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Kinetic Modeling and Simulation of Metallocene
Catalyzed Olefin Polymerization
THESIS
Submitted in the partial fulfillment of the
requirements for the degree of
DOCTOR OF PHILOSOPHY
by
NIKHIL PRAKASH
Under the Supervision of
Dr Arvind Kumar Sharma
and
Dr Sushil Kumar
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE (BITS), PILANI
Pilani Campus, Rajasthan, India
2013
DEDICATED
To
My Parents, Wife & Daughter
iv
ACKNOWLEDGEMENTS
It gives me a deep sense of gratitude and an immense pleasure to sincerely thank my
supervisor Dr Arvind Kumar Sharma Assistant Professor, Chemical Engineering
Department (BITS-Pilani, Pilani campus) and co-supervisor Dr Sushil Kumar, Assistant
Professor, Chemical Engineering Department (MNNIT, Allahabad) for their constant
encouragement, constructive and valuable suggestions, and moral support throughout the
period of this research work. It has been a privilege for me to work under their valuable
guidance.
I thank the members of Doctoral Advisory Committee, Dr Pratik N. Sheth, Assistant
Professor, Chemical Engineering Department, and Dr Smita Raghuvanshi, Assistant
Professor, Chemical Engineering Department (BITS-Pilani, Pilani campus) for their
support and suggestions to carry out this work effectively.
My sincere thanks go to Prof B N Jain, Vice-Chancellor, BITS-Pilani for giving me the
opportunity to carry out the PhD work in BITS-Pilani. I am thankful to Prof G
Raghurama, Director (Pilani Campus), Prof. Sanjeev K Aggarwal, Director (Goa
Campus), Prof V S Rao, Director (Hyderabad Campus), Prof R K Mittal, Director, (Dubai
Campus), Prof R N Saha, Deputy Director (Pilani Campus), Prof S K Verma, Dean,
Academic Research Division (PhD Programme), Dr H R Jadhav, Professor-in-charge,
Academic Research Division (PhD Programme) for providing the necessary facilities and
infrastructure to carry out this work.
I extend my sincere thanks to Prof R P Vaid, Prof T N S Mathur, Prof B R Natrajan, Prof
B V Babu, Prof A K Sarkar and Prof B B Gulyani for their motivation with affectionate
enquiries about the status of my PhD work.
I extend my special thanks to Dr Suresh Gupta, Head (Chemical Engineering
Department), Dr Harekrishna Mohanta, Convener (Departmental Research Comittee), Dr
Pradipta Chattopadhaya, Mr Amit Jain, Mr Ajaya K Pani, Ms Priya C Sande, Mr Utkarsh
Maheshwari, Mr Subhajit Majumder, Dr Banasri Roy and Dr Sonal Majumder of
v
Chemical Engineering Department for their valuable advice and moral support throughout
the work.
I would like to express the earnest appreciation to Dr Ashish M Gujrathi, Dr Saptarshi
Chatopadyay and Mr Basheer Ahmed for their contribution in stimulating suggestions
and encouragement.
I would also take this opportunity to thank Mr Babu Lal Saini, Mr Jangvir, Mr Ashok
Saini, Mr Jeevan Verma and Mr Subodh Kumar Azad for their good wishes and
cooperation during my PhD work.
I would also like to convey my special thanks to all my students, for extending their
constant support in various ways.
This work could not have been completed without the moral support I got from my loving
parents Mr Dhwaj Prakash Saxena and Mrs Madhvi Saxena, in-laws Dr Subhash Chand
Jauhari and Mrs Neerja Jauhari, my dear sister Mrs Tulika Saxena and my loving wife
Dr Shikha Jauhari. Their unconditional love, constant encouragement, moral support and
immense confidence in me made this work possible. I would like to express my love and
affections to my daughter Vedanshi Saxena for being cheerful all the way and
surprisingly understanding at such a little age.
Thanks are due, to my computing machines (Laptop and Desktop) for being incessant
companion of mine till today and for running day and night during simulation studies
without a single failure or crash.
Last but not the least, I pray and thank to almighty God for showering His blessings and
giving me the inner strength and patience.
NIKHIL PRAKASH
vi
ABSTRACT
Metallocene catalyst system refers to the combination of bis(cyclopentadienyl)metal
complexes of Group 4 (IVB) or cyclopentadienyl-substituted derivatives thereof, and a
cocatalyst, typically methylalumoxane (MAO). Metallocene catalysts have in general
demonstrated high productivity, narrow molecular weight distribution (MWD), greater
efficiency in using comonomer to reduce the density, capability of producing polymer
with varying molecular weights and controlled stereoregularity.
Metallocene based catalyst technology is expected to revolutionize the polyolefin
industry immensely, particularly in polyethylene and polypropylene markets. Metallocene
polyolefins are projected to penetrate a broad array of polymer markets. First with the
higher priced specialty markets, followed by the high volume and commodity markets.
New markets are also expected to be created with the development of new classes of
polymer those are not possible with conventional Ziegler-Natta technologies.
In the present work, the mechanistic aspects of Ziegler-Natta and metallocene
catalyst systems have been studied in detail and used in building up mathematical models
for α-olefin polymerization using metallocene catalysts. Based on the interpretations of
mechanisms for metallocene-catalyzed polymerization, an ecumenical reaction set for
ethylene and propylene polymerization that includes reactions corresponding to all types
of metallocenes is proposed. Thereafter, mathematical models for ethylene and α-olefin
polymerization in a batch/semi-batch/constant-stirred-tank reactor are built up. The
models developed are capable of predicting polymerization kinetics and polymer
properties (viz. number-average- & weight-average molecular weights and polydispersity
index) in general. In addition, mole fraction of dead polymer chains with terminal double
bond and density of long-chain branches & short-chain branches may be determined with
ethylene polymerization model. Fraction of vinyl-terminated chains, butenyl-terminated
chains, isobutyl-terminated chains and vinylidene-terminated chains relative to the total
unsaturated termination may be determined with propylene polymerization model.
Model equations developed include a set of coupled, nonlinear and stiff ordinary
differential equations (ODEs) for the dynamic polymerization. To estimate the kinetic
parameters and to study the effect of parameters, model ODEs are solved with ODE-15s
function provided MATLAB™
7.0.1 software.
In this study, a novel natural logarithmic differential evolution (NLDE) approach
of optimization, a remediated edition of original differential evolution algorithm is
proposed and used to solve parameter estimation problem. Proposed NLDE algorithm is
capable of handing multiple objective functions simultaneously, providing room to admit
objective functions based on polymerization rate, molecular weights, PDI, fraction of
dead polymer chains with terminal double bond, fraction of vinyl-terminated chains,
butenyl-terminated chains, isobutyl-terminated chains and vinylidene-terminated chains
etc. if experimental data are available.
vii
Ethylene polymerization model is applied to gas phase polymerization with silica
supported, bridged Me2Si[Ind]2ZrCl2 catalyst and to solution phase polymerization with
in-situ-silica supported, bridged Et[Ind]2ZrCl2 catalyst and MAO. Propylene
polymerization model is applied to the solution phase production of polypropylene
catalyzed with Me2Si[Ind]2ZrCl2, Et(Ind)2ZrCl2, Me2Si(Ind)2HfCl2, Et(Ind)2HfCl2, (2,4,6-
Me3Ind)2ZrCl2, (2,4,7-Me3Ind)2ZrCl2 and Me2Si[2,4,6-Me3Ind]2ZrCl2 catalyst and MAO.
Models are validated with experimental data available in open literature and
model kinetic parameters are estimated for each catalytic system. Further, parametric
study is carried out for all the polymerization systems in order to examine the effect of
variation in monomer pressure (concentration), polymerization temperature, initial
amount of catalyst and cocatalyst to catalyst mole ratio on polymerization kinetics and
macro- and microstructural properties of polymer.
Set of kinetic parameters determined is catalyst dependent and unique. In general,
it was found that chain initiation, propagation and spontaneous catalyst deactivation are
the essential reactions in olefin polymerization and strongly affect the polymerization
kinetics. Various termination reactions are dependent on actual polymerization process
employed and reaction conditions. Termination reactions are found to affect molecular
weight distribution and microstructural properties of the final product in different ways.
Keywords:
Modeling and simulation; Mechanism; Metallocene; Polymerization; Gas phase;
Solution phase; Ethylene; Propylene; Natural logarithmic differential evolution;
Optimization; Estimation; Kinetic parameters; Kinetics; Microstructural properties.
viii
TABLE OF CONTENTS
Certificate iii
Acknowledgements iv
Abstract vi
Table of Contents viii
List of Figures xi
List of Tables xvi
Nomenclature xvii
1. Introduction 1
1.1 Motivation 1
1.1.1 Metallocene catalyst systems 3
1.1.2 Evolution of the metallocenes 4
1.1.3 Mathematical modeling and simulation 6
1.2 Objectives of research 8
1.3 Organization of thesis 9
2. Literature Review 10 2.1 Ziegler-Natta polymerization 10
2.1.1 Ziegler-Natta catalysts 11
2.1.2 Mechanism of Ziegler-Natta polymerization 11
2.2 Metallocene polymerization 15
2.2.1 Metallocenes catalyst system 16
Metallocenes in ethylene polymerization 20
Metallocenes in propylene polymerization 20
2.2.2 Mechanism of metallocene polymerization 24
Mechanism for activation 24
Mechanisms for propagation 26
Mechanisms for termination 30
2.3 Experimental studies 33
2.4 Mechanistic, modeling & simulation studies 49
2.5 Gaps in research 59
2.6 Scope of the work 59
3. Mathematical Model Development and Simulation 61
3.1 Metallocene polymerization kinetics and model development 62
3.1.1 Mathematical treatment of polymerization kinetics 62
3.2 Modeling of ethylene polymerization 65
3.2.1 Kinetics 65
3.2.2 Model development for ethylene polymerization 70
3.3 Modeling of propylene polymerization 75
3.3.1 Kinetics 75
3.3.2 Model development for propylene polymerization 81
3.4 Simulation methodology 86
3.4.1 Numerical solution procedure 87
3.4.2 Objective function formulation 88
3.4.3 Optimization approach 91
3.4.3.1 Differential evolution 92
Natural logarithmic differential evolution 93
ix
Summary of the chapter 97
4. Results and Discussion 98 4.1 Ethylene polymerization 101
4.1.1 Ethylene polymerization with Me2Si[Ind]2ZrCl2/MAO 101 Estimated parameters and effect of temperature 103
Effect of ethylene pressure 105 Effect of catalyst amount 110
4.1.2 Ethylene polymerization with in-situ-supported Et[Ind]2ZrCl2/MAO 112 Estimated parameters and effect of temperature 113
Effect of ethylene pressure 117 Effect of catalyst amount 117
Effect of cocatalyst to catalyst mole ratio 117
Polyethylene properties 121
4.2 Propylene polymerization 124
4.2.1 Propylene polymerization with Me2Si[Ind]2ZrCl2/MAO 126 Estimated parameters and effect of temperature 126
Effect of pressure 135
Effect of catalyst concentration 138
4.2.2 Propylene polymerization with Et[Ind]2ZrCl2/MAO 140 Estimated parameters and effect of temperature 140 Effect of pressure 147
Effect of catalyst concentration 150
4.2.3 Propylene polymerization with Me2Si[Ind]2HfCl2/MAO 153 Estimated parameters and effect of temperature 153 Effect of pressure 160
Effect of catalyst concentration 163
4.2.4 Propylene polymerization with Et[Ind]2HfCl2/MAO 166 Estimated parameters and effect of temperature 166 Effect of pressure 174
Effect of catalyst concentration 174
4.2.5 Propylene polymerization with [2,4,6-Me3Ind]2ZrCl2/MAO 179 Estimated parameters and effect of temperature 179
Effect of pressure 184
Effect of catalyst concentration 186
4.2.6 Propylene polymerization with [2,4,7-Me3Ind]2ZrCl2/MAO 188
Estimated parameters and effect of Al/Zr mole ratio 188
Effect of pressure 194
Effect of catalyst concentration 196
4.2.7 Propylene polymerization with Me2Si[2,4,6-Me3Ind]2ZrCl2/MAO 197
Estimated parameters and effect of temperature 197
Effect of pressure 203
Effect of catalyst concentration 206
Summary of the chapter 208
5. Concluding Remarks 209
5.1 Summary 209
5.1.1 Introduction 209
5.1.2 Gaps in research 211
5.1.3 Scope of the work 211
5.1.4 Model development and simulation 212
5.1.5 Results and discussion 214
x
5.2 Conclusions 228
5.3 Major contributions 234
5.4 Future scope of research 235
References 236
List of Publications 251
Appendix I: Code in MATLAB to Estimate the Kinetic Parameters
in Ethylene Polymerization with Me2Si[Ind]2ZrCl2/MAO 255
Appendix II: Code in MATLAB to Estimate the Kinetic Parameters
in Ethylene Polymerization in-situ-supported-
Et[Ind]2ZrCl2/MAO 264
Appendix III: Code in MATLAB to Estimate the Kinetic Parameters
in Propylene Polymerization with Me2Si[Ind]2ZrCl2/MAO 275
Biographies 287
xi
LIST OF FIGURES
Figure
No.
Title
Page
No.
1.1 Generic structure of metallocene catalyst. 5
1.2 Various ligands of metallocene. 5
2.1 Bimetallic mechanism of Z-N polymerization by Natta. 12
2.2 Bimetallic mechanism of Z-N polymerization by Patat and Sinn. 12
2.3 Monometallic mechanism of Z-N polymerization by Cossee. 13
2.4 Trigger mechanism of Z-N polymerization by Ystenes. 13
2.5 Chain termination by β-H transfer to monomer (β-H elimination). 14
2.6 Chain termination by spontaneous intramolecular β-H transfer. 14
2.7 Chain termination by molecular hydrogen. 14
2.8 Chain termination to the Group I-III metal alkyl. 14
2.9 Partial hydrolysis of trimethylaluminum to form MAO. 17
2.10 (a) Linear and (b) cyclic structures of MAO
(c) two-dimensional ladder and (d) three-dimensional cage structures
of MAO oligomers.
17
2.11 (a) Chain-end and (b) enantiomorphic site mechanisms of
stereocontrol.
18
2.12 Unbridged (a) and bridged (b) catalysts used in ethylene
polymerization.
22
2.13 General symmetry classifications. 23
2.14 General metallocene symmetry classifications. 23
2.15 Activation of a metallocene complex by methylaluminoxane (MAO).
M = transition metal atom and □ = vacant coordination site
25
2.16 Cossee-Arlman mechanism of propagation with a metallocene
catalyst.
28
2.17 Green-Rooney mechanism of propagation with a metallocene catalyst. 28
2.18 Modified Green-Rooney mechanism of propagation with a
metallocene catalyst.
29
2.19 Transition state α-agostic mechanism of propagation with a
metallocene catalyst
29
2.20 β-Hydrogen transfer to monomer. 32
2.21 β-hydrogen elimination (Spontaneous chain transfer). 32
2.22 β-methyl elimination (Spontaneous chain transfer). 32
2.23 Chain transfer to cocatalyst. 32
3.1 Determination of population for sequent generation in DE. 94
3.2 Flow sheet: Differential evolution optimization procedure. 95
4.1 Ethylene polymerization rate vs. time with Me2Si[Ind]2ZrCl2
(E1)/MAO.
[Catalyst (E1) = 0.2 g; P = 5 bar; Al/Zr = 386]
104
4.2 Polymerization rate vs. time: Effect of pressure.
[Catalyst (E1) = 0.2 g, Al/Zr = 386, T = (a) 50 °C, (b) 60 °C,
(c) 70 °C]
108
4.3 Effect of pressure on average molecular weights.
[Catalyst (E1) = 0.2 g, Al/Zr = 386, T = (a) 50 °C, (b) 60 °C,
(c) 70 °C]
110
xii
4.4 Polymerization rate vs. time: Effect of catalyst amount.
[P = 5 bar, Al/Zr = 386, T = (a) 50 °C, (b) 60 °C, (c) 70 °C]
112
4.5 Effect of temperature on ethylene polymerization rate with in-situ-
supported Et[Ind]2ZrCl2 (E2)/MAO.
[Catalyst (E2) = 6 μmol, Al/Zr = 500 and P = 80 psig]
116
4.6 Active catalyst sites vs. reaction time.
[Catalyst (E2) = 6 μmol, Al/Zr = 500 and P = 80 psig]
116
4.7 Effect of pressure on ethylene polymerization rate.
[Catalyst (E2) = 6 μmol, Al/Zr = 500 and 60 °C]
118
4.8 Active catalyst sites vs. reaction time
[Catalyst (E2) = 6 μmol, Al/Zr = 500 and 60 °C]
118
4.9 Effect of catalyst amount on ethylene polymerization rate.
[Al/Zr = 500, 60 °C, and 80 psig]
119
4.10 Active catalyst sites vs. reaction time.
[Al/Zr = 500, 60 °C and 80 psig]
119
4.11 Effect of Al/Zr mole ratio on ethylene polymerization rate.
[Catalyst (E2) = 6 μmol, 60 °C and 80 psig]
120
4.12 Active catalyst sites vs. reaction time
[Catalyst (E2) = 6 μmol, 60 °C and 80 psig]
120
4.13 Effect of Al/Zr mole ratio on propylene polymerization rate.
[Catalyst (P1) = 10 μM, T = 25 °C and P = 30 psi]
128
4.14 Effect of Al/Zr mole ratio on propylene polymerization rate.
[Catalyst (P1) = 10 μM, T = 75 °C and P = 30 psi]
128
4.15 Effect of temperature on propylene polymerization rate.
[Catalyst (P1) = 10 μM, Al/Zr = 500 and P = 30 psi]
129
4.16 Active catalyst site concentration vs. time.
[Catalyst (P1) = 10 μM, Al/Zr = 500, 75 °C and 30 psi]
131
4.17 Hydride actived complex concentration vs. time.
[Catalyst (P1) = 10 μM, Al/Zr = 500, T = 25 °C and P = 30 psi]
132
4.18 Hydride activated complex concentration vs. time.
[Catalyst (P1) = 10 μM, Al/Zr = 500, T = 75 °C and P = 30 psi]
132
4.19 Methyl activated complex concentration vs. time.
[Catalyst (P1) = 10 μM, Al/Zr = 500 and P = 30 psi]
133
4.20 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P1) = 10 μM, Al/Zr = 500, and T = 25 °C]
136
4.21 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P1) = 10 μM, Al/Zr = 500 and T = 75 °C]
136
4.22 Effect of pressure on average molecular weights.
[Catalyst (P1) = 10 μM, Al/Zr = 500 and T = 25 °C]
137
4.23 Effect of pressure on average molecular weights
[Catalyst (P1) = 10 μM, Al/Zr = 500 and T = 75 °C]
137
4.24 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Zr = 500, T = 25 °C and P = 30 psi]
138
4.25 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Zr = 500, T = 75 °C and P = 30 psi]
139
4.26 Effect of catalyst concentration on average molecular weights.
[Al/Zr = 500, T = 25 °C and P = 30 psi]
139
4.27 Effect of catalyst concentration on average molecular weights.
[Al/Zr = 500, T = 75 °C and P = 30 psi]
140
4.28 Polymerization rate vs. time.
[Catalyst (P2) = 10 μM, Al/Zr = 2000, T = 25 °C and P = 30 psi]
142
xiii
4.29 Polymerization rate vs. time.
[Catalyst (P2) = 10 μM, T = 75 °C and P = 30 psi]
142
4.30 Active catalyst site concentration vs. time.
[Catalyst (P2) = 10 μM, Al/Zr = 2000 and P = 30 psi]
144
4.31 Hydride actived complex concentration vs. time.
[Catalyst (P2) = 10 μM, Al/Zr = 2000, T = 25 °C and P = 30 psi]
145
4.32 Hydride actived complex concentration vs. time.
[Catalyst (P2) = 10 μM, Al/Zr = 2000, T = 75 °C and P = 30 psi]
145
4.33 Methyl actived complex concentration vs. time.
[Catalyst (P2) = 10 μM, Al/Zr = 2000 and P = 30 psi]
146
4.34 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P2) = 10 μM, Al/Zr = 2000 and T = 25 °C]
148
4.35 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P2) = 10 μM, Al/Zr = 500 and 75 °C]
149
4.36 Effect of pressure on average molecular weights.
[Catalyst (P2) = 10 μM, Al/Zr = 2000 and T = 25 °C]
149
4.37 Effect of pressure on average molecular weights.
[Catalyst (P2) = 10 μM, Al/Zr = 500 and T = 75 °C]
150
4.38 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Zr = 2000, T = 25 °C and P = 30 psi]
151
4.39 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Zr = 500, T = 75 °C and P = 30 psi]
151
4.40 Effect of catalyst concentration on average molecular weights.
[Al/Zr = 2000, T = 25 °C and P = 30 psi]
152
4.41 Effect of catalyst concentration on average molecular weights.
[Al/Zr = 500, T = 75 °C and P = 30 psi]
152
4.42 Effect of Al/Hf mole ratio on propylene polymerization rate.
[Catalyst (P3) = 10 μM, T = 40 °C and P = 30 psi]
155
4.43 Effect of Al/Hf mole ratio on propylene polymerization rate.
[Catalyst (P3) = 10 μM, T = 80 °C and P = 30 psi]
156
4.44 Active catalyst site concentration vs. time.
[Catalyst (P3) = 10 μM, Al/Hf = 2000 and P = 30 psi]
156
4.45 Hydride actived complex concentration vs. time.
[Catalyst (P3) = 10 μM, Al/Hf = 2000 and P = 30 psi]
157
4.46 Methyl actived complex concentration vs. time.
[Catalyst (P3) = 10 μM, Al/Hf = 2000, T = 40 °C and P = 30 psi]
158
4.47 Methyl actived complex concentration vs. time.
[Catalyst (P3) = 10 μM, Al/Hf = 2000, T = 80 °C and P = 30 psi]
159
4.48 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P3) = 10 μM, Al/Hf = 2000 and T = 40 °C]
161
4.49 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P3) = 10 μM, Al/Hf = 2000 and T = 80 °C]
161
4.50 Effect of pressure on average molecular weights.
[Catalyst (P3) = 10 μM, Al/Hf = 2000 and T = 40 °C]
162
4.51 Effect of pressure on average molecular weights.
[Catalyst (P3) = 10 μM, Al/Hf = 2000 and T = 80 °C]
162
4.52 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Hf = 2000, T = 40 °C and P = 30 psi]
164
4.53 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Hf = 2000, T = 80 °C and P = 30 psi]
164
4.54 Effect of catalyst concentration on average molecular weights.
[Al/Hf = 2000, T = 40 °C and P = 30 psi]
165
xiv
4.55 Effect of catalyst concentration on average molecular weights.
[Al/Hf = 2000, T = 80 °C and P = 30 psi]
165
4.56 Effect of Al/Hf mole ratio on propylene polymerization rate.
[Catalyst (P4) = 10 μM, T = 40 °C and P = 30 psi]
167
4.57 Effect of Al/Hf mole ratio on propylene polymerization rate.
[Catalyst (P4) = 10 μM, T = 80 °C and P = 30 psi]
167
4.58 Active catalyst site concentration vs. time.
[Catalyst (P4) = 10 μM, Al/Hf = 500 and P = 30 psi]
170
4.59 Hydride actived complex concentration vs. time.
[Catalyst (P4) = 10 μM, Al/Hf = 500, T = 40 °C and P = 30 psi]
170
4.60 Hydride actived complex concentration vs. time.
[Catalyst (P4) = 10 μM, Al/Hf = 500, T = 80 °C and P = 30 psi]
171
4.61 Methyl actived complex concentration vs. time.
[Catalyst (P4) = 10 μM, Al/Hf = 500, T = 40 °C and P = 30 psi]
172
4.62 Methyl actived complex concentration vs. time.
[Catalyst (P4) = 10 μM, Al/Hf = 500, T = 80 °C and P = 30 psi]
172
4.63 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P4) = 10 μM, Al/Hf = 500 and T = 40 °C]
175
4.64 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P4) = 10 μM, Al/Hf = 500 and T = 80 °C]
175
4.65 Effect of pressure on average molecular weights.
[Catalyst (P4) = 10 μM, Al/Hf = 500 and T = 40 °C]
176
4.66 Effect of pressure on average molecular weights.
[Catalyst (P4) = 10 μM, Al/Hf = 500 and T = 80 °C]
176
4.67 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Hf = 500, T = 40 °C and P = 30 psi]
177
4.68 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Hf = 500, T = 80 °C and P = 30 psi]
178
4.69 Effect of catalyst concentration on average molecular weights.
[Al/Hf = 500, T = 40 °C and P = 30 psi]
178
4.70 Effect of catalyst concentration on average molecular weights.
[Al/Hf = 500, T = 80 °C and P = 30 psi]
179
4.71 Effect of Al/Zr mole ratio on propylene polymerization rate.
[Catalyst (P5) = 20 μM, T = 0 °C and P = 0.98 atm]
180
4.72 Active catalyst site concentration vs. time.
[Catalyst (P5) = 20 μM, Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
182
4.73 Hydride actived complex concentration vs. time.
[Catalyst (P5) = 20 μM, Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
183
4.74 Methyl actived complex concentration vs. time.
[Catalyst (P5) = 20 μM, Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
183
4.75 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P5) = 20 μM, Al/Zr = 2000 and T = 0 °C]
185
4.76 Effect of pressure on average molecular weights.
[Catalyst (P5) = 20 μM, Al/Zr = 2000 and T = 0 °C]
186
4.77 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
187
4.78 Effect of catalyst concentration on average molecular weights.
[Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
187
4.79 Effect of Al/Zr mole ratio on propylene polymerization rate.
[Catalyst (P6) = 20 μM, T = 0 °C and P = 0.98 atm]
189
4.80 Active catalyst site concentration vs. time.
[Catalyst (P6) = 20 μM, Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
192
xv
4.81 Hydride actived complex concentration vs. time.
[Catalyst (P6) = 20 μM, Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
192
4.82 Methyl actived complex concentration vs. time.
[Catalyst (P6) = 20 μM, Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
193
4.83 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P6) = 20 μM, Al/Zr = 2000 and T = 0 °C]
195
4.84 Effect of pressure on average molecular weights.
[Catalyst (P6) = 20 μM, Al/Zr = 2000 and T = 0 °C]
195
4.85 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
196
4.86 Effect of catalyst concentration on average molecular weights.
[Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
197
4.87 Effect of temperature on propylene polymerization rate.
[Catalyst (P7) = 20 μM, Al/Zr = 2000 and P = 0.98 atm]
199
4.88 Active catalyst site concentration vs. time.
[Catalyst (P7) = 20 μM, Al/Zr = 2000 and P = 0.98 atm]
201
4.89 Hydride actived complex concentration vs. time. (a) 30 °C, (b) 70 °C
[Catalyst (P7) = 20 μM, Al/Zr = 2000 and P = 0.98 atm]
202
4.90 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P7) = 20 μM, Al/Zr = 2000 and T = 30 °C]
204
4.91 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P7) = 20 μM, Al/Zr = 2000 and T = 70 °C]
204
4.92 Effect of pressure on average molecular weights.
[Catalyst (P7) = 20 μM, Al/Zr = 2000 and T = 30 °C]
205
4.93 Effect of pressure on average molecular weights.
[Catalyst (P7) = 20 μM, Al/Zr = 2000 and T = 70 °C]
205
4.94 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Zr = 2000, T = 30 °C and P = 0.98 atm]
206
4.95 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Zr = 2000, T = 70 °C and P = 0.98 atm]
207
4.96 Effect of catalyst concentration on average molecular weights.
[Al/Zr = 2000, T = 30 °C and P = 0.98 atm]
207
4.97 Effect of catalyst concentration on average molecular weights.
[Al/Zr = 2000, T = 70 °C and P = 0.98 atm]
208
xvi
LIST OF TABLES
Table
No.
Title
Page
No.
1.1 World Commodity Polymers Consumption Estimate 2
1.2 Scales of Modeling Polyolefin Processes 7
2.1 Generations of Ziegler-Natta Catalyst 11
2.2 Representative Examples of Metallocenes 19
2.3 Metallocene Catalyst Systems 20
2.4 Metallocene Catalyzed Ethylene Polymerization, Experimental Studies 40
2.5 Metallocene Catalyzed Propylene Polymerization, Experimental
Studies
45
2.6 Metallocene Catalyzed Olefin Polymerization, Mechanistic, Modeling
& Simulation Studies
55
3.1 Reactions Conceived in Ethylene Polymerization 69
3.2 Reactions Conceived in Propylene Polymerization 80
4.1 Sections discussing results of ethylene polymerization 100
4.2 Sections discussing results of propylene polymerization 100
4.3 Reactions Considered in Ethylene Polymerization 102
4.4 Estimated Parameters for Me2Si[Ind]2ZrCl2 (E1)/MAO 106
4.5 Predicted Molecular Weights & PDI with Me2Si[Ind]2ZrCl2 (E1)/MAO 106
4.6 Reactions Considered in Ethylene Polymerization 114
4.7 Estimated Parameters for Et[Ind]2ZrCl2 (E2)/MAO 115
4.8 Polyethylene Properties with Et[Ind]2ZrCl2 (E2)/MAO 122
4.9 Reactions Considered in Propylene Polymerization 125
4.10 Estimated Parameters for Me2Si[Ind]2ZrCl2 (P1)/MAO 130
4.11 Predicted Properties with Me2Si[Ind]2ZrCl2 (P1)/MAO 130
4.12 Estimated Parameters for Et(Ind)2ZrCl2 (P2)/MAO 143
4.13 Predicted Polypropylene Properties with Et(Ind)2ZrCl2 (P2)/MAO 143
4.14 Estimated Parameters for Me2Si (Ind)2HfCl2 (P3)/MAO 154
4.15 Predicted Properties with Me2Si (Ind)2HfCl2 (P3)/MAO 155
4.16 Estimated Parameters for Et(Ind)2HfCl2 (P4)/MAO 169
4.17 Predicted Properties with Et(Ind)2HfCl2 (P4)/MAO 169
4.18 Estimated Parameters for [2,4,6-Me3Ind]2ZrCl2 (P5)/MAO 181
4.19 Predicted Properties with [2,4,6-Me3Ind]2ZrCl2 (P5)/MAO 181
4.20 Estimated Parameters for [2,4,7-Me3Ind]2ZrCl2) (P6)/MAO 190
4.21 Predicted Properties with [2,4,7-Me3Ind]2ZrCl2) (P6)/MAO 190
4.22 Estimated Parameters for Me2Si[2,4,6-Me3Ind]2ZrCl2) (P7)/MAO 200
4.23 Predicted Properties with Me2Si[2,4,6-Me3Ind]2ZrCl2) (P7)/MAO 200
xvii
NOMENCLATURE
Cat Catalyst (-)
Cocat Cocatalyst (-)
Cp Cyclopentadienyl (-)
CR Cross over frequency in DE (-)
)(iD Dead polymer chain containing i segments (-)
)(iD
Dead polymer chain detached from the catalyst (-)
nDP Number average degree of polymerization (-)
wDP Weight average of degree of polymerization (-)
f Mole fraction of dead polymer chains with terminal double bond (-)
F Weighing factor (-)
)(kF Rate based objective function (-)
)(kG Molecular weight based objective function (-)
)(kH Microstructure based objective function (-)
ka Rate constant for catalyst activation (L.mol-1
.s-1
)
kin Rate constant for chain initiation (L.mol-1
.s-1
)
kp Rate constant for chain propagation (L.mol-1
.s-1
)
kd Rate constant for spontaneous catalyst deactivation (s-1
)
ktH Rate constant for chain transfer to hydrogen (L.mol-1
.s-1
)
ktCo Rate constant for chain transfer to cocatalyst (L.mol-1
.s-1
)
kβ Rate constant for β-hydride elimination (s-1
)
klcb Rate constant for incorporation of polymer chains with terminal
double bonds
(L.mol-1
.s-1
)
ktM Rate constant for chain transfer to monomer (L.mol-1
.s-1
)
kscb Rate constant for short chain branching (s-1
)
kβ,H Rate constant for spontaneous chain transfer to catalyst (s-1
)
rk
Rate constant for reinitiation after chain transfer to catalyst (L.mol-1
.s-1
)
kβ,Me Rate constant for β-methyl elimination (s-1
)
ks Rate constant for secondary insertion (L.mol-1
.s-1
)
ksp Rate constant for propagation after secondary insertion (L.mol-1
.s-1
)
ksM Rate constant for transfer to monomer after secondary insertion (L.mol-1
.s-1
)
ktAl Rate constant for transfer to cocatalyst (L.mol-1
.s-1
)
krAl Rate constant for reactivation after transfer to cocatalyst (L.mol-1
.s-1
)
Em Molar mass of ethylene (g/mol)
SRUm
Molecular weight of the structural repeat unit (g/mol)
M Monomer (-)
nM Number average molecular weight (g/mol)
wM Weight average molecular weight (g/mol)
NP Number of population in DE (-)
P Pressure (atm, bar)
0P Active catalyst site capable of polymerization (-)
xviii
iP Polymer chain attached to a catalyst site containing i monomer
segment
(-)
0*
HP Hydride activated complex (-)
0*
MeP Methyl activated complex (-)
0dP
Deactivated catalyst (-)
A
nP
Active site of type A with a growing polymer chain of length n (-)
)(iR Secondary inserted chains (-)
Rp,max Maximum polymerization rate (mol/L/s)
t Time (sec, min)
T Temperature (°C)
rV
reactor volume (m3)
X Halogen (-)
Greek Symbols l
n n
th moment of the molecular weight distribution of live chains
attached to catalyst site
n n
th moment of the molecular weight distribution of dead chains
n n
th moment of the molecular weight distribution of dead chains
with terminal double bond
m
n n
th moment of the molecular weight distribution of secondary
inserted chains
v0 Zeroth moment for vinylidene-terminated dead chains
'0 v
Zeroth moment for vinyl-terminated dead chains
b0 Zeroth moment for butenyl-terminated dead chains
i0 Zeroth moment for isobutyl-terminated dead chains
E Density of ethylene
Abbreviations
ACO Ant colony optimization
ADM Advection dispersion model
ANN Artificial neural network
a-PP Atactic polypropylene
CAGR Compound annual growth rate
CGC Constrained geometry catalyst
CLD Chain length distribution
CSTR Constant stirred tank reactor
DE Differential evolution
DGM Dusty gas model
Et Ethyl
Flu Fluxional
GA Genetic algorithm
Ind Indenyl
i-PP Isotactic polypropylene
LCB Long chain branching
MAO Methyl aluminoxane
Me Methyl
xix
MGM Multi grain model
mPE Metallocene polyethylene
MW Molecular weight
MWD Molecular weight distribution
NLDE Natural logarithmic differential evolution
PDI Poly dispersity index
PE Polyethylene
PFFDM Polymeric flow Fick's diffusion model
PFM Polymeric flow model
PP Polypropylene
PSO Particle swarm optimization
SA Simulated annealing
SANS Small angle neutron scattering
SCB Short chain branching
TIBA Tri-isobutyl-aluminium
TMA Triethylaluminium
Y Yield
Z-N Ziegler Natta
1
CHAPTER – 1
INTRODUCTION
1.1 Motivation
Polyolefins are the largest group of thermoplastics, often referred to as commodity
thermoplastics, these are polymers of olefins such as ethylene, propylene, butenes, isoprene, and
pentenes, and their copolymers. Polyolefins consist only of carbon and hydrogen atoms and are
non-aromatic. Two most important commodity polyolefins are polyethylene and polypropylene
and they are very popular due to their low cost and wide range of applications.
The term polyethylene (PE) describes a huge family of resins obtained by polymerizing
ethylene gas, CH2=CH2, and it is by far the largest volume commercial polymer. This
thermoplastic is available in a range of flexibilities and other properties depending on the
production process, with high density materials being the most rigid. Polyethylene can be formed
by a wide variety of thermoplastic processing methods and is particularly useful where moisture
resistance at low cost is required. Low density polyethylene typically has a density value ranging
from 0.91 to 0.925 g/cm³, linear low density polyethylene is in the range of 0.918 to 0.94 g/cm³,
while high density polyethylene ranges from 0.935 to 0.96 g/cm³ and above (Odian, 2004).
Polyethylene finds variety of applications in aerospace and automotive applications, batteries,
bearings, building materials, blending, bags, containers, coating, compounding, cosmetics,
membrane, medical/healthcare applications, prosthetics, packaging and irrigation etc.
Polypropylene (PP) is produced by polymerizing propylene with suitable catalysts.
Polypropylene has demonstrated certain advantages in improved strength, stiffness and higher
temperature capability over polyethylene. Polypropylene has been successfully applied to the
2
forming of fibers due to its good specific strength and is one of the lightest plastics available
with a density of 0.905 g/cm3. Polypropylene is used in aerospace and automotive applications,
bags, batteries, bottles, coating, computer components and data storage, microwave cookware,
cosmetics, eyeglasses, films, fibers, fuel tanks, insulation, medical applications, membrane,
food/medical/pharmaceutical packaging, solar panels, tapes, tableware/disposables, sealants etc
(http://www.ides.com; 26_Dec_2012).
Polyethylene and polypropylene are increasingly replacing other materials because of
their versatile properties, low cost, reduced environmental impact, and easy recycling.
World commodity polymers consumption is estimated to reach 214 million tonnes by
2015, with polyethylene and polypropylene accounting for the largest share. An estimate of
compound annual growth rate (CAGR) of commodity polymers during 2011-2015 is given in
Table 1.1 (http://www.icis.com; 27/Dec/2012).
