� Et�i � kitma½C2� þ kitH½H2� þ kitAl½Al� þ�

dli1dt

¼ kii ½C2� þ ½C4�ð Þ½Ti� Et�i � kitma½C2� þ kitmb½C4� þ kitH½H2��

þ kitmb½C4�li0 þ kirib½C4�½Ti�H�i þ kipa½C2� þ kipb½C4��

li0

dmij

dt¼ kitma½C2� þ kitmb½C4� þ kitH½H2� þ kitAl½Al� þ kid þ kipAl½Al�

� �



rate, molecular weight distribution and chemical composi-

tiondistribution.Thebalanceequations formomentsof live

chains, ln and dead chains, mn are also given in Table 3.

Using well-established particle models at industrial opera-

tion conditions, it has been proved that mass- and heat-

transfer resistances might be neglected in the slurry

polymerization of ethylene.[8] It can be assumed that all

chemical components present in the gas and liquid phases

are in thermodynamic equilibrium and, especially because

the catalyst particles used in this work had no defined

spherical morphology, concentrations at the surface of

active centers might be equal to the concentrations at

the bulk of liquid phase. Since the range of obtained

comonomer contents were around HDPE grades, the

polymer swelling can be ignored and the effect of the

formed polymer on thermodynamic equilibrium could be

neglected. Consequently, the initial concentrations of

polymerization components were calculated using the

Peng-Robinson equation of state.[24]

Deconvolution of the MWD into the weighted summa-

tion of its Flory distribution components is, without doubt,

the basis for simulation of olefin polymerization using

multi-site Ziegler-Natta catalysts. This idea was first

suggested by Vickroy et al.[25] and used extensively since

its successful application by Kissin[13] and a detailed

explanation of modeling procedure by Soares and Hamie-

lec.[26] In this technique, the difference between measured

and predictedMWD should beminimized bymanipulation

ts.

(1)

�i½Al� (2)

t�i(3)

�H�i(4)

� (5)

(6)

kid þ kipAl½Al��li0 þ kirib½C4�½Ti�H�i

(7)

þ kitAl½Al� þ kid þ kipAl½Al��li1

(8)

lij

(9)

DOI: 10.1002/mren.200900039


Figure 1. GPC curve and deconvolution results of polyethyleneprepared in Run 8.

of three series of variables using numerical optimization

methods. The three series of variables are Flory parameters

of each site, their weight fractions and the number of sites.

It is typical to consider five active centers for deconvolution

ofMWD[27]while fewernumbershavealso appliedwithout

problems.[8] In this paper, no differences were noticed

between errors come from considering four or five sites, so

four siteswere taken into account. Figure 1 shows a typical

MWD and its deconvolution results.

In all parameter estimation steps, including deconvolu-

tion of MWD or estimation of kinetic rate constants, the

Nelder-Mead simplex method was used for automatic

manipulation of variables. Likewise, in all parameter

estimation steps, the error between measured and pre-

dicted values, such as MWD or polymerization rate

curves, was calculated using the following defined error

function:

Ratei

Macrom

� 2010

F ¼ 1�

Pni¼1

Pmeasured � Ppredicted� �2

Pni¼1

P2measured � 1n

Pni¼1

Pmeasured

� �2 (1)

where P refers to the target parameter and i refers to

number of reported points, such as the number of

molecular weights in a MWD graph or the number

of times in a polymerization rate curve. According to the

population balance equations of Table 3, one can

calculate the modeling results including polymerization

rate (g � s�1), yield (g), Flory parameter and weight

fraction of each site:

Figure 2. Measured (dots) and predicted (lines) polymerizationyield at each active center as a function of time, Run 1 to 4.