Table 1.1 World Commodity Polymers Consumption Estimate
Compound Annual Growth Rate (CAGR) [% / year]
2011 - 2015
Polyethylene
4.5
Polypropylene
5.8
Polyvinyl chloride
4.2
Polystyrene
2.9
Metallocene catalyzed olefin polymerization has recently attracted research interest since
these catalysts allow the production of tailored macromolecules with properties those can be
accurately designed. A broad spectrum of properties and applications of the polyolefins can be
3
attained with metallocenes due to their single types of sites. Kinetic studies of catalytic
polymerization provide considerable insight into the mechanism of the reactions and scale-up or
commercialization of a polymerization process staggeringly depends on the understanding of the
kinetic behavior of the system under various operating conditions.
1.1.1 Metallocene catalyst systems
Metallocenes belong to a relatively old class of organometallic complexes, with ferrocene being
the first discovered in 1951 (Kauffman, 1983). At that time the term metallocene was used to
describe a complex with a metal sandwiched between two η5-cyclopentadienyl (Cp) ligands.
Since the discovery of ferrocene, a large number of metallocenes have been prepared and the
term has evolved to include a wide variety of organometallic structures including those with
substituted Cp rings, those with bent sandwich structures, and even the half-sandwich or mono-
Cp complexes. Metallocene catalyst system refers to the combination of
bis(cyclopentadienyl)metal complexes of Group 4 (IVB) [especially zirconium, titanium and
hafnium], or cyclopentadienyl-substituted derivatives thereof, and a cocatalyst, typically
methylalumoxane (MAO). Titanocene and zirconocene dichlorides were the first metallocenes
studied (Natta et al. 1957(a), Breslow and Newburg, 1957).
A generic structure of metallocene catalyst is shown in Figure 1.1. Where M is the
transition metal of group IV, normally Zr, Ti and Hf, A is the optional bridging atom usually Si
or C. R is a σ – homoleptic (a metal compound with all ligands identical) hydrocarbyl such as H,
alkyl or other hydro groups and X is chlorine or other halogen from group VII A or an alkyl
group. The simplest metallocene precursor has the formula Cp2MX2.
4
Many variants of metallocenes are also available with different ligands. The most studied ligands
are η5-cyclopentadienyl (Cp) and various substituted cyclopentadienyls, including alkyl-
substituted η5-cyclopentadienyls, 1-indenyl (Ind), 4,5,6,7-tetrahydro-1-indenyl (H4Ind), and 9-
fluorenyl (Flu) ligands as shown in Figure 1.2 (Odian, 2004).
MAO is an oligomeric compound described by the formula (CH3AlO)n, structure of
which is not yet fully understood. MAO plays several roles: it alkylates the metallocene
precursor by replacing halogen atoms with methyl groups; it produces the catalytic active ion
pair Cp2MCH3+/MAO
−, where the cationic moiety is considered responsible for polymerization
and MAO− acts as weakly coordinating anion.
1.1.2 Evolution of the metallocenes
The first step towards controlled polyolefin polymerization was taken by Karl Ziegler and his
group in 1953 (Ziegler, 1963). While investigating ethylene oligomerization in the presence of
aluminium alkyls, they discovered that transition metal compounds were efficient catalysts. In
the presence of aluminium alkyl activators, zirconium and titanium halides catalyzed the
polyinsertion process, which yielded high molecular weight and high density linear polyethylene.
One year later Natta introduced the process of stereoselective α-olefin and diene polymerization.
The discovery of Ziegler-Natta catalysts together with Phillips-type (activator-alkyl-free
SiO2/CrO3) catalysts initiated a rapid growth of polyolefin technology and the production of
polyolefin materials exhibiting a broad range of properties. In 1963, Ziegler and Natta were
awarded the Nobel prize in chemistry (Tynys, 2007).
The polymerization of ethylene with a single-site, metallocene-type catalyst was reported
for the first time in 1957. Initially, these catalysts showed very low polymerization activity due
5
to the cocatalyst employed [Et2AlCl or Et3Al] (Natta et al., 1957(b), Breslow and Newburg,
1957).
AR
R
R
R
RR
X
X
M
M
X
X
Figure 1.1 Generic structure of metallocene catalyst.
CH•
Indenyl
CH•
tetrahydroIndenyl
H•C
fluorenyl
Figure 1.2 Various ligands of metallocene.
In 1973, Reichert and Meyer reported that small amount of water, remarkably improved
the activity of the catalyst system Cp2TiEtCl/EtAlCl2. They proposed a stabilized catalyst
complex resulting from an increase in Lewis acidity (Reichert and Meyer, 1973). Prior to this
water was considered to be a catalyst poison. Next important step was made using racemic
ethylene-bis(4,5,6,7-tetrahydro-l-indenyl)titanium dichloride synthesized by Wild et al. in 1982
for stereospecific polymerization of propene.
6
Ewen et al. in 1988 synthesized a Cs-symmetric zirconocene ([Me2C(Flu)(Cp)]ZrCl2)
capable of producing crystalline syndiotactic polypropylene in high yields at conventional
polymerization conditions.
Since late 1980s, a worldwide, industrial and academic research & development in the
field of metallocene catalysts has been continued.
1.1.3 Mathematical modeling and simulation
Many problems encountered in industrial polymerization processes are associated with inherent
complexities in polymerization kinetics and mechanisms, physical changes and transport effects
(e.g., viscosity increase, particle formation, precipitation, interfacial mass and heat transfer
limitations), non-ideal mixing and conveying, and strong process nonlinearity (potential thermal
runaway, limit cycles, multiple steady states).
Models in polymer reaction engineering involve phenomena at different scales, which
can be classified as macro-; meso- and microscale (Ray, 1988; McKenna and Soares, 2001). The
features of various scales of modeling are presented in Table 1.2.
In the recent yesteryears, modeling of polymerization processes has become more
demanding. Success of meso- and macro-level models greatly relies upon the understandings at
micro-level. Detailed physical properties and thermodynamic data on the partitioning of species
among phases are required to quantify the concentrations of reactants at the loci of
polymerization and valid kinetic rate constants are required for calculating rates and polymer
properties. Rate constants are related to the structure of reactants used in polymerization which
render the effective use of process models in state estimation and control.
7
Table 1.2 Scales of Modeling Polyolefin Processes
Scale Feature
Macroscale (> 1 m) Detailed description of reactor hydrodynamics in order to model
mixing and reactor stability, reactor and particle size distributions,
particle entrainment, etc.
Mesoscale (> 10−3–10−
2 m) Modeling of interparticle, intraparticle, and particle–wall
interactions, especially in terms of heat and mass transfer. This in
turn requires models for particle morphology evolution, and
monomer adsorption. This is also the interface between the
continuum approach used at the macroscale and the discrete
approach needed at the microscale.
Microscale (< 10-3
m) Modeling of polymerization kinetics, the nature of active sites,
diffusion of monomer in the polymer and crystallization of
polymer molecules.
Mathematical modeling is a powerful tool not only for the development of process
understanding, but also for the design of advanced process technology. In particular, a kinetic
model plays an important role in designing polymerization conditions to tailor a polymer’s
molecular architecture. A comprehensive kinetic study of polymerization process helps
developing effective models at meso- and macro-levels, so this study has been focused on the
estimation of kinetic parameters and prediction of polymer properties through modeling at
micro-level.
8
The two most representative objectives in modeling polymerization reactions are to compute (1)
polymerization rate and (2) polymer properties (molecular level and microscopic level) for
various reaction conditions.
In general, polymerization models are derived from the fundamental chemistry and
physics of the polymerization processes to calculate reaction rates and polymer architectural
parameters. Such models are called the first principles models. For certain polymerization
systems, complex molecular structures are not appropriate for the first-principles modeling and
hence empirical or semi-empirical models are the practical alternatives (Yoon et al., 2004).
Determining the parameters of a kinetic model by using laboratory, pilot plant, or plant
data is the most critical step for the successful development of a process model. Practically, it is
not always possible to design experiments to determine all the relevant kinetic parameters.
Therefore, in modern kinetic modeling, pseudo-rate constant methods and computer aided
parameter estimation techniques are widely used.
In transition metal catalyzed olefin polymerizations, the kinetic parameters are catalyst
dependent. Therefore, whenever a new catalyst is employed, a new set of kinetic parameters
must be determined. Considering the fact that the properties of polyolefins are mostly dictated by
the nature of catalyst being used and that a large number of different types of catalysts is used for
different polymer grades, it becomes very important to have a well-established parameter
estimation procedure that can be applied to any catalyst systems.
1.2 Objectives of research
The objectives of the present research are to
1. Study the mechanistic aspects of Ziegler-Natta and metallocene catalyst systems and
various polymerization mechanisms of olefins using metallocene catalysts.
9
2. Develop kinetic models for the polyolefin synthesis using metallocene catalysts.
3. Validate the developed models with available experimental data in literature and to
estimate the kinetic model parameters.
4. Study the effects of various parameters like polymerization time, temperature, monomer
pressure (concentration), catalyst concentration, co-catalyst to catalyst mole ratio and
concentration of transfer agents etc.
1.3 Organization of thesis
The thesis is presented in five chapters. An exhaustive review of literature on olefin
polymerization using various metallocene catalysts is given in Chapter 2. Mathematical models
developed for ethylene and propylene polymerization with metallocene catalysts and simulation
methodology are discoursed in Chapter 3. The obtained simulation results are discussed in detail
in Chapter 4. In Section 4.1, results obtained from simulations of ethylene polymerization model
with different metallocene catalysts are discussed. Model simulation results of solution phase
propylene homopolymerization with various metallocene catalysts are presented and discussed in
Section 4.2. Chapter 5 deals with the summary of the work and important conclusions drawn
from the present study.
10
CHAPTER – 2
LITERATURE REVIEW
Over the past two decades new catalyst technologies have reinvigorated polyolefin industry by
rapidly expanding new polyolefin materials and technology. Catalysts based on group 4 metals
exhibit the most attractive combination of activity, selectivity and generality to a wide variety of
α-olefins. In all polymerization reactions, the phases of chain growth include initiation,
propagation, and termination and the corresponding elementary rate laws and kinetic constants
completely describe the catalytic kinetics and the distribution of polymer products. In this
chapter, the literature on olefin polymerization catalyzed by transition metal catalysts is
summarized and the existing literature on metallocene catalyzed polymerization is discussed
mainly with regard to ethylene and propylene.
Olefin polymerization with Ziegler-Natta and metallocene catalysts and important
mechanisms available in literature are presented in Sections 2.1 and 2.2 respectively. Various
studies reported in the literature on ethylene and propylene polymerization using metallocene
catalysts are discussed in detail in Sections 2.3 (experimental) and 2.4 (theoretical and modeling)
of this chapter.
2.1 Ziegler-Natta polymerization
Ziegler–Natta catalysts have been used in the commercial manufacture of various polyolefins
since 1956 (Heinen, 2012). Usually Ziegler catalysts refer to Ti-based systems for
polymerization of ethylene and Ziegler–Natta catalysts refer to systems for polymerization of
propylene (Cerruti, 1999).
11
2.1.1 Ziegler-Natta catalysts
The Ziegler-Natta catalyst is the combination of a transition-metal salt whose metal is from
groups IV to VII of the Periodic Table, and a metal alkyl whose metal is from groups I to III of
the Table. Different generations of Z-N catalyst are shown in Table 2.1 (Suba et al., 2007).
Table 2.1 Generations of Ziegler-Natta Catalyst
Generation Catalyst composition Productivity
(kg/g)
Isotactic
Index
I. δ-TiCl3, 0.33AlCl3 + (Et)2AlCl 1.5 90-94
II. γ-TiCl3 + (Et)2AlCl 4.0 94-97
III. TiCl4/monoester (ID)/MgCl2 + (Et)3Al/ester (ED) < 20 90-95
IV. TiCl4/diester (ID)/MgCl2 + (Et)3Al /silane (ED) > 25 95-99
V. TiCl4/diether, succinate (ID)/MgCl2 + (Et)3Al > 50 95-99
ID: internal (electron) donor; ED: external (electron) donor
2.1.2 Mechanism of Ziegler-Natta polymerization
Various mechanisms have been proposed to explain the olefin polymerization catalyzed by
Ziegler-Natta initiators and several good reviews are available in literature (Fontana and
Osborne, 1960; Boor, 1979; Corradini et al., 1982; Cavallo et al., 1998). As part of an effort to
unite the cognition in this field, several attempts have been made to propose a mechanism that
could be applied to all Ziegler-Natta catalyzed polymerizations but the mechanism of
polymerization through Z-N catalyst is still not absolutely clear. Out of several proposed
mechanisms, the most recognized ones are summarized in Figure 2.1 through Figure 2.8
(Goodman, 1967; Boor, 1979; Ystenes, 1991; Castonguay and Rappe, 1992; Hamielec and
Soares, 1996; Margl et al., 1999).
12
Ti
H2C
R
H3C
Al
CH2 CHR
Ti
H2C
R
H3C
Al
Ti
H2
C
R
CH3
Al
CH2 CHR
Ti
H2C
R
RHC
Al
CH2 CH3
Figure 2.1 Bimetallic mechanism of Z-N polymerization by Natta.
Ti
H2C
R
H3C
Al
CH2 CHR
Ti
H2C
R
H3C
Al
Ti
CH2
R
CH3
Al
H2C
RHC CH2
HRC
Ti
H2C
R
CH3
Al
H2C
HRC
Ti
H2C
R
RHC
Al
CH2CH3
Figure 2.2 Bimetallic mechanism of Z-N polymerization by Patat and Sinn.
13
M
P
+ CH3 M
P
CH3
M
P
CH3
M
CH3
P
Vacant coordination site P Growing polymer chain
Figure 2.3 Monometallic mechanism of Z-N polymerization by Cossee.
Figure 2.4 Trigger mechanism of Z-N polymerization by Ystenes.
14
M
H2C CH
R
H
CH2H2C
M
H2C CH
R
CH3H2C
+
Figure 2.5 Chain termination by β-H transfer to monomer (β-H elimination).
M
H2
C
CH R
HM
H2C CH
R
H
+
Figure 2.6 Chain termination by spontaneous intramolecular β-H transfer.
M
H2
C CH2
R
HM
H3C CH2
R
H
+H
Figure 2.7 Chain termination by molecular hydrogen.
M
H2
C CH2
R
MH2
C CH2
RCH2CH3 ++ Al(C2H5)3 (C2H5)2Al
Figure 2.8 Chain termination to the Group I-III metal alkyl.
15
The comparative extents of these reactions depend on various factors such as monomer, the
initiator components, temperature, concentrations and other reaction conditions. Under normal
conditions of polymerization, intramolecular hydride transfer is negligible and termination
occurs mainly by transfer reactions (Chanda, 2013).
2.2 Metallocene polymerization
There are three different commercial olefin polymerization processes where metallocene
catalysts can be used viz solution, gas phase and slurry process. Homogeneous catalysts are used
in the solution process. The first commercial metallocene polyethylene (mPE) was introduced to
the market as differentiated or specialty product for applications like food packaging and impact
modifiers, available with the densities ranging from 0.86 to 0.91 g/cm3. Typical examples are
ExactTM
plastomers by Exxon commercialized in 1991 and the AffinityTM
and EngageTM
products by Dow in 1993. In 1995 the first commodity mPE products were commercialized by
Exxon as ExceedTM
polyethylene with densities ranging from 0.915 to 0.930 g/cm3 (Lue, 1999).
Metallocene catalysts are inherently soluble (homogeneous) catalysts, therefore, the
solution process was the first commercial process to use metallocene catalyst to produce
polyethylenes. Gas phase and Slurry process require heterogeneous catalyst. Metallocene
catalysts need to be supported so that they can be employed in gas phase or slurry phase olefin
polymerization processes. They can be supported in three different ways: 1) The metallocene
compound is reacted with supported MAO to produce a supported catalyst. 2) A cross-linked
MAO, which is insoluble in hydrocarbon, is reacted with a metallocene compound to produce a
supported catalyst. The cross-linked MAO may or may not contain inert supports. 3) To support
the metallocene compound first and then react it with soluble MAO.
16
2.2.1 Metallocene catalyst system
Generic structure and features of metallocene catalyst are discussed in Section 1.1.1. Neutral
metallocene compound is inactive without an activator and requires a strong Lewis acid to form
a cationic metal center, which is active in α-olefin polymerization. The predominating cocatalyst
for metallocene activation is methylaluminoxane (MAO). The aluminoxane is understood to be
involved in site activation by alkylation of the metallocene, prevention of site deactivation, and
impurity scavenging (Hamielec and Soares, 1996). The cocatalyst concentration affects the
productivity of the polymerization and the molecular weight of the polymer. MAO is prepared
by controlled hydrolysis of trimethylaluminum (TMA) as shown in Figure 2.9. The structure of
the MAO is not known with certainty, Figure 2.10 unveils various structural suggestions of
MAO in literature (Zohuri et al., 2012).
A number of other activators have been developed as of late. Some organic boron
compounds, such as trisphenylmethyltetrakis-(pentafluorophenyl)borate [Ph3C]+ [B(C6F5)4]
-,
especially fulfill a role as noncoordinating, non-nucleophilic counter anion to the active cationic
species (Chen and Marks, 2000).
Both, the ligand set of a single-site catalyst and the growing polymer chain are found to
influence the stereochemistry. In a chain-growth polymerization reaction a polymer chain
remains bound to the active metal center during monomer enchainment and the stereogenic
center from the last enchained monomer unit influences the stereochemistry of monomer
addition; if this influence is significant, the mode of stereochemical regulation is referred to as
“polymer chain-end control”, whereas if the ligand set is chiral and overrides the influence of the
polymer chain end, the mechanism of stereochemical direction is termed “enantiomorphic-site
control” as shown in Figure 2.11 (Coates, 2000).
17
nH2O + (n+1) AlMe
Me
Me
Al
Me
Me
O Al
Me
Men
+ 2n CH4
Figure 2.9 Partial hydrolysis of trimethylaluminum to form MAO.
Al
Me
Me
O Al
Me
On
Al
Me
Me(a)
Al
Me
O Al
Me
On
Al
Me
(b)
O
Al
O
Me
O
Al
Me
Al
O
Me
O
Al
O
Al
Me
O
Al
MeMe
(c)
Al
OO
Al Al
O
Al
O O
O
AlAl
O
Al Al
O
Al Al
(d)
Figure 2.10 (a) Linear and (b) cyclic structures of MAO.,
(c) Two-dimensional ladder and (d) three-dimensional cage structures of MAO oligomers.
18
MP
MP
LnM
m mm mmmr
End controls stereochemistry
isotacticStereoerror
Pm
m
MP
MP
LnM
m mm mmmr
End controls stereochemistry
isotacticStereoerror
Pr
r
(a)
M P MP
LnMP
r mmmr
Ligand controls stereochemistry
isotacticStereoerror
mm
(b)
Figure 2.11 (a) Chain-end and (b) Enantiomorphic site mechanisms of stereocontrol
19
The number of known metallocene complexes is very large. Various types of
metallocenes are generally categorized as nonstereorigid, nonstereorigid ring substituted,
stereorigid, cationic and supported metallocenes. Representative examples of each
category are shown in Table 2.2 (Gupta et al. 1994).
Table 2.2 Representative Examples of Metallocenes
[a] Nonstereorigid metallocenes:
(i) Cp2MCl2 (M = Zr, Ti, Hf)
(ii) Cp2ZrR2 (R = CH3, Ph, CH2Ph, CH2SiMe3)
(iii) (Ind)2ZrMe2
[b] Nonstereorigid ring substituted metallocenes:
(i) (Me5C5)2MCl2 (M = Zr, Ti, Hf)
(ii) (Me3SiCp)2ZrCl2
[c] Stereorigid metallocenes:
(i) Et(Ind)2ZrCl2
(ii) Et(Ind)2ZrMe2
(iii) Et(IndH4)2ZrCl2
[d] Cationic metallocenes:
(i) Cp2MR(L)+[BPh4]
- (M = Zr, Ti)
(ii) [Et(Ind)2ZrMe]+[B(C6F5)4]
-
(iii) [Cp2ZrMe]+[C2B9H11)2M]
- (M = Co)
[e] Supported metallocenes:
(i) Al2O3-Et[IndH4]2ZrCl2
(ii) MgCl2.Cp2ZrCl2
(iii) SiO2.Et[Ind]2ZrCl2
Many metallocene systems are found to be active for both ethylene and propylene
polymerization. Activities may vary from high to moderate depending upon the structure
of the catalyst, moreover stereorigid catalysts are found to be stereo- and regiospecific
towards propylene polymerization. Different metallocene systems used in ethylene and
propylene polymerization are summarized in Table 2.3.
20
Table 2.3 Metallocene Catalyst Systems
Catalyst Cocatalyst Remarks
[a] Chlorocyclopentadienyl
derivatives of Ti,
e.g. Cp2TiCl2
Dialkyl aluminium
chloride
Active for ethylene; inactive
for propylene
polymerization.
[b] Nonstereorigid,
e.g. Cp2MX2
MAO Highly active for ethylene;
active for propylene
(atactic).
Stereorigid,
e.g. Et(Ind)2MCl2
MAO Active for stereo- and
regiospecific propylene
polymerization.
Supported,
e.g. SiO2.Et[Ind]2MCl2
MAO /
Alkylaluminium
Active for ethylene and
propylene polymerization.
[c] Ionic Metallocenes
e.g. Cp2MR(L)+[BPh4]
-
- Active for ethylene and
propylene polymerization.
Metallocenes in ethylene polymerization
The metallocenes generally used for ethylene polymerization are unbridged, bridged,
substituted and half-sandwich complexes. Polyethylene is most commonly polymerized
with cyclopentadiene metallocenes of Ti, Zr, and Hf, with zirconocene preferred for their
stability. The metallocenes generally used for ethylene polymerization are achiral.
Examples of unbridged and bridged catalysts are shown in Figure 2.12.
Metallocenes in propylene polymerization
Examination of the outcomes of many separate investigations reveals a predictable
relationship between metallocene complex symmetry and polypropylene tacticity (Leek et
al., 1997; Ewen, 1998). Achiral metallocenes yield atactic polypropylene. Stereoregular
polypropylene requires the use of chiral metallocene catalysts. Metallocene catalyst
21
symmetries are usually either oblique (C1, C2) or rectangular (Cs, C2v) as shown in Figure
2.13. Single-site polymerization catalysts, based on ligand geometries of catalysts and
their stereoselectivities for polyolefin synthesis, can be divided into four main symmetry
categories viz C2v, Cs, C2, C1 as shown in Figure 2.14.
Catalysts exhibiting C2v symmetry typically produce atactic polymers or
moderately stereoregular polymers by chain-end control mechanisms. Cs-symmetric
catalysts that have mirror planes containing the two diastereotopic coordination sites
behave similarly. Cs-symmetric catalysts that have a mirror plane reflecting two
enantiotopic coordination sites frequently produce syndiotactic polymers. C2-symmetric
complexes, both racemic mixtures and enantiomerically pure ones, typically produce
isotactic polymers via a site-control mechanism. Stereoselectivities of asymmetric (C1)
complexes are unpredictable and have been reported to produce polymer architectures
ranging from highly isotactic, to atactic, including isotactic-atactic stereoblock and
hemiisotactic (Razavi et al., 2006; Chen, 2009).
Supported metallocene catalyst systems are preferred to soluble versions in
conventional polyolefin plants, which were designed to use supported Ziegler- Natta or
Cr2O3-based catalysts. Metallocenes can be supported on a number of substrates, such as
SiO2, MgCl2 or Al2O3. Supported catalysts also provide polypropylene with fewer
stereochemical defects (Rudin, 1999).
22
R1 R2
R3
R4
R5M
XR1
R5
R4
R3
R2X
Zr
ClCl
X
M
ClCl
X
Zr
ClCl
X
R1
R2
R2
R1
M
ClCl
X
R
Zr
ClCl
X
Figure 2.12 Unbridged (a) and bridged (b) catalysts used in ethylene polymerization.
M: Zr, Ti, Hf
X: Cl, CH3
R: H, CH3
X: C2H4, (CH3)2Si; R1: CH3; R2: CH3
X: (CH3)2Si; R1: Ph, Naph; R2: H
M: Zr, Hf;
X: C2H4, (CH3)2Si X: C2H4, (CH3)2Si M: Zr; X: (CH3)2C, Ph2C; R: H, CH3,
tBu
M: Hf; X: (CH3)2C; R: H
X: (CH3)2Si, C2H4
(b)
(a)
23
C1 C2Cs C2vOblique Rectangular
Figure 2.13 General symmetry classifications.
MP
C2v
MP MP MP MP
meso-Cs C2 C1Cs
Figure 2.14 General metallocene symmetry classifications.
24
2.2.2 Mechanism of metallocene polymerization
Understanding the mechanisms and kinetics involved in the polymerization process enables to
predict the structure of the polymer formed. Propagation and termination rates determine
molecular weight, molecular weight distribution while catalyst initiation and deactivation
processes have an influence on the kinetics, and the cocatalyst may have an effect on the extent
of the prevailing mechanisms (Alt and Köppl, 2000; Resconi et al., 2000; Imanishi and Naga,
2001). Various mechanisms of olefin polymerization with metallocene catalyst are discussed in
this section.
Mechanisms for activation
Studies reveal that besides acting as a scavenger for impurities in the reaction medium and
alkylating agent, methylaluminoxane (MAO) is involved in the formation of a cationic Group 4
metal center with a vacant coordination site (Brintzinger et al., 1995; Ystenes et al., 2000;
Estenoz and Chiovetta, 2001; Takeuchi, 2010). MAO is considered to act as a Lewis acid,
abstracting chloride/methyl groups from metallocene, and thus enabling the formation of active
species (Pѐdeutour et al., 2001).
The formation of the metallocenium center, which contains a vacant coordination site,
takes place during a fast ligand exchange between methyl groups of MAO and chlorine in the
metallocene catalyst. After methylating the catalyst, MAO abstracts a methide ligand with active
metallocenium catalyst formation as shown in Figure 2.15.
25
M
Cl
ClMAO
M
Cl
Me MAOM
Me
M
Me
Me
MAO MAO
Open coordination
site
MAO
Figure 2.15 Activation of a metallocene complex by methylaluminoxane (MAO).
M = transition metal atom and □ = vacant coordination site.
26
Mechanisms for propagation
Several mechanisms have been proposed for propagation in olefin polymerization with
metallocenes. This has been generally accepted that propagation proceeds by α-olefin
coordination and insertion via a transition state (Resconi et al., 2000). Important mechanisms
are discussed in the following subsections.
Cossee Arlman mechanism
Cossee-Arlman mechanism for propagation in metallocene catalyzed polymerization of α-olefins
is inspired by and quite so similar to the one proposed by Cossee for Ziegler-Natta olefin
polymerization. Figure 2.16 explains the mechanism schematically.
Green-Rooney mechanism
The mechanism proposed by Rooney and Green, involves an oxidative 1,2-hydrogen shift from
the α-carbon of the polymer chain, generating a metal-alkylidene hydride. This species then
reacts with an olefin to generate a metallacyclobutane, and reductive elimination completes the
propagation sequence as schematized in Figure 2.17 (Grubbs and Coates, 1996). Green-Rooney
mechanism, involving metathesis like step was not accepted and refuted convincingly by
Clawson and coworkers (Clawson et al., 1985).
Modified Green-Rooney Mechanism
Modified Green-Rooney mechanism, proposed by Green, Rooney and Brookhart is an
intermediate version to the Cossee-Arlman and Green-Rooney mechanisms, where a hydrogen
atom on the α-carbon of the growing polymer chain interacts with the metal center all over the
catalytic cycle. This three-center, two-electron covalent bond, termed as 'agostic interaction',
27
occurs when the hydrogen atom is simultaneously bonded to both a carbon and a metal atom as
shown in Figure 2.18.
Transition State α-agostic Mechanism
This mechanism is a hybrid of the Cossee-Arlman and modified Green-Rooney mechanisms.
This meshanism suggests the olefin insertion, where an α-hydrogen interacts with the metal
center only during the transition state of the C-C bond formation (shown in Figure 2.19).
The presence of α-agostic interaction in the transition state has been observed
experimentally in mechanistic studies (Leclerc and Brintzinger, 1995, 1996) and is consistent
with the proposed conformation adopted by the growing polymer chains as a means of affecting
enentiaoselective propylene insertion. These experimental results support the modified Green-
Rooney mechanism as the most likely mechanism for propylene insertion.
28
M
Me
M
Me
M
Me
n
M
Men
Figure 2.16 Cossee-Arlman mechanism of propagation with a metallocene catalyst.
M
CH2P
M
CHP
H
M
CHP
H
R
M
PHC
H
R
M
PH2C R
Figure 2.17 Green-Rooney mechanism of propagation with a metallocene catalyst.
29
M
H
P
H
M
H
P
H
M
H
P
H
M
H
P
H
M
H
H
P
Figure 2.18 Modified Green-Rooney mechanism of propagation with a metallocene
catalyst.
M
H
P
H
M
H
P
H
M
H
P
H
M
H
P
H
M
H H
P
Figure 2.19 Transition state α-agostic mechanism of propagation with a metallocene
catalyst.
30
Mechanisms for termination
Termination usually occurs via transfer mechanisms. Each chain transfer reaction results in
dissociation of the chemical bond between the metal atom in a metallocene active center and the
last monomer unit in the growing polymer chain. A number of chain termination modes are
possible, some of the accomplished mechanisms are described below:
β-Hydrogen transfer to monomer
In this mechanism, β-hydrogen is transferred from the growing polymer chain to an incoming
monomer as depicted in Figure 2.20. This is the prevalent chain termination mechanism under
the common experimental conditions (Margl et al., 1999).
This reaction results in the formation of polymer molecules with the terminal -vinyl
group in ethylene polymerization and vinylidene group in homo- and co-polymerization of α-
olefins.
β-Hydrogen elimination (Spontaneous chain transfer)
Chain termination may also take place via spontaneous β-hydrogen transfer to the transition
metal atom of metallocene as shown in Figure 2.21.
The preference for a β-hydrogen transfer to monomer vs. transition metal is often
determined by spacial conditions in the vicinage of the transition metal atom in the active center.
Since the transfer to monomer is preceded by coordination of a monomer molecule at the metal
atom, it is favoured when the active center is more open sterically. For example, β-hydrogen
transfer to monomer is preferred chain transfer reaction in propylene polymerization with the
metallocene catalyst based on the meso-isomer of Me2Si(3-Me-Ind)(Ind)ZrCl2 (one of its lateral
31
side is open). On the other hand, the active center based on the racemic isomer of the same
complex has both its coordination positions sterically crowded, so spontaneous chain transfer is
more preferred in this case (Kissin, 2008).
β-Methyl elimination
Chain termination via β-methyl elimination occurs only in special cases like at high
polymerization temperatures with hafnocenes (Figure 2.22). The mechanism of transfer is
similar to the β-hydrogen elimination. A polymer chain coordinated to the sterically more
hindered site predominantly undergoes a unimolecular β-methyl elimination reaction and leads to
ethenyl (allylic) end groups (Hajela and Bercaw, 1994; Guo et al., 1994; Resconi et al., 1996;
Schöbel et al., 2013).
Chain transfer to aluminium
MAO usually contains leftover Al(CH3)3, which may also render the chain termination via
transfer to aluminium as shown in Figure 2.23. Chain transfer to Al is more commonly observed
at lower propylene concentration (Resconi et al., 1990; Naga and Mizunuma, 1998).
32
M
CH2
H
P
H
MM
CH2
H
P
HM
CH2
H
P
HM
H
P
H2C
H
H
H
Figure 2.20 β-Hydrogen transfer to monomer.
M
CH2
H
P
H
MM
H
PHM
CH2
H
P
H
Figure 2.21 β-hydrogen elimination (Spontaneous chain transfer).
M
CH2
H
Me
P
MM
Me
P MeM
CH2
Me
P
H
M
CH2
Me
P
H
Figure 2.22 β-methyl elimination (Spontaneous chain transfer).
M
CH2
H
P
Me
M RM Al
CH2
R
R
R
+ Al
R
R
R
CHMeP
+ Al
R
CH2
R
CHMeP
Figure 2.23 Chain transfer to cocatalyst (aluminium).
33
2.3 Experimental studies
Ethylene polymerization studies on nonstereorigid, stereorigid and supported metallocene
catalysts are generally carried out to evaluate the activity of the catalysts and product
properties. Tacticity has been the additional interest of investigation in propylene
polymerization apart from kinetic studies. Among many metallocene catalysts, zirconium
based catalysts have been extensively studied in both ethylene and propylene
polymerization. Salient features of some of the experimental studies reported in the
literature on metallocene catalyzed ethylene and propylene polymerization are presented
below.
Several researchers (Rieger and Jainik, 1994; Charpentier et al., 1997; Chakravarti
and Ray (2001); Young and Ma, 2002; Marques and Alcantara, 2004; Zohuri et al., 2005;
Sarzotti et al., 2007) have studied experimental aspects of ethylene polymerization with
zirconocene dichloride/methylalumoxane/trimethylaluminum (Cp2ZrCl2/MAO/TMA)
catalyst systems. In their work, the investigators have examined the effects of variation
of the monomer concentration, catalyst concentration, Al/Zr ratio, polymerization
temperature on the catalyst activity and polymer properties.
Agnillo et al. (1998) in addition to Cp2ZrCl2, investigated its titanium and
hafnium analogues (Cp2TiCl2 and Cp2HfCl2), as well as rac-
ethylenebis(indenyl)zirconium dichloride (Et(Ind)2ZrCl2) and rac-ethylenebis(4,5,6,7-
tetrahydroindenyl)zirconium dichloride (Et(H4Ind)2ZrCl2) for ethylene polymerization.
Reactors with different reaction environments like solution phase (Charpentier et
al., 1997; Zohuri et al., 2005; Sarzotti et al., 2007), slurry process (Chakravarti and Ray
(2000); Young and Ma, 2002) and alumina as support to catalysts (Marques and
Alcantara, 2004) were used.
34
Chu et al. (2000 a) carried out ethylene polymerization with a novel in-situ-
supported metallocene catalyst that eliminated the need for a supporting step before
polymerization. In their sequential study Chu et al. (2000 b) proposed a polymerization
mechanism for the in situ supported Et[Ind]2ZrCl2 catalyst suggesting that during
polymerization, the in situ supported metallocene catalysts may deactivate, but
homogeneous metallocene species present in the reactor may form new active sites and
compensate for deactivated sites.
Mehdiabadi and Soares (2009) studied the solution polymerization of ethylene
using rac-Et(Ind)2ZrCl2/MAO and dimethylsilyl(tert-butylamido)(tetramethyl-
cyclopentadienyl)titanium Dichloride (CGC-Ti)/MAO in a semi-batch reactor and
investigated how polymerization conditions affect the polymerization kinetics with two
metallocene catalysts.
ethylenebis(indenyl)zirconium dichloride (EtInd2ZrCl2) with silica support (Tissea
et al., 2010 a) and the role of morphological properties of different silica used as supports
(Tissea et al., 2010 b) were investigated to infer the effects of monomer concentration,
temperature and alkyl aluminium concentration, upon the reaction rate and the
polyethylene properties.
Firme et al. (2005) carried out ethylene and propylene polymerization using
Ind2ZrCl2 and Ind2Zr(CH3)2/MAO catalytic systems modified by the sterically demanding
bridged alicyclic alcohols, adamantan-1-ol, adamantan-2-ol, 2-methyladamantan-2-ol,
and fenchyl alcohol. Polymers with higher molecular weights were obtained with
modifiers as compared to the non modified systems, but no structural changes in the
polyethylenes were observed.
The initial stages of gas-phase polymerizations of ethylene and propylene were
analyzed by Machado et al. (2011) using a fixed bed stopped flow reactor. The very early
35
development of particle morphology and polymer properties were analyzed for three
different commercial catalyst systems: MgCl2- and SiO2-supported Ziegler–Natta and
SiO2-supported metallocene.
Bridged metallocene catalysts have attracted extensive research interest due to
their high activity towards ethylene polymerization and control over chain branching.
Roos et al. (1997) carried out ethylene polymerization in a stirred powder bed reactor
with silica supported rac-Me2Si[Ind]2ZrC12/methylaluminoxane (MAO) with the
objective to study the influence of temperature on the gas phase polymerization of
ethylene. Authors also modeled deactivation as a first order dependence with respect to
the polymerization rate.
Wang et al. (1998) used high-temperature and high-pressure continuous stirred-
tank reactor (CSTR) for the polymerization of ethylene with the constrained geometry
metallocene system, [C5-Me4(SiMe2NtBu)]TiMe2(CGC-Ti)/tris(pentafluorophenyl)boron
(TPFPB)/ modified methylaluminoxane (MMAO) and synthesized polyethylenes with
long chain branching (LCB) densities up to 0.44 carbons/10000 carbons, and narrow
polydispersity indices about 2.
Studies on ethylene polymerization with cycloalkylidenebridged cyclopentadienyl
metallocene (Wang et al. 2005), 4,4 -bis(methylene)biphenylene bridged homodinuclear
titanocene and zirconocene (Sun et al., 2006), Ph2C(Cp)(Flu)ZrCl2 (Freitas et al., 2011),
N,N-ethylenebis(3-methoxysalicylideneiminato)titanium dichloride (Pietruszka et al.,
2012) and bridged cyclopentadienyl indenyl (fluorenyl) zirconocene complexes (Huang et
al., 2010) activated by MAO were focused to investigate the effects of various reaction
conditions on polymerization rate and polyethylene properties.
Petitjean et al. (1999) through density functional calculations described possible
mechanisms of ethylene polymerization in the presence of zirconocene catalysts.
36
Ramachandran et al. (2009) employed small-angle neutron scattering (SANS) to
investigate the structure and long chain branch (LCB) content of metallocene-catalyzed
polyethylene and applied a scaling approach to SANS data to determine the mole fraction
branch content of LCBs in PE.
Literature on propylene polymerization shows a huge concern on tacticity aspects in
addition to the other research interests. A great attention has been given to the isospecific
polymerization of propylene.
The kinetics of propylene polymerization initiated by rac-(EBI)Zr(NMe2)2) /
MAO (Kim and Hwang, 1998), rac-Me2Si(1-C5H2-2-Me-4-tBu)2Zr(NMe2)2 / MAO (Kim
et al., 1999), rac-(EBI)Zr(NC4H8)2) / MAO (Kim, 1999) were studied by changing
various experimental parameters. The molecular weight of isotactic polymer produced
was, in general, found to decrease with an increase in [Al]/[Zr] ratio, polymerization
temperature, and catalyst concentration, whereas a reverse trend was observed for catalyst
activity.
Resconi et al. (1999) analyzed the chemical structures of end groups of medium-
low molecular weight atactic and isotactic polypropylenes (a-PP and i-PP), produced with
zirconocene/MAO catalysts and used to infer the chain-transfer reaction mechanisms,
which were subsequently correlated with the zirconocene ligand structure and the
polymerization conditions. For the chiral, isospecific ansa-zirconocenes such as rac-
[ethylenebis(1-indenyl)]ZrCl2 / MAO and rac-[ethylenebis(4,7-dimethyl-1-indenyl)]ZrCl2
/ MAO catalysts, i-PP molecular weight was observed dependent on the regiospecificity
of the catalyst.
Lin et al. (2000) investigated the kinetics of propylene polymerization in toluene
solution by bis(2-phenylindenyl)zirconium dichloride, (2-PhInd)2ZrCl2/MAO at 20 °C.
37
Polymerization rates were found to be increasing with increase in monomer and
zirconium concentrations and the activity decreased faster at higher monomer
concentrations.
A good deal of experimental studies on various isospecific metallocene catalysts
for propylene polymerization has been carried out and is available in literature (Schmidt
and Alt, 2001; Meier et al., 2001; Belelli et al., 2001; Marques et al., 2002; Marques et
al., 2003; Song et al., 2003; Song et al., 2004; Palza et al., 2006).
The kinetic behaviour of propene polymerization in heptane using
bis(cyclopentadienyl)zirconium dichloride / MAO as catalyst system was studied by
Ochoteco et al. (2001). Investigations were made for both homogeneous and
heterogeneous systems and the effect of the process variables such as temperature,
pressure, Al/Zr ratio and catalyst concentration on the catalytic performance (activity and
polymer properties) were investigated.
Marques et al. (2002) investigated the effect of temperatures and Al/Mt on the
catalyst activity and the polymer characteristics. Higher catalyst activity for the
zirconocenes was observed, while the hafnocenes produced polypropylene with higher
molecular weight. The complexes with dimethylsilane bridge produced polypropylene
with higher molecular weight, stereoregularity and higher melting temperature in
comparison with the corresponding polymers using the ethylidene bridge.
Yasin et al. (2004) synthesized an unbridged metallocene catalyst bis(2,4,6-
trimethylindenyl)zirconium dichloride. They carried out propylene polymerization with
this catalyst and compared the results with bis(2,4,7-trimethylindenyl) zirconium
dichloride to investigate the steric effects of substituents on the catalytic activity and
microstructure of the resulting polymer. In their subsequent study, Yasin et al. (2005)
synthesized a chiral ansa-metallocene catalyst, namely rac-dimethylsilyl-bis(2,4,6-
38
trimethyl-1-indenyl)zirconium dichloride and used for isospecific polymerization of
propylene, with methyl aluminoxane (MAO) as the cocatalyst. The influences of
polymerization temperature on the polymerization activity, polypropylene microstructure
and polymer properties were investigated, and the results were compared with rac-
dimethylsilyl-bis(indenyl)zirconium dichloride under identical conditions.
Syndiospecific polymerization of propylene has attracted relatively little attention
by researchers. Such studies were generally channeled to understand the kinetics and
effect of reaction parameters on catalyst activity and stereoregularity.
Ko and Woo (2003) performed kinetic studies on the syndiospecific
polymerizations of propylene with iPr(Cp)(Flu)ZrCl2 / MAO at 20, 40 and 70 0C and at 5
atm with various Al/Zr molar ratios. It was concluded that active site isomerization was
dominant source for stereoirregularity and that was strongly dependent on the
polymerization temperature.
Marques and Conte (2006) produced syndiotactic and isotactic polypropylene
using the metallocene compounds Ph2C(Flu)(Cp)ZrCl2 and SiMe2(2-Me,4-Ph-Ind)2ZrCl2
in homogeneous system and supported on silica/MAO. These catalysts were evaluated
either isolated or as a binary system. They observed that at all the studied polymerization
temperatures, the binary catalyst produced polypropylenes with lower melting
temperatures in comparison with those obtained when the mixture of isolated supported
syndio- and isospecific catalysts was employed.
Sanginov et al. (2006) reported a marked rise in the efficiency for syndiospecific
Ph2CCpFluHfMe2 and isospecific rac-Me2SiInd2ZrMe2, upon introduction of Lewis bases
into a reaction medium.
Use of a mixture of racemic metallocenes and Ziegler–Natta catalysts (Lisovskii
et al., 1998), heterogeneous metallocene catalysts on clay minerals (Weiss et al., 2002)
39
and effects of various trialkylaluminiums with tBuNSiMe2C5Me4TiMe2/MAO (Dare et
al., 2004) are some examples of the various dimensions of studies being carried out in
propylene polymerization. A summary of the experimental studies on ethylene and
propylene polymerization is presented in Table 2.4 and Table 2.5 respectively.
40
Table 2.4 Metallocene Catalyzed Ethylene Polymerization, Experimental Studies
SL
No.
Catalyst Cocatalyst(s) Parameters studied
Important Findings References
1.
Cp2ZrCl2
MAO/TMA Effect of [Zr], Al/Zr ratio
and addition of TMA Catalyst productivity (CP) increased (↑)
and molecular weight (MW) decreased
(↓) with small additions of TMA (up to
AlTMA:AlMAO = 1.4)
Rieger and Jainik,
1994
2. MMAO/TMA Effect of [Zr] and
temperature (T) With ↑ in [Zr], MW ↓ and the catalyst
activity (CA) ↑. With ↑ in T between 140
and 200 °C, MW ↓ and polydispersity ↑.
Charpentier et al.,
1997
3. MAO Reaction time (t), [C2H4],
[Zr], and Al/Zr ratio Yield (Y) ↑ with ↑ in [C2H4], t, [Zr], and
the Al/Zr ratio.
Young and Ma, 2002
4. MAO/TMA Comparison of
homogeneous & alumina
supported systems
MW with supported catalysts was higher
than that obtained with the homogeneous
system
Marques and
Alcantara, 2004
5. MAO [C2H4], T, Al/Zr ratio and
H2 as a chain transfer
agent
Y ↑ with ↑ in Al/Zr ratio to a limiting
value. CA ↑ with T to 60 °C and slightly
↓ with more ↑ in T. CP ↑ with ↑ in
[C2H4]. MW ↓ with ↑ in Al/Zr ratio, T
and [H2].
Zohuri et al., 2005
6. MAO, MMAO
and poly(MAO)
Cocatalyst type and
Al/Zr ratio
Existence of two or more active site
types. Proposed a model to explain broad
MWDs
Sarzotti et al., 2007
41
Table 2.4 Metallocene Catalyzed Ethylene Polymerization, Experimental Studies (continued...)
SL
No.
Catalyst Cocatalyst(s) Parameters studied Important Findings References
7. Et[Ind]2ZrCl2 TMA, MAO, silica
treated with MAO
(SMAO)
t, [Zr], T, Al/Zr
ratio, TMA/MAO
ratio, TMA/SMAO
ratio
In-situ-supported catalyst did not show rate
decay with t. With ↑ in Al/Zr ratio CA ↑ and
MW and PDIs were unaffected. With ↑ in T,
MW ↓. Proposed a polymerization mechanism
for the in-situ supported catalyst
Chu et al.,
2000 a, b
8. Et[Ind]2ZrCl2
and CGC-Ti
MAO [C2H4], [Zr]
and [Ti]
First order kinetics with rac-Et(Ind)2ZrCl2 for
polymerization and catalyst deactivation.
Mehdiabadi
and Soares,
2009
9. Et[Ind]2ZrCl2 SMAO [C2H4], T, [Al] and
silica properties
[C2H4] has positive but T has negative effect on
MW. Influence of the physical nature of the
silica support on polymerization.
Tisse et al.
2010 a, b
10. [Ind]2ZrCl2
and
[Ind]2Zr(CH3)2
MAO modified
with bridged
alicyclic alcohols
Modifiers Addition of modifiers to catalyst did not alter
the structure of the polyethylenes but slightly ↑ MW.
Firme et al.,
2005
11. SiO2-supported
Et(Ind)2ZrCl2
- [C2H4], T Development of the morphology of the
polymer particles was analyzed.
Machado et
al. 2011
42
Table 2.4 Metallocene Catalyzed Ethylene Polymerization, Experimental Studies (continued...)
SL
No.
Catalyst Cocatalyst(s) Parameters studied Important Findings References
12. Me2Si[Ind]2ZrC12 MAO T Polymerization and
deactivation rate ↑ with ↑ in T.
Roos et al.
1997
13. [C5-Me4(SiMe2NtBu)]TiMe2
and CGC-Ti
tris(pentafluoroph
enyl)boron
(TPFPB) and
MMAO
[C2H4], T;
kinetics of LCB Branching ↑ with [C2H4]. Low
LCB density at elevated T.
Kinetic rate constants were
estimated graphically.
Wang et al.
1998
14. (CH2)nC(C5H4)2MCl2; M =
Ti, Zr, Hf; n = 4,5,6
MAO structure-activity
relationship
much higher activities with
cycloalkylidene-bridged
titanocene catalystst than the
corresponding zirconocene and
hafnocene analogues.
Wang et al.
2005
15 (CpTiCl2)2[C5H4CH2C6H4-p-
C6H4CH2C5H4] and
(CpZrCl2)2[C5H4CH2C6H4-p-
C6H4CH2C5H4]
MAO [Ti], [Zr], T, Al/Zr
ratio CAs were compared. MW ↑
with t.
Sun et al.
2006
16. Ph2C(Cp)(Flu)ZrCl2 MAO Compared
homogeneous and
heterogeneous
systems; T
Homogeneous polymerizations
were more active. CA ↓
significantly with ↑ in T under
homogeneous conditions
Freitas et al.
2011
43
Table 2.4 Metallocene Catalyzed Ethylene Polymerization, Experimental Studies (continued...)
SL
No.
Catalyst Cocatalyst(s) Parameters studied Important Findings References
17. N,N-ethylenebis(3-
methoxysalicylideneiminato)titan
ium dichloride immobilized on
Mg support
Me3Al and
MAO
T, [C2H4], and t CA ↑ with ↑ in T and [C2H4].
Bulk density ↑ with T and t.
Broad MWD was obtained.
Pietruszka et
al. 2012
18. [(p-CH3-
Ph)2C(C5H4)(C9H6)]ZrCl2,
{(p-CH3-Ph)[p-C(CH3)3-
Ph]C(C5H4)(C9
H6)}ZrCl2,
[(p-CH3-
Ph)2C(C5H4)(C13H8)]ZrCl2,
{(p-CH3-Ph)[p-C(CH3)3-
Ph]C(C5H4)(C13H8)}ZrCl2
MAO Various catalysts Effect on CA and MW was
discussed.
Huang et al.
2010
19. Cp2ZrCl2, Cp2TiCl2, Cp2HfCl2,
Et(Ind)2ZrCl2 and
Et(H4Ind)2ZrCl2
MAO, TMA Various catalysts, T,
[Zr], [Ti], [Hf],
[MAO], chain
transfer agent (CTA),
and substitution of
MAO with TMA
↑ in T and [catalyst] or [MAO]
caused a ↓ in MW.
Replacement of TMA with
MAO or addition of H2 (CTA)
induced a drastic ↓ in MW. For
the Zr catalysts, transfer to
C2H4 was the main chain
transfer mechanism.
Agnillo et al.
1998
44
Table 2.4 Metallocene Catalyzed Ethylene Polymerization, Experimental Studies (continued...)
SL
No.
Catalyst Cocatalyst(s) Parameters studied Important Findings References
20. Zirconocene MAO Mechanisms of
polymerization through
density functional calculations
and energetics were
discussed.
- Petitjean et al.
1998
21. Unbridged supported
zirconocene
MAO T and [C2H4] ↑ in CA and decay with ↑
in T. Polymerization
rates were first order in
ethylene.
Chakravarti
and Ray 2001
22. Metallocene - - Structure and LCB
content of Dow HDB
samples were determined
using small-angle
neutron scattering
(SANS)
Ramachandran
et al. 2009
45
Table 2.5 Metallocene Catalyzed Propylene Polymerization, Experimental Studies
SL
No.
Catalyst Cocatalyst(s) Parameters studied Important Findings References
1. rac-(EBI)Zr(NMe2)2 MAO [Zr], T and Al/Zr ratio ↓ in MW with ↑ in Al/Zr ratio, T
and [Zr]. Maximum CA found
at Al/Zr = 2000, [Zr] = 137.1
μM and T = 30 °C
Kim and
Hwang, 1998
2. rac-Me2Si(1-C5H2-2-Me-4-tBu)2Zr(NMe2)2,
rac-Me2Si(1-C5H2-2-Me-4-tBu)2ZrMe2
MAO [Zr] and T ↑ in polymerization rate (Rp)
and ↓ in stereoregularity, Tm,
MW, and MWD with ↑ in T.
Linear ↑ in Rp with ↑ in [Zr]
Kim et al. 1999
3. rac-(EBI)Zr(NC4H8)2) MAO [Zr] and Al/Zr ratio CA ↑ with ↑ in Al/Zr ratio. MW
↓ and PDI insensitive with ↑ in
Al/Zr ratio and [Zr]. iPP with a
meso pentad value of 94.7%
produced.
Kim, 1999
4. rac-(EBI)ZrCl2, rac-[EB(4,7-
dimethyl-1-indenyl)]ZrCl2
MAO Analyzed chemical
structures of end
groups of aPP and iPP
Proposed chain transfer reaction
mechanism
Resconi et al.,
1999
5. (2-PhInd)2ZrCl2 MAO [C3H6] and [Zr] ↑ of the isotactic dyads and
pentads and Rp with ↑ in [C3H6],
β-hydride elimination dominant.
PDI = 2.0-2.6
Lin et al., 2000
46
Table 2.5 Metallocene Catalyzed Propylene Polymerization, Experimental Studies (continued...)
SL
No.
Catalyst Cocatalyst(s) Parameters studied Important Findings References
6. Ind'2ZrCl2 (Ind' = 2-alkyl- or
arylalkyl-substituted indenyl)
MAO Catalyst structure and T Productivity, MW and
isotacticity ↓ with ↑ in T
Schmidt and
Alt, 2001
7. Me2Si[Ind]2ZrCl2 MAO/SiO2 [C3H6], T,
Compared gas and
liquid phase
polymerization
Lower reaction rates were
found in the gas phase
compared to liquid phase.
Rp,max ↑ linearly with
[C3H6]. Deactivation rate ↑
with ↑ in T.
Meier et al.
2001
8. SiMe2(Ind)2ZrCl2,
Et(Ind)2ZrCl2,
SiMe2(Ind)2HfCl2,
Et(Ind)2HfCl2
MAO Catalyst type, T, and
Al/Mt ratio MW ↓ with ↑ in Al/Zr ratio
or T. CA, MW and tacticity
were compared for different
catalysts.
Marques et
al., 2002
9. Cp2ZrCl2 and
SiMe2(Ind)2ZrCl2
MAO, different
immobilization
methods
Homogeneous vs.
heterogeneous systems
High CA and low MW with
homogeneous systems
Marques et
al., 2003
10. rac-Me2Si(1-Indenyl)2ZrMe2
and rac-Me2Si(1-
Indenyl)2ZrCl2
AliBu3/[Ph3C] and
MAO
Initial [C3H6], [Zr];
Quenched-flow kinetic
study
First-order dependence of
yield (Y) on [M] and [Zr].
Song et al.,
2003
47
Table 2.5 Metallocene Catalyzed Propylene Polymerization, Experimental Studies (continued...)
SL
No.
Catalyst Cocatalyst(s) Parameters studied Important Findings References
11. rac-Me2Si(1-Ind)2ZrCl2 CPh3[B(C6F5)4]/TIBA
and MAO
Comparison of activators
through quenched-flow
kinetic study
CA with MAO was 20
times less than borate
activators
Song et al.,
2004
12. (2,4,6-Me3Ind)2ZrCl2
and (2,4,7-Me3)2ZrCl2
MAO, TIBA T, [Zr], Al/Zr ratio and
[MAO], [TIBA] MW ↑ with ↑ in T and ↓ in
Al/Zr ratio. CA ↓ with ↑ in
[Zr]; High CA obtained
with addition of TIBA in
MAO. Isotacticity ↑ with ↓
in T
Yasin et al.
2004
13. rac- Me2Si(2,4,6-
Me3Ind)2ZrCl2 and rac-
Me2Si(Ind)2ZrCl2
MAO T and catalyst type Very high CA at T = 50 -
70 °C and P = 1 atm. rac-
Me2Si(2,4,6-Me3Ind)2ZrCl2
more isospecific.
Yasin et al.
2005
14. Cp2ZrCl2 MAO P, T, [MAO], Al/Zr ratio
and [Zr] CA ↑ with ↑ in P, T and
[MAO]. MW ↑ with ↑ in P
and ↓ in Al/Zr ratio.
Ochoteco et
al. 2001
15. Me2Si(2-Me-Ind)2ZrCl2 MAO P, T, Al/Zr ratio and [Zr] Linear relationship of Y
with [Zr] and P. With ↑ in
T or Al/Zr ratio, Y ↑ to a
maximum, then ↓.
Palza et al.
2006
48
Table 2.5 Metallocene Catalyzed Olefin Polymerization, Experimental Studies (continued...)
SL
No.
Catalyst Cocatalyst(s) Parameters studied Important Findings References
16. iPr(Cp)(Flu)ZrCl2 MAO T and Al/Zr ratio Syndiotactic PP. CA ↓ with
↑ in Al/Zr ratio.
Stereoerrors were
influenced by T and not by
Al/Zr ratio.
Ko and Woo,
2003
17. Ph2C(Flu)(Cp)ZrCl2 and
SiMe2(2-Me,4-Ph-
Ind)2ZrCl2
MAO T and type of catalyst CA ↑ with ↑ in T. CA of
supported catalyst 50% less
than homogeneous.
Marques and
Conte, 2006
18. Ph2CCpFluHfMe2 and
rac-Me2SiInd2ZrMe2
AliBu
3, CPh3B(C6F5)4 Influence of the external
Lewis base as
modifying reagents.
10 times ↑ in CA for
syndiospecific and 15-30
times ↑ in CA for isospecific
catalyst.
Sanginov et
al. 2006
19. tBuNSiMe2C5Me4TiMe2 MAO, Oct3Al, Me3Al,
Et3Al, iBu3Al
Effect of various alkyl
aluminiums CA ↑ with Oct3Al and iBu3Al (possesing more
bulky groups) and CA ↓
with Me3Al, Et3Al.
Dare et al.
2004
20. C2H4(Ind)2ZrCl2,
Me2Si(Ind)2ZrCl2, ZN
catalysts (Ti on MgCl2)
MAO Mixed catalysts MW, Tm and melt flow
index (MFI) were
determined for one step and
two step polymerization.
Lisovskii et
al. 1998
49
2.4 Mechanistic, modeling & simulation studies
Information on mechanistic aspects for olefin polymerization is essential to develop
realistic kinetic models. The mechanisms with special focus on stereochemical control,
homo- and copolymerization of α-olefins with variety of metallocene catalysts have been
extensively studied and published in literature.
Brintzinger et al. (1995) reviewed stereospecific α-olefin polymerization with
chiral metallocene catalysts encompassing correlations between catalyst structures &
stereoselectivities and mechanisms of stereochemical control.
Chien (1999) presented a comprehensive and comparative review on applications
of inorganic support materials, MAO-free supported catalysts, modified silica support and
polymeric support to metallocene catalyst for olefin polymerization.
Mechanism of alkene polymerization reactions with metallocene catalysts has
been discussed by Kissin and Goldman (2009). In their study, ethylene was
homopolymerized and copolymerized with four 1-alkenes—propene, hex-1-ene, hept-1-
ene, and 3,3-dimethylbut-1-ene using (n-Bu-Cp)2ZrCl2 and (Me-Cp)2ZrCl2 complexes
activated with MAO.
Petoff et al. (1998) used bis(2-arylindenyl)zirconium dichlorides and rac- &
meso-dimethylsilyl(bis(2-phenylindenyl))zirconium dichloride activated by
methylaluminoxane (MAO) produce elastomeric polypropylene and provided mechanistic
insight of polymerization.
Several reviews have been published on mechanisms of polymerization via
metallocene catalysts (Busico, 1998; Kaminsky and Strübel, 1998; Kleinschmidt et al.,
1999; Chen and Marks, 2000; Jongsomjit et al., 2005; Bochmann, 2006). Very recently
Kaminsky (2009) and Takeuchi (2010) have presented state of the art mechanistic
50
reviews on a number of transition metal catalysts including early and late transition
metals applied in olefin polymerization.
Modeling studies have been carried out to study the kinetics and mechanism of
polymerizations and the structure of polymer chains, such as the distribution of molecular
weight, branching, stereoregularity and chain topology in different polymer systems.
Kinetics and transport effects were studied and modeled (Bhagwat et al., 1994;
Zarand and Mortazavi, 2005; Kanellopoulos et al., 2007) for slurry polymerization of
ethylene with Ziegler-Natta Catalysts.
Sarzotti et al. (2007) proposed a model for ethylene polymerization with
homogeneous Cp2ZrCl2/aluminoxane catalysts to fit the experimental rate data of as a
function of polymerization time and [Al]/[Zr] ratio. They considered the existence of two
types of active sites and assumed that the active complex formed when the metallocene
precursor is activated, PA, is reversibly transformed to an intermediate species, P*, and
irreversibly to a second active site type, PB.
xAPP n
A
n * (2.1)
B
nn PP * (2.2)
where A
nP is an active site of type A with a growing polymer chain of length n, *
nP an
intermediate catalyst complex with a polymer chain of length n.
The rate constant for initiation and propagation of active site type A were assumed to be
same to reduce the number of parameters. Catalyst deactivation was also not considered.
Instantaneous rate of polymerization was determined by the following expression:
rEE
B
pB
A
pAP VmMPkPkR1
(2.3)
where E : density of ethylene, Em : molar mass of ethylene and rV : reactor volume
51
Iedema (2012) described a semi-analytical approach to model simultaneous chain
scission and branching that assumes the separation of the scission and the branching
problem.
Yiannoulakis et al. (2000) developed a dynamic model for the calculation of the
molecular weight and long chain branching distributions in a continuous solution
metallocene-catalyzed ethylene polymerization reactor.
Roos et al. (1997) studied ethylene polymerization in a stirred powder bed reactor
with silica supported rac-Me2Si[Ind]2ZrC12/methylaluminoxane (MAO) and modeled
deactivation as a first order dependence with respect to the polymerization rate as given
by Equation 2.5.
*CCkR mpP (2.4)
Pd Rkdt
dC
*
(2.5)
Where mC : monomer concentration, *C : concentration of active sites.
Chakravarti and Ray (2001) developed a slurry reactor model to predict the
polymerization behaviour under various reaction conditions. They considered simple
kinetic scheme in their model including activation, propagation and deactivation only as
below:
*
0 Activation CMCpot (2.6)
*
1
* n Propagatio nn CMC (2.7)
ndn DCC * on Deactivati (2.8)
Mehdiabadi and Soares (2009) suggested that kinetics of ethylene polymerization
with rac-Et(Ind)2ZrCl2/MAO can be described with first order reactions for
polymerization and catalyst deactivation. Focusing on catalyst deactivation studies they
52
only considered initiation, propagation and deactivation steps in their kinetic model and
also simplified it by assuming same values of the initiation and propagation constants.
Zeigler-Natta catalysts remained democratic in various modeling studies for both
ethylene and propylene polymerization. Various researchers (Neto and Pinto, 2001; Pater
et al., 2003; Veera, 2003; Reginato, 2003; Luo, 2008), through their modeling efforts,
have enriched the literature to understand propylene polymerization with Z-N catalysts. In
their studies, morphological or transport models like Polymeric flow (PF) model,
multigrain (MG) model, polymeric flow Fick’s diffusion model (PF FDM), Multigrain
Fick’s diffusion model (MG FDM), advection-dispersion model (ADM) and the dusty gas
model (DGM) were generally developed and improved.
Kinetic modeling studies on propylene polymerization with metallocene catalysts were
rare till recent past. Few studies available in literature are discussed hereunder.
Nele et al. (2001) proposed a two state kinetic model to describe the propylene
polymerization behavior of ansa-metallocene catalysts. They applied the model to
describe the polymerization behavior of some simple symmetrical [Me2C(Cp)(Flu)MCl2;
M = Zr, Hf] and unsymmetrical [Me2Y(Cp)(Ind)MCl2; M = Zr, Hf; Y = C, Si] catalysts,
activated with MAO.
A kinetic model was proposed by Ochoteco et al. (2001) to explain the
experimental evolution of catalyst activity at different Al/Zr ratios and catalyst
concentrations for the homogeneous system. In their model they considered catalyst
activation, chain initiation, chain propagation and a reversible second order catalyst
deactivation step followed by an irreversible deactivation process forming inactive
species.
53
*
0 Activation CMAOC ik (2.9)
*
1
*
01 Initiation CMC pk
(2.10)
*
1
* 2 n Propagatio n
k
n CMC p (2.11)
inactivedormant
inactive
k
CC
CC
,2,2
*
n
,2
*
0
2C
2 on Deactivati'1
(2.12)
Belelli et al. (2001) proposed a mathematical model for a semibatch laboratory
polymerization reactor using ethylenbisindenylzirconium dichloride (EtInd2ZrCl2) / MAO
that predicted reactor productivity and the molecular properties of the product. SRK
equation of state was employed to estimate the equilibrium concentration of propylene in
toluene (solvent) at the gas-liquid interface for different pressures at polymerization
temperature which was fixed. In their model they considered the existence of different
types of catalyst sites with reactions quoted below:
kkk PMPki
10 Initiation (2.13)
k
j
kk
j PMPkp
1n Propagatio (2.14)
k
j
kkk
j DPHPk
H- sfer toChain tran (2.15)
kkk PMPHkr
1on Reactivati (2.16)
k
jd
kk
j DCPkd 1
ondeactivatiorder First (2.17)
k
m
k
jd
kk
m
k
j DDCPPkd 2
ondeactivatiorder Second
2
(2.18)
54
Lahelin et al. (2003) prepared polypropylene with rac-SiMe2(2-Me-4-
PhInd)2ZrMe2/MAO (rac-dimethylsilylbis(2-methyl-4-phenylindenyl)dimethylzirconium/
methylaluminoxane) in heptane solution at temperatures from 50 °C to 80 °C with
varying concentrations of monomer, hydrogen, triisobutylaluminium (TIBA) and MAO.
Authors developed kinetic model for low propylene concentrations on the basis of
polymerization data.
Nele et al. (2005) modeled stereospecific polymerization of propylene with C1-
symmetric Me2Si(Ind)(Flu)ZrCl2 complex, activated with methyl aluminoxane. They
proposed the existence of asymmetric ansa and fluxional metallocene catalysts in (at
least) two different states during the lifetime of the growing polymer chain. Authors
employed the Coleman–Fox model in their study, which incorporates the benefits of both
kinetic and probabilistic modeling approaches.
Palza et al. (2006) developed a mathematical model for Me2Si(2-Me-Ind)2ZrCl2
catalyzed propylene polymerization based on the method of the moments. Authors
observed a special effect with respect to the co-catalyst (MAO) on productivity and
improved their model incorporating deactivation-reactivation mechanisms associated with
MAO.
A brief review of the mechanistic and modeling studies on ethylene and propylene
polymerization is summarized in Table 2.6.
55
Table 2.6 Metallocene Catalyzed Olefin Polymerization, Mechanistic, Modeling & Simulation Studies
SL
No.
Monomer Catalyst Cocatalyst(s) Study / Model Remarks References
1. α-Olefin Chiral Metallocene MAO Reviewed kinetics and
mechanism
- Brintzinger et
al. 1995
2. α-Olefin Metallocene MAO Reviewed applications of
inorganic support materials,
MAO-free supported catalysts,
modified silica and polymeric
support.
- Chien, 1999
3. Ethylene (n-Bu-Cp)2ZrCl2 and (Me-
Cp)2ZrCl2
MAO Mechanism and kinetics of
chain growth and chain
transfer reactions were
discussed.
- Kissin and
Goldman,
2009
4. Propylene bis(2-arylindenyl)zirconium
dichlorides and
rac- & meso-dimethylsilyl(bis(2-
phenylindenyl))zirconium
dichloride
MAO Proposed mechanism for the
production of elastomeric
polypropylene.
- Petoff et al.,
1998
5. Ethylene Z-N - Modeling for isothermal slurry
polymerization
Rp and Polymer
properties
Bhagwat et
al., 1994
6. Ethylene Z-N - Combined polymeric
multigrain (PMGM) and
polymeric multilayer (PMLM)
models
Simulation with
parameters taken
from literature.
Zarand and
Mortazavi,
2005
56
Table 2.6 Metallocene Catalyzed Olefin Polymerization, Modeling Studies (continued...)
SL
No.
Monomer Catalyst Cocatalyst(s) Study / Model Remarks References
7. Olefin Z-N - Unsteady-state diffusion
model
Pore size distribution
and crystallinity
Kanellopoulos
et al. 2007
8. Ethylene Cp2ZrCl2 MAO / MMAO Kinetic modeling Rate data prediction Sarzotti et al.,
2007
9. Ethylene Metallocene - Semi-Analytical Model for
simultaneous chain scission
and branching
Comparison of semi-
analytical model with
Monte Carlo simulations
Idema, 2012
10. Ethylene Metallocene
- Dynamic modeling,
continuous reactor
MW and long chain
branching (LCB)
distributions
Yiannoulakis
et al. 2000
11. Ethylene rac- Me2Si[Ind]2ZrC12 MAO Modeling of deactivation Rate data prediction Roos et al.,
1997
12. Propylene Z-N TEA Kinetic modeling Kinetic parameter
estimation
Pater et al.
2003
13. Propylene Z-N TEA Kinetic modeling for slurry
and bulk polymerization
Prediction of MWD,
chain composition
distribution (CCD),
particle size distribution
(PSD)
Netoa and
Pinto, 2001
57
Table 2.6 Metallocene Catalyzed Olefin Polymerization, Modeling Studies (continued...)
SL
No.
Monomer Catalyst Cocatalyst(s) Study / Model Remarks References
14. Propylene Z-N - Modeling for liquid-phase
polymerization in loop
reactor
Polymer properties Reginato et
al., 2003
15. Olefin Z-N - Advection-dispersion model
(ADM) and the dusty gas
model (DGM)
Comparison with
Polymeric flow (PF)
Fick’s diffusion (FDM),
multigrain FDM (MG
FDM) has been made.
Veera, 2003
16. Propylene Me2Si(Ind)(Flu)ZrCl2 MAO Applied Extended
Coleman–Fox Model
T effects on the
microstructure of
polymer
Nele et al.,
2005
17. Propylene SiO2 supported Cp2ZrCl2 MAO Kinetic model for catalyst
activity at different Al/Zr
ratios and [Zr]
Explanation of their
experimental evolution
Ochoteco et
al., 2001
18. Propylene EtInd2ZrCl2 MAO Modeling for homogeneous
polymerization in semibatch
reactor
Prediction of
productivity
Belelli et al.,
2001
19. Propylene Me2Si(2-Me-Ind)2ZrCl2 MAO Modeling for homogeneous
polymerization in semibatch
reactor
Prediction of Y Palza et al.,
2006
58
Table 2.6 Metallocene Catalyzed Olefin Polymerization, Modeling Studies (continued...)
SL
No.
Monomer Catalyst Cocatalyst(s) Study / Model Remarks References
20. Propylene Silica supported
metallocene and Z-N
MAO / MgCl2 Modeling for liquid-phase
polymerization in stirred
tank reactor using Monte
Carlo simulation.
Prediction of Y and MW Luo et al.,
2008
21. Propylene Me2C(Cp)(Flu)MCl2, M
= Zr; Hf and
Me2Y(Cp)(Ind)MCl2, M
= Zr, Hf, Y = C, Si
MAO Two state kinetic model for
liquid-phase polymerization
in semibatch reactor.
Stereosequence
distributions
Nele et al.,
2001
22. Propylene rac-SiMe2(2-Me-4-
PhInd)2ZrMe2
MAO / TIBA Polymer characterization,
mechanisms and kinetic
models.
Prediction of MW and
end groups
Lahelin et al.,
2003
59
2.5 Gaps in research
The existing literature on polymerization with metallocene catalyst systems suggests that
efforts have been made in understanding the mechanisms and work performance of
metallocene based catalyst systems with different co-catalysts. These systems allow tailor
making of polymers and offer other process advantages such as ease of handling of
metallocene systems and favourable conditions for polymerization from a commercial
point of view, which has evoked the examen of the commercial potential of such catalyst
systems. The commercial exploitation of such systems has, however, started in a limited
way due to prohibitive cost of the catalyst and the ambiguity associated with the
aluminoxanes (co-catalysts).
Various homo- and copolymers have been synthesized using metallocene catalyst
systems and most of the work is of experimental nature at either laboratory scale or the
pilot scale with more or less common objectives like investigating catalyst activity,
product properties and effect of parameters thereon.
Very little attempts have been made in modeling and simulation related studies for the
polymerization process. Majorly Z-N catalyzed polymerization of olefins was on focus
for modeling the morphological and transport related phenomena. Modeling efforts on
olefin polymerization with metallocene catalysts are as less as negligible when compared
to the other catalyst systems. Further, kinetic modeling and simulation of metallocene
catalyzed olefin polymerization is in its dissilient stage and provides a huge opportunity
to address the understanding of kinetics comprehensively.
2.6 Scope of the work
Significant development in the synthesis of new metallocenes and co-catalysts is
anticipated in near future leading to tailor-made polymers, including functionalized
60
polyolefins with predictable properties. In view of this, it is imperative to model the
polymerization of olefins involving different metallocene catalyst systems. A lot of scope
exists for theoretical as well as computation studies on metallocene catalyzed olefin
polymerization and hence developing a kinetic model and simulation of the same
unquestionably is a task not only for research but also of industrial importance.
In this work, the mechanistic aspects of Ziegler-Natta and metallocene catalyst
systems have been studied in detail and used in building up mathematical models for
ethylene and propylene polymerization using metallocene catalysts. Developed models
are validated with the experimental data available in literature and kinetic parameters are
estimated using differential evolution (DE) approach of optimization. Study on the effects
of various parameters like monomer concentration, polymerization temperature, catalyst
concentrations, and cocatalyst to catalyst molar ratio etc. upon rate of polymerization,
molecular weights and poly dispersity index and stereoregularity is carried out.
The outcomes of this study will help in better understanding of the chemistry and
process of the olefinic polymerization with these revolutionary catalyst systems.
61
CHAPTER – 3
MATHEMATICAL MODEL DEVELOPMENT
AND
SIMULATION
Understanding the fundamentals of reactions that are at the heart of industrial catalytic olefin
polymerization processes yields information that is important for achieving improvements in
these processes, and can also pioneer pathways to new processes and materials. Mathematical
modeling is a mighty tool for the development of process understanding and advanced reactor
technology in the polymer industry. Recent advances in theory, computational software, and
hardware have made it possible to complement experiments and empiricism with sound
mathematical models. These mathematical models, which are constructed on a first principles
basis and have been validated with experimental data, may be used as a surrogate of the real
process for a variety of applications where it may be costly, inconvenient, impractical, or unsafe
to use the actual process.
A kinetic model consists of mass and population balance equations, which are derived
based on elementary reactions proposed in the reaction mechanism. This usually yields a system
of differential and/or algebraic equations that can be solved using various numerical methods
e.g., numerical discretization.
The aim of this chapter is to discuss the development of mathematical models for
metallocene catalyzed ethylene & propylene polymerization and kinetic parameter estimation
methodology used in this work.
62
3.1 Metallocene polymerization kinetics & model development
It is apparent that a kinetic model capable of predicting the molecular development in a
polymerization reactor in terms of the process operating conditions should include appropriate
representations of all chemical and physical phenomena occurring at micro-scale. Kinetic models
vary in their complexity but in general following issues should be addressed while modeling a
polymerization reaction (inferred from Kiparissides, 1996).
(i) Identification of the elementary reactions from the polymerization kinetic mechanism.
(ii) Derivation of polymerization rate functions for all reacting species [e.g. monomer(s),
catalyst, chain transfer agent(s), "live" polymer chains and "dead" polymer chains].
(iii) Specification of the number of phases in the reactor. Homogeneous vs. heterogeneous
kinetics.
(iv) Necessary thermodynamic models for the: (a) calculation of monomer(s) concentration(s)
in the different phases, (b) calculation of enthalpies of the different reacting species, etc.
(v) Selection of reactor configuration. (e.g. batch, semi-batch, continuous, etc).
(vi) Derivation of all necessary reactor design equation for: (a) reaction mixture and (b)
coolant/heating fluids.
(vii) Choosing numerical method for solving the model equations. Estimation of unknown
model parameters. Performing sensitivity studies. Investigation of dynamic model
behavior, etc.