¼ M0 kii½Ti� Et�i þ kipali0 þ kitmal

i0 þ kiria½Ti�H�i

� �C2

(2)

Experimental points are the product of weight fraction of eachsite and the measured yield at each run.
Yieldi ¼ M0:ðli1 þ mi
1Þ (3)

ol. React. Eng. 2010, 4, 135–144

WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

i li0 þ mi0
t ¼
li1 þ mi1

(4)

mi ¼ Yieldi

PYieldi

(5)

Results and Discussion

Homopolymerization Analysis I: Estimation ofYield-Affecting Rate Constants

In homopolymerization, the yield-affecting rate constants,

kia, kii, k

ipa and kid, can be determined using experimental

polymerization rate curves and without considering

MWDs.[13] Four homopolymerization reactions at similar

experimental conditions but different polymerization

times, of 15, 30, 60 and 120min, were carried out for this

purpose. Initial guesses for rate constants of all sites were

generated by modeling of the polymerization rate curve at

120min using a simple single-site model. Next, the yield-

affecting rate constants of all sites were calculated by

means of the numerical optimization method. The target

function involved error functions defined for the yield of

each site at four times and the associated polymerization

rate curves at four runs. Figure 2 shows the calculated yield

profiles, while the corresponding estimated rate constants

are listed in Table 4.

Since the difference between initiation and propagation

steps would not depicted in homopolymerization results,

the calculated initiation rate constant has no physical

support and are just initial guesses. This rate constant



Table 4. Estimated yield-affecting rate constants.

Site ka ki kpa kd

L �mol�1 � s�1 L �mol�1 � s�1 L �mol�1 � s�1 s�1

1 186.900 1.686 200.580 0.002

2 370.200 0.655 259.800 0.001

3 128.380 4.050 293.390 3.5� 10�4

4 497.970 0.204 15.807 8.7� 10�10

140

should be determined using copolymerization results

where it has direct effects on the chemical composition

distribution. The logical trend in deactivation rate con-

stants shows that Site 1 is responsible for polymerization

behaviorat initial stagesofpolymerization,while Site3and

4 are responsible for polymerization behavior in the final

stages of polymerization, when Site 1 and 2 have been

deactivated. Because polymerization rate curves are all of

thedecreasing type, no coherent trend canbedistinguished

between activation or initiation rate constants.

Homopolymerization Analysis II: Estimation ofMWD-Affecting Rate Constants

Inhomopolymerization, theMWD-affecting rate constants,

kitma, kitH and kitAl, can be determined using experimental

polymerization rate curves andMWDs. Since chain transfer

to hydrogen forms a dormant active center, considering re-

initiation reaction for estimation procedure is inevitable in

this step. In addition, one should consider the poisoning

effect of excess amount of TEAL in polymerization runs

where theTEAL concentrationmayvary. Therefore, in three

seriesofhomopolymerization reactions, theconcentrations

of chain transfer agents, including ethylene, hydrogen and

TEAL, were varied (Run 5 to 10).

According to the steady-state kinetic approach,[28] the

instantaneous kinetic chain length is the reciprocal of the

Flory parameter:

Macrom

� 2010

ti ¼ 1

ni¼ kitma½C2� þ kitH½H2� þ kitAl½Al�

kipa½C2�(6)

Figure 3. Measured (dots) and predicted (lines) polymerizationrate and MWD of each site, Run 2.

Thus, if one cankeepall polymerization conditions stable

and change the concentration of each chain transfer agent

one at a time, it is possible to determine the ratio of the rate

constantof chaintransfer to theconsideredagent totherate

constant of the propagation reaction.[18] Since the propaga-

tion rate constants have been estimated in the previous

section, one can determine the absolute value of the chain

transfer rate constants individually for each site. The

ol. React. Eng. 2010, 4, 135–144


obtained sets of chain transfer rate constants were used as

the initial guesses for the numerical optimization proce-

dure. Because no changes were observed in the relative

contribution of active centers, in runs where the TEAL

concentrationwas variable, it was assumed that the excess

amount of TEALpoisons all sites approximately to the same

degree. The target function involvederror functionsdefined

for the weight fraction of each site and the corresponding

Flory parameters from deconvolution results and the

associated polymerization rate curves for all runs con-

sidered for this step, plus the previous four time variable

homopolymerization runs. Figures 3 and 4 compare the

measured and predicted polymerization rate curves and

corresponding MWDs for two typical polymerization runs,

related to steps one and two of the homopolymerization

analysis procedure, respectively. The estimated rate con-

stants of chain transfer reactions, poisoningbyTEALand re-

initiation by ethylene are listed in Table 5.