3.1.1 Mathematical treatment of polymerization kinetics
Population balance approach is used to rescript the kinetic equations for each polymerization
step in terms of the leading moments of the molecular-weight distribution of the polymer. This
63
treatment is a statistical technique that enables us to track various polymer chain properties
without the need to include the very large number of equations and unknowns required to
account for chains of every possible length. Characterization of polymer properties is modeled
using the population balances and method of moments. Construction of moment balances allows
the tracking of average polymer properties like number and weight average molecular weights,
chain-length distributions, type and frequency of chain branching, and content of terminal double
bonds etc.
Leading moments of the molecular weight distribution
The moments are averages of the concentrations of polymer molecules that are weighted by their
chain lengths. The moment expression for live polymer chains is given by Equation 3.1.
1
)(i
nl
n iPi (3.1)
where l
n is the nth
moment of the molecular weight distribution of live chains attached to
catalyst site and , )(iP is the molar concentration of the corresponding live polymer chains.
Similar expression for dead chains can be written as Equation 3.2.
2
)(i
n
n iDi (3.2)
where n is the nth
moment of the molecular weight distribution of dead chains and )(iD is the
molar concentration of the corresponding dead chains.
The three leading moments, namely, the zeroth, first, and second, are adequate for depicting the
molecular weight distribution of most commercial polymers.
64
Polymer properties in terms of moment expressions
A variety of common polymer properties those can be deduced using the moment expressions are
given below.
The number-average degree of polymerization is given by Equation 3.3.
00
11
l
l
nDP (3.3)
The weight-average degree of polymerization is given by Equation 3.4.
11
22
l
l
wDP (3.4)
The number-average and weight average molecular weights can be calculated from Equations
3.5 & 3.6.
00
11
l
l
SRUn mM (3.5)
11
22
l
l
SRUw mM (3.6)
where SRUm is the molecular weight of the structural repeat unit.
The polydispersity index is defined as the ratio of weight average to number average molecular
weight and can be computed using Equation 3.7.
211
0022
l
ll
n
w
M
MPDI (3.7)
65
3.2 Modeling of ethylene polymerization
3.2.1 Kinetics
Several authors have recently presented kinetic schemes for metallocene-catalyzed ethylene
polymerization (Wang et al., 1998; Yiannoulakis et al., 2000; Jongsomjit et al., 2005; Bochmann
et al., 2006; Kaminsky et al., 2009). Based on the understanding of mechanisms for metallocene-
catalyzed polymerization developed, a general reaction set for ethylene polymerization that
includes reactions corresponding to all types of metallocenes is proposed in this section. All of
them may not be applicable to every system, e.g. in absence of hydrogen as a chain transfer
agent, chain transfer to hydrogen is not possible and hence should not be considered in scheme.
Catalyst activation by cocatalyst
The catalyst is activated by the cocatalyst.
)0(PCocatCat ak (3.8)
where Cat is the catalyst, Cocat is the cocatalyst species, 0P represents an active catalyst site
that is capable of polymerization, and ka is the rate constant for catalyst activation. MAO is used,
often as cocatalyst species for many metallocene catalysts.
Chain initiation
The active sites react with monomer to initiate chain growth.
10 PMP ink (3.9)
66
where M is the monomer, 1P is a polymer chain attached to catalyst containing one monomer
segment, and kin is the rate constant for chain initiation.
Chain propagation
This step is the addition of monomer species to the growing chains. The addition of monomer
species involves a complexation of the double-bond of the monomer at the catalyst site.
1 iPMiP pk (3.10)
where iP is a polymer chain attached to a catalyst site and kp is the rate constant for chain
propagation. This reaction controls the monomer conversion in the reactor.
Spontaneous catalyst deactivation
A propagating chain converts into a dead polymer chain due to spontaneous deactivation of
active site.
iDPiP d
kd 0
00 d
k PP d
(3.11)
(3.12)
where 0dP is deactivated catalyst, and kd is the rate constant for spontaneous catalyst
deactivation.
Chain transfer to hydrogen
Hydrogen acts as a strong chain transfer agent for metallocene catalysts. Chain transfer to an
external chain transfer agent, such as hydrogen results in a saturated chain end.
67
0)(2 PiDHiP tHk (3.13)
where )(iD is a dead chain containing i segments, and ktH is the rate constant for chain transfer to
hydrogen.
Chain transfer to monomer
Monomer acts as a chain-transfer agent as well.
1)( PiDMiP tMk
(3.14)
where )(iDis a dead polymer chain detached from the catalyst, and contains a terminal double
bond and ktM is the rate constant for chain transfer to monomer. Chain transfer to the ethylene
leads to the formation of vinyl (CH2=CH−) end group.
Chain transfer to cocatalyst
In some cases, cocatalyst also acts as a chain-transfer agent. Chain transfer to the aluminum
(cocatalyst) is usually of minor importance in ethylene polymerization but comes out to be more
important in propene polymerization (Naga and Mizunuma, 1998).
0)( PiDCocatiP tCok (3.15)
where ktCo is the rate constant for chain transfer to cocatalyst.
β-Hydride elimination (Spontaneous chain transfer to the metal)
In β-hydride elimination, a polymer chain can detach from the active site, leaving it with a
terminal double bond. The catalyst site remains active for reinitiation and polymerization. This
68
reaction is important for the incorporation of long chain branches, which requires chains with
terminal double bonds. β-H elimination (chain transfer to the metal) leads to the formation of
vinyl (CH2=CH−) end group in ethylene polymerization.
0)( PiDiPk
(3.16)
where kβ is the rate constant for β-hydride elimination.
Long-chain branching (Transfer to polymer)
A live chain can react with a dead chain containing a terminal double bond to form a single chain
with a long branch.
jiPjDiP lcbk )( (3.17)
where klcb is the rate constant for incorporation of polymer chains with terminal double bonds.
Constrained geometry metallocene catalysts are able to utilize this reaction to produce polymer
with long-chain branches at moderate reactor conditions.
Short-chain branching (Backbiting or intramolecular H-abstraction)
Short chain branching is not applicable to ethylene homopolymerization but important for
copolymerization with α-olefins.
iPiP scbk ' (3.18)
where kscb is the rate constant for short chain branching.
69
Comonomers are normally used to produce HDPE products of varying densities. The
introduction of α-olefins, such as propylene, 1-butene, and 1- hexene, creates short-chain
branching along the polymer backbone, lowering the crystallinity of the polymer.
Table 3.1 summarizes the reactions considered in the kinetic mechanism.
Table 3.1 Reactions Conceived in Ethylene Polymerization
Reaction Stoichiometry Description
1. )0(PCocatCat ak Catalyst activation
2. 10 PMP ink Chain initiation
3. 1 iPMiP pk Chain propagation
4. iDPiP d
kd 0 {chain}
Spontaneous catalyst deactivation 5. 00 d
k PP d {site}
6. 0)(2 PiDHiP tHk Chain transfer to hydrogen
7. 1)( PiDMiP tMk
Chain transfer to monomer
8. 0)( PiDCocatiP tCok
Chain transfer to cocatalyst
9. 0)( PiDiPk
β-hydride elimination
10. jiPjDiP lcbk )( Long-chain branching
11. iPiP scbk ' Short-chain branching
70
3.2.2 Model development for ethylene polymerization
Mathematical model building based on the reactions considered in Table 3.1 for ethylene
polymerization is discussed in this section. Semi-batch reactor has been commonly used for
kinetic studies of polymerization by various researchers.
Material balances
In order to obtain the model equations in terms of various moments, material balance has been
written on different species available in the reactor, which is described in the following sections.
Unactivated catalyst, 'Cat'
CocatCatkr aCat (3.19)
Vacant activated catalyst sites, 'P(0)'
ll
tCo
l
tHdinaP kCocatkHkPkPMkCocatCatkr 0002)0( )0()0( (3.20)
Deactivated catalyst sites, 'Pd(0)'
l
dP Pkrd 0)0( )0( (3.21)
Cocatalyst, 'Cocat'
l
tCoaCocat CocatkCocatCatkr 0 (3.22)
71
Monomer, 'M'
l
tM
l
pinM MkMkPMkr 00)0( (3.23)
Hydrogen, 'H2'
l
tHH Hkr 022 (3.24)
Active chains, 'P(i)'
)1()1()1()1(
)1()1()1()1()0(
0
02)1(
PkPkPkPCocatk
MkPMkPHkPkPMkPMkr
scblcbtCo
l
tMtMtHdpinP
(3.25)
)()()()(
)()()()1()(
0
2)(
iPkiPkiPkiPCocatk
iPMkiPHkiPkiPMkiPMkr
scblcbtCo
tMtHdppiP
(3.26)
where
n is the nth
moment of the molecular weight distribution of dead chains with terminal
double bond, given by Equation 3.25.
2
)(i
n
n iDi (3.27)
)(iD is the molar concentration of dead chains with terminal double bond.
Dead chains, 'D(i)'
)()( 2)( iPHkiPkr tHdiD (3.28)
72
Dead chains with terminal double bond, 'D=(i)'
0)()()()()( iPkiPCocatkiPkiPMkr lcbtCotMiD
(3.29)
Live chains with long chain branches, 'P(i+j)'
0)( )( iPkr lcbjiP (3.30)
Live chains with short chain branches, 'P'(i)'
)()('
iPkr scbiP (3.31)
Moments of chain length distribution (CLD) of living polymer chains
Zeroth moment, ' l
0 '
l
tHdin kHkkPMkr l 02)0(0
(3.32)
First moment, 'l
1 '
l
scblcbptCotM
l
tCotHdtMin
kkMkCocatkMk
kCocatkHkkMkPMkr l
01
12)0(1
(3.33)
Second moment, 'l
2 '
l
scbtCotHdtM
l
lcbp
l
lcbptCotMin
kkCocatkHkkMk
kMkkMkCocatkMkPMkr l
22
1102 2)0(2
(3.34)
73
Moments of CLD of dead polymer chains
Zeroth moment, ' 0 '
l
tHd Hkkr 020
(3.35)
First moment, ' 1 '
l
tHd Hkkr 121
(3.36)
second moment, ' 2 '
l
tHd Hkkr 222
(3.37)
Moments of CLD of dead polymer chains with terminal double bond
Zeorth moment, '
0 '
l
lcbtCotM kkCocatkMkr 000
(3.38)
First moment, '
1 '
l
lcb
l
tCotM kkCocatkMkr 0111
(3.39)
Second moment, '
2 '
l
lcb
l
tCotM kkCocatkMkr 0222
(3.40)
Polymer properties in terms of moment expressions
A variety of common polymer properties can be deduced using the moment expressions. The
number-average degree of polymerization is given by Equation 3.41.
74
000
111
l
l
nDP (3.41)
The weight-average degree of polymerization is given by Equation 3.42.
111
222
l
l
wDP (3.42)
The MWD of the polymer chains is characterized by number-average, weight-average molecular
weights and PDI calculated from Equations 3.43, 3.44 and 3.45.
000
11105.28
l
l
nM (3.43)
111
22205.28
l
l
wM (3.44)
2111
000222
l
ll
n
w
M
MPDI
(3.45)
Mole fraction of dead polymer chains with terminal double bond can be computed by Equation
3.46.
00
0
f
(3.46)
Number of long-chain branches and short-chain branches per 103 carbon atoms can be calculated
using Equations 3.47 and 3.48 respectively (Pladis and Kiparissides, 1998).
111
5001000/l
LCBCLCB
(3.47)
75
111
5001000/l
SCBCSCB
(3.48)
3.3 Modeling of propylene polymerization
3.3.1 Kinetics
A general reaction set for propylene polymerization that includes reactions corresponding to all
types of metallocenes is proposed in this section. As in the case of ethylene polymerization
kinetics all of them may not be relevant to every system.
Catalyst activation by cocatalyst
The catalyst gets activated by the cocatalyst.
)0(PCocatCat ak (3.49)
where Cat is the catalyst, Cocat is the cocatalyst species, 0P represents an active catalyst site
that is capable of polymerization, and ka is the rate constant for catalyst activation. MAO is used
as cocatalyst species for many metallocene catalysts.
Chain initiation
The active sites react with monomer to initiate chain growth.
10 PMP ink (3.50)
where M is the monomer, 1P is a polymer chain attached to catalyst containing one monomer
segment, and kin is the rate constant for chain initiation.
76
Chain propagation
This step is the addition of monomer species to the growing chains. The addition of monomer
species involves a complexation of the double-bond of the monomer at the catalyst site.
1 iPMiP pk (3.51)
where iP is a polymer chain attached to a catalyst site and kp is the rate constant for chain
propagation. This reaction controls the monomer conversion in the reactor.
Spontaneous catalyst deactivation
A propagating chain converts into a dead polymer chain due to spontaneous deactivation of
active site.
iDPiP d
kd 0
00 d
k PP d
(3.52)
(3.53)
where 0dP is deactivated catalyst, and kd is the rate constant for spontaneous catalyst
deactivation. Spontaneous deactivation leads to the formation of vinylidene (CH2=C<) end group
in propylene polymerization.
Chain transfer to monomer (β hydrogen transfer)
Monomer also acts as a chain-transfer agent.
1)( PiDMiP tMk (3.54)
77
where )(iD is a dead polymer chain detached from the catalyst, and ktM is the rate constant for
chain transfer to monomer. Chain transfer to the monomer leads to the formation of vinylidene
(CH2=C<) end group in propylene polymerization.
β-Hydride elimination (Spontaneous chain transfer to the catalyst)
A propagating chain may terminate by transferring its β-H to the catalyst, carrying unsaturation
at its end. The catalyst site remains active for reinitiation and polymerization.
0)( *,
H
kPiDiP H (3.55)
where 0*
HP is a hydride activated complex, and kβ,H is the rate constant for spontaneous chain
transfer to catalyst. Chain transfer to catalyst also contributes to the formation of vinylidene
(CH2=C<) end group in propylene polymerization.
Reinitiation after β-Hydride elimination
Hydride catalyst activated complex 0*
HP reinitiates a new chain with monomer
)1(0* PMP rk
H (3.56)
where rk is the rate constant for reinitiation after chain transfer to catalyst.
β-Methyl elimination (Spontaneous chain transfer to the metal)
A propagating chain may also terminate by transferring its β-CH3 to the catalyst. This
mechanism of termination is very unusual and applies to special cases where highly substituted
Cp rings are involved (Resconi et al., 1992, Ethuis et al.,1992, Lin and Tsai, 2008).
78
0)( *,
Me
kPiDiP Me (3.57)
where 0*
MeP is a regenerated methyl complex, and kβ,Me is the rate constant for β-methyl
elimination. Chain transfer to catalyst by this mechanism leads to the formation of vinyl
(CH2=CH—) end group in propylene polymerization.
Secondary (2, 1) insertion
In propylene polymerization, a monomer can insert either by primary (1, 2) or secondary (2, 1)
fashion. With early transition metal complexes, insertion generally occurs in a (1, 2) - fashion
resulting in less bulky group on the metal (Makio and Fujita, 2008). A secondary (2, 1) insertion
is unusual for Group 4 transition metal mediated polymerization but can have a significant
influence on polymerization kinetics. Secondary insertion gives a dormant site which has a low
activity for further propene insertion and leads to regioirregularities in the chain.
)1( iRMiP sk (3.58)
where )1( iR is a living polymer chain with (2, 1) insertion, and ks is the rate constant for
secondary insertion.
Propagation after secondary (2, 1) insertion
Chain continues propagating after a secondary insertion occurs.
)1( iPMiR spk (3.59)
where ksp is the rate constant for propagation after secondary insertion.
79
Chain transfer to monomer after secondary (2, 1) insertion
β-hydrogen transfer is a considerable path of termination for living chains after (2, 1) insertion.
)1()( PiDMiR sMk (3.60)
where ksM is the rate constant for transfer to monomer after secondary insertion.
Chain transfer to cocatalyst MAO
MAO usually contains leftover Al(CH3)3. Chain transfer to Al is more common at lower propene
concentration. (Resconi et al., 1990; Naga and Mizunuma, 1998). Chain transfer to Al is
included to describe the influence of cocatalyst on molecular weight and the percentage of
different end groups.
0)( *,
Me
kPiDCocatiP Alt (3.61)
A methylated catalyst activated complex 0*
MeP is formed after chain transfer to cocatalyst. kt,Al
is the rate constant for transfer to cocatalyst.
Reactivation after chain transfer to cocatalyst MAO
The reactivation of methylated catalyst complex is treated as an elementary reaction different
from normal catalyst activation because the two activated complexes are raised in dissimilar
chemical environments.
10* PMP rAlk
Me (3.62)
krAl is the rate constant for reactivation after transfer to cocatalyst.
80
Table 3.2 summarizes the reactions considered in the kinetic mechanism of propylene
polymerization.
Table 3.2 Reactions Conceived in Propylene Polymerization
Reaction Stoichiometry Description
1. )0(PCocatCat ak Catalyst activation
2. 10 PMP ink Chain initiation
3. 1 iPMiP pk Chain propagation
4. iDPiP d
kd 0 {chain}
Spontaneous catalyst deactivation 5. 00 d
k PP d {site}
6. 1)( PiDMiP tMk Chain transfer to monomer
7. 0)( *,
H
kPiDiP H β-Hydride elimination
8. )1(0* PMP rk
H Reinitiation after β-H elimination
9. 0)( *,
Me
kPiDiP Me β-Methyl Elimination
10. )1( iRMiP sk Secondary (2, 1) insertion
11. )1( iPMiR spk Propagation after (2, 1) insertion
12. )1()( PiDMiR sMk Chain transfer after (2, 1) insertion
13. 0)( *,
Me
kPiDCocatiP Alt Chain transfer to cocatalyst
14. 10* PMP rAlk
Me Reactivation transfer to cocatalyst
81
3.3.2 Model development for propylene polymerization
In this section, mathematical model development based on the reactions considered in Table 3.2
for propylene polymerization has been discussed. The model developed is applicable for a batch,
semi-batch or constant stirred tank reactor.
Material balances
Model equations in terms of various moments are obtained for which material balance has been
written on different species present in the reactor. Description follows, in forthcoming sections.
Unactivated catalyst, 'Cat'
CocatCatkr aCat (3.63)
Vacant activated catalyst sites, ' 0P , 0*
HP , 0*
MeP '
)0()0()0( PkPMkCocatCatkr dinaP (3.64)
)]0(][[ *
0,)0(* Hr
l
HPPMkkr
H (3.65)
)]0(][[ *
,0,0,)0(* MeAlr
l
Alt
l
MePPMkCocatkkr
Me
(3.66)
Deactivated catalyst sites, 'Pd(0)'
l
dP Pkrd 0)0( )0( (3.67)
82
Cocatalyst, 'Cocat'
l
AltaCocat CocatkCocatCatkr 0, (3.68)
Monomer, 'M'
)]0(][[][
][][)]0(][[)0(
*
,0
00
*
00
MeAlr
m
sM
m
sp
l
sHr
l
tM
l
pinM
PMkMk
MkMkPMkMkMkPMkr
(3.69)
where m
n is the nth
moment of the molecular weight distribution of secondary (2,1) inserted
chains given by Equation 3.59.
1
)(i
nm
n iRi (3.70)
)(iR is the molar concentration of secondary (2,1) inserted chains.
Active chains, 'P(i)'
)0()1()1()1(
)0()1()1()1()1()0(
*
,,0,
*
,)1(
MeAlrAlt
m
sMsMe
HrHtMdpinP
PMkCocatPkMkPMkPk
PMkPkPMkPkPMkPMkr
(3.71)
CocatPkPMk
PkPkPMkPkPPMkr
Alts
MeHtMdpP
)2()2(
)2()2()2()2()2()1(
,
,,)2(
(3.72)
)()1()(
)()()()()()1(
,
,,)(
iPCocatkiRMkiPMk
iPkiPkiPMkiPkiPiPMkr
Altsps
MeHtMdpiP
(3.73)
Dead chains, 'D(i)'
CocatiPk
iRMkiPkiPkiPMkiPkr
Alt
sMMeHtMdiD
)(
)()()()()(
,
,,)(
(3.74)
83
Secondary (2,1) inserted chains, 'R(i)'
)()()1()( iRMkiRMkiPMkr sMspsiR (3.75)
Moments of chain length distribution (CLD) of living polymer chains
Zeroth moment, ' l
0 '
)0(
)0()0(
*
,0
*
0,,,0
MeAlr
m
sMsp
Hr
l
AltsMeHdin
PMkMkk
PMkCocatkMkkkkPMkr l
(3.76)
First moment, 'l
1 '
)0()0(
)0(
*
,
*
010
101,,,01
MeAlrHr
m
sM
mm
sp
ll
tM
l
AltsMeHd
l
pin
PMkPMkMkMk
MkCocatkMkkkkMkPMkr l
(3.77)
Second moment, 'l
2 '
)0()0(2
2)0(
*
,
*
021020
2,,,102
MeAlrHr
m
sM
mmm
sp
ll
tM
l
AltsMeHd
ll
pin
PMkPMkMkMkMk
CocatkMkkkkMkPMkr l
(3.78)
Moments of CLD of dead polymer chains
Zeroth moment, ' 0 '
In order to reckon the percentage chain termination by different routes, the zeroth moment for
dead chains is dissevered into:
(i) vinylidene-terminated chains ( 0 )v, due to termination via spontaneous deactivation, chain
transfer to monomer (β-H transfer) and chain transfer to catalysts (β-H elimination);
(ii) vinyl-terminated chains ( 0 )v', due to chain transfer to catalysts via β-CH3 elimination);
84
(iii) butenyl-terminated chains ( 0 )b due to chain transfer after secondary (2,1) insertion and
(iv) isobutyl-terminated chains ( 0 )i due to chain transfer to cocatalyst.
l
HtMd kMkkrv
0,)( 0
(3.79)
l
Mekrv
0,)( '0
(3.80)
m
sM Mkrb
0)( 0
(3.81)
l
Alt Cocatkri
0,)( 0
(3.82)
ibvvrrrrr
)()(')()( 00000 (3.83)
First moment, ' 1 '
m
sM
l
AlttMMeHd MkCocatkMkkkkr 11,,,1
(3.84)
Second moment, ' 2 '
m
sM
l
AlttMMeHd MkCocatkMkkkkr 22,,,2
(3.85)
Moments of CLD of secondary (2,1) inserted chains
Zeorth moment, ' m
0 '
m
sMsp
l
s MkkMkr m 000
(3.86)
First moment, 'm
1 '
m
sMsp
ll
s MkkMkr m 1101
(3.87)
85
Second moment, 'm
2 '
m
sMsp
lll
s MkkMkr m 2210 22
(3.88)
Polymer properties in terms of moment expressions
The number-average degree of polymerization is given by Equation 3.89.
ml
ml
nDP000
111
(3.89)
The weight-average degree of polymerization is given by Equation 3.90.
ml
ml
wDP111
222
(3.90)
The MWD of the polymer chains is characterized by number-average and weight-average
molecular weights calculated from Equations 3.91 and 3.92 respectively.
Number average molecular weight, ' nM '
ml
ml
nM000
11108.42
(3.91)
Weight average molecular weight, ' wM '
ml
ml
wM111
22208.42
(3.92)
The polydispersity index (PDI) is calculated from Equation 3.93.
2111
000222
ml
mlml
PDI
(3.93)
86
Micro-structural properties of polypropylene in terms of moment expressions
The developed model is capable of predicting the fractions of end groups generated by various
modes of chain transfer. Fraction of vinyl-terminated chains, butenyl-terminated chains,
isobutyl-terminated chains and vinylidene-terminated chains relative to the total unsaturated
termination is calculated from Equations 3.94, 3.95, 3.96 and 3.97 respectively.
i0b0v'0v0
v'0'
) () () () (
) (
vf
(3.94)
i0b0v'0v0
b0
) () () () (
) (
bf
(3.95)
i0b0v'0v0
i0
) () () () (
) (
if
(3.96)
ibvv ffff '1 (3.97)
3.4 Simulation methodology
The proper estimation of kinetic parameters is a cardinal step in the modeling. Model validation
is associated with the determination of a set of kinetic parameters for which the model
predictions match with the experimental behavior of different catalysts in polymerization. Unlike
in other polymerization systems such as free-radical polymerizations, the kinetic parameters in
catalytic olefin polymerization are strongly dependent on the nature of the catalyst. For a kinetic
model developed, relevant kinetic parameters must be determined for each catalyst system (Choi
et al., 1997).
Parameter estimation is a crucial yet difficult task, it is uncommon to find models with
parameters that have been estimated using experimental data. Most modelers obtain approximate
87
values of model parameters from the literature (Xie et al., 1995; Soares and Hamielec, 1996;
Park et al., 2003; Iedema, 2004; Lo and Ray, 2005; Park et al., 2005; Luo et al., 2007; Luo et al.,
2010). These models allowed only qualitative explanations of important phenomena during
olefin polymerization and qualitative previsions of the relationships between operating
conditions and polymer properties, no experiments have been used to verify these explanations
and predictions.
The models developed in this study are grounded on kinetics of ethylene and propylene
polymerization with various metallocene catalysts. For the validation of these models, the
experimental data have been taken from literature (Roos et al., 1997; Chakravarti, et al., 2001;
Marques et al., 2002;Yasin et al., 2004; Yasin et al., 2005).
The previously applied procedures of manual, graphical or trial and error based
estimation of kinetics parameters require significant time and efforts (Huang and Rempel, 1997;
Ochoteco et al., 2001; Nele et al., 2001; Matos et al., 2001; Khare et al., 2002; Lo and Ray,
2005; Neto et al., 2005; Kou et al., 2005; Hagen, 2006; Mehdiabadi and Soares, 2013).
Moreover these techniques are only reliable for small number of parameters.
This section presents the methodology for estimation of kinetic parameters that is based
on a systematic optimization strategy known as differential evolution (DE).
3.4.1 Numerical solution procedure
Model equations developed in previous chapter include a set of coupled, nonlinear and stiff
ordinary differential equations for the dynamic polymerization. These ordinary differential
equations (ODEs) were solved with MATLAB™
7.0.1 software (MATLAB version 7.0.1, 2004).
88
For numerically solving stiff differential equations, certain implicit methods, in particular
backward differentiation methods, perform much better than explicit ones. Explicit numerical
methods usually experience instability on a stiff equations.
The MATLAB ODE suite contains two implicit methods for stiff systems:
The implicit Runge–Kutta pair ode23s of orders 2 and 3,
The implicit numerical differentiation formulas ode15s of orders 1 to 5.
Both the methods have a built-in local error estimate to control the step size. Moreover
ode15s is a variable-order packages which use higher order methods and smaller step size when
the solution varies rapidly. The code ode15s for stiff systems is a quasi-constant step size
implementation of the numerical differentiation formulas of order 1 to 5 in terms of backward
differences. In this study, the model equations were worked out with ODE-15s function provided
in MATLAB™
7.0 software.
3.4.2 Objective function formulation
The goal of the optimization process is to find the parameter values that result in a maximum or
minimum of a function called the objective function. Objective function is a mathematical
expression describing a relationship of the optimization parameters or the result of an operation,
such as simulation that uses the optimization parameters as inputs. In this section, the
development of expression for objective functions on different basis is discussed.
89
Polymerization rate
Overall rate of polymerization is governed by the kinetic rates of monomer trapping and
monomer insertion. The models assume that the intrinsic rates of monomer trapping and
insertion are independent of active site location on the polymer chain. Experimental evidences
justify this assumption for monomer trapping (Simon et al., 1999; Roos et al., 1997; Chakravarti,
et al., 2001; Marques et al., 2002; Yasin et al., 2004; Yasin et al., 2005). Thus the rate of
polymerization can be viewed as the rate of disappearance of the monomer in ethylene (Equation
3.23) and propylene (Equation 3.69) polymerization under steady as well as transient states.
Further, since there is a negligible consumption of monomer in initiation and chain transfer
reactions as compared to the propagation reaction (long chain assumption), the rate of
polymerization can be calculated by Equation 3.98 for both ethylene and propylene
polymerization.
l
pMp Mkrr 0 (3.98)
The values of rate of polymerization calculated from Equation 3.98 are compared with
experimental ones at each measured point. All deviations between experimental and calculated
values (errors) are squared and summed up to form an objective function given in Equation 3.99.
n
j jp
jpjp
r
rrkF
1
2
exp,,
mod,,exp,,)(
(3.99)
90
Molecular weights
If the experimental, end of the run values, of number average and weight average molecular
weights are available, additional objective function can be formulated as given by Equation
3.100.
2
exp,
mod,exp,
2
exp,
mod,exp,)(
w
ww
n
nn
M
MM
M
MMkG
(3.100)
where mod,nM and mod,wM are the values calculated from Equations 3.43 & 3.44 for ethylene and
Equations 3.91 & 3.92 for propylene polymerization respectively.
Microstructural data
In propylene polymerization, when experimental microstructural (methyl pentad distribution
etc.) data are available, fractions of vinyl-terminated chains, butenyl-terminated chains, isobutyl-
terminated chains calculated from model (Equations 3.94 through 3.96) facilitate formulating yet
another objective function as shown in Equation 3.101.
2
exp,
mod,exp,
2
exp,
mod,exp,
2
exp,'
mod,'exp,')(
i
ii
b
bb
v
vv
f
ff
f
ff
f
ffkH
(3.101)
The parameter vector k, sought to be estimated involves the kinetic rate constants for the
reactions asserted in the model. These parameters have been determined by minimizing the
above-mentioned objective function equation(s). Let alone that the model equations are the
constraints to this optimization problem. Inclusion of multiple objective functions render more
appropriate estimation of kinetic parameters as they are large in number, interrelated and pose
different magnitudes of effect on polymerization rate and polymer properties.
91
3.4.3 Optimization approach
Minimization of the objective function in parameter estimation problems, particularly in the field
of polymer engineering, may lead to difficult numerical problems related to the large number of
model parameters, high correlativity among model parameters and multimodal nature of the
objective function. In order to overcome these difficulties, the use of heuristic optimization
method, differential evolution (DE) is proposed and worked out in this study.
Various established methods are being used as parameter estimation techniques, such as
the graphical method and the gradient-based non-linear optimization method (Kenny, 1994; Klar
et al., 2002; Di et al., 2008). The graphical method is limited and it does not have precision to
calculate the parameters. The graphical method can only take on those problems that can be
converted to linear regression problems, while the gradient-based nonlinear optimization method
is easy to trap into local optima.
Recent studies on optimization technique have developed an attractive class of
algorithms, viz. evolutionary algorithms. Evolutionary algorithms deal simultaneously with a set
of possible solutions (the so-called population) and have been received increasing attention due
to their powerful capability for global search. Some popular evolutionary algorithms are genetic
algorithm (GA), ant colony optimization (ACO), particle swarm optimization (PSO) and
differential evolution (DE). Differential evolution has been successfully applied to solve a wide
range of optimization problems such as optimization of non-linear functions (Angira and Babu,
2003), optimal design of shell and tube heat exchangers (Babu and Munawar, 2007),
optimization of process synthesis and design problems (Angira and Babu, 2006), optimization of
non-linear chemical processes (Babu and Angira, 2006) and Estimation of heat transfer
parameters in a trickle-bed reactor (Babu and Sastry, 1999).
92
3.4.3.1 Differential evolution (DE)
DE is a stochastic, population-based optimization algorithm introduced by Storn and Price in
1996 developed to optimize real parameter, real valued functions. Genetic algorithms (GA),
artificial neural networks (ANN), simulated annealing (SA) and differential evolution (DE)
algorithms are among the most famous nontraditional optimization methods used in the
combinatorial optimization field.
DE belongs to the class of genetic algorithms (GAs) which use biology-inspired
operations of crossover, mutation, and selection on a population in order to minimize an
objective function over the course of successive generations. Differential Evolution is a parallel
direct search method which utilizes NP, D-dimensional parameter vectors xi,G (i = 1; 2; . . .; NP)
as a population for each generation G. NP does not change during the minimization process. The
initial vector population is chosen randomly and in such a way that it should cover the entire
parameter space. DE generates new parameter vectors by adding the weighted difference
between two population vectors to a third vector. This operation is called mutation. The mutated
vector’s parameters are then mixed with the parameters of another predetermined vector, the
target vector, to yield the trial vector. Parameter mixing is referred to as crossover (Storn and
Price, 1997). The cost (in case of minimization) of the objective function is evaluated with target
and trial vectors and compared. The vector giving smaller cost secures a place in the population
of next generation. Same procedure is repeated NP times to decide the next generation of vectors
(Figure 3.1) and continues from one generation to another till some convergence criterion is met
as described in a flow sheet in Figure 3.2.
The key parameters of control in DE are: NP- population size, CR- cross over constant,
and F - weight applied to random differential (scaling factor). The details of DE algorithm and
93
pseudo code are available in the literature (Goldberg, 1989; Price and Storn, 1997; Onwubolu
and Babu, 2004; Babu and Angira, 2006; Angira and Babu, 2006). These key parameters of DE
are problem dependent. However, certain guidelines and heuristics are available in literature for
the choice of these parameters.
Natural logarithmic differential evolution (NLDE)
Differential evolution is a floating-point encoding evolutionary algorithm for global optimization
over continuous spaces, which also works well with discrete variables (Price et al., 2005;
Feoktistov, 2006). Since inception, algorithm has been modified and extended several times with
new versions being proposed (Fan and Lampinen, 2003; Ali and Törn, 2004; Rakesh and
Santosh, 2007; Ali, 2007; Liu and Wang, 2009; Pham, 2012).
In simple DE, linear mapping rule applies, in the initialization of normalized population
according to Equation 3.102 and in the mutation operation according to Equation 3.103.
New variable = Min value of the variable +
Random number (max value of the variable - Min value of the
variable)
(3.102)
Mutant vector = Base vector + F (Difference of two randomly chosen vectors) (3.103)
For a very wide range of variable values, linear rule of mapping may not be able to
exploit the entire search domain to generate the initial population which poses a problem of
improper population distribution. Similar trouble also arises across the generations due to
mutation operator (Sheth and Babu, 2008).
94
Figure 3.1 Determination of population for sequent generation in DE.
95
Figure 3.2 Flow sheet: Differential evolution optimization procedure.
96
In order to avert the problem of improper population distribution, natural logarithmic
mapping rule is proposed for initialization of normalized population and mutation as given by
Equation 3.104 and Equation 3.105 respectively.
New variable = exp {loge(Min value)+Random no. [loge(Max value)-loge(Min
value)]} (3.104)
Mutant vector = exp {loge(variable x3) + F [ loge (variable x1)- loge (variable x2)]} (3.105)
Natural logarithmic transformation of variables provide additional advantage by ensuring
that all kinetic parameter estimates are positive (since negative values do not make physical
sense). Natural logarithmic differential evolution (NLDE), an amended version of simple DE, is
used by incorporating natural log initialization and natural log mutation to take care of wide
ranges of variable values. This approach of optimization is applied by taking Equation(s) 3.99
through 3.101 as objective function to be minimized to find the globally optimum set of kinetic
parameters. Theoretical values were calculated from model equations for the objective function
and latter was minimized iteratively till convergence. Based on heuristics the values of DE key
parameters in this study were set as follows:
The population size (NP) was taken as fifty times the size of parameter vector,
Weighing factor (F) = 0.7 and
Cross over constant (CR) = 0.9
The ranges of the kinetic parameters for simulation were chosen based on the reported values in
the literature.
97
Summary of the chapter: Comprehensive mathematical models developed for ethylene and
propylene polymerization catalyzed with metallocene catalyst are discussed in detail. Developed
models are able to capture essential polymer properties, including number- and weight average
molecular weights, polydispersity index (PDI); fraction of polymer chains with terminal double
bond and frequency of long chain branching in ethylene polymerization; fraction of vinyl-
terminated chains, butenyl-terminated chains, isobutyl-terminated chains and vinylidene-
terminated chains in propylene polymerization.
Numerical solution procedure of the modeling equations and objective function
formulation are discussed in the later sections. Subsequently, a detailed overview of optimization
techniques used for kinetic parameter estimation is provided with special attention to differential
evolution approach. A remediated version of simple DE, viz. natural logarithmic differential
evolution (NLDE) is proposed, which is applied in parameter estimation in this study.
Methodology presented was applied to simulate the developed models with data reported in
literature. Obtained results are discussed in the next chapter.
98
CHAPTER – 4
RESULTS AND DISCUSSION
In this chapter, the simulation results obtained for the ethylene and propylene polymerization
using different metallocene catalyst systems are presented in sections 4.1 and 4.2 respectively.
Sections 4.1.1 and 4.1.2 detail the results obtained for ethylene polymerization in gas phase
with silica supported and in solution phase with in-situ-supported zirconocene catalysts
respectively. Results of solution phase propylene homopolymerization with various
metallocene catalysts are presented and discussed in Sections 4.2.1 through 4.2.7. Models
developed in Chapter 3 are simulated for validation and study of effects of reaction
parameters with data reported in open literature. Experimental conditions and corresponding
references are briefed in Table 4.1 and Table 4.2. All polymerizations in referenced works
were carried out in a stirred, semi-batch autoclave reactor with a flowmeter unit.
In all ethylene and propylene polymerization studies carried out in this work, model
equations are first solved analytically. To obtain analytical solution, the reactions of chain
propagation, spontaneous deactivation and chain transfer to monomer are considered. It was
possible to solve the model equations analytically since the concentration of monomer was
held constant (by maintaining its pressure). Differential equations describing moments of
living and dead polymer chain length distribution were analytically solved yielding Equation
4.1 through Equation 4.6.
tkP d
l exp).0(0 (4.1)
tMkktk
k
PkktMdd
tM
tMpl
expexp)0(
1 (4.2)
99
tMkkt
k
MkPkk
tMkktkkkk
Pkk
tMd
tM
ptMp
tMddtMp
tM
tMpl
exp.)0(2
expexp2)0(
22
(4.3)
tk
k
PMkkd
d
tMd
exp1)0(
0
(4.4)
dtMdd
d
tMd
tMd
tM
tMdtMp
kMkkk
tk
Mkk
tMkk
k
PMkkkk
11exp
exp
)0(1
(4.5)
tMkkt
MkkMkk
tMkk
k
MkPkk
kMkk
k
tk
Mkk
tMkk
k
kkPMkkkk
trd
tMdtMd
tMd
tM
ptMp
dtMd
d
d
tMd
tMd
tM
tMptMdtMp
exp.
1exp)0(2
11
expexp
2)0(22
(4.6)
Analytical solution described above is only possible for semibatch reactor with
invariable monomer concentration throughout the polymerization. For the parameters
estimation in all the subsequent studies in this work, this approach is utilized along with
NLDE to determine the coarse values of kinetic parameters in order to judge the range of
parameters those were required in fine optimization.
100
Table 4.1 Sections discussing results of ethylene polymerization
Sec. Catalyst system Reaction conditions Solvent Reference
4.1.1 E1: Silica supported (Me2Si[Ind]2ZrCl2) / MAO P =5 bar; Catalyst: 0.2 g; Al/Zr = 386;
T = 40 °C, 50
°C, 60
°C and 70
°C
- Roos et al. (1997)
4.1.2 E2: In-situ-supported (Et[Ind]2ZrCl2) / MAO P = 80 psig; Catalyst: 6 μmol; Al/Zr = 500;
T = 40 °C, 60
°C, 80
°C, 100
°C, 120
°C.
Hexane Chu et al. (2000)
* For the sake of brevity, above two catalysts shall be denoted as E1 and E2 respectively from here onwards.
Table 4.2 Sections discussing results of propylene polymerization
Sec. Catalyst Reaction conditions Solvent Reference
4.2.1 P1: (Me2Si[Ind]2ZrCl2) / MAO P = 2 bar; [Zr] = 10 μmol/L; T = 25 °C and 75
°C;
Al/Zr 500 and 2000
Toluene Marques et al.
(2002)
4.2.2 P2: (Et(Ind)2ZrCl2) / MAO - Do - - Do -
4.2.3 P3: (Me2Si(Ind)2HfCl2) / MAO P = 2 bar; [Hf] = 10 μmol/L; T = 40 °C and 80
°C;
Al/Hf 500 and 2000
- Do -
4.2.4 P4: (Et(Ind)2HfCl2) / MAO - Do - - Do -
4.2.5 P5: ([2,4,6-Me3Ind]2ZrCl2) / MAO P = 0.98 atm; [Zr] = 20 μmol/L; T = 0 °C;
Al/Zr = 2000 and 4000
- Do - Yasin et al.
(2004)
4.2.6 P6: ([2,4,7-Me3Ind]2ZrCl2) / MAO P = 0.98 atm; [Zr] = 20 μmol/L; T = 0 °C;
Al/Zr = 1000, 2000 and 4000
- Do -
4.2.7 P7: (Me2Si[2,4,6-Me3Ind]2ZrCl2) / MAO P = 0.98 atm; [Zr] = 20 μmol/L; Al/Zr = 2000;
T = 30 °C, 50
°C and 70
°C
- Do - Yasin et al.
(2005)
* For the sake of brevity, above seven catalysts shall be denoted as P1, P2, P3, P4, P6 and P7 respectively from here onwards.
101
4.1 Ethylene polymerization
Ethylene polymerization model is applied to gas phase polymerization with silica supported,
bridged Me2Si[Ind]2ZrCl2 catalyst and to solution phase polymerization with in-situ-silica
supported, bridged Et[Ind]2ZrCl2 catalyst. Based on experimental conditions, suitable
assumptions are made and a truncated form of comprehensive model is applied to both the
cases. The set of reactions considered are described in corresponding sections ahead.
4.1.1 Ethylene polymerization with Me2Si[Ind]2ZrCl2 (E1)/MAO
Ethylene polymerization model (EPM), discussed in Section 3.2.2, is applied to gas phase
polymerization of ethylene with silica- supported Me2Si[Ind]2ZrCl2 catalyst and kinetic
parameters are obtained. Experimental data for model validation are taken from Roos et al.
(1997). Simulations are carried out (with ODE-15s function provided in MATLAB™ 7.0
software) using natural logarithmic differential evolution approach of optimization to estimate
the kinetic parameters.
Model description
The gas phase production of polyethylene by silica supported Me2Si[Ind]2ZrCl2 catalyst is a
multistep process that necessarily includes initiation, propagation, and termination. Following
assumptions are made while employing the comprehensive ethylene polymerization model:
Assumptions
(i) Instantaneous formation of active sites (reaction between catalyst and cocatalyst).
(ii) First-order propagation with respect to monomer and the active site is assumed, and
the reactivity of the complex species does not depend on the length of the polymeric
chain.
102
(iii) Chain transfer takes place to monomer only and follows first order kinetics. This
assumption is induced, as the reactor was flushed with ethylene several times after
scavenging and catalyst introduction. This precludes the possibility of chain transfer to
anything but ethylene.
(iv) The same type of site generates after transfer as is primitively formed by activation of
catalyst with cocatalyst. So chain transfer step does not change the number of active
sites.
(v) Monomer consumption in initiation and chain transfer reactions is negligible as
compared to the propagation reaction (long chain assumption).
(vi) First-order deactivation of active sites.
Ethylene polymerization reactions considered based on above assumptions are
described in Table 4.3.
Table 4.3 Reactions Considered in Ethylene Polymerization
Reaction Stoichiometry Description
1. )0(PCocatCat Instantaneous catalyst activation
2. 10 PMP ink Chain initiation
3. 1 iPMiP pk Chain propagation
4. iDPiP d
kd 0 {chain} Spontaneous catalyst
deactivation 5. 00 d
k PP d {site}
6. 1)( PiDMiP tMk
Chain transfer to monomer
103
Estimated parameters and effect of temperature
Figure 4.1 shows the polymerization rates predicted by the model proposed by Roos et al.
(Section 2.4; Page 51) and ethylene polymerization model (Section 3.2.2) developed in this
study. Roos et al. modeled the polymerization rate assuming that deactivation of the catalyst
increases with increasing polymerization rate under isothermal conditions. Since they did not
consider initiation or any chain transfer reaction, the rate profiles are obtained exponentially
decreasing from a maximum value. For the same reason, polymerization rates are under
predicted at low temperatures (50 °C and 60
°C) and over predicted at 70
°C.
On the other hand, polymerization rates predicted by EPM exhibit a good agreement
with experimental data, at all the temperatures (40 °C, 50
°C, 60
°C and 70
°C). Estimated
kinetic parameters and objective function values F(k) are shown in Table 4.4. Close range of
objective function values (from 0.29934 to 0.6735) obtained, shows good fit with
experimental observations. Rates of initiation and propagation are increasing with increase in
temperature, as inferred from the estimated rate constants for these reactions. At the beginning
of polymerization, also referred to as induction period, the monomer is mainly consumed in
initiation followed by propagation reaction. During this period, the slope of polymerization
rate vs. time curve is indicative of the rate of initiation. In Figure 4.1, both, the initial
experimental polymerization rates and corresponding EPM predictions, clearly evince that
initiation and propagation rates are increasing with increase in temperature. Frequency of
spontaneous deactivation and chain transfer to monomer are also increasing with increase in
temperature, moreover the latter termination mechanism dominates over the former.
Increasing rates of deactivation and transfer to monomer at 50 °C, 60 °C and 70 °C are
responsible for a steeper decay in polymerization rate after reaching a maximum value.
104
0 100 200
0.0
8.0x10-3
1.6x10-2
40 0C
50 0C
60 0C
70 0C
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
Figure 4.1 Ethylene polymerization rate vs. time with Me2Si[Ind]2ZrCl2 (E1)/MAO;
Symbols: Exp data; Lines: Model Prediction ‐‐‐ (Roos et al.), — (EPM; Section 3.2.2).
[Catalyst (E1) = 0.2 g; P = 5 bar; Al/Zr = 386]
105
Model proposed by Roos et al. is not capable to predict average molecular weights and PDIs.
Molecular weight distribution obtained with EPM is shown in Table 4.5. Average molecular
weights are decreasing with increase in temperature. For an increase of 10 °C, from 40 °C to
70 °C, the weight average molecular weight ( wM ) is found to decrease by 67.40 %, 70.47 %
and 81.67 % respectively. This inverse trend is normal in chain growth polymerization and is
usually ascribed to the high activation energies for chain transfer reactions as compared to the
propagation reaction (Rudin, 1999). With increase in temperature, the rate of chain
termination to monomer increases rapidly as compared to the rate of propagation, which
results in decreasing molecular weights. Polymers prepared with metallocene catalysts
possess narrow molecular weight distribution, with polydispersities ranging from 2.0 to 2.5,
which is a consequence of the single-site feature of the metallocenes (Stevens; 1999).
Polydispersity indices of polyethylene prepared with Me2Si[Ind]2ZrCl2/MAO catalyst system
are found to be 1.999 irrespective of temperature.
Simulations are carried out further, to study the effects of ethylene pressure and
catalyst amount on polymerization rate and average molecular weights.
Effect of ethylene pressure
Polymerization rate is linearly increasing with ethylene pressure at all the temperatures as
shown in Figures 4.2. Low pressures (1-3 bar), i.e., low concentrations of monomer, reasons
low rates that are steady and maintained (due to negligible transfer reaction). At higher
pressures (5-7 bar), higher polymerization rates are obtained, but at the same time, transfer to
monomer also increases with high monomer concentrations resulting in steeper decay in
polymerization rates.
106
Table 4.4 Estimated Parameters for Me2Si[Ind]2ZrCl2 (E1)/MAO
T (°C) 40 50 60 70
ink ×10-5
(M-1
.s-1
) 8.4474 26.975 35.832 44.951
pk ×10-3
(M-1
.s-1
) 4.3141 7.2207 42.568 195.420
dk ×10-3
(s-1
) 42.032 45.265 54.526 94.095
tMk (M-1.s
-1) 4.9690 5.5374 6.4339 7.0468
F(k) (-) 0.29934 0.56928 0.67350 0.45939
Table 4.5 Predicted Molecular Weights & PDI with Me2Si[Ind]2ZrCl2 (E1)/MAO
T (°C)
nM (g/mol) wM (g/mol) PDI
40 9.628938 × 105 1.925759 × 10
6 1.999970
50 3.139822 × 105 6.276838 × 10
5 1.999106
60 9.268304 × 104 1.853654 × 10
5 1.999990
70 1.698616 × 104 3.397203 × 10
4 1.999982
107
0 50 100 150 200
0.0
3.0x10-3
6.0x10-3
9.0x10-3
1.2x10-2
1.5x10-2
7 bar
5 bar
3 bar
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
1 bar
(a)
0 50 100 150 200
0.0
4.0x10-3
8.0x10-3
1.2x10-2
1.6x10-2
2.0x10-2
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
7 bar
5 bar
3 bar
1 bar
(b)
108
0 50 100 150 200
0.0
6.0x10-3
1.2x10-2
1.8x10-2
2.4x10-2
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
7 bar
5 bar
3 bar
1 bar
(c)
Figure 4.2 Polymerization rate vs. time: Effect of pressure.
[Catalyst (E1) = 0.2 g, Al/Zr = 386, T = (a) 50 °C, (b) 60 °C (c) 70 °C]
As shown in Figures 4.3(a), at 50 °C, average molecular weights are increasing
slightly and not affected much with a change in ethylene pressure. A similar trend is obtained
at 60 °C as shown in Figure 4.3(b). However, at 70 °C [Figure 4.3(c)], an increase in wM from
2.03457 × 104 to 3.0561 × 10
4 is obtained for a change in pressure from 1 bar to 3 bar. With
further increase in pressure till 7 bar, a slight increase in average molecular weights are
observed. These results come along to be consistent with the assumption that the chain
transfer to monomer reaction is bimolecular giving a unvarying wM with respect to ethylene
concentration (pressure).
109
0 2 4 6 8
2.0x105
4.0x105
6.0x105
A
ve
rag
e m
ole
cu
lar
we
igh
t (g
.mo
l-1)
Pressure (bar)
Mn
Mw
T = 50 0C
(a)
1 2 3 4 5 6 7
0.00
7.50x104
1.50x105
2.25x105
T = 60 0C
Ave
rag
e M
ole
cu
lar
we
igh
t g
.mo
l-1
Pressure (bar)
Mn
Mw
(b)
110
1 2 3 4 5 6 7
0.0
2.0x104
4.0x104
T = 70 0C
Ave
rag
e M
ole
cu
lar
we
igh
t (g
.mo
l-1)
Pressure (bar)
Mn
Mw
(c)
Figure 4.3 Effect of pressure on average molecular weights.
[Catalyst (E1) = 0.2 g, Al/Zr = 386, T = (a) 50 °C, (b) 60 °C (c) 70 °C]
Effect of catalyst Amount
A steady increase in polymerization rate with increase in catalyst amount at constant
temperature and pressure is observed as shown in Figures 4.4. At higher temperatures, the rate
reaches to a maximum within no time and then decreases steeply, suggesting that both, the
initiation and the termination rates increase staggeringly with catalyst amount. With variation
in initial catalyst amount, average molecular weights and PDIs are not affected and the values
are upheld to those given in Table 4.5. In literature, for metallocene catalysts average
molecular weights are reported to be decreasing with increase in catalyst concentration
[Estrada and Hamielec (1994); Rieger and Janiak (1994); Kaminsky (1996); Zohuri et al.
(2005); Cheng and Tang (2010)]. This phenomenon is commonly explicated by the fact that
there is an increase in rate of chain transfer reactions which are dependent on the active site
concentration i.e. chain transfer to monomer, chain transfer to cocatalyst and β-hydride
111
elimination. The model applied to the present case is based on the assumption that the chain
transfer takes place to monomer only. Increase in catalyst amount seems to affect both, the
propagation rate and chain transfer rate equivalently at a given temperature and ethylene
pressure, due to which an invariant MWD is obtained.
0 50 100 150 200
0.0
4.0x10-3
8.0x10-3
1.2x10-2
1.6x10-2
2.0x10-2
0.5 g
0.4 g
0.3 g
0.2 gPo
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
0.1 g
Zr
(a)
0 50 100 150 200
0.0
5.0x10-3
1.0x10-2
1.5x10-2
2.0x10-2
2.5x10-2
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
0.5 g
0.4 g
0.3 g
0.2 g
0.1 g
Zr
(b)
112
0 50 100 150 200
0.0
6.0x10-3
1.2x10-2
1.8x10-2
2.4x10-2
3.0x10-2
3.6x10-2
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
0.5 g
0.4 g
0.3 g
0.2 g
0.1 g
Zr
(c)
Figure 4.4 Polymerization rate vs. time: Effect of catalyst amount.
[P = 5 bar, Al/Zr = 386, T = (a) 50 °C, (b) 60 °C (c) 70 °C]
4.1.2 Ethylene polymerization with in-situ-supported Et[Ind]2ZrCl2 (E2)/MAO
Ethylene polymerization model is applied to solution phase polymerization of ethylene with
in-situ-silica supported Et[Ind]2ZrCl2 catalyst and kinetic parameters are obtained. Data for
the model validation are taken from Chu et al. (2000).
Model description
Some rudimentary assumptions made while employing the model are described hereunder:
Assumptions
(i) Constant and uniform temperature and ethylene concentration inside the reactor.
(ii) Instantaneous vapor-liquid equilibrium. In solution polymerization, the concentration
of monomer in the liquid phase is dependent upon the solubility of the gas in the
solvent. Atiqullah et al. (1998), compared various equations of state for predicting
113
ethylene solubility in toluene under similar reaction conditions and observed that
Peng-Robinson (PR) and Soave–Redlich–Kwong (SRK) equations of state are equally
good for the purpose. Here, Peng-Robinson equation of state (PR-EOS) is used to
determine the concentration of ethylene in solvent at different temperatures and
monomer pressures.
(iii) Negligible polymer volume as compared to the total volume of the reactor, i.e.
constant gas-phase volume in the reactor.
(iv) Instantaneous activation of catalyst sites.
(v) First-order propagation with respect to monomer and the active site, and the reactivity
of the activated complex species does not devolve on the length of the polymer chain.
(vi) First-order deactivation of active sites. Spontaneous catalyst deactivation in a
propagating chain produces dead chain without terminal double bond.
(vii) Chain transfer to monomer, catalyst and cocatalyst produce dead chains with terminal
double bond.
(viii) Chain transfer to a dead polymer with terminal double bond produces long chain
branching.
Ethylene polymerization reactions considered based on above assumptions are
described in Table 4.6.
Estimated parameters and effect of temperature
Model is simulated in order to estimate the kinetic parameters at different temperatures using
ethylene flow rate as a measure of polymerization rate. A very large population size (120
times the dimension) is used to make certain of receiving optimized estimates of parameters.
Table 4.7 summarizes the parameters estimates at 40 °C, 60 °C, 80 °C, 100 °C and 120° C at
fixed catalyst amount 6μmols, Al/Zr = 500 and ethylene pressure 80 psig. Figure 4.5 shows
114
the model predictions of polymerization rate at different temperatures, which are very close to
the experimental values.
Table 4.6 Reactions Considered in Ethylene Polymerization
Reaction Stoichiometry Description
1. )0(PCocatCat Instantaneous catalyst activation
2. 10 PMP ink Chain initiation
3. 1 iPMiP pk Chain propagation
4. iDPiP d
kd 0 {chain} Spontaneous catalyst
deactivation 5. 00 d
k PP d {site}
6. 1)( PiDMiP tMk
Chain transfer to monomer
7. 0)( PiDCocatiP tCok
Chain transfer to cocatalyst
8. 0)( PiDiPk
β-hydride elimination
9. jiPjDiP lcbk )( Long-chain branching
As defined in Equation 3.99 and reported in Table 4.7, the values of objective function
indicate that the model is capable of predicting kinetic behaviour for all the temperatures
expeditiously. Function values obtained are ranging closely, with a least value of 1.1186 at
80 °C representing the best fit to experimental observations as compared to highest value of
1.6063 at 120 °C. Significantly lower values of propagation rate constants at 40
°C and 60
°C
are obtained relating to very low polymerization rate and catalyst activity with respect to those
at higher temperatures. At 80 °C, high propagation rate as compared to those at lower
temperatures trace higher polymerization rate with highest activity of the catalyst.
Polymerization rates are increasing with increase in temperature from 40 °C to 120 °C.
115
Declining rate profiles at 100 °C and 120
°C can be explained by serious catalyst deactivation,
transfers to monomer and β-hydride elimination at higher temperatures as pointed out by the
values of these parameters obtained.
Figure 4.6 shows the predicted decrease in concentration of active catalyst sites with
time at different temperatures. At 40 °C, 60 °C and 80 °C, all the active catalyst sites are
occupied within initial 15 minutes, whereas at higher temperatures (100 °C and 120 °C)
certain fraction of those could not attach a monomer to initiate the chain.
Experimental kinetic data at different operating conditions, like different catalyst
amount, ethylene pressure and cocatalyst to catalyst mole ratio, are utilized to validate the
model at fixed temperature of 60 °C. Parameters estimated at 60 °C, 80 psig and 6 μmol
catalyst amount with Al/Zr = 500 are used to verify the model responses at various conditions.
Table 4.7 Estimated Parameters for Et[Ind]2ZrCl2 (E2)/MAO
T
(°C)
ink ×103
(M-1
.s-1
)
pk
(M-1
.s-1
)
dk ×105
(s-1
)
tMk ×103
(M-1
.s-1
)
tCok ×106
(M-1
.s-1
)
k ×106
(s-1
)
lcbk ×105
(M-1
.s-1
)
F(k)
(-)
40 7.9660 17.1863 1.5377 3.3232 78.1863 1.3426 1.6298 1.3832
60 12.2164 44.9804 2.4541 19.5830 43.1982 7.9277 5.6526 1.1674
80 14.8433 188.9502 4.2504 35.4385 4.1940 8.6007 14.1722 1.1186
100 21.4832 529.4216 8.5169 56.6386 1.6294 1107.979 16.3020 1.3586
120 63.7560 938.188 74.0553 86.9489 1.4610 1307.70 30.3491 1.6063
116
0 10 20 30 40 50 60
0.0
1.0x10-3
2.0x10-3
3.0x10-3
40 0C
60 0C
T 80 0C
100 0C
120 0C
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
Figure 4.5 Effect of temperature on ethylene polymerization rate with in-situ-supported
Et[Ind]2ZrCl2 (E2)/MAO; solid lines are model predictions.
[Catalyst (E2) = 6 μmol, Al/Zr = 500 and P = 80 psig]
0 10 20 30 40 50 60
0
2
4
6
40 0C
60 0C
80 0C
100 0C
120 0C
Active
ca
taly
st site
s (m
ol)
Time (minutes)
Figure 4.6 Active catalyst sites vs. reaction time.
[Catalyst (E2) = 6 μmol, Al/Zr = 500 and P = 80 psig]
117
Effect of ethylene pressure
The model also predicted the proportional changes in polymerization rate at different ethylene
pressures as shown in Figure 4.7. The trend vindicates the first order dependence of rate on
ethylene concentration as no declining regions are seen. Figure 4.8 shows that, at 40 psig
pressure active catalyst sites stayed available for all the polymerization time suggesting the
non-initiation of some active site. At 80 psig pressure, active site were occupied within 10
minutes and for higher pressures within 20 minutes.
Effect of catalyst amount
Figure 4.9 shows a good match between experimental observations and model predictions at
3, 12 and 18 μmol of initial catalyst amount taken. The model adequately captures the features
of polymerization with in-situ-supported metallocene catalyst by following the sustained
polymerization rate with time and increase in the same with the increase in catalyst amount.
Figure 4.10 portrays that all the active catalyst sites were occupied within 10 minutes,
irrespective of initial amount of catalyst used.
Effect of cocatalyst to catalyst mole ratio
Effect of cocatalyst to catalyst mole ratio on the model predictions of polymerization rate are
shown in Figure 4.11. Use of high Al/Zr ratio brings in higher polymerization rate and for the
entire range of ratios, the maximum rate was reached within 10 minutes. Figure 4.12 depicts
that for lower ratios (250 and 500), active catalyst sites disappeared within first 10 minutes,
whereas for higher ratios (above 500) these decreased with time but remained available for
entire polymerization time. This is possible for high amount of cocatalyst delays the
spontaneous catalyst deactivation and also facilitates the regeneration of deactivated catalyst.
118
0 10 20 30 40 50 60
0.0
6.0x10-4
1.2x10-3
1.8x10-3
2.4x10-3
3.0x10-3
40 psig
80 psig
120 psig
160 psig
Time (minutes)
Po
lym
eri
za
tio
m r
ate
(m
ole
s/L
/s)
Figure 4.7 Effect of pressure on ethylene polymerization rate; solid lines are model
predictions.
[Catalyst (E2) = 6 μmol, Al/Zr = 500 and 60 °C]
0 10 20 30 40 50 60
0.0
1.5
3.0
4.5
6.0
7.5
40 psig
80 psig
120 psig
160 psig
Activa
red
ca
taly
st site
s (m
ol)
Time (minutes)
Figure 4.8 Active catalyst sites vs. reaction time.
[Catalyst (E2) = 6 μmol, Al/Zr = 500 and 60 °C]
119
0 10 20 30 40 50 60
0.0
4.0x10-4
8.0x10-4
1.2x10-3
1.6x10-3
2.0x10-3
18 mol
12 mol
6 mol
3 mol
Initial catalyst amount
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
Figure 4.9 Effect of catalyst amount on ethylene polymerization rate; solid lines are
model predictions.
[Al/Zr = 500, 60 °C, and 80 psig]
0 10 20 30 40 50 60
0
5
10
15
20
Active
ca
taly
sts
site
s (m
ol)
Time (minutes)
3 mol
6 mol
12 mol
18 mol
Initial catalyst amount
Figure 4.10 Active catalyst sites vs. reaction time.
[Al/Zr = 500, 60 °C and 80 psig]
120
0 10 20 30 40 50 60
0.0
5.0x10-4
1.0x10-3
1.5x10-3
2.0x10-3
2.5x10-3
3.0x10-3
Al / Zr
250
500
1000
2000
4000P
oly
me
riza
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
Figure 4.11 Effect of Al/Zr mole ratio on ethylene polymerization rate; solid lines are
model predictions.
[Catalyst (E2) = 6 μmol, T = 60 °C and P = 80 psig]
0 10 20 30 40 50 60
0.0
1.5
3.0
4.5
6.0
Al / Zr
Active
ca
taly
st site
s (m
ol)
Time (minutes)
250
500
1000
2000
4000
Figure 4.12 Active catalyst sites vs. reaction time.
[Catalyst (E2) = 6 μmol, T = 60 °C and P = 80 psig]
121
Polyethylene properties
Table 4.8 presents model cyphered properties of polyethylene at different sets of operating
conditions. Following experimental observations, average molecular weights of polyethylene
obtained from the model are found to be decreasing with increase in temperature, whereas the
change in catalyst amount, cocatalyst to catalyst mole ratio or ethylene pressure brought
insignificant effect.
The calculated number average molecular weight values of polyethylene are ranging
from 35127.01 to 38166.63 for varation in catalyst amount, from 35127.01 to 39089.35, for
varation in cocatlyst to catalyst mole ratio and from 35127.01 to 37860.78 for varation in
ethylene pressure, which advises that change in catalyst amount, cocatalyst to catalyst mole
ratio or ethylene pressure have insignificant effect on molecular weight and its distribution of
the polymer produced by in-situ supported Et[Ind]2ZrCl2 catalyst.
For homogeneous and otherwise supported metallocene catalysts, average molecular
weights have been reported to be decreasing with increase in catalyst concentration or
cocatalyst to catalyst mole ratio by many researchers [Estrada and Hamielec (1994); Rieger
and Janiak (1994); Kaminsky (1996); Zohuri et al. (2005); Cheng and Tang (2010)]. This
phenomenon is usually explained by the fact that there is an increase in rate of chain transfer
reactions which are dependent on the active site concentration. It is worth noticing that, the
rate of propagation is also dependent on the active site concentration and when the active site
concentration is maintained during polymerization (as for in-situ supported catalyst),
propagation rate increases proportionally, resulting effectively no change in average
molecular weight.
122
Table 4.8 Polyethylene Properties with Et[Ind]2ZrCl2 (E2)/MAO
S. No. T
(°C)
Zr
(μmoles)
Al/Zr
(-)
P
(psig) nM
(Model)
wM
(Exp)
wM
(Model)
PDI
(Exp)
PDI
(Model)
f (=)
(Model)
lcb/1000C
(Model)
1. 60 3 500 80 38166.63 74038.10 76256.93 2.50 1.998 0.870 4.633 × 10-9
2. 60 6 500 80 35127.01 74234.10 70211.57 2.59 1.998 0.986 7.238 × 10-10
3. 60 12 500 80 35127.08 67110.50 70211.94 2.22 1.998 0.986 1.447 × 10-9
4. 60 18 500 80 35135.82 70916.20 70230.15 2.26 1.998 0.986 2.168 × 10-9
5. 60 6 250 80 38143.88 78666.70 76211.49 2.60 1.998 0.986 5.428 × 10-10
6. 60 6 500 80 35127.01 74234.10 70211.57 2.59 1.998 0.986 7.238 × 10-10
7. 60 6 1000 80 39089.35 80000.00 78022.34 2.13 1.996 0.987 2.501 × 10-14
8. 60 6 2000 80 38969.59 72000.00 77783.32 2.40 1.996 0.864 2.244 × 10-15
9. 60 6 4000 80 36296.29 69333.30 72483.69 2.20 1.997 0.781 9.308 × 10-16
10. 40 6 500 80 56975.08 - 112696.71 - 1.978 0.909 1.439 × 10-7
11. 60 6 500 80 35140.93 74234.10 70211.57 2.59 1.998 0.986 7.238 × 10-10
12. 80 6 500 80 29273.46 - 53102.05 - 1.814 0.972 1.265 × 10-11
13. 100 6 500 80 19669.60 - 39299.87 - 1.998 0.985 3.034 × 10-12
14. 120 6 500 80 13965.66 - 27931.31 - 2.000 0.992 5.615 × 10-12
15. 60 6 500 40 36419.99 59165.70 69598.61 2.44 1.911 0.887 8.061 × 10-12
16. 60 6 500 80 35127.01 74234.10 70211.57 2.59 1.998 0.986 7.238 × 10-10
17. 60 6 500 120 37412.19 75418.70 72579.66 2.61 1.940 0.845 3.105 × 10-8
18. 60 6 500 160 37860.78 73472.40 75607.98 2.62 1.997 0.965 5.798 × 10-9
123
As monomer is also acting as a chain transfer agent, the insight of the effect of ethylene pressure
on molecular weight may be gained by probing Equation 4.7 for number average degree of
polymerization. Estimated parameters (Table 4.7) indicate that chain transfer to monomer is
dominating over other transfer reactions, for which the contribution of other transfer reactions
may be neglected in the denominator of Equation 4.7. For such a situation, degree of
polymerization and so the molecular weight show up to be independent of monomer
concentration.
0 lcbtCotM
pn
kkCocatkMk
MkDP (4.7)
The effect of temperature is widely acknowledged with the reason that the activation
energy for chain transfer is greater than that for propagation. Consequently, early termination of
the chains results in lower average moelcular weight. Polydispersity in all the cases is obtained
very close to 2.
As inferred from calculated fraction of dead chains with double bond at the end, major
modes of chain termination, nearly for all sets of conditions, are believed to be chain transfer to
monomer, chain transfer to cocatalyst and β-hydride elimination. Relative magnitude of different
transfer reactions is evident from estimated parameters at various temperatures. Long chain
branching frequency is detected to be negligibly low, except only for low (40 °C) temperature
and high (120 and 160 psig) ethylene pressures, suggesting that the product is comprising of
linear chains and posseses high density.
124
4.2 Propylene polymerization
Model description
Propylene polymerization model with some elementary assumptions is applied to the solution
phase production of polypropylene catalyzed with various zirconium and hafnium based catalyst
systems. In all the studies hereinafter, simulations are carried out using 'natural logarithmic
differential evolution' approach of optimization.
Assumptions
(i) Instantaneous formation of active sites.
(ii) First-order propagation with respect to monomer and the active site, and the reactivity of
the activated complex species does not devolve on the length of the polymer chain.
(iii) First-order deactivation of active sites.
(iv) Chain transfer following a propagation by primary (1, 2) insertion takes place by β-
hydride transfer to the monomer, producing a vinylidene-terminated dead chain and
liberating an active center.
(v) Chain transfer to catalyst (Zr) takes place by β-hydride elimination, producing a dead
chain and a hydride activated complex which reinitiates and follows the features of a
primitive propagating chain.
(vi) A secondary (2, 1) insertion brings about a mis-inserted chain, which can undergo
propagation and terminate by β-hydride transfer to the monomer to produce a dead chain
with butenyl-end group.
(vii) Monomer consumption in the initiation, secondary (2, 1) insertion and all chain transfer
reactions is negligible as compared to the propagation reaction via primary (1, 2)
insertion.
125
(viii) Chain transfer to cocatalyst (MAO) occurs, producing a dead chain and a methyl
activated complex which reinitiates a new chain.
Polymerization reactions considered based on the above assumptions are described in
Table 4.9.
Table 4.9 Reactions Considered in Propylene Polymerization
Reaction Stoichiometry Description
1. )0(PCocatCat Instantaneous catalyst activation
2. 10 PMP ink Chain initiation
3. 1 iPMiP pk Chain propagation
4. iDPiP d
kd 0 {chain}
Spontaneous catalyst deactivation 5. 00 d
k PP d {site}
6. 1)( PiDMiP tMk Chain transfer to monomer
7. 0)( *,
H
kPiDiP H
β-Hydride elimination
8. )1(0* PMP rk
H Reinitiation after β-H elimination
9. )1( iRMiP sk Secondary (2, 1) insertion
10. )1( iPMiR spk Propagation after (2, 1) insertion
11. )1()( PiDMiR sMk Chain transfer after (2, 1) insertion
12. 0)( *,
Me
kPiDCocatiP Alt Chain transfer to cocatalyst
13. 10* PMP rAlk
Me Reinitiation after transfer to
cocatalyst
126
4.2.1 Propylene polymerization with Me2Si[Ind]2ZrCl2 (P1)/MAO
Propylene polymerization model is applied to solution phase polymerization of propylene with
Me2Si[Ind]2ZrCl2/MAO catalyst system and kinetic parameters are obtained. Data for the model
validation are taken from Marques et al. (2002).
Propylene concentration in toluene is calculated from Equation 4.8 for a specific temperature
(in °C) and pressure. [Bravakis et al. (1998) referred and used in Marques et al. (2002)].
psi 45at 1049.5 1001.1536.0
psi 30at 1066.3 1075.6354.0
psi 15at 1083.1 1037.3179.0
252
253
253
63
TT
TT
TT
X HC (4.8)
Estimated parameters and effect of temperature
Kinetic parameters for Me2Si[Ind]2ZrCl2 (P1)/MAO catalyst system are estimated by simulating
the model with experimental data at 25 °C and 75 °C with Al/Zr molar ratio of 500.
Experimental data for Al/Zr ratio of 2000 are used to validate the model at both temperatures.
Figure 4.13 and Figure 4.14 present experimental and model predicted propylene polymerization
rates at 25 °C and 75 °C respectively. Experimental observations reveal that both, the
temperature and MAO to catalyst molar ratio have a significant effect on polymerization rate.
Model predictions are in very close agreement with experimental observations at both the
temperatures and Al/Zr molar ratios of 500 & 2000.
Figure 4.13 and Figure 4.14 show that with an increase in Al/Zr molar ratio,
polymerization rate increases at both the temperatures considered. Larger Al/Zr ratio gives rise to
higher concentration of activated complex at the beginning of the reaction which increases the
rate of initiation and consequently the rate of propagation. At very high Al/Zr ratio, the
127
decreasing polymerization rate after reaching a maximum may be explained by large rate of
chain transfer to cocatalyst.
Figure 4.15 explains that the temperature has a profound effect on polymerization rate.
Maximum rate is seen to increase four folds at 75 °C when compared with that at 25 °C, at fixed
catalyst concentration, Al/Zr ratio and pressure. Increase in propagation rate constant pk with
temperature is also in coherence with this observed fact. As understood by dk and tMk values
obtained, spontaneous deactivation and chain transfer to monomer are activated hugely with
increase in temperature. Ascribing to which, the polymerization rate is decreasing steeply at 75
°C after reaching a maximum.
Kinetic parameters and objective function [F(k)] values which indicate the extent of fit of
experimental data with model prediction are given in Table 4.10.
Active catalyst site concentration drops to near zero within 5 minutes and 10 minutes at
25 °C and 75 °C respectively as shown in Figure 4.16. Solubility of propylene in solvent
decreases with increase in temperature at fixed pressure (Equation 4.8), thereby reducing the
concentration of monomer in the reaction mixture. Further, the fact, ].[Mk in = 2.2388×10-3
s-1
at
25 °C vs. ].[Mk in = 0.2753×10-3
s-1
at 75 °C explains that chain initiation rate is slower at 75
°C. Active sites are deactivated spontaneously as well, frequency of which is obtained higher (cf.
dk , Table 4.10) at 75 °C.
128
0 10 20 30 40 50 60
0.0
0.5
1.0
1.5
2.0
2.5
Po
lym
eri
za
tio
n r
ate
(m
ol/L
/s)
Time (minutes)
1:500
1:2000
Zr:Al
Figure 4.13 Effect of Al/Zr mole ratio on propylene polymerization rate; solid lines are
model predictions.
[Catalyst (P1) = 10 μM, T = 25 °C and P = 30 psi]
0 10 20 30 40 50 60
0
2
4
6
Po
lym
eri
za
tio
n r
ate
(mo
les / L
/ s
)
Time (minutes)
1:500
1:2000
Zr:MAO
Figure 4.14 Effect of Al/Zr mole ratio on propylene polymerization rate; solid lines are
model predictions.
[Catalyst (P1) = 10 μM, T = 75 °C and P = 30 psi]
129
0 10 20 30 40 50 60
0.0
1.5
3.0
4.5
6.0
Po
lym
eri
za
tio
n r
ate
(m
ol/L
/s)
Time (minutes)
T
25 °C
75 °C
Figure 4.15 Effect of temperature on propylene polymerization rate; solid lines are model
predictions.
[Catalyst (P1) = 10 μM, Al/Zr = 500 and P = 30 psi]
The overall rate of disappearance of active sites is higher at 25 °C due to which these are
exhausted little early. Fractional disappearance of active sites by spontaneous deactivation may
be determined by ].[/ Mkkk indd . This fraction is found less at lower temperatures and as per
calculations 10.28% active sites at 25 °C whereas, 53.1% active sites at 75 °C are figured to be
deactivated spontaneously.
Maximum concentration of hydride activated complex 0*
HP is noted to be
3.5409×10-9
μM (in 7 minutes) at 25 °C and 4.0866×10-7
μM (in 14 minutes) at 75 °C as shown
in Figure 4.17 and Figure 4.18 respectively.
130
Table 4.10 Estimated Parameters for Me2Si[Ind]2ZrCl2
(P1)/MAO
T (°C) 25 75
ink ×103
(M-1
.s-1
) 5.1845 10.7586
pk × 10-4
(M-1
.s-1
) 8.566 39.4588
dk × 104
(s-1
) 2.5665 3.1176
tMk (M-1
.s-1
) 2.5699 10.2576
Hk , ×106
(s-1
) 1.55277 11.7565
rk ×10-2
(M-1
.s-1
) 1.5294 152.700
sk ×107
(M-1
.s-1
) 1.3249 11.6351
spk ×104
(M-1
.s-1
) 1.4583 118.967
sMk (M-1
.s-1
) 0.047 1.0394
Altk , (M-1
.s-1
) 426.688 141.243
rAlk (M-1
.s-1
) 8.574 13.0285
F(k) (-) 0.272 0.03713
Table 4.11 Predicted Properties with Me2Si[Ind]2ZrCl2
(P1)/MAO
T (°C) 25 75
Exp Model Exp Model
nM ×10-4
(g/mol)
- 13.549 - 0.9925
wM ×10-4
(g/mol)
12.3 27.257 1.5 1.9984
PDI 1.8 2.011 2.8 2.013
vf (%) - 74.1 - 86.3
bf (%) - 5.2 - 3.4
if (%) - 20.7 - 10.3
131
0 10 20 30 40 50 60
0
5
10
A
ctive
ca
taly
st site
s (m
ole
s/L
)
Time (minutes)
T
25 °C
75 °C
Figure 4.16 Active catalyst site concentration vs. time.