According to Table 5, it can be concluded that transfer

reactions are more favored in Site 1 and 2. Figures 3 and 4

show that Site 1 and 2, which are responsible for the

formation of shorter chains, are active in early stages of

polymerization, while Site 3 and 4, which are stable until

DOI: 10.1002/mren.200900039



the last stages of the polymerization, are responsible for the

formation of longer chains.

Copolymerization Analysis

The remained kinetic rate constants, kipb, kitmb and kirib, have

influence on the copolymerization results. The rate con-

stants for initiation with 1-butene were assumed to be

equal to that of initiation with ethylene. Four copolymer-

ization runs at similar operational conditions but varying

polymerization times and 1-butene concentrations were

carried out in order to study the copolymerization behavior

(Run 11 to 14). The initial guess for chain transfer to

comonomer was made using the steady-state kinetic

approach, as described in the previous section:

Tab

Sit

1

2

3

4

Macrom

� 2010

ti ¼ 1

ni¼ kitma½C2� þ kitmb½C4� þ kitH½H2� þ kitAl½Al�

kipa½C2� þ kipb½C4�(7)

Ignoring the propagation-with-comonomer term com-

pared to the propagation-with-ethylene term in the

denominator, one can calculate the chain transfer to

comonomer rate constants of each site from the slope of

le 5. Estimated MWD-affecting rate constants.

e ktma ktH

L �mol�1 � s�1 L �mol�1 � s�1

0.986 28.791

0.303 4.9� 10�4

0.270 0.002

0.005 2.7� 10�6

ol. React. Eng. 2010, 4, 135–144


a plot of their Flory parameters versus the ratio of

comonomers-to-ethylene concentration. Figure 5 shows

these plots for four considered sites as an example.

The target function for numerical optimization involved

error functions defined for the weight fraction of each site

and the corresponding Flory parameters from deconvolu-

tion of MWDs, polymerization rate curves and the

measured weight fractions of 1-butene for all copolymer-

ization runs considered in this step. Figure 6 showsa typical

copolymerization rate curve and the corresponding

MWD and Figure 7 compares the measured and predicted

1-buteneweight fractions. Theestimated copolymerization

related rate constants are listed in Table 6.

It should be noted that precise determination of the

comonomer incorporation of each site needs experimental

results for the comonomer composition distribution

from TREF or CRYSTAF analysis; here, only the average

comonomer contents were evaluated and therefore the

absolute value of rate constants for propagation with or

chain transfer to comonomer obtained for different

sites are not quantitatively reliable. However qualitative

conclusions can be made by investigating the deconvolu-

tion results. Table 7 shows the deconvolution results

as 1-butene concentration increases. Flory parameters of

all sites increase continuously as 1-butene content

increases, as it plays the role of a chain transfer agent.

But there is also an interesting change in the fraction of

different sites. The contribution of two components with

the highest Flory components (lowest molecular weights),

Site 1 and 2, increases at higher 1-butene concentrations,

while the fraction of two components with lowest Flory

parameters (highest molecular weights), Site 3 and 4,

decreases. The obtained results indicate that the Flory

components with high comonomer contents are formed in

the early stages of reactionwhen Site 1 and 2 dominate the

kinetics whereas longer chains with lower comonomer

contents are formed in the late stages of polymerization

when Site 3 and 4 are dominant. The higher comonomer

reactivity of Site 1 and 2 has been translated into higher

rate constants of propagation with and chain transfer to

1-butene reactions in comparison to Site 3 and 4, as shown

in Table 6.

ktAl kpAl kria

L �mol�1 � s�1 L �mol�1 � s�1 L �mol�1 � s�1

0.890 3.8� 10�9 521.500

1.1� 10�8 3.8� 10�9 3 488.800

0.154 3.8� 10�9 0.002

8.2� 10�4 3.8� 10�9 0.366



Figure 5. Calculation of initial guesses for chain transfer to 1-butene for each active center, Run 3, 11 and 12; [C4]¼0, 0.152 and 0.301M.