[Catalyst (P1) = 10 μM, Al/Zr = 500 and P = 30 psi]
With P1/MAO catalyst system, β-H elimination is observed to occur negligibly ( Hk , of
the order of 10-6
) as compared to all other modes of chain termination. Increase in temperature
enhances the frequency of β-H elimination. Due to higher concentration of monomer at 25 °C
than that at 75 °C, reinitiation rate after β-H elimination is higher, which is reflected by the
falling concentration of hydride activated complex after reaching a maximum. At 75 °C, due to
lower reinitiation rate, 0*
HP is almost steady beyond maximum.
Concentration of methyl activated complex 0*
MeP is found to reach a maximum of
0.8142 μM (1.8 min) followed by a steep decrease at 25 °C as compared to a maximum of
0.2329 μM (3.58 min) at 75 °C followed by slow decrease as shown in Figure 4.19.
132
0 10 20 30 40 50 60
0.0
1.0x10-9
2.0x10-9
3.0x10-9
4.0x10-9
Hyd
rid
e a
ctiva
ted
co
mp
lex (m
ole
s/L
)
Time (minutes)
Figure 4.17 Hydride actived complex concentration vs. time.
[Catalyst (P1) = 10 μM, Al/Zr = 500, T = 25 °C and P = 30 psi]
0 10 20 30 40 50 60
0.0
2.0x10-7
4.0x10-7
Hyd
rid
e a
ctiva
ted
co
mp
lex (m
ole
s/L
)
Time (minutes)
Figure 4.18 Hydride activated complex concentration vs. time.
[Catalyst (P1) = 10 μM, Al/Zr = 500, T = 75 °C and P = 30 psi]
133
Chain transfer to cocatalyst is decreasing with increase in temperature. This trend is also
observed with the subsequent cases studied in this work. MAO is an oligomer and consists
mainly of units of the basic structure [Al4O3Me6], which contains four aluminium, three oxygen
atoms and six methyl groups. As the aluminium atoms in this structure are co-ordinatively
unsaturated, the basic units join together and give rise to different structures as described in
Figure 2.10. The observed trend of decrease in chain transfer to MAO with increase in
temperature suggests the existence of different molecular structures of MAO at different
temperatures. It seems that MAO changes its structure from simple (linear or cyclic) at low
temperatures to a congested one (ladder or cage) at high temperatures and thereby offering a
steric hindrance for chain transfer. At 75 °C, the rectivation rate, after transfer to cocatalyst is
slow due to low concentrations of monomer and methyl activated complex, for which the
concentration of 0*
MeP is seen decreasing slowly opposite to that at 25 °C.
0 10 20 30 40 50 60
0.0
0.3
0.6
0.9
Me
thyl a
ctiva
ted
co
mp
lex (m
ole
s/L
)
Time (minutes)
T
25 °C
75 °C
Figure 4.19 Methyl activated complex concentration vs. time.
[Catalyst (P1) = 10 μM, Al/Zr = 500 and P = 30 psi]
134
Average molecular weights, PDI and percentage of butenyl- & isobutyl-terminated chains
predicted by the model are given in Table 4.11. Degree of polymerization depends upon relative
rate of propagation against various termination rates as given by Equation 4.9.
MkkkCocatkMk
MkDP
sdHAlttM
pn
,,
(4.9)
Agreement between experimental observations and predicted results for MWD is shown
in Table 4.9. Low molecular weights obtained at higher (75 °C) temperature, may be attributed to
the relatively high frequency of termination via transfer to monomer, spontaneous deactivation
and β-H transfer. The model predicts Schulz-Flory distribution with a polydispersity index very
around 2 for all reaction conditions.
For P1/MAO, chain termination is understood to take place majorly via spontaneous
catalyst deactivation, transfer to monomer and β-Hydride transfer because vf values in Table
4.11 point the dominant presence of chains with vinylidene end group. Low bf values suggest
that the fraction of chains with butenyl end group which are produced by termination after
secondary insertion is miserable, which is expected for highly isotactic polypropylene. However,
with increase in temperature, bf is observed to decrease. This is due to relatively higher increase
in pk than the increase in sk values with increase in temperature. Further, it has been noted that
increased regioirregular (2,l) insertions slow down chain propagation and inhibit chain transfer to
the monomer. This effect is consistent with previous results reported in the literature for the
studied catalyst system [Busico et al. (1998)].
135
if value (Table 4.11) shows, 20.7% terminated chains bear isobutyl end group at 25 °C, which
represent a significant chain transfer to cocatalyst. Further, increase in temperature results in a
decreased rate of chain transfer to cocatalyst which is indicated by the decrease in if (10.3% at
75 °C). This chain transfer dominates at lower temperature, due to depressed, competing β-H
elimination rates [Amin (2007)].
Effect of Pressure
Figure 4.20 shows that an increase in monomer pressure results in a steady increase in
polymerization rate up to a maximum ranging in between 0.685 moles/L/s (15 psi) to 2.056
moles/L/s (45 psi) at 25 °C and fixed Zr = 10 μM, Al/Zr = 500. Increase in pressure seems to
increase the monomer concentration at reaction site and thereby increasing the propagation rate
proportionally as expected and obtained from simulation results. At 75 °C, higher polymerization
rates are observed {2.489 moles/L/s (15 psi) to 8.487 moles/L/s (45 psi)}, which do not continue
maintained due to high rates of termination at this temperature (Figure 4.21).
A marginal increase of 2.22 % (15-30 psi) and 0.87 % (30-45 psi) in weight average
molecular weight wM is observed with increase in pressure at 25 °C (Figure 4.22). Similarly,
at 75 °C, a maximum increment of 10.31 % in wM is seen (Figure 4.23). Therefore, an increase
in monomer pressure is believed to increase the polymerization rate appreciably with an incresed
intake of monomer but brings negligible effect on polymer molecular weights.
136
0 10 20 30 40 50 60
0.0
0.5
1.0
1.5
2.0
2.5
45 psi
30 psiP
oly
me
riza
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
15 psi
Figure 4.20 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P1) = 10 μM, Al/Zr = 500, and T = 25 °C]
0 10 20 30 40 50 60
0
2
4
6
8
45 psi
30 psi
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
15 psi
Figure 4.21 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P1) = 10 μM, Al/Zr = 500 and T = 75 °C]
137
15 30 45
1x105
2x105
3x105
Mo
lecu
lar
we
igh
t (g
/mo
l)
Pressure (psi)
Mn
Mw
Figure 4.22 Effect of pressure on average molecular weights.
[Catalyst (P1) = 10 μM, Al/Zr = 500 and T = 25 °C]
15 30 45
5.0x103
1.0x104
1.5x104
2.0x104
2.5x104
Mo
lecu
lar
we
igh
t (g
/mo
l)
Pressure (psi)
Mn
Mw
Figure 4.23 Effect of pressure on average molecular weights.
[Catalyst (P1) = 10 μM, Al/Zr = 500 and T = 75 °C]
138
Effect of catalyst concentration
Polymerization rates at various catalyst concentrations (10, 20, 40 and 80 μM) at 25 °C and 75
°C are shown in Figure 4.24 and Figure 4.25 respectively. On doubling the catalyst
concentration, maximum polymerization rate is almost doubled, showing a linear proportional
dependence. Higher polymerization rates with decreasing trend after reaching a maximum are
obtained at 75 °C, which can be explained with the fact that along with propagation, termination
rates also increase profusely at high temperatures. Average molecular weights are found to be
insensitive to catalyst concentration as shown in Figure 4.26 and Figure 4.27. Catalysts
concentration may affect the degree of polymerization via β-H transfer only. Significantly low
values of Hk , (Table 4.10), as compared to other transfer rate constants, warrant that change in
catalyst concentration has no effect on molecular weights.
0 10 20 30 40 50 60
0
2
4
6
8
10
12
[Zr] = 80M
[Zr] = 40M
[Zr] = 20M
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
[Zr] = 10M
Figure 4.24 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Zr = 500, T = 25 °C and P = 30 psi]
139
0 10 20 30 40 50 60
0
10
20
30
40
50
[Zr] = 80M
[Zr] = 40M
[Zr] = 20M
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
[Zr] = 10M
Figure 4.25 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Zr = 500, T = 75 °C and P = 30 psi]
0 20 40 60 80
1x105
2x105
3x105
Mo
lecu
lar
we
igh
t (g
/mo
l)
[Zr] (M)
Mn
Mw
Figure 4.26 Effect of catalyst concentration on average molecular weights.
[Al/Zr = 500, T = 25 °C and P = 30 psi]
140
0 20 40 60 80
5.0x103
1.0x104
1.5x104
2.0x104
2.5x104
Mo
lecu
lar
we
igh
t (g
/mo
l)
[Zr] (M)
Mn
Mw
Figure 4.27 Effect of catalyst concentration on average molecular weights.
[Al/Zr = 500, T = 75 °C and P = 30 psi]
4.2.2 Propylene polymerization with Et(Ind)2ZrCl2 (P2)/MAO
Propylene polymerization model is applied to solution phase polymerization of propylene with
Et(Ind)2ZrCl2/MAO catalyst system and kinetic parameters are obtained. Simulations are carried
out numerically using 'natural logarithmic differential evolution' approach of optimization. Data
for the model validation are taken from Marques et al. (2002).
Estimated parameters and effect of temperature
Model predictions of reaction rate captures the typical behavior of experimental rate profiles for
propylene polymerization with Et(Ind)2ZrCl2/MAO catalyst system as shown in Figures. 4.28
and 4.29. Estimated kinetic parameters and objective function values are given in Table 4.12. At
75 °C, parameters are evaluated at an Al/Zr molar ratio of 500 and the experimental data at a
141
ratio of 2000 are used to verify the model prediction for polymerization rate. Excellent
predictions are obtained as shown in Figures. 4.28 and 4.29.
Figure 4.30 presents the concentration profiles of active catalyst site at 25 °C and 75 °C.
Active sites are found to decrease exponentially from 10 μM to almost nil in 27 minutes at 25 °C
and within 15 minutes at 75 °C at fixed pressure 30 psi and Al/Zr = 2000. ].[Mk in = 3.61×10-4
at 25 °C vs. ].[Mk in = 9.39×10-4
at 75 °C suggest that chain initiation rate is faster at 75 °C and
therefore active catalyst sites take lesser time to exhaust than that at 25 °C. Spontaneous
deactivation of catalyst site is also increased staggeringly at 75 °C with aids to lower
concentration of active sites observed. As low as 0.35% active catalyst sites at 25 °C against
23.9% at 75 °C are disappearing by spontaneous deactivation.
142
0 10 20 30 40 50 60
0
1
2
Po
lym
eri
za
tio
n r
ate
(m
ol/L
/s)
Time (minutes)
Figure 4.28 Polymerization rate vs. time; solid lines are model predictions.
[Catalyst (P2) = 10 μM, Al/Zr = 2000, T = 25 °C and P = 30 psi]
0 10 20 30 40 50 60
0
1
2
3
Zr:MAO
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
1:500
1:2000
Figure 4.29 Polymerization rate vs. time; solid lines are model predictions.
[Catalyst (P2) = 10 μM, T = 75 °C and P = 30 psi]
143
Table 4.12 Estimated Parameters for Et(Ind)2ZrCl2 (P2)/MAO
T (°C) 25 75
ink ×103
(M-1
.s-1
) 1.7348 1.7686
pk ×10-5
(M-1
.s-1
) 1.1175 2.0135
dk ×106
(s-1
) 1.2638 294.8404
tMk ×104
(M-1
.s-1
) 9.4135 536.7625
Hk , ×106
(s-1
) 6.9652 9.6698
rk ×10-2
(M-1
.s-1
) 2.8886 3.4166
sk ×105
(M-1
.s-1
) 9.9996 11.6170
spk (M-1
.s-1
) 4.5639 68.0812
sMk ×104
(M-1
.s-1
) 4.2017 46.3481
Altk , ×10-4
(M-1
.s-1
) 6.4558 1.5515
rAlk (M-1
.s-1
) 12.9809 93.7468
F(k) (-) 0.2995 0.3615
Table 4.13 Predicted Polypropylene Properties with Et(Ind)2ZrCl2 (P2)/MAO
T (°C) 25 75
Al/Zr 2000 500 2000
Exp Model Exp Model Exp Model
nM ×10-4
(g/mol) - 2.5785 - 0.49795 - 0.3316
wM ×10-4
(g/mol) 5.4 5.3249 1.2 1.0353 0.8 0.7434
PDI (-) 2.2 2.065 2.1 2.079 1.9 2.242
vf (%) - 51.2 - 71.2 - 64.9
bf (%) - 4.6 - 4.1 - 3.8
if (%) - 44.2 - 24.7 - 31.3
144
0 10 20 30 40 50 60
0
4
8
12
Active
ca
taly
st site
s (m
ole
s/L
)
Time (minutes)
T
25 °C
75 °C
Figure 4.30 Active catalyst site concentration vs. time.
[Catalyst (P2) = 10 μM, Al/Zr = 2000 and P = 30 psi]
Like P1/MAO catalyst system, with Et(Ind)2ZrCl2 (P2)/MAO also, β-H elimination is not
observed significant at both the temperatures considered. Maximum concentration of hydride
activated complex 0*
HP is got to be at 2.2354×10-8 μM (in 14 minutes) at 25 °C and
1.7542×10-7 μM (in 27 minutes) at 75 °C as shown in Figure 4.31 and Figure 4.32 respectively.
Hk , values (Table 4.10) suggest that the rate of β-H elimination is low at 25 °C, for which the
concentration of 0*
HP is low at 25 °C as compared to that at 75 °C. Higher reinitiation rate after
β-H elimination at 25 °C is obtained due to which 0*
HP decreases after reaching the maxima.
145
0 10 20 30 40 50 60
0.00
7.50x10-9
1.50x10-8
2.25x10-8
Hyd
rid
e a
ctiva
ted
co
mp
lex (m
ole
s/L
)
Time (minutes)
Figure 4.31 Hydride actived complex concentration vs. time.
[Catalyst (P2) = 10 μM, Al/Zr = 2000, T = 25 °C and P = 30 psi]
0 10 20 30 40 50 60
0.0
6.0x10-8
1.2x10-7
1.8x10-7
Hyd
rid
e a
ctiva
ted
co
mp
lex (m
ole
s/L
)
Time (minutes)
Figure 4.32 Hydride actived complex concentration vs. time.
[Catalyst (P2) = 10 μM, Al/Zr = 2000, T = 75 °C and P = 30 psi]
146
Chain transfer to cocatalyst is found to be very high at Al/Zr = 2000 at both the temperatures. At
25 °C, concentration of methyl activated complex is found to reach a maximum of 5.484 μM
(3.37 min) as compared to 3.932 μM (4 min) at 75 °C as shown in Figure 4.33. Methyl activated
complex concentration after reaching the maximum is decreasing steeply to a negligible value at
6.2 min (25 °C) and 7.63 min (75 °C). The rectivation rate constant, after transfer to cocatalyst
is high but monomer concentration is less at 75 °C. Consequently, the reactivation rates are
comparable and high at both the temperatures ( 7013.2MkrAl at 25 °C, 9779.4MkrAl at
75 °C), which clarifies the quick decrease in concentration of 0*
MeP with time.
0 10 20 30 40 50 60
0
2
4
6
Me
thyl a
ctiva
ted
co
mp
lex (m
ole
s/L
)
Time (minutes)
T
25 °C
75 °C
Figure 4.33 Methyl actived complex concentration vs. time.
[Catalyst (P2) = 10 μM, Al/Zr = 2000 and P = 30 psi]
147
Average molecular weights, PDI and percentage of vinylidene-, butenyl- & isobutyl-terminated
chains predicted by the model for Et(Ind)2ZrCl2/MAO catalyst system are presented in Table
4.13. Model predictions are in very close correspondence with the experimental values. Effect of
temperature on MWD is obtained similar to that discussed for P1/MAO system in Section
4.2.1.1, however P2 yields low molecular weight product at matched reaction conditions. PDIs
at all reaction conditions are obtained near 2.0.
For P2/MAO, vf values in Table 4.13 indicate that chain termination takes place mainly
via spontaneous catalyst deactivation, transfer to monomer and β-Hydride and increasing with an
increase in temperature. With an increase in Al/Zr ratio, vf decreases due to enhanced chain
transfer to cocatalyst. Low bf values obtained advise that the fraction of chains with butenyl
end group is very less as compared to others. bf is noted to decrease with increase in
temperature as well as Al/Zr mole ratio. 44.2 % chains at 25 °C and 31.3 % chains at 75 °C are
found to terminate via chain transfer to cocatalyst as indicated by if values (Table 4.13, Al/Zr =
2000). This trend of decreasing rate of chain transfer to cocatalyst with increase in temperature is
consistent with that obtained with P1/MAO system. The rate of chain transfer to cocatalyst is
proportional to the catalyst concentration. if value is found to increase from 23.6 % to 43.2 %
with increase in Al/Zr mole ratio from 500 to 2000 respectively at 75 °C.
Effect of Pressure
With increase in monomer pressure an increase in polymerization rate up to a stabilized
maximum of 0.8808 moles/L/s (15 psi) and 2.678 moles/L/s (45 psi) at 25 °C and fixed Zr = 10
μM, Al/Zr = 2000 is observed as shown in Figure 4.34. At 75 °C, little higher polymerization
148
rates {Rpmax: 1.083 moles/L/s (15 psi) and 3.96 moles/L/s (45 psi)} with decreasing trend are
observed (Figure 4.35). The changes in average molecular weights with variation in pressures at
25 °C (Figure 4.36) and at 75 °C (Figure 4.37) are trifling. The discussion on the trends of
resultant rate profiles and average molecular weights at varied pressures given in Section 4.2.1.1
adjudges equivalently appropriate for the results obtained for Et(Ind)2ZrCl2/MAO catalyst
system.
0 10 20 30 40 50 60
0
1
2
3
45 psi
30 psi
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
15 psi
Figure 4.34 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P2) = 10 μM, Al/Zr = 2000 and T = 25 °C]
149
0 10 20 30 40 50 60
0
1
2
3
4
45 psi
30 psi
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
15 psi
Figure 4.35 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P2) = 10 μM, Al/Zr = 500 and T = 75 °C]
15 30 45
1.5x104
3.0x104
4.5x104
6.0x104
Mo
lecu
lar
we
igh
t (g
/mo
l)
Pressure (psi)
Mn
Mw
Figure 4.36 Effect of pressure on average molecular weights.
[Catalyst (P2) = 10 μM, Al/Zr = 2000 and T = 25 °C]
150
15 30 45
3.0x103
6.0x103
9.0x103
1.2x104
Mo
lecu
lar
we
igh
t (g
/mo
l)
Pressure (psi)
Mn
Mw
Figure 4.37 Effect of pressure on average molecular weights.
[Catalyst (P2) = 10 μM, Al/Zr = 500 and T = 75 °C]
Effect of catalyst concentration
Polymerization rates at different catalyst (Et(Ind)2ZrCl2) concentrations at 25 °C and 75 °C are
shown in Figure 4.38 and Figure 4.39 respectively. Like Me2Si[Ind]2ZrCl2 (P1), with this catalyst
system also, polymerization rate is found to be linearly dependent on the catalyst concentration.
However P2 offers a lower polymerization rates than P1 under similar reaction conditions. To
compare, at Zr = 10 μM and 75 °C, a maximum rate of 2.45 moles/L/s with P2 is obtained
against 8.475 mole/L/s with P1. Additionally, average molecular weights are observed to be
unaffected by catalyst concentration as shown in Figure 4.40 and Figure 4.41, which can be
attributed to very low values of Hk , (Table 4.11), as compared to other transfer rate constants.
This has been predicted that catlyst P2 produces lower molecular weight polypropylene when
compared to P1 for alike reaction conditions (Table 4.9 and Table 4.11). The effect of catalyst
151
concentration upon rate and molecular weights are qualitatively similar to that discussed for
catalyst P1 in Section 4.2.1.1.
0 10 20 30 40 50 60
0
4
8
12
16
[Zr] = 80M
[Zr] = 40M
[Zr] = 20M
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
[Zr] = 10M
Figure 4.38 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Zr = 2000, T = 25 °C and P = 30 psi]
0 10 20 30 40 50 60
0
5
10
15
20
[Zr] = 80M
[Zr] = 40M
[Zr] = 20MPo
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
[Zr] = 10M
Figure 4.39 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Zr = 500, T = 75 °C and P = 30 psi]
152
0 20 40 60 80
2.0x104
3.0x104
4.0x104
5.0x104
6.0x104
Mo
lecu
lar
we
igh
t (g
/mo
l)
[Zr] (M)
Mn
Mw
Figure 4.40 Effect of catalyst concentration on average molecular weights.
[Al/Zr = 2000, T = 25 °C and P = 30 psi]
0 20 40 60 80
6.0x103
9.0x103
1.2x104
Mo
lecu
lar
we
igh
t (g
/mo
l)
[Zr] (M)
Mn
Mw
Figure 4.41 Effect of catalyst concentration on average molecular weights.
[Al/Zr = 500, T = 75 °C and P = 30 psi]
153
4.2.3 Propylene polymerization with Me2Si(Ind)2HfCl2 (P3)/MAO
Simulation results obtained with propylene polymerization model applied to solution phase
polymerization of propylene with Me2Si(Ind)2HfCl2/MAO catalyst system are presented and
discussed in this section. Data for the model validation were taken from Marques et al. (2002).
Estimated parameters and effect of temperature
Experimental rate data at Al/Hf = 2000 are regressed with the model to determine kinetic
parameters. Further, the model is simulated to predict the polymerization rate at Al/Hf = 500.
Estimated kinetic parameters and objective function [F(k)] values are provided in Table 4.14.
Figures. 4.42 and 4.43 depict a good agreement between the experimental data and model
predictions of polymerization rate with Me2Si(Ind)2HfCl2 (P3)/MAO catalyst system at 40 °C
[F(k) = 1.5175] and 80 °C [F(k) = 0.3161] respectively. Experimental observations reveal that at
40 °C, polymerization rate is weakly affected by a change in Al/Hf ratio as shown in Figure 4.42.
Whereas at 80 °C, polymerization rate increases with increase in cocatalyst to catalyst mole ratio
(Figure 4.43).
Polymerization rate is observed to increase with increase in temperature. At fixed catalyst
concentration, Al/Hf ratio (2000) and monomer pressure, maximum rate is found to be 0.4068
moles/L/s (29 minutes) at 40 °C against 0.9378 moles/L/s (7 minutes) at 80 °C. A rapid gain in
propagation rate at 80 °C is evident from lower frequency of spontaneous catalyst deactivation
(Table 4.12) and a higher initiation rate at this temperature, which provides high concentration of
initiated chains during initial period. Hafnium based metallocene catalysts are known for
producing high molecular weight polymers in comparison to their zirconium analogues, but at
154
the expense of substantially reduced catalytic activity [Ewen et al. (1987); Nakayama and Shiono
(2005)].
Figure 4.44 lays out the concentration profiles of active catalyst site at 40 °C and 80 °C.
Active sites are found to decrease slowly from 10 μM to 1.37 μM in entire polymerization time
at 40 °C and from 10 μM to 0.0124 μM within 19.6 minutes at 80 °C at fixed pressure 30 psi
and Al/Zr = 2000. Chain initiation rate is faster at 80 °C ( ].[Mk in = 1.0898×10-4
s-1
vs. 4.34×10-
5 s
-1 at 40 °C, Table 4.14) and the frequency of spontaneous deactivation of catalyst sites is also
found increasing with temperature (cf. dk , Table 4.14). Therefore active catalyst sites are
consumed in very less time at 80 °C than that at 40 °C.
Table 4.14 Estimated Parameters for Me2Si(Ind)2HfCl2 (P3)/MAO
T (°C) 40 80
ink ×103 (M
-1.s
-1) 0.3045 2.2591
pk ×10-4
(M-1
.s-1
) 4.0398 5.5376
dk ×104 (s
-1) 1.8879 5.3001
tMk ×102 (M
-1.s
-1) 0.2771 8.7269
Hk , ×106
(s-1
) 9.2795 188.81
rk ×10-3
(M-1
.s-1
) 1.4199 22.8680
sk ×108 (M
-1.s
-1) 0.2002 1.4798
spk ×105
(M-1
.s-1
) 1.1091 684.3200
sMk ×108 (M
-1.s
-1) 1.2084 102.4800
Altk , (M-1
.s-1
) 183.5400 99.5750
rAlk ×10-3
(M-1
.s-1
) 1.0293 2.4608
F(k) (-) 1.5175 0.3161
155
Table 4.15 Predicted Properties with Me2Si(Ind)2HfCl2 (P3)/MAO
T (°C) 40 80
Al/Zr 500 2000 500 2000
Exp Model Exp Model Exp Model Exp Model
nM ×10-4
(g/mol) - 10.7115 - 12.1355 - 2.4346 - 1.6478
wM ×10-4
(g/mol) 34.6 30.0459 27.1 24.3074 6.2 4.8698 3.9 3.2919
PDI 2.9 2.8050 2.8 2.003 1.9 2.0002 1.9 1.9977
vf (%) - 86.6 - 87.2 - 90.2 - 90.7
bf (%) - 4.2 - 3.1 - 3.5 - 2.4
if (%) - 9.2 - 9.7 - 6.3 - 6.9
0 10 20 30 40 50 60
0.0
0.2
0.4
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
Hf:MAO
1:500
1:2000
Figure 4.42 Effect of Al/Hf mole ratio on propylene polymerization rate; solid lines are
model predictions.
[Catalyst (P3) = 10 μM, T = 40 °C and P = 30 psi]
156
0 10 20 30 40 50 60
0.0
0.2
0.4
0.6
0.8
1.0
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
Hf:MAO
1:500
1:2000
Figure 4.43 Effect of Al/Hf mole ratio on propylene polymerization rate; solid lines are
model predictions.
[Catalyst (P3) = 10 μM, T = 80 °C and P = 30 psi]
0 10 20 30 40 50 60
0
5
10
Active
ca
taly
st site
s (m
ole
s/L
)
Time (minutes)
T
40 °C
80 °C
Figure 4.44 Active catalyst site concentration vs. time.
[Catalyst (P3) = 10 μM, Al/Hf = 2000 and P = 30 psi]
157
Spontaneous deactivation rate of active catalyst sites is high with comparison to the rate of
initiation at both temperatures. 81.3% active sites at 40 °C and 82.9% at 80 °C are calculated to
disappear due to spontaneous deactivation.
With Me2Si(Ind)2HfCl2 (P3)/MAO system, frequency of β-H elimination is found to increase
from 9.2795×10-6
s-1
at 40 °C to 1.8881×10-4
s-1
at 80 °C (Table 4.14).
At 40 °C, concentration of hydride activated complex reaches a maximum of 1.906×10-7
μM (in 28 minutes) and decreases with a slower rate, on the other hand, at 80 °C it reaches a
maximum of 2.327×10-6 μM (in 8.85 minutes) and decreases appreciably as shown in Figure
4.45. Reinitiation rate after β-H elimination is high ( rk >103) at both the temperatures, so
considerable decrease in 0*
HP is seen at higher concentration.
0 10 20 30 40 50 60
0.0
5.0x10-7
1.0x10-6
1.5x10-6
2.0x10-6
2.5x10-6
3.0x10-6
T
40 °C
80 °C
Hyd
rid
e a
ctiva
ted
co
mp
lex (m
ole
s/L
)
Time (minutes)
Figure 4.45 Hydride actived complex concentration vs. time.
[Catalyst (P3) = 10 μM, Al/Hf = 2000 and P = 30 psi]
158
Chain transfer to cocatalyst is seen significant at Al/Hf = 2000 at both the temperatures. At 40
°C, concentration of methyl activated complex is observed to reach a maximum of 6.96×10-4
μM
(26.8 min) as compared to 1.836×10-5
μM (26.4 min) at 80 °C as shown in Figure 4.46 and
Figure 4.47 respectively. Reactivation rates are high and comparable at both the temperatures
hence methyl activated complex concentration after reaching the maximum is decreasing
appreciably.
Molecular weight distribution and percentage of vinylidene-, butenyl- & isobutyl-
terminated chains predicted by the model for Me2Si(Ind)2HfCl2 (P3)/MAO system are given in
Table 4.15.
0 10 20 30 40 50 60
0.0
2.0x10-4
4.0x10-4
6.0x10-4
8.0x10-4
Me
thyl a
ctiva
ted
co
mp
lex (m
ole
s/L
)
Time (minutes)
Figure 4.46 Methyl actived complex concentration vs. time.
[Catalyst (P3) = 10 μM, Al/Hf = 2000, T = 40 °C and P = 30 psi]
159
0 10 20 30 40 50 60
0.0
4.0x10-6
8.0x10-6
1.2x10-5
1.6x10-5
2.0x10-5
Me
thyl a
ctiva
ted
co
mp
lex (m
ole
s/L
)
Time (minutes)
Figure 4.47 Methyl actived complex concentration vs. time.
[Catalyst (P3) = 10 μM, Al/Hf = 2000, T = 80 °C and P = 30 psi]
Model predictions are in good agreement with the experimental values. Effect of
temperature on MWD is obtained qualitatively similar to that observed for P1/MAO and
P2/MAO systems (Sections 4.2.1.1 and 4.2.2.1). It is worth noting that Me2Si(Ind)2HfCl2
(P3)/MAO system yields a way higher molecular weight polypropylene when compared to its Zr
analogue i.e. P1/MAO. Model predicted value of PDI at 40 °C with Al/Hf molar ratio is 2.8 (exp.
value = 2.9) and for all other reaction conditions is about 2.0.
Like other catalyst systems discussed so far, with P3/MAO also, the major pathway of
chain termination is via spontaneous catalyst deactivation, transfer to monomer and β-Hydride
elimination. Among these β-Hydride elimination is found increasing dominantly whereas
spontaneous deactivation and transfer to monomer are seen to decrease a bit, with an increase in
160
temperature. vf values are increasing with increase in temperature at a constant Al/Hf molar ratio
as shown in Table 4.15, but there is negligible increase with an increase in Al/Hf ratio at a given
temperature. Fraction of chains with butenyl end group is very less as compared to others as
inferred by low bf values obtained, which are also noticed to decrease with increase in
temperature as well as Al/Hf mole ratio.
9.2 % chains at 40 °C and 6.3 % chains at 80 °C ( if values at Al/Hf = 500) are found to
terminate via chain transfer to cocatalyst which increases with increase in Al/Hf molar ratio. The
trend of decreasing rate of chain transfer to cocatalyst with increase in temperature is similar to
that obtained with its Zr analogue (P1/MAO system) but the amount of this termination is quite
less with P3/MAO system.
Effect of Pressure
Polymerization rates up to a maximum of 0.131 moles/L/s (15 psi), 0.38 moles/L/s (30 psi) and
0.693 moles/L/s (45 psi) at 40 °C and fixed Zr = 10 μM, Al/Hf = 2000 are obtained as shown in
Figure 4.48. At 80 °C, maximum rates of 0.382 moles/L/s at 15 psi, 0.855 moles/L/s at 30 psi
and 1.350 moles/L/s at 45 psi are obtained (Figure 4.49). A steady increase in polymerization
rate with an increase in monomer pressure is seen at both the temperatures. Higher initiation rate
at 80 °C, provide larger concentration of live chains which in turn alleviate to reach high
polymerization rate in less time.
Effect of monomer pressure on molecular weight is found potential at lower temperature.
A significant increase of 71.6 % (15-30 psi) and 30.72 % (30-45 psi) in weight average
molecular weight wM is observed with increase in pressure at 40 °C (Figure 4.50). At 80 °C,
161
relatively lower increase of 18.8 % (15-30 psi) and 6.73 % (30-45 psi) in wM is observed with
increase in pressure (Figure 4.51).
0 10 20 30 40 50 60
0.00
0.25
0.50
0.75
45 psi
30 psi
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
15 psi
Figure 4.48 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P3) = 10 μM, Al/Hf = 2000 and T = 40 °C]
0 10 20 30 40 50 60
0.00
0.25
0.50
0.75
1.00
1.25
1.50
45 psi
30 psi
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
15 psi
Figure 4.49 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P3) = 10 μM, Al/Hf = 2000 and T = 80 °C]
162
15 30 45
7.0x104
1.4x105
2.1x105
2.8x105
3.5x105
Mo
lecu
lar
we
igh
t (g
/mo
l)
Pressure (psi)
Mn
Mw
Figure 4.50 Effect of pressure on average molecular weights.
[Catalyst (P3) = 10 μM, Al/Hf = 2000 and T = 40 °C]
15 30 45
1.0x104
2.0x104
3.0x104
4.0x104
Mo
lecu
lar
we
igh
t (g
/mo
l)
Pressure (psi)
Mn
Mw
Figure 4.51 Effect of pressure on average molecular weights.
[Catalyst (P3) = 10 μM, Al/Hf = 2000 and T = 80 °C]
163
Effect of catalyst concentration
Polymerization rates at different catalyst concentrations (10, 20, 30, 40 and 50 μM) at 40 °C and
80 °C are shown in Figure 4.52 and Figure 4.53 respectively. With increase in catalyst
concentration, more number of active catalyst sites are rendered, which increase the rate of
initiation even if monomer concentration is invariant. Propagation rate is also dependent of
concentration of live chains. On this account, a higher polymerization rate is expected with
increase in catalyst concentration, which is correctly predicted by the model at both the
temperatures. Due to higher initiation rate at 80 °C, maximum polymerization rate is achieved
earlier than that at 40 °C at corresponding catalyst concentrations. Since high catalyst
concentration promotes β-H elimination, polymerization rate diminishes at a faster rate as seen in
Figure 4.52 and Figure 4.53 respectively. This effect is particularly pronounced at 80 °C due to
very high Hk , value. Further, the model predicts a proportional dependence of propagation rate
on catalyst concentration.
With P3/MAO catalyst system, average molecular weights are noticed to be decreasing
with increase in catalyst concentration as shown in Figure 4.54 and Figure 4.55. An inverse
relationship between polymer molecular weight and catalyst (Z-N/metallocene) concentration
have been obtained and reported in literature [Breslow and Newburg (1959), Chien (1959),
Brintzinger (1995)]. With P3/MAO, this effect is more influencing at 40 °C, where higher
molecular weights are obtained. A decrease of 35.1% (Hf = 10-20 μM), 21.5% (Hf = 20-30
μM), 14% (Hf = 30-40 μM) and 9.1% (Hf = 40-50 μM) in wM at 40 °C (Figure 4.54), whereas
at 80 °C, 2.1 - 2.3% (Figure 4.55) decrease is obtained.
164
0 10 20 30 40 50 60
0.0
0.5
1.0
1.5
2.0
[Zr]
50M
10MPo
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
40M
30M
20M
Figure 4.52 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Hf = 2000, T = 40 °C and P = 30 psi]
0 10 20 30 40 50 60
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
[Zr]
50M
10MPo
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
40M
30M
20M
Figure 4.53 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Hf = 2000, T = 80 °C and P = 30 psi]
165
0 10 20 30 40 50
0.0
7.0x104
1.4x105
2.1x105
2.8x105
Mn
Mw
Mo
lecu
lar
we
igh
t (g
/mo
l)
[Hf] (M)
Figure 4.54 Effect of catalyst concentration on average molecular weights.
[Al/Hf = 2000, T = 40 °C and P = 30 psi]
0 10 20 30 40 50
1.0x104
2.0x104
3.0x104
4.0x104
Mn
Mw
Mo
lecu
lar
we
igh
t (g
/mo
l)
[Hf] (M)
Figure 4.55 Effect of catalyst concentration on average molecular weights.
[Al/Hf = 2000, T = 80 °C and P = 30 psi]
166
4.2.4 Propylene polymerization with Et(Ind)2HfCl2 (P4)/MAO
Propylene polymerization model is applied to solution phase polymerization of propylene with
Et(Ind)2HfCl2 catalyst and kinetic parameters are obtained. Data for the model validation were
taken from Marques et al. (2002).
Estimated parameters and effect of temperature
Kinetic parameters for Et(Ind)2HfCl2 (P4)/MAO catalyst system are estimated by simulating the
model with experimental data at 40 °C and 80 °C with Al/Hf molar ratio of 500. The model
predictions are affirmed with experimental data for Al/Hf ratio of 2000 at both temperatures.
Kinetic parameters and objective function [F(k)] values obtained for P4/MAO system are given
in Table 4.16. Figure 4.56 and Figure 4.57 present experimental and model predicted propylene
polymerization rates at 40 °C and 80 °C respectively. Model predictions are in conformation
with experimental observations at both the temperatures and Al/Hf molar ratios of 500 & 2000.
Figure 4.56 and Figure 4.57 show that with an increase in Al/Hf molar ratio,
polymerization rate increases at both the temperatures. The trend observed is coherent with the
previously discussed studies. At high Al/Hf ratio, large rate of chain transfer to cocatalyst
induces the decrease in polymerization rate after reaching a maximum. In general, increase in
temperature is found to upshoot in higher polymerization rate.
With P4/MAO system, Rp,max is found to increase from 0.5216 moles/L/s at 40 °C to
1.092 moles/L/s at 80 °C at Al/Hf molar ratio of 2000, following the general trend. But with this
catalyst system at Al/Hf ratio of 500, Rp,max is found to decrease meagerly from 0.1952 moles/L/s
at 40 °C to 0.1148 moles/L/s at 80 °C.
167
0 10 20 30 40 50 60
0.0
0.2
0.4
0.6
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
Hf:MAO
1:500
1:2000
Figure 4.56 Effect of Al/Hf mole ratio on propylene polymerization rate; solid lines are
model predictions.
[Catalyst (P4) = 10 μM, T = 40 °C and P = 30 psi]
0 10 20 30 40 50 60
0.0
0.5
1.0
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
Hf:MAO
1:500
1:2000
Figure 4.57 Effect of Al/Hf mole ratio on propylene polymerization rate; solid lines are
model predictions.
[Catalyst (P4) = 10 μM, T = 80 °C and P = 30 psi]
168
This surprising drift indicates that Al/Hf ratio of 500 is too low to efficiently activate this catalyst
at 80 °C. Owing to this fact a lower value of propagation rate constant ( pk ) is obtained at 80 °C
than that at 40 °C. The model is able to capture the experimental rate profile adequately well at
Al/Hf mole ratio of 2000 with the kinetic parameters determined at a ratio of 500.
Figure 4.58 shows that active catalyst site concentration decreases to negligible value
within 2 minutes at 80 °C, on the contrary, at 40 °C, active catalyst site concentration decreases
slowly. This fact is also evident from very high ink value at 80 °C (Table 4.16). Despite a
decrease in spontaneous deactivation frequency of active catalyst with temperature, extremely
high initiation rate at 80 °C ( ].[Mkin = 1.018×10-3 vs. 4.08×10
-5 at 40 °C ) copiously contribute
for overall rapid decrease in active catalyst site concentration. For the same reasons, fractional
disappearance of active sites by spontaneous deactivation is found to be 93.8% at 40 °C and 16.4
% at 80 °C.
Maximum concentration of hydride activated complex 0*
HP is obtained to be
7.0652×10-3
μM (in 33.5 minutes) at 40 °C and 1.9034 μM (in 17 minutes) at 80 °C as shown in
Figure 4.59 and Figure 4.60 respectively. With P4/MAO catalyst system, β-H elimination is
observed to occur at a higher frequency when compared to other catalyst systems discussed so
far and following the previous trends, increases with increase in temperature.
Reinitiation rate after β-H elimination is higher at 40 °C, which is evident from the
falling concentration of hydride activated complex after reaching a maximum. At 80 °C, due to
lower reinitiation rate, 0*
HP is almost stabilized after maximum.
169
Table 4.16 Estimated Parameters for Et(Ind)2HfCl2 (P4)/MAO
T (°C) 40 80
ink ×104 (M
-1.s
-1) 2.8629 211.0300
pk ×10-3
(M-1
.s-1
) 3.4717 28.5890
dk ×104 (s
-1) 2.0049 6.1684
tMk ×104 (M
-1.s
-1) 3.4470 5.3048
Hk , ×105 (s
-1) 3.0296 127.7600
rk ×103 (M
-1.s
-1) 1.6685 7.7346
sk ×109 (M
-1.s
-1) 1.2524 37.0180
spk ×105 (M
-1.s
-1) 5.6388 17.9600
sMk ×103 (M
-1.s
-1) 5.6042 10.6520
Altk , (M-1
.s-1
) 177.79 34.7210
rAlk ×10-3
(M-1
.s-1
) 2.4626 3.3866
F(k) (-) 1.3866 0.5143
Table 4.17 Predicted Properties with Et(Ind)2HfCl2 (P4)/MAO
T (°C) 40 80
Al/Zr 500 2000 500 2000
Exp Model Exp Model Exp Model Exp Model
nM ×10-8
(g/mol) - 8.2808 - 8.0613 - 1.3440 - 0.5421
wM ×10-8
(g/mol) 22.1 20.8312 17.7 16.1219 2.9 2.6891 2.0 1.0805
PDI 1.83 2.5156 1.85 1.9999 2.78 2.0008 2.18 1.9931
vf (%) - 81.20 - 80.50 - 86.10 - 83.90
bf (%) - 6.30 - 5.40 - 4.60 - 3.90
if (%) - 12.50 - 14.10 - 9.30 - 12.2
170
0 10 20 30 40 50 60
0
3
6
9
12
Active
ca
taly
st site
s (m
ole
s/L
)
Time (minutes)
T
40 °C
80 °C
Figure 4.58 Active catalyst site concentration vs. time.
[Catalyst (P4) = 10 μM, Al/Hf = 500 and P = 30 psi]
0 10 20 30 40 50 60
0.0
2.0x10-3
4.0x10-3
6.0x10-3
8.0x10-3
Hyd
rid
e a
ctiva
ted
co
mp
lex (m
ole
s/L
)
Time (minutes)
Figure 4.59 Hydride actived complex concentration vs. time.
[Catalyst (P4) = 10 μM, Al/Hf = 500, T = 40 °C and P = 30 psi]
171
0 10 20 30 40 50 60
0.0
0.5
1.0
1.5
2.0
Hyd
rid
e a
ctiva
ted
co
mp
lex (m
ole
s/L
)
Time (minutes)
Figure 4.60 Hydride actived complex concentration vs. time.
[Catalyst (P4) = 10 μM, Al/Hf = 500, T = 80 °C and P = 30 psi]
With P4/MAO system, chain transfer to cocatalyst is obtained very high at 40 °C than
that at 80 °C. Concentration of methyl activated complex 0*
MeP is found to reach a maximum
of 0.4011 μM (17.3 min) followed by a steep decrease at 40 °C as compared to a maximum of
1.8862×10-5
μM (1.4 min) at 80 °C followed by slow decrease as shown in Figure 4.61 and
Figure 4.62 respectively. Due to low concentrations of monomer and methyl activated complex
at 80 °C, the rectivation rate, after transfer to cocatalyst is slow and therefore the concentration
of 0*
MeP is decreasing slowly. At 40 °C, high concentration of methyl activated complex
enhances the reactivation rate for which a rapid decrease in 0*
MeP is observed after reaching a
maximum.
172
0 10 20 30 40 50 60
0.0
0.2
0.4
Me
thyl a
ctiva
ted
co
mp
lex (m
ole
s/L
)
Time (minutes)
Figure 4.61 Methyl actived complex concentration vs. time.
[Catalyst (P4) = 10 μM, Al/Hf = 500, T = 40 °C and P = 30 psi]
0 10 20 30 40 50 60
0.0
5.0x10-6
1.0x10-5
1.5x10-5
2.0x10-5
Me
thyl a
ctiva
ted
co
mp
lex (m
ole
s/L
)
Time (minutes)
Figure 4.62 Methyl actived complex concentration vs. time.
[Catalyst (P4) = 10 μM, Al/Hf = 500, T = 80 °C and P = 30 psi]
173
For P4/MAO system, model predicted molecular weight distribution and percentage of vinyl-,
butenyl- and isobutyl-terminated chains are given in Table 4.17. A good match between
experimental and predicted results for MWD is obtained.
Polymerization rates obtained with this catalyst system are comparable with P3/MAO but
very less when compared with P1/MAO and P2/MAO. P4/MAO yielded very high molecular
weights (~ 108 g/mol) of polypropylene as compared to its zirconium analogue (P2) as well as
other catalysts (P1 and P3) discussed heretofore.
Molecular weight is found to decrease with increase in temperature (from 40 °C to 80 °C)
and also with increase in Al/Hf molar ratio (from 500 to 2000). At 80 °C, chain termination via
β-H elimination is highly dominating followed by the transfer to monomer, than that at 40 °C,
which causes a significant drop in molecular weight at higher temperature. Polydispersity index
predicted by the model for Al/Hf ratio of 500 at 40 °C is 2.5156, whereas for all other reaction
conditions it is around 2.0.
At all the temperatures and Al/Hf ratios considered, the highest percentage of terminated
chains hold vinylidene end group ( vf > 80%), followed by isobutyl end group (9.3 < if < 14.1)
and butenyl end group (3.9 < bf < 6.3) as shown in Table 4.17. At a fixed Al/Hf molar ratio,
with increase in temperature, vf is increasing whilst bf and if are found decreasing. Increase in
Al/Hf ratio increases the rate of chain transfer to cocatalyst and thereby if increases at a given
temperature.
Though these trends for P4/MAO are consistent with those catalyst systems discussed
earlier but all the termination rates are notably less, due to which high molecular weight product
is raised.
174
Effect of Pressure
Figure 4.63 and Figure 4.64 show the effect of monomer pressure on polymerization rate at 40
°C and 80 °C respectively with P4/MAO system at fixed Zr = 10 μM, Al/Zr = 500. For a change
in monomer pressure of 30 psi to 45 psi, at 40 °C, Rp,max increases from 0.1720 mol/L/s to 0.3186
mol/L/s and at 80 °C, Rp,max increases from 0.07838 mol/L/s to 0.1176 mol/L/s. This indicates
that an increase in monomer pressure results in linear increase in polymerization rate. As
observed experimentally and discussed earlier, this catalyst system affords lower polymerization
rates at 80 °C with an Al/Hf molar ratio of 500.
With increase in pressure, molecular weights are found to increase. 2.92 fold increase in
wM for a change in pressure from 15 to 30 psi and 1.86 fold increase for a change from 30 to 45
psi is found at 40 °C (Figure 4.65). Similarly, at 80 °C, 1.94 fold increase in wM for a change in
pressure from 15 to 30 psi and 2.82 fold increase for a change from 30 to 45 psi is observed
(Figure 4.66).
Effect of catalyst concentration
Polymerization rates at different catalyst concentrations (10, 20, 30, 40 and 50 μM) at 40 °C and
80 °C with P4/MAO system are shown in Figure 4.67 and Figure 4.68 respectively.
Polymerization rate is linearly increasing with increase in catalyst concentration at both the
temperatures and in tune with the discussion for P3/MAO system.
With P4/MAO catalyst system also, an inverse relationship between polymer molecular
weight and catalyst concentration is obtained as shown in Figure 4.69 and Figure 4.70.
175
0 10 20 30 40 50 60
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
45 psi
30 psi
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
15 psi
Figure 4.63 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P4) = 10 μM, Al/Hf = 500 and T = 40 °C]
0 10 20 30 40 50 60
0.00
0.05
0.10
0.15
45 psi
30 psi
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
15 psi
Figure 4.64 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P4) = 10 μM, Al/Hf = 500 and T = 80 °C]
176
15 30 45
0.0
1.5x109
3.0x109
4.5x109
Mo
lecu
lar
we
igh
t (g
/mo
l)
Pressure (psi)
Mn
Mw
Figure 4.65 Effect of pressure on average molecular weights.
[Catalyst (P4) = 10 μM, Al/Hf = 500 and T = 40 °C]
15 30 45
0.0
1.5x108
3.0x108
4.5x108
Mo
lecu
lar
we
igh
t (g
/mo
l)
Pressure (psi)
Mn
Mw
Figure 4.66 Effect of pressure on average molecular weights.
[Catalyst (P4) = 10 μM, Al/Hf = 500 and T = 80 °C]
177
At 40 °C, wM is decreasing from 2.0831 × 109
to 1.757 × 109
(15.6%) for a change in catalyst
concentration from 10 μM to 20 μM. On further increase in catalyst concentration (30, 40 and 50
μM), wM is not observed to decrease much. This outcome again emphasizes that at this
temperature, Al/Hf ratio of 500 is not sufficient to activate catalyst efficaciously and therfore
increase in catalyst concentration does not really increase the number of active chains. On the
contrary, at 80 °C wM is found to decrease with each increment in catalyst concentration. As
shown in Figure 4.70, for every 10 μM increase in catalyst concentration, 44% (10-20 μM),
28.1% (20-30 μM), 19.95% (30-40 μM) and 17.94% (40-50 μM) decrease in wM is obtained.
0 10 20 30 40 50 60
0.00
0.25
0.50
0.75
1.00
[Zr]
50M
10MPo
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
40M
30M
20M
Figure 4.67 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Hf = 500, T = 40 °C and P = 30 psi]
178
0 10 20 30 40 50 60
0.0
0.1
0.2
0.3
0.4
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
[Zr]
50M
10M
40M
30M
20M
Figure 4.68 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Hf = 500, T = 80 °C and P = 30 psi]
0 10 20 30 40 50
0.0
5.0x108
1.0x109
1.5x109
2.0x109
2.5x109
Mo
lecu
lar
we
igh
t (g
/mo
l)
[Hf] (M)
Mnbar
Mwbar
Figure 4.69 Effect of catalyst concentration on average molecular weights.
[Al/Hf = 500, T = 40 °C and P = 30 psi]
179
0 10 20 30 40 50
0.0
1.0x108
2.0x108
3.0x108
Mo
lecu
lar
we
igh
t (g
/mo
l)
[Hf] (M)
Mnbar
Mwbar
Figure 4.70 Effect of catalyst concentration on average molecular weights.
[Al/Hf = 500, T = 80 °C and P = 30 psi]
4.2.5 Propylene polymerization with (2,4,6-Me3Ind)2ZrCl2 (P5)/MAO
Propylene polymerization model is applied to solution phase polymerization of propylene with
an ansa-metallocene catalyst and kinetic parameters are obtained. Data for the model validation
were taken from Yasin et al. (2004).
Estimated parameters and effect of Al/Zr mole ratio
Model predicted and experimental polymerization rate profiles for propylene polymerization
with [2,4,6-Me3Ind]2ZrCl2(P5)/MAO catalyst system at Al/Zr molar ratios of 2000 and 4000 are
presented in Figure 4.71. Kinetic parameters are estimated at 0 °C with experimental rate data at
0.98 atm pressure, 20 μM catalyst and Al/Zr molar ratio of 2000. Experimental rate data at Al/Zr
mole ratio of 4000 are employed to verify the model prediction for polymerization rate. Model
180
predictions of polymerization rate are in good agreement with the experimental polymerization
rate at both the cocatalyst to catalyst mole ratios considered. Values of estimated kinetic
parameters and objective function F(k) are given in Table 4.18.
When compared with P1-P4/MAO systems, polymerization rates with P5/MAO system are
found very less, so is the pk value estimated by the model. At 0 °C, Low polymerization rates
(Rp,max = 2.593 × 10-3
moles/L/s) at Al/Zr mole ratio of 2000 and high rates (Rp,max = 4.921× 10-3
moles/L/s) at Al/Zr mole ratio of 4000 are echoing the trend obtained with other catalyst
systems. Also a sharp decrease in polymerization rate after reaching a maximum is seen at high
Al/Zr ratio implying a larger frequency of chain transfer to cocatalyst.
0 10 20 30 40 50 60
0.0
2.0x10-3
4.0x10-3
6.0x10-3
Zr:MAO
1:2000
1:4000
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
Figure 4.71 Effect of Al/Zr mole ratio on propylene polymerization rate; solid lines are
model predictions.
[Catalyst (P5) = 20 μM, T = 0 °C and P = 0.98 atm]
181
Table 4.18 Estimated Parameters for [2,4,6-Me3Ind]2ZrCl2 (P5)/MAO
T 0 °C
ink (M-1
.s-1
) 7.3213 × 10-3
pk (M-1
.s-1
) 71.2665
dk (s-1
) 3.8988 × 10-4
tMk (M-1
.s-1
) 2.6484
Hk , (s-1
) 8.2268 × 10-3
rk (M-1
.s-1
) 628.343
sk (M-1
.s-1
) 8.8514 × 10-4
spk (M-1
.s-1
) 1.6957 × 10-4
sMk (M-1
.s-1
) 0.12125
Altk , (M-1
.s-1
) 7.9402 × 103
rAlk (M-1
.s-1
) 32.4558
F(k) (-) 0.27301
Table 4.19 Predicted Properties with [2,4,6-Me3Ind]2ZrCl2 (P5)/MAO
T (°C) 0
Al/Zr 2000 4000
Exp Model Exp Model
nM (g/mol) 3.02 × 104 2.6715 × 10
4 2.71 × 10
4 2.3537 × 10
4
wM (g/mol) 7.7 × 104 6.2516 × 10
4 6.77 × 10
4 5.4211 × 10
4
PDI 2.55 2.34 2.50 2.30
vf (%) - 81.2671 - 76.4243
bf (%) - 4.4303 - 4.8034
if (%) - 14.3026 - 18.7723
182
Active catalyst site concentration decreases to 0.03 μM from 20 μM in 8.6 minutes as shown in
Figure 4.72. Further, the values, ].[Mk in = 1.2792×10-2
s-1
vs. dk = 3.8988 × 10-4
s-1
suggest
that 2.96 % active sites are deactivating spontaneously and rest are utilized to initiate the chains.
With [2,4,6-Me3Ind]2ZrCl2 (P5)/MAO system, frequency of β-H elimination is found to
be 8.2268 × 10-3
(Table 4.16). Concentration of hydride activated complex reaches a maximum
of 1.4165×10-4 μM rapidly (in 4.6 minutes) and decrease thereafter as shown in Figure 4.73.
Fast reinitiation rate after β-H elimination ( rk = 628.343) causes the decrease in 0*
HP .
Figure 4.74 shows that the concentration of methyl activated complex reaches a
maximum of 8.5575 μM within 1.37 minutes followed by a sharp decrease to a negligible value.
The rate of chain transfer to cocatalyst and reactivation after transfer both are quite high with this
catalyst system at 0 °C.
0 10 20 30 40 50 60
0
5
10
15
20
25
Active
ca
taly
st co
nce
ntr
atio
n (m
ole
s/L
)
Time (minutes)
Figure 4.72 Active catalyst site concentration vs. time.
[Catalyst (P5) = 20 μM, Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
183
0 10 20 30 40 50 60
0.0
5.0x10-5
1.0x10-4
1.5x10-4
Hyd
rid
e a
ctive
d c
ata
lyst co
nce
ntr
atio
n (m
ole
s/L
)
Time (minutes)
Figure 4.73 Hydride actived complex concentration vs. time.
[Catalyst (P5) = 20 μM, Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
0 10 20 30 40 50 60
0
3
6
9
Me
thyl a
ctive
d c
ata
lyst co
nce
ntr
atio
n
(m
ole
s/L
)
Time (minutes)
Figure 4.74 Methyl actived complex concentration vs. time.
[Catalyst (P5) = 20 μM, Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
184
Table 4.19 presents the molecular weights of polypropylene and percentage of variously
terminated chains predicted by the model for [2,4,6-Me3Ind]2ZrCl2 (P5)/MAO system at 0 °C. At
Al/Zr mole ratio of 2000, wM is found to be 6.2516 × 104
which is decreased by 13.3% on
raising the ratio to 4000. Model predicted molecular weights and PDIs closely match with the
experimentally determined values. Predicted value of PDI is almost unchanged with an increase
in Al/Zr mole ratio, and is found to be 2.34 and 2.30 at an Al/Zr molar ratio of 2000 and 4000
respectively.
Chain termination is via spontaneous catalyst deactivation, transfer to monomer and β-Hydride
elimination remain in force followed by chain transfer to cocatalyst as inferred by vf and if
values obtained (Table 4.19). With increase in Al/Zr ratio, if is found to increase while vf
decreases, showing higher transfer to cocatalyst at higher ratio. Secondary insertions are found to
be less with this catalyst system and unresponsive to Al/Zr ratio as evident from small
percentage (4.43 - 4.8 %) of butenyl end groups obtained (Table 4.19).
Effect of Pressure
Monomer pressure is doubled and tripled from 0.98 atm (experimental) and polymerization rates
are prognosticated with the model as given in Figure 4.75. The maximum rate observed at 0.98
atm, 1.97 atm and 2.96 atm are 0.0022 moles/L/s, 0.0061 moles/L/s and 0.0133 moles/L/s
respectively at 0 °C and fixed Zr = 20 μM, Al/Zr = 2000. With increase in monomer pressure, a
higher initiation and polymerization rates are noted, which is consistent with trend observed
earlier with any catalyst system.
Figure 4.76 depicts that no appreciable increase in molecular weight is obtained with increase in
monomer pressure with this catalyst system. 0.9% and 0.4% increase in weight average
185
molecular weight is recorded for a change of 0.98 to 1.97 atm and from 1.97 to 2.96 atm in
monomer pressure. High monomer concentration not only increases the initiation and
propagation rates but also raises various chain termination rates in which monomer is involved.
Therefore eventually for this catalyst system, despite high polymerization rate no considerable
change in molecular weight is observed.
0 10 20 30 40 50 60
0.0
5.0x10-3
1.0x10-2
1.5x10-2
2.96 atm
1.97 atm
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
0.98 atm
Figure 4.75 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P5) = 20 μM, Al/Zr = 2000 and T = 0 °C]
186
1 2 3
0
3x104
6x104
8x104
Mo
lecu
lar
we
igh
t (g
/mo
l)
Pressure (atm)
Mn
Mw
Figure 4.76 Effect of pressure on average molecular weights.
[Catalyst (P5) = 20 μM, Al/Zr = 2000 and T = 0 °C]
Effect of catalyst concentration
Polymerization rates at different catalyst concentrations (10, 20, 40 and 80 μM) at 0 °C are
shown in Figure 4.77. Increase in catalyst concentration is resulting in increasead polymerization
rate which reaches to a maximum and decreases afterwards due to elevated termination rates. On
doubling the catalyst concentration, maximum polymerization rate is found to be doubled,
showing a linear proportional dependence. The trend is similar to those obtained with other
catalyst systems discussed earlier.
187
0 10 20 30 40 50 60
0.00
2.50x10-3
5.00x10-3
7.50x10-3
1.00x10-2
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
[Zr]
10M
80M
40M
20M
Figure 4.77 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
0 20 40 60 80
2x104
4x104
7x104
Mo
lecu
lar
we
igh
t (g
/mo
l)
[Zr] (M)
Mn
Mw
Figure 4.78 Effect of catalyst concentration on average molecular weights.
[Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
188
With P5/MAO catalyst system, average molecular weights are noticed to be unaltered with
increase in catalyst concentration as shown in Figure 4.78. An inverse relationship between
polymer molecular weight and catalyst concentration is obtained with hafnium based
metallocenes (P3 and P4), whereas Zr based metallocenes (P1 and P2) have shown a similar
trend as obtained for P5.
4.2.6 Propylene polymerization with [2,4,7-Me3Ind]2ZrCl2) (P6)/MAO
Propylene polymerization model is applied to solution phase polymerization of propylene with
an ansa-metallocene catalyst and kinetic parameters are obtained. Data for the model validation
were taken from Yasin et al. (2004).
Estimated parameters and effect of Al/Zr mole ratio
Figure 4.79 shows the model prediction for the experimental behaviour of [2,4,7-Me3Ind]2ZrCl2
catalyst in propylene polymerization for different Zr/MAO ratios. Experimental polymerization
rate data at Al/Zr molar ratio of 2000 are regressed to determine the kinetic parameters at 0 °C
and other sets of rate data are used to verify the simulation results. A good match between the
model predicted and experimental rate is obtained at all the Al/Zr ratios studied. Values of
estimated kinetic parameters and objective function [F(k)] are given in Table 4.20.
189
0 10 20 30 40 50 60
0.0
1.0x10-3
2.0x10-3
3.0x10-3
Zr:MAO
1:1000
1:2000
1:4000
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
Figure 4.79 Effect of Al/Zr mole ratio on propylene polymerization rate; solid lines are
model predictions.
[Catalyst (P6) = 20 μM, T = 0 °C and P = 0.98 atm]
Polymerization rates with P6/MAO are observed very less when compared with P1-
P4/MAO systems, but comparable with P5/MAO system at identical reaction conditions. The
pk value estimated by the model is slightly higher (106.06 M-1
.s-1
vs. 71.27 M-1
.s-1
) than
P5/MAO system. At Al/Zr mole ratio of 1000, polymerization rate is all less (Rp,max = 1.24 × 10-3
moles/L/s) than those observed at higher ratios. At Al/Zr ratio of 2000, polymerization rate is
found to reach a maximum of 2.9389 × 10-3
moles/L/s, but with further increase in Al/Zr ratio to
4000 polymerization rate is not observed to increase and yields a maximum rate of 2.6332 × 10-3
moles/L/s. This suggests that Al/Zr ratio as high as 4000 is sufficient to activate this catalyst and
higher ratios would bring no further increase in polymerization rate.
190
Table 4.20 Estimated Parameters for [2,4,7-Me3Ind]2ZrCl2) (P6)/MAO
T 0 °C
ink (M-1
.s-1
) 6.2848 × 10-3
pk (M-1
.s-1
) 1.0606 × 102
dk (s-1
) 3.4959 × 10-4
tMk (M-1
.s-1
) 1.0114
Hk , (s-1
) 9.8943 × 10-4
rk (M-1
.s-1
) 3.1658 × 102
sk (M-1
.s-1
) 1.0107 × 10-3
spk (M-1
.s-1
) 2.0210 × 10-5
sMk (M-1
.s-1
) 0.2694
Altk , (M-1
.s-1
) 5.4686 × 102
rAlk (M-1
.s-1
) 4.2644
F(k) (-) 0.92561
Table 4.21 Predicted Properties with [2,4,7-Me3Ind]2ZrCl2) (P6)/MAO
T 0 °C
Al/Zr 1000 2000 4000
Exp Model Exp Model Exp Model
nM (g/mol) - 4.9605 × 104 - 2.8396 × 10
4 - 2.9229 × 10
4
wM (g/mol) - 1.2994 × 105 - 7.8196 × 10
4 - 6.1089 × 10
4
PDI - 2.62 - 2.75 - 2.09
vf (%) - 86.4288 - 83.4627 - 77.8977
bf (%) - 6.2102 - 6.3333 - 6.4323
if (%) - 7.361 - 10.204 - 15.670
191
Active catalyst site concentration decreases from 20 μM to 0.0522 μM in 9.3 minutes as shown
in Figure 4.80. Further, the values, ].[Mk in = 2.1687×10-3
s-1
vs. dk = 3.4959 × 10-4
s-1
suggest
that 13.88 % active sites are deactivating spontaneously while rest are utilized to initiate the
chains.
With [2,4,7-Me3Ind]2ZrCl2 (P6)/MAO system, frequency of β-H elimination is found to
be 9.8943 × 10-4
(Table 4.18). Concentration of hydride activated complex reaches a maximum
of 3.0554×10-5 μM in 13 minutes and decrease thenceforth as shown in Figure 4.81. Due to
rapid reinitiation rate after β-H elimination ( rk =316.58 M-1
.s-1
), 0*
HP decreases after
reaching the maximum.
The rate of chain transfer to cocatalyst and reactivation after transfer both are quite high
with P6/MAO catalyst system at 0 °C. Altk , and rAlk values with this catalyst system are much
higher than those obtained with (P1-P4)/MAO but less than that obtained with P5/MAO system.
Figure 4.82 points that the concentration of methyl activated complex reaches a maximum of
11.84 μM within 3.58 minutes followed by a sharp decrease to a negligible value.
Table 4.21 gives the molecular weights of polypropylene and percentage of variously
terminated chains predicted by the model for [2,4,7-Me3Ind]2ZrCl2 (P6)/MAO system at 0 °C
and different Al/Zr ratios.
192
0 10 20 30 40 50 60
0
5
10
15
20
Active
ca
taly
st co
nce
ntr
atio
n
(m
ole
s/L
)
Time (minutes)
Figure 4.80 Active catalyst site concentration vs. time.
[Catalyst (P6) = 20 μM, Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
0 10 20 30 40 50 60
0.0
1.0x10-5
2.0x10-5
3.0x10-5
Hyd
rid
e a
ctiva
ted
co
mp
lex c
on
ce
ntr
atio
n
(m
ole
s/L
)
Time (minutes)
Figure 4.81 Hydride actived complex concentration vs. time.
[Catalyst (P6) = 20 μM, Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
193
0 10 20 30 40 50 60
0
3
6
9
12
Me
thyl a
ctiva
ted
co
mp
lex c
on
ce
ntr
atio
n
(m
ol/L
)
Time (minutes)
Figure 4.82 Methyl actived complex concentration vs. time.
[Catalyst (P6) = 20 μM, Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
Increase in Al/Zr molar ratio is resulting in decrease in molecular weights. At Al/Zr mole
ratio of 1000, wM is found to be 1.2994 × 105
which is decreased by 39.82% on raising the ratio
to 2000. On increasing Al/Zr ratio from 2000 to 4000, wM is further decreased by 21.88%.
Weight average molecular weights obtained with P6/MAO are very close to those obtained with
P5/MAO catalyst system at identical conditions, however, polydispersity indices are little high
with P6/MAO evincing broader molecular weight distribution.
vf values in Table 4.21 indicate that chain termination is majorly via spontaneous
catalyst deactivation, transfer to monomer and β-Hydride elimination. With increase in Al/Zr
ratio, if is found increasing while vf decreasing, exhibiting the trends similar to that discovered
with P5/MAO. Secondary insertions with P6/MAO catalyst system are higher than those with
194
P5/MAO but less in comparison with (P1-P4)/MAO systems. Percentage of butenyl end groups
are almost unchanged with variation in Al/Zr ratio.
Effect of Pressure
Like P5/MAO and other catalysts studied up to now, P6/MAO also evidences a higher initiation
rate and increased polymerization rate with increase in monomer pressure. Polymerization rates
are predicted with the model at 0.98, 1.97 and 2.96 atm monomer pressures as shown in Figure
4.83. The maximum rates observed at 0.98 atm, 1.97 atm and 2.96 atm are 0.0030 moles/L/s,
0.0081 moles/L/s and 0.0179 moles/L/s respectively at 0 °C and fixed Zr = 20 μM, Al/Zr = 2000.
Polymerization rates at studied monomer pressures are very comparable with those obtained with
P1/MAO system.
Figure 4.84 shows that a little increase in molecular weight is obtained with increase in monomer
pressure with P6/MAO catalyst system. In weight average molecular weight, a 5.18% and 2.97%
increase is seen respectively for a change of 0.98 to 1.97 atm and from 1.97 to 2.96 atm in
monomer pressure. The small increase in wM conforms to the effect of pressure discussed for
P5/MAO system.
195
0 10 20 30 40 50 60
0.0
6.0x10-3
1.2x10-2
1.8x10-2
2.96 atm
1.97 atmPo
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
0.98 atm
Figure 4.83 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P6) = 20 μM, Al/Zr = 2000 and T = 0 °C]
1 2 3
0
3x104
5x104
8x104
1x105
Mo
lecu
lar
we
igh
t (g
/mo
l)
Pressure (atm)
Mn
Mw
Figure 4.84 Effect of pressure on average molecular weights.
[Catalyst (P6) = 20 μM, Al/Zr = 2000 and T = 0 °C]
196
Effect of catalyst concentration
With P6/MAO also, polymerization rates are increasing linearly with increase in catalyst
concentration. Figure 4.85 shows polymerization rates at different catalyst concentrations (10,
20, 40 and 80 μM) at 0 °C. Since high initiation rates prevail at high catalyst concentrations,
maximum rate is achieved shortly at higher catalyst concentration.
Like P5/MAO catalyst system, average molecular weights with P6/MAO are also not changing
with increase in catalyst concentration as shown in Figure 4.86.
0 10 20 30 40 50 60
0.0
5.0x10-3
1.0x10-2
1.5x10-2
Zr
80M
40M
20M
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
10 M
Figure 4.85 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
197
0 20 40 60 80
2x104
4x104
6x104
8x104
1x105
Mo
lecu
lar
we
igh
t (g
/mo
l)
[Zr] (M)
Mn
Mw
Figure 4.86 Effect of catalyst concentration on average molecular weights.
[Al/Zr = 2000, T = 0 °C and P = 0.98 atm]
4.2.7 Propylene polymerization with Me2Si[2,4,6-Me3Ind]2ZrCl2) (P7)/MAO
Propylene polymerization model is applied to solution phase polymerization of propylene with
an ansa-metallocene catalyst and kinetic parameters are obtained. Simulations are carried out
using natural logarithmic differential evolution and the experimental data for the model
validation are taken from Yasin et al. (2005).
Estimated parameters and effect of temperature
Experimental and model predicted polymerization rate profiles for propylene polymerization
with Me2Si[2,4,6-Me3Ind]2ZrCl2(P7)/MAO catalyst system at 30 °C, 50 °C and 70 °C are
presented in Figure 4.87. Kinetic parameters are estimated at different temperatures with
198
experimental rate data at 0.98 atm pressure, 20 μM catalyst and Al/Zr molar ratio of 2000.
Estimated kinetic parameters and objective function F(k) values are given in Table 4.22.
A decent agreement between experimental and model predicted polymerization rates is
received at 30 °C and 70 °C. At 50 °C from 2.75 minutes to 12 minutes, very few experimental
data points are available and a large number of data is available for regression only after 12
minutes. For this reason, model is found to under predict the polymerization rate in this time
range which includes the peak corresponding to maximum polymerization rate. Due to possible
unreliableness, model outputs at 50 °C are excluded in further discussion.
Figure 4.87 shows that model predicted maximum polymerization rates at 30 °C and 70
°C are 7.103 × 10-3
(experimental: 8.38 × 10-3
) and 8.747.103 × 10-3
(experimental: 8.95 × 10-3
)
respectively which explicates that polymerization rate with P7/MAO catalyst system is increased
to a little extent with increase in temperature. Despite an increase in pk value with increase in
temperature, the low increase in polymerization rate may be explained with the decreased
solubility of propylene in toluene at higher temperature at fixed monomer pressure. Temperature
is rather found to affect initiation rates, which are increasing with increase in temperature and
thereby achieve maximum rate earlier. Chain termination rates via different routes are
sufficiently high at all the temperatures to induce a descending polymerization rate profile after a
maximum.
Both initiation rate ( ].[Mkin = 2.441×10-2
s-1
vs. 4.849×10-3
s-1
at 30 °C) and frequency of
spontaneous catalyst deactivation ( dk = 4.0949×10-4
s-1
vs. 6.1255×10-5
s-1
at 30 °C) are high at
70 °C. That's why active catalyst site concentration decreases to a negligible value within 3.5
minutes at 70 °C, whereas at 30 °C, it takes 19.8 minutes as shown in Figure 4.88.
199
0 10 20 30 40 50 60
0.0
2.0x10-3
4.0x10-3
6.0x10-3
8.0x10-3
1.0x10-2
30 0C
50 0C
70 0C
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
Figure 4.87 Effect of temperature on propylene polymerization rate; solid lines are model
predictions.
[Catalyst (P7) = 20 μM, Al/Zr = 2000 and P = 0.98 atm]
200
Table 4.22 Estimated Parameters for Me2Si[2,4,6-Me3Ind]2ZrCl2) (P7)/MAO
T (°C) 30 70
ink (M-1
.s-1
) 8.075708 × 10-3
9.985286 × 10-2
pk (M-1
.s-1
) 9.361976 × 102 2.211042 × 10
3
dk (s-1
) 6.125482 × 10-5
4.094922 × 10-4
tMk (M-1
.s-1
) 0.2562587 12.5457
Hk , (s-1
) 1.955819 × 10-4
2.307301 × 10-3
rk (M-1
.s-1
) 101.5226 407.3533
sk (M-1
.s-1
) 1.747163 × 10-3
1.255236 × 10-2
spk (M-1
.s-1
) 1.264525 × 10-4
1.523233 × 10-2
sMk (M-1
.s-1
) 1.33222 × 10-3
1.001232 × 103
F(k) (-) 0.5969 1.0339
Table 4.23 Predicted Properties with Me2Si[2,4,6-Me3Ind]2ZrCl2) (P7)/MAO
T (°C) 30 70
Exp Model Exp Model
nM (g/mol) - 3.9672 × 104 - 7.9313 × 10
3
wM (g/mol) - 7.9344 × 104 - 1.5823 × 10
4
PDI (-) - 2.000 - 1.995
vf (%) - 99.8648 - 99.9097
bf (%) 0.37 0.1352 0.10 0.0903
201
Fractional disappearance of active sites by spontaneous deactivation is as low as 1.24% at 30 °C
and 1.6% at 70 °C.
0 10 20 30 40 50 60
0
5
10
15
20
Active
ca
taly
st site
s (m
ole
s/L
)
Time (minutes)
T
30 0C
70 0C
Figure 4.88 Active catalyst site concentration vs. time.
[Catalyst (P7) = 20 μM, Al/Zr = 2000 and P = 0.98 atm]
Concentration of hydride activated complex 0*
HP is found to increase to a maximum
followed by a decrease. Maximum 0*
HP achieved is 4.112×10-5
μM (in 6.4 minutes) at 30 °C
and 3.367 μM (in 5.85 minutes) at 70 °C as shown in Figure 4.89 (a) and (b) respectively. With
P7/MAO catalyst system, frequency of β-H elimination is substantial and increases with increase
in temperature from 30 °C to 70 °C. Reinitiation rate after β-H elimination is much lower at 70
°C than that at 30 °C due to which a high value of maximum concentration of hydride activated
complex is obtained.
202
0 10 20 30 40 50 60
0.0
2.0x10-5
4.0x10-5
Hyd
rid
e a
ctiva
ted
ca
taly
st site
s
(m
ole
s/L
)
Time (minutes)
(a)
0 10 20 30 40 50 60
0
1
2
3
Hyd
rid
e a
ctiva
ted
ca
taly
st site
s
(m
ole
s/L
)
Time (minutes)
(b)
Figure 4.89 Hydride actived complex concentration vs. time. (a) 30 °C, (b) 70 °C
[Catalyst (P7) = 20 μM, Al/Zr = 2000 and P = 0.98 atm]
203
Model predicted properties of polypropylene synthesized with Me2Si[2,4,6-Me3Ind]2ZrCl2)
(P7)/MAO are given in Table 4.23. Molecular weights are found to decrease with increase in
temperature. wM estimated at 30 °C is 7.9344 × 104 which is decreased by 80% at 70 °C.