Figure 7. Predicted vs. measured 1-butene weight fractions,Run 11 to 14.

142Macromol. React. Eng. 2010, 4, 135–144

� 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim DOI: 10.1002/mren.200900039


Table 8. Final estimated rate constants.

Rate constant Site 1

ka(L �mol�1 � s�1) 117.497 52

ki(L �mol�1 � s�1) 31.615

kpa(L �mol�1 � s�1) 489.945 1

kd(s�1) 0.004

ktma(L �mol�1 � s�1) 0.915

kth(L �mol�1 � s�1) 26.133

ktal(L �mol�1 � s�1) 0.584 2.

kpal(L �mol�1 � s�1) 4.0� 10�10 4.0

kria(L �mol�1 � s�1) 48.893 10

kpb(L �mol�1 � s�1) 2.001

ktmb(L �mol�1 � s�1) 1.043

krib(L �mol�1 � s�1) 1 001.001

Table 6. Estimated copolymerization-affecting rate constants.

Site kpb ktmb krib

L �mol�1 � s�1 L �mol�1 � s�1 L �mol�1 � s�1

1 2.001 1.126 1 001.000

2 2.699 0.057 23.748

3 0.103 0.058 1.8� 105

4 1.676 0.003 4 284.600

Table 7. Effects of 1-butene concentration on deconvolutionresults.

[1-butene] Site Flory parameter Contribution

mol � L�1 wt.-%

0 1 0.00745 8.881

2 0.00202 43.024

3 0.00075 36.090

4 0.00021 12.005

0.152 1 0.00761 11.722

2 0.00214 46.444

3 0.00082 33.224

4 0.00025 8.610

0.301 1 0.00936 11.129

2 0.00251 46.343

3 0.00099 34.450

4 0.00032 8.078



Finally, since the yield-affecting rate constants have

some effects on MWD, the MWD-affecting rate constants

have some effects of rate curves and, likewise, the

copolymerization-affecting rate constants have similar

effects on both, the final parameter estimation step was

carried out by fine tuning of all rate constants using all

experimental data. The final estimated rate constants are

listed in Table 8.

Conclusion

A simplified comprehensive kinetic scheme has been

developed by combining different mechanisms developed

by Soares, Thompson and Kissin.[5,12,13] The number of

unknown rate constants has been reduced by disregarding

different selectivity forgrowingpolymerchainsendedwith

ethylene or 1-butene. The stability of the Ti–Et compound,

as described by Kissin, was also taken into account. This

mechanism is able to explain the specific features observed

in ethylene/1-butene copolymerization, including the

synergistic effect of the comonomer and the effect of

hydrogen on increasing comonomer content. A systematic

algorithm was designed to estimate each group of kinetic

constants separately. First, initial guesses for different type

of rate constants were generated based on analysis of

deconvolution results; then, the exact rate constants were

estimated by means of the Nelder-Mead numerical

optimization algorithm. It was shown that short chains

with higher 1-butene content are formed in the earlier

stages of reaction,while longer chainswith fewer 1-butene

units are formed in the later stages of polymerization.

Site 2 Site 3 Site 4

3.16041 73.750 168.603

3.714 2 770.435 3.473

40.762 320.480 26.888

0.001 2.8� 10�4 1.2� 10�6

0.281 0.249 0.005

0.002 5.2� 10�4 7.0� 10�6

6� 10�8 0.136 0.001

� 10�10 4.0� 10�10 4.0� 10�10

432.420 8.9� 10�6 0.031

2.360 0.112 1.907

0.064 2.1� 10�5 0.001

81.870 30 695.490 6 468.946



144

Acknowledgements: The authors would like to thank JamPetrochemical Co. for financial and technical support of this work.

Received: June 24, 2009; Revised: August 24, 2009; Publishedonline: October 27, 2009; DOI: 10.1002/mren.200900039

Keywords: kinetics (polym.); modeling; olefin copolymerization;parameter estimation; Ziegler-Natta polymerization

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DOI: 10.1002/mren.200900039

a simplified comprehensive kinetic scheme for modeling of ethylene/1-butene copolymerization using...

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