PDIs
at all reaction conditions are obtained almost 2.0 representing standard molecular weight
distribution.
Polypropylene synthesized using P7/MAO system is highly isotactic. Experimental
values of bf , which are closely predicted by the model also (Table 4.23) suggest that the fraction
of dead chains with butenyl end group is very less (< 0.5%) speculating high isotacticity. Since
transfer to cocatalyst is not considered in the model applied here, high vf values account for all
other types of chain termination i.e. via spontaneous catalyst deactivation, transfer to monomer
and β-Hydride elimination.
Effect of Pressure
Monomer pressure is varied from 0.98 atm to 1.97 atm and 2.96 atm to study the effect on
polymerization rate and polymer molecular weight at fixed Zr = 20 μM, Al/Zr = 2000. Figure
4.90 and Figure 4.91 depict the rate profiles at 30 °C and 70 °C respectively. Following the
earlier trends, polymerization rate is found increasing with increase in monomer pressure.
With increase in monomer pressure a linear increase in polymerization rate with a
maximum of 8.61142 × 10-3
moles/L/s (0.98 atm), 1.9996 × 10-2
moles/L/s (1.97 atm) and
3.3598 × 10-2
moles/L/s (2.96 atm) at 30 °C is observed as shown in Figure 4.90. At 70 °C also
a linear relationship between rate and monomer pressure is seen, with a maximum rate of 9.0595
× 10-3
moles/L/s (0.98 atm), 2.4396 × 10-2 moles/L/s (1.97 atm) and 4.0954 × 10
-2 moles/L/s
(2.96 atm). Higher rates are predicted at 70 °C at corresponding monomer pressures.
204
No appreciable change in average molecular weights with variation in monomer pressures is
observed at both 30 °C (Figure 4.92) and 70 °C (Figure 4.93) temperature.
0 10 20 30 40 50 60
0.0
1.0x10-2
2.0x10-2
3.0x10-2
2.96 atm
1.97 atm
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
0.98 atm
Figure 4.90 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P7) = 20 μM, Al/Zr = 2000 and T = 30 °C]
0 10 20 30 40 50 60
0.0
1.0x10-2
2.0x10-2
3.0x10-2
4.0x10-2
2.96 atm
1.97 atm
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
0.98 atm
Figure 4.91 Polymerization rate vs. time: Effect of pressure.
[Catalyst (P7) = 20 μM, Al/Zr = 2000 and T = 70 °C]
205
1 2 3
4.0x104
6.0x104
8.0x104
Mn
Mw
Mo
lecu
lar
we
igh
t (g
/mo
l)
Pressure (atm)
Figure 4.92 Effect of pressure on average molecular weights.
[Catalyst (P7) = 20 μM, Al/Zr = 2000 and T = 30 °C]
1 2 3
5x103
1x104
2x104
2x104
Mo
lecu
lar
we
igh
t (g
/mo
l)
Pressure (atm)
Mn
Mw
Figure 4.93 Effect of pressure on average molecular weights.
[Catalyst (P7) = 20 μM, Al/Zr = 2000 and T = 70 °C]
206
Effect of catalyst concentration
Polymerization rates at different catalyst (P7) concentrations at 30 °C and 70 °C are shown in
Figure 4.94 and Figure 4.95 respectively. On doubling the catalyst concentration, predicted
maximum polymerization rate is found to be doubled showing a linear dependence at 30 °C as
well as at 70 °C. However at 70 °C, polymerization rate is noted to be 23% higher than that at 30
°C at corresponding catalyst concentration. All the termination reactions are highly activated at
70 °C, therefore polymerization rate is decreasing steeply after reaching a maximum at all
catalyst concentrations.
Frequency of β-H elimination is calculated to be negligibly low as compared to the chain transfer
to monomer ( Hk , vs. MktM in Table 4.22) at both the temperatures considered. For this
reason, average molecular weights are unchanged with increase in catalyst concentration as
shown in Figure 96 and Figure 97.
0 10 20 30 40 50 60
0.0
1.0x10-2
2.0x10-2
3.0x10-2
[Zr]
80M
40M
20M
Po
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
10M
Figure 4.94 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Zr = 2000, T = 30 °C and P = 0.98 atm]
207
0 10 20 30 40 50 60
0.0
1.0x10-2
2.0x10-2
3.0x10-2
4.0x10-2
[Zr]
80M
40M
20MPo
lym
eri
za
tio
n r
ate
(m
ole
s/L
/s)
Time (minutes)
10M
Figure 4.95 Polymerization rate vs. time: Effect of catalyst concentration.
[Al/Zr = 2000, T = 70 °C and P = 0.98 atm]
0 20 40 60 80
4.0x104
6.0x104
8.0x104
Mo
lecu
lar
we
igh
t (g
/mo
l)
[Zr] (M)
Mn
Mw
Figure 4.96 Effect of catalyst concentration on average molecular weights.
[Al/Zr = 2000, T = 30 °C and P = 0.98 atm]
208
0 30 60 90
5x103
1x104
2x104
2x104
Mo
lecu
lar
we
igh
t (g
/mo
l)
[Zr] (M)
Mn
Mw
Figure 4.97 Effect of catalyst concentration on average molecular weights.
[Al/Zr = 2000, T = 70 °C and P = 0.98 atm]
Summary of the chapter: Models developed for ethylene and propylene polymerization in
previous chapter have simulated and results were discussed in this chapter. Gas phase and
solution phase ethylene polymerization with silica-supported, bridged zirconocene catalysts were
discussed first. Solution phase propylene polymerization with various bridged/unbridged
zirconocene and hafnocene catalysts were discussed in ulterior sections. Kinetic parameters were
determined for each catalyst system with 'natural logarithmic differential evolution' approach of
optimization.
209
CHAPTER – 5
CONCLUDING REMARKS
Kinetic models for olefin polymerization with metallocene catalysts were developed and
validated with experimental data. Kinetic model parameters were estimated using novel
natural logarithmic differential evolution approach of optimization. Parametric studies
were carried out to investigate the effect of various reaction conditions on polymerization
kinetics and polymer properties. This chapter presents a brief summary of the work
followed by conclusions, major contributions and future scope for research in this area.
5.1 Summary
5.1.1 Introduction
Polyolefins are the largest group of thermoplastics, often referred to as commodity
thermoplastics. Two most important and common polyolefins are polyethylene and
polypropylene and they are very popular due to their low cost and wide range of
applications. Metallocene catalyzed olefin polymerization has recently attracted research
interest since these catalysts allow the production of tailored macromolecules with
properties that can be accurately designed due to their single types of sites. Kinetic
studies of catalytic polymerization provide considerable insight into the mechanism of the
reactions and scale-up or commercialization of a polymerization process staggeringly
depends on the understanding of the kinetic behavior of the system under various
operating conditions.
Many problems encountered in industrial polymerization processes are associated
with inherent complexities in polymerization kinetics and mechanisms, physical changes
and transport effects, non-ideal mixing and conveying, and strong process nonlinearity.
210
Mathematical modeling is a powerful tool not only for the development of process
understanding, but also for the design of advanced process technology. In particular, a
kinetic model plays an important role in designing polymerization conditions to tailor a
polymer’s molecular architecture. Valid kinetic rate constants are required for calculating
polymerization rates and polymer properties. A comprehensive kinetic study of
polymerization process helps developing effective models at meso- and macro-levels.
Therefore, this study is focused on the estimation of kinetic parameters and prediction of
polymer properties through modeling at micro-level. The two most representative
objectives in modeling polymerization reactions are to compute polymerization rate and
polymer properties (molecular level and microscopic level) for various reaction
conditions.
Determining the parameters of a kinetic model by using laboratory, pilot plant, or
plant data is the most critical step for the successful development of a process model. It is
not always possible to design experiments to determine all the relevant kinetic
parameters. Therefore, in modern kinetic modeling, pseudo-rate constant methods and
computer aided parameter estimation techniques are widely used.
In transition metal catalyzed olefin polymerizations, the kinetic parameters are
catalyst dependent. Therefore, whenever a new catalyst is employed, a new set of kinetic
parameters must be determined. Considering the fact that the properties of polyolefins are
mostly dictated by the nature of catalyst being used and that a large number of different
types of catalysts are used for different polymer grades, it becomes very important to
have a well-established parameter estimation procedure that can be applied to any catalyst
systems.
211
5.1.2 Gaps in research
The existing literature on polymerization with metallocene catalyst systems suggests that
efforts have been made in understanding the mechanisms and work performance of
metallocene based catalyst systems. These systems allow tailor making of polymers and
offer other process advantages such as ease of handling of metallocene systems and
favourable conditions for polymerization from a commercial point of view, which has
evoked the examination of the commercial potential of such catalyst systems. The
commercial exploitation of such systems has, however, started in a limited way due to
prohibitive cost of the catalyst and the ambiguity associated with the aluminoxanes (co-
catalysts).
Various homo- and copolymers have been synthesized using metallocene catalyst
systems and most of the work is of experimental nature at either laboratory scale or the
pilot scale with more or less common objectives like investigating catalyst activity,
product properties and effect of parameters thereon.
Very little attempts have been made in modeling and simulation related studies for the
polymerization process. Majorly Z-N catalyzed polymerization of olefins was on focus
for modeling the morphological and transport related phenomena. Modeling efforts on
olefin polymerization with metallocene catalysts are as less as negligible when compared
to the other catalyst systems. Further, kinetic modeling and simulation of metallocene
catalyzed olefin polymerization is in its dissilient stage and provides a huge opportunity
to address the understanding of kinetics comprehensively.
5.1.3. Scope of the work
Significant development in the synthesis of new metallocenes and co-catalysts is
anticipated in near future leading to tailor-made polymers, including functionalized
polyolefins with predictable properties. In view of this, it is imperative to model the
212
polymerization of olefins involving different metallocene catalyst systems. A lot of scope
exists for theoretical as well as computation studies on metallocene catalyzed olefin
polymerization and hence developing a kinetic model and simulation of the same
unquestionably is a task not only for research but also of industrial importance.
In this work, the mechanistic aspects of Ziegler-Natta and metallocene catalyst
systems have been studied in detail and used in building up mathematical models for
ethylene and propylene polymerization using metallocene catalysts. Developed models
are validated with the experimental data available in literature and kinetic parameters are
estimated using differential evolution (DE) approach of optimization. Study on the effects
of various parameters like monomer concentration, polymerization temperature, catalyst
concentrations, and cocatalyst to catalyst molar ratio etc. upon rate of polymerization,
molecular weights and poly dispersity index and stereoregularity is carried out.
The outcomes of this study will help in better understanding of the chemistry and
process of the olefinic polymerization with these revolutionary catalyst systems.
5.1.4. Model development and simulation
In this study, mathematical models for metallocene catalyzed ethylene and propylene
polymerization are developed on a first principles basis and have been validated with
experimental data. These models may be used as a surrogate of the real olefin
polymerization process where the use of actual process may be costly or inopportune.
Comprehensive kinetic models consisting of mass and population balance equations, are
developed based on elementary reactions proposed in the reaction mechanism.
Founded on the interpretations of mechanisms for metallocene-catalyzed
polymerization, an ecumenical reaction set for ethylene and propylene polymerization
that includes reactions corresponding to all types of metallocenes, is proposed. Thereafter,
mathematical models for ethylene and propylene polymerization in a batch/semi-
213
batch/constant stirred tank reactor are built up based on the reactions considered. The
models are capable of predicting polymerization rate, and polymer properties (viz.
number-average- & weight-average molecular weight and PDI) in general. In addition,
mole fraction of dead polymer chains with terminal double bond and number of long-
chain branches & short-chain branches per 103 carbon atoms may be determined with
ethylene polymerization model. And fraction of vinyl-terminated chains, butenyl-
terminated chains, isobutyl-terminated chains and vinylidene-terminated chains relative to
the total unsaturated termination may be ascertained with propylene polymerization
model.
Model equations developed include a set of coupled, nonlinear and stiff ordinary
differential equations (ODEs) for the dynamic polymerization. To estimate the kinetic
parameters and to study the effect of parameters, these ODEs are solved with ODE-15s
function provided MATLAB™
7.0.1 software (MATLAB version 7.0.1, 2004).
Various established methods that are being used as parameter estimation
techniques, such as the graphical method and the gradient-based non-linear optimization
method either do not have precision to calculate the parameters or are easily get trapped
into local optima. In this study, a novel natural logarithmic differential evolution (NLDE)
approach of optimization, a remediated version of differential evolution algorithm (Price
and Storn, 1997) is proposed and used to solve parameter estimation problem. Proposed
NLDE algorithm is capable of handing multiple objectives simultaneously, providing
room to admit objective functions based on polymerization rate, molecular weights, PDI,
fraction of dead polymer chains with terminal double bond, fraction of vinyl-terminated
chains, butenyl-terminated chains, isobutyl-terminated chains and vinylidene-terminated
chains etc. if experimental data are available.
214
5.1.5 Results and discussion
In the following sections, the simulation results obtained for the ethylene and propylene
polymerization using different metallocene catalyst systems are summarized.
5.1.5.1 Ethylene polymerization
A. Silica supported (Me2Si[Ind]2ZrCl2)/MAO
Ethylene polymerization model is applied to gas phase polymerization of ethylene with
silica-supported Me2Si[Ind]2ZrCl2 catalyst and kinetic parameters are obtained.
Simulations are carried out analytically as well as numerically (with ODE-15s function
provided in MATLAB™ 7.0 software) using natural logarithmic differential evolution
approach of optimization to estimate the kinetic parameters. Predicted polymerization
rates exhibit a good agreement with experimental data, at all the temperatures (40 °C, 50
°C, 60
°C and 70
°C). Estimated kinetic parameters and objective function values F(k) are
shown in Table 4.4. Close range of objective function values (from 0.29934 to 0.6735)
obtained, shows good fit with experimental observations. Rates of initiation and
propagation are increasing with increase in temperature, as inferred from the estimated
rate constants for these reactions. For an increase of 10 °C, from 40 °C to 70 °C, the
weight average molecular weight ( wM ) is found to decrease by 67.40 %, 70.47 % and
81.67 % respectively. Polydispersity indices of polyethylene prepared with
Me2Si[Ind]2ZrCl2/MAO catalyst system are found to be 1.999 irrespective of temperature.
Polymerization rate is linearly increasing with ethylene pressure at all the
temperatures as shown in Figures 4.2. Low pressures (1-3 bar), i.e., low concentrations of
monomer, reasons low rates that are steady and maintained (due to negligible transfer
reaction). At higher pressures (5-7 bar), higher polymerization rates are obtained, but at
215
the same time, transfer to monomer also increases with high monomer concentrations
resulting in steeper decay in polymerization rates.
A steady increase in polymerization rate with increase in catalyst amount at
constant temperature and pressure is observed. At higher temperatures, the initiation and
termination rates also increase staggeringly with catalyst amount. Molecular weights and
PDI are not appreciably affected by changing catalyst amount.
B. In-situ-supported Et[Ind]2ZrCl2 (E2)/MAO
Ethylene polymerization model is applied to solution phase polymerization of ethylene
with in-situ-silica supported Et[Ind]2ZrCl2 catalyst and kinetic parameters are obtained. A
very large population size (120 times the dimension) is used to make certain of receiving
optimized estimates of parameters. Table 4.7 summarizes the parameters estimates at 40°,
60°, 80
°, 100
° and 120
° C with F(k) values. The model predictions of polymerization rate
obtained at different temperatures, are real close to the experimental values. F(k) values
obtained are ranging closely, with a least value of 1.1186 at 80 °C representing the best fit
to experimental observations as compared to highest value of 1.6063 at 120 °C.
Significantly lower values of propagation rate constants at 40 °C and 60
°C are obtained
relating to very low polymerization rate and catalyst activity with respect to those at
higher temperatures. At 80 °C, high propagation rate but lower deactivation and touching
transfer rates as compared to those at lower temperatures trace higher polymerization rate
with highest activity of the catalyst. At 40 °C, 60
°C and 80
°C, all the active catalyst sites
were occupied within initial 15 minutes, whereas at higher temperatures certain fraction
of those could not attach a monomer to initiate the chain.
Experimental kinetic data at different operating conditions, like different catalyst
amount, ethylene pressure and cocatalyst to catalyst mole ratio, were utilized to validate
216
the model at fixed temperature of 60 °C. Parameters estimated at 60
°C, 80 psig and
6 μmol catalyst amount with Al/Zr = 500 were used to verify the model responses at
various conditions.
The model predicts a proportional change in polymerization rate at different
ethylene pressures vindicating first order dependence of rate on ethylene concentration.
At 40 psig pressure, active catalyst sites stayed available for all the polymerization time
suggesting the non-initiation of some active site. At 80 psig pressure, active site were
occupied within 10 minutes and for higher pressures within 20 minutes.
A good match between experimental rate observations and model predictions at 3,
12 and 18 μmol of initial catalyst amount taken is obtained. The model adequately
captured the features of polymerization with in-situ-supported metallocene catalyst by
following the sustained polymerization rate with time and increase in the same with the
increase in catalyst amount. All the active catalyst sites were occupied within 10 minutes,
irrespective of initial amount of catalyst used.
Use of high Al/Zr ratio brings in higher polymerization rate and for the entire
range of ratios. For lower ratios (250 and 500), active catalyst sites disappeared within
first 10 minutes, whereas for higher ratios (above 500) these decreased with time but
remained available for entire polymerization time.
Average molecular weights of polyethylene obtained from the model are found to
be decreasing with increase in temperature, whereas the change in catalyst amount,
cocatalyst to catalyst mole ratio or ethylene pressure brought insignificant effect.
Estimated parameters indicate that chain transfer to monomer is dominating over
other transfer reactions, for which degree of polymerization and so the molecular weight
show up to be independent of monomer concentration. Polydispersity in all the cases are
obtained very close to 2. As inferred from calculated fraction of dead chains with double
bond at the end, major modes of chain termination, nearly for all sets of conditions, are
217
believed to be chain transfer to monomer, chain transfer to cocatalyst and β-hydride
elimination. Long chain branching frequency is detected to be negligibly low, except only
for low (40 °C) temperature and high (120 and 160 psig) ethylene pressures, suggesting
that the product is comprising of linear chains and posseses high density.
5.1.5.2 Propylene polymerization
A. Me2Si[Ind]2ZrCl2 (P1)/MAO
Propylene polymerization model is applied to solution phase polymerization of propylene
with Me2Si[Ind]2ZrCl2/MAO catalyst system and kinetic parameters are obtained.
Kinetic parameters are estimated by simulating the model with experimental data
at 25 °C and 75 °C with Al/Zr molar ratio of 500. Experimental data for Al/Zr ratio of
2000 are used to validate the model at both temperatures. Kinetic parameters and F(k)
values are given in Table 4.10.
Experimental observations reveal that both, the temperature and MAO to catalyst
molar ratio have a significant effect on polymerization rate. Model predictions are in very
close agreement with experimental observations at both the temperatures (25 °C and 75
°C) and Al/Zr molar ratios of 500 & 2000. With an increase in Al/Zr molar ratio,
polymerization rate increases at both the temperatures considered. At very high Al/Zr
ratio, the decreasing polymerization rate after reaching a maximum is observed.
Maximum polymerization rate is seen to increase four folds at 75 °C when
compared with that at 25 °C, at fixed catalyst concentration, Al/Zr ratio and pressure.
Increase in propagation rate constant pk with temperature is also in coherence with this
observed fact. As understood by dk and tMk values obtained, spontaneous deactivation
and chain transfer to monomer are activated hugely with increase in temperature.
218
Ascribing to which, the polymerization rate is decreasing steeply at 75 °C after reaching a
maximum.
With P1/MAO catalyst system, β-H elimination is observed to occur negligibly
(Hk ,
of the order of 10-6
) as compared to all other modes of chain termination. Increase in
temperature enhances the frequency of β-H elimination.
Chain transfer to cocatalyst is found to be very high at 25 °C than that at 75 °C. At
75 °C, the rectivation rate, after transfer to cocatalyst is slow due to low concentrations of
monomer and methyl activated complex.
A good agreement between experimental observations and predicted results for
MWD is obtained The model predicts Schulz-Flory distribution with a polydispersity
index around 2 for all reaction conditions. The dominant presence of chains with
vinylidene end group over the chains with butenyl end group is predicted by the model
suggesting highly isotactic polypropylene. A significant chain transfer to cocatalyst is
predicted at 25 °C. Increase in temperature (75 °C) results in a decreased rate of chain
transfer to cocatalyst.
An increase in monomer pressure results in a steady increase in polymerization
rate up to a maximum ranging in between 0.685 moles/L/s (15 psi) to 2.056 moles/L/s (45
psi) at 25 °C and fixed Zr = 10 μM, Al/Zr = 500. At 75 °C, higher polymerization rates
are observed {2.489 moles/L/s (15 psi) to 8.487 moles/L/s (45 psi)}, which decline after
reaching a maximum. Polymer molecular weights are negligibly increased with pressure.
Polymerization rates at various catalyst concentrations (10, 20, 40 and 80 μM) at
25 °C and 75 °C show that on doubling the catalyst concentration, maximum
polymerization rate is almost doubled, expressing a linear proportional dependence.
Average molecular weights are found to be insensitive to catalyst concentration.
219
B. Et(Ind)2ZrCl2 (P2)/MAO
Propylene polymerization model is applied to solution phase polymerization of propylene
with Et(Ind)2ZrCl2/MAO catalyst system and kinetic parameters are obtained.
Model predictions of reaction rate capture the typical behavior of experimental
rate profiles for propylene polymerization with Et(Ind)2ZrCl2/MAO catalyst system.
Estimated kinetic parameters and F(k) values are given in Table 4.12.
Like P1/MAO catalyst system, with Et(Ind)2ZrCl2 (P2)/MAO also, β-H
elimination is not observed significant at both the temperatures considered. Chain transfer
to cocatalyst is found to be very high at Al/Zr = 2000 at both the temperatures (25 °C and
75 °C).
Model predictions are in very close correspondence with the experimental values
of weight average molecular weight and PDI. Effect of temperature on MWD is obtained
similar to that obtained for P1/MAO system, however P2 yields low molecular weight
product at matched reaction conditions. PDIs at all reaction conditions are obtained near
2.0.
For P2/MAO, chain termination takes place mainly via spontaneous catalyst
deactivation, transfer to monomer and β-Hydride and increasing with an increase in
temperature. With an increase in Al/Zr ratio chain transfer to cocatalyst is enhanced. Low
bf values obtained advise that the fraction of chains with butenyl end group is very less
as compared to others. bf is noted to decrease with increase in temperature as well as
Al/Zr mole ratio. A trend of decreasing rate of chain transfer to cocatalyst with increase
in temperature is observed consistent with that obtained with P1/MAO system.
With increase in monomer pressure an increase in polymerization rate up to a
stabilized maximum of 0.8808 moles/L/s (15 psi) and 2.678 moles/L/s (45 psi) at 25 °C
and fixed Zr = 10 μM, Al/Zr = 2000 is observed. At 75 °C, little higher polymerization
rates {Rpmax: 1.083 moles/L/s (15 psi) and 3.96 moles/L/s (45 psi)} with decreasing trend
220
are observed. The changes in average molecular weights with variation in pressures at 25
°C and at 75 °C are trifling.
Like Me2Si[Ind]2ZrCl2 (P1), with this catalyst system also, polymerization rate is
found to be linearly dependent on the catalyst concentration. However P2 offers a lower
polymerization rates than P1 under similar reaction conditions. Additionally, average
molecular weights are observed to be unaffected by catalyst concentration, which is to be
attributed to very low values of Hk ,
, as compared to other transfer rate constants. Model
prediction suggests that catlyst P2 produces lower molecular weight polypropylene when
compared to P1 for alike reaction conditions. The effect of catalyst concentration upon
rate and molecular weights is qualitatively similar to that obtained for catalyst P1.
C. Me2Si(Ind)2HfCl2 (P3)/MAO
Simulation results obtained with propylene polymerization model applied to solution
phase polymerization of propylene with Me2Si(Ind)2HfCl2/MAO catalyst system are
summarized here. Estimated kinetic parameters and objective function [F(k)] values are
provided in Table 4.14. A good agreement between the experimental data and model
predictions of polymerization rate with (P3)/MAO catalyst system at 40 °C (F = 1.5175)
and 80 °C (F = 0.3161) is found. Polymerization rate is observed to increase with increase
in temperature. At fixed catalyst concentration, Al/Hf ratio (2000) and monomer pressure,
maximum rate is found to be 0.4068 moles/L/s (29 minutes) at 40 °C against 0.9378
moles/L/s (7 minutes) at 80 °C. Hafnium based metallocene catalysts are known for
producing high molecular weight polymers in comparison to their zirconium analogues,
but at the expense of substantially reduced catalytic activity [Ewen et al. (1987);
Nakayama and Shiono (2005)].
221
With (P3)/MAO system, frequency of β-H elimination is found to increase from
9.2795×10-6
s-1
at 40 °C to 1.8881×10-4
s-1
at 80 °C. Chain transfer to cocatalyst is seen
significant at Al/Hf = 2000 at both the temperatures.
Model predictions for molecular weight distribution are in good agreement with
the experimental values. Effect of temperature on MWD is obtained qualitatively similar
to that observed for P1/MAO and P2/MAO systems. It is worth noting that (P3)/MAO
system yields a way higher molecular weight polypropylene when compared to its Zr
analogue i.e. P1/MAO.
Like other catalyst systems discussed so far, with P3/MAO also, the major
pathway of chain termination is via spontaneous catalyst deactivation, transfer to
monomer and β-Hydride elimination. Among these β-Hydride elimination is found
increasing dominantly whereas spontaneous deactivation and transfer to monomer are
seen to decrease a bit, with an increase in temperature. vf values are increased with
increase in temperature at a constant Al/Hf molar ratio, but there is a negligible increase
with an increase in Al/Hf ratio at a given temperature. Fraction of chains with butenyl end
group is very less as compared to others and is noticed to decrease with increase in
temperature as well as Al/Hf mole ratio. The trend of decreasing rate of chain transfer to
cocatalyst with increase in temperature is similar to that obtained with its Zr analogue
(P1/MAO system) but the amount of this termination is quite less with P3/MAO system.
Polymerization rate up to a maximum of 0.131 moles/L/s (15 psi),
0.380 moles/L/s (30 psi) and 0.693 moles/L/s (45 psi) at 40 °C and fixed Zr = 10 μM,
Al/Hf = 2000 is obtained. At 80 °C, maximum rate of 0.382 moles/L/s at 15 psi, 0.855
moles/L/s at 30 psi and 1.350 moles/L/s at 45 psi is obtained. A steady increase in
polymerization rate with an increase in monomer pressure is seen at both the
temperatures.
222
Effect of monomer pressure on molecular weight is found potential at lower temperature.
A significant increase of 71.6 % (15-30 psi) and 30.72 % (30-45 psi) in weight average
molecular weight wM is observed with increase in pressure at 40 °C. At 80 °C,
relatively lower increase of 18.8 % (15-30 psi) and 6.73 % (30-45 psi) in wM is observed
with increase in pressure.
Polymerization rates predicted at different catalyst concentrations (10, 20, 30, 40
and 50 μM) at 40 °C and 80 °C show that polymerization rate increases with increase in
catalyst concentration. Since high catalyst concentration promotes β-H elimination,
polymerization rate diminishes at a faster rate. This effect is found particularly loud at 80
°C due to very high Hk ,
value.
With P3/MAO catalyst system, average molecular weights are noticed to be
decreasing with increase in catalyst concentration and this effect is more influencing at 40
°C, where higher molecular weights are obtained. A decrease of 35.1% (Hf = 10-20 μM),
21.5% (Hf = 20-30 μM), 14% (Hf = 30-40 μM) and 9.1% (Hf = 40-50 μM) in wM at 40
°C is observed.
D. Et(Ind)2HfCl2 (P4)/MAO
Propylene polymerization model is applied to solution phase polymerization of propylene
with Et(Ind)2HfCl2 catalyst and kinetic parameters are obtained.
Kinetic parameters and objective function [F(k)] values obtained for P4/MAO system are
given in Table 4.16. Model predictions are in conformation with experimental
observations showing an increase in polymerization rate at both the temperatures (40 °C
and 80 °C) and Al/Hf molar ratios of 500 & 2000. With P4/MAO system, Rp,max is found
to increase from 0.5216 moles/L/s at 40 °C to 1.092 moles/L/s at 80 °C at Al/Hf molar
ratio of 2000, following the general trend. But at an Al/Hf ratio of 500, Rp,max is
discovered to decrease meagerly from 0.1952 moles/L/s at 40 °C to 0.1148 moles/L/s at
223
80 °C. This surprising drift indicates that Al/Hf ratio of 500 is too low to efficiently
activate this catalyst at 80 °C. Owing to this fact a lower value of propagation rate
constant (pk ) is obtained at 80 °C than that at 40 °C. Chain transfer to cocatalyst is
obtained much higher at 40 °C than that at 80 °C. A good match in experimental and
predicted results for MWD is obtained. Polymerization rates obtained with this catalyst
system are comparable with P3/MAO but very less when compared with P1/MAO and
P2/MAO.
P4/MAO yielded very high molecular weights (~ 108 g/mol) of polypropylene as
compared to its zirconium analogue (P2) as well as other catalysts (P1 and P3) discussed
heretofore. Molecular weight is found to decrease with increase in temperature (from 40
°C to 80 °C) and also with increase in Al/Hf molar ratio (from 500 to 2000). At 80 °C,
chain termination via β-H elimination is highly dominating followed by the transfer to
monomer, than that at 40 °C, which causes a significant drop in molecular weight at
higher temperature. Polydispersity index predicted by the model for Al/Hf ratio of 500 at
40 °C is 2.5156, whereas for all other reaction conditions it is about 2.0. At all the
temperatures and Al/Hf ratios considered, the highest percentage of terminated chains
hold vinylidene end group ( vf > 80%), followed by isobutyl end group (9.3 < if < 14.1)
and butenyl end group (3.9 < bf < 6.3). At a fixed Al/Hf molar ratio, with increase in
temperature, vf is increasing whilst bf and if are found decreasing. Increase in Al/Hf
ratio increases the rate of chain transfer to cocatalyst and thereby if increases at a given
temperature. Though these trends for P4/MAO are consistent with those catalyst systems
discussed earlier but all the termination rates are notably less, due to which high
molecular weight product is raised.
224
For a change in monomer pressure of 30 psi to 45 psi, at 40 °C, Rp,max increases from
0.1720 mol/L/s to 0.3186 mol/L/s and at 80 °C, Rp,max increases from 0.07838 mol/L/s to
0.1176 mol/L/s. This indicates that an increase in monomer pressure results in linear
increase in polymerization rate.
With increase in pressure, molecular weights are found to increase. 2.92 fold
increase in wM for a change in pressure from 15 to 30 psi and 1.86 fold increase for a
change from 30 to 45 psi is found at 40 °C. Similarly at 80 °C, 1.94 fold increase in wM
for a change in pressure from 15 to 30 psi and 2.82 fold increase for a change from 30 to
45 psi is observed.
Polymerization rate is linearly increasing with increase in catalyst concentration at
both the temperatures and in tune with the discussion for P3/MAO system.
With P4/MAO catalyst system also, an inverse relationship between polymer molecular
weight and catalyst concentration is obtained.
E. [2,4,6-Me3Ind]2ZrCl2 (P5)/MAO
Propylene polymerization model is applied to solution phase polymerization of propylene
with an ansa-metallocene catalyst and kinetic parameters are obtained.
Model predictions of polymerization rate are in good agreement with the experimental
polymerization rate at both the cocatalyst to catalyst mole ratios (2000 and 4000)
considered. Values of estimated kinetic parameters and objective function F(k) are given
in Table 4.18.
When compared with P1-P4/MAO systems, polymerization rates with P5/MAO
system are found very less, so is the pk value estimated by the model. At 0 °C, Low
polymerization rates (Rp,max = 2.593 × 10-3
moles/L/s) at Al/Zr mole ratio of 2000 and
high rates (Rp,max = 4.921× 10-3
moles/L/s) at Al/Zr mole ratio of 4000 are echoing the
trend obtained with other catalyst systems. Also a sharp decrease in polymerization rate
225
after reaching a maximum is seen at high Al/Zr ratio implying a larger frequency of chain
transfer to cocatalyst.
At Al/Zr mole ratio of 2000, wM is found to be 6.2516 × 104
which is decreased
by 13.3% on raising the ratio to 4000. Model predicted molecular weights and PDIs
closely match with the experimentally determined values. Predicted value of PDI is
almost unchanged with an increase in Al/Zr mole ratio, and is found to be 2.34 and 2.30
at an Al/Zr molar ratio of 2000 and 4000 respectively.
Chain termination is via spontaneous catalyst deactivation, transfer to monomer
and β-Hydride elimination remain in force followed by chain transfer to cocatalyst as
inferred by vf and if values obtained. With increase in Al/Zr ratio, if is found increasing
while vf decreasing. Secondary insertions are found to be less with this catalyst system
and unresponsive to Al/Zr ratio.
With increase in monomer pressure, a higher initiation rate and increased
polymerization rate is noted, which is consistent with trend observed earlier with any
catalyst system. No appreciable increase in molecular weight is obtained with increase in
monomer pressure with this catalyst system.
Increase in catalyst concentration results in linearly increasing polymerization rate
which reaches to a maximum and decreases afterwards due to elevated termination rates.
With P5/MAO catalyst system, average molecular weights are noticed to be unaltered
with increase in catalyst concentration. An inverse relationship between polymer
molecular weight and catalyst concentration are obtained with hafnium based
metallocenes (P3 and P4), whereas Zr based metallocenes (P1 and P2) have shown a
similar trend as obtained for P5.
226
F. [2,4,7-Me3Ind]2ZrCl2) (P6)/MAO
Propylene polymerization model is applied to solution phase polymerization of propylene
with an ansa-P6 catalyst and kinetic parameters are obtained.
A good match between the model predicted and experimental rate is obtained at
all the Al/Zr ratios (1000, 2000 and 4000) studied. Values of estimated kinetic parameters
and objective function [F(k)] are given in Table 4.20.
Polymerization rates with P6/MAO are observed really less when compared with
P1-P4/MAO systems, but comparable with P5/MAO system at identical reaction
conditions. The pk value estimated by the model is slightly higher (106.06 M
-1.s
-1 vs.
71.27 M-1
.s-1
) than P5/MAO system. It is found that Al/Zr ratio as high as 4000 is
sufficient to activate this catalyst and higher ratios bring no further increase in
polymerization rate.
With (P6)/MAO system, frequency of β-H elimination is found to be
9.8943 × 10-4
. The rate of chain transfer to cocatalyst and reactivation after transfer both
are quite high with P6/MAO catalyst system at 0 °C. Altk ,
and rAlk values with this
catalyst system are much higher than those obtained with (P1-P4)/MAO but less than that
obtained with P5/MAO system.
Increase in Al/Zr molar ratio is resulting in decrease in molecular weights. At
Al/Zr mole ratio of 1000, wM is found to be 1.2994 × 105
which is decreased by 39.82%
on raising the ratio to 2000. On increasing Al/Zr ratio from 2000 to 4000, wM is further
decreased by 21.88%. Weight average molecular weights obtained with P6/MAO are very
close to those obtained with P5/MAO catalyst system at identical conditions, however,
polydispersity indices are little high with P6/MAO evincing broader molecular weight
distribution.
227
Chain termination is majorly taking place via spontaneous catalyst deactivation, transfer
to monomer and β-Hydride elimination. Secondary insertions with P6/MAO catalyst
system are higher than P5/MAO but less in comparison with (P1-P4)/MAO systems.
Like P5/MAO and other catalysts studied up to now, P6/MAO also evidences a higher
initiation rate and increased polymerization rate with increase in monomer pressure.
Polymerization rates at studied monomer pressures are very comparable with those
obtained with P1/MAO system.
A little increase in molecular weight is obtained with increase in monomer
pressure with P6/MAO catalyst system similar to as obtained with P5/MAO system.
With P6/MAO also, polymerization rates are increasing linearly with increase in catalyst
concentration. Like P5/MAO catalyst system, average molecular weights with P6/MAO
are also not changing with increase in catalyst concentration.
G. Me2Si[2,4,6-Me3Ind]2ZrCl2) (P7)/MAO
Propylene polymerization model is applied to solution phase polymerization of propylene
with an ansa-metallocene catalyst and kinetic parameters are obtained.
Estimated kinetic parameters and objective function F(k) values are given in Table 4.22.
A decent agreement between experimental and model predicted polymerization rates is
received at 30 °C and 70 °C.
With P7/MAO catalyst system, frequency of β-H elimination is substantial and
increases with increase in temperature from 30 °C to 70 °C. Reinitiation rate after β-H
elimination is much lower at 70 °C than that at 30 °C.
Molecular weights are found to decrease with increase in temperature. wM
estimated at 30 °C is 7.9344 × 104 which is decreased by 80% at 70 °C.
PDIs at all
reaction conditions are obtained almost 2.0 representing standard molecular weight
distribution. Polypropylene synthesized using P7/MAO system is highly isotactic.
228
Following the earlier trends, polymerization rate is found increasing linearly with increase
in monomer pressure. No appreciable change in average molecular weights with variation
in monomer pressures is observed at both 30 °C and 70 °C temperature.
Polymerization rates at different catalyst (P7) concentrations are showing a linear
dependence at 30 °C as well as at 70 °C. However at 70 °C polymerization rate is noted to
be 23% higher than that at 30 °C at corresponding catalyst concentration. Frequency of β-
H elimination is found to be negligibly low as compared to the chain transfer to monomer
and therefore, average molecular weights are unchanged with increase in catalyst
concentration.
5.2 Conclusions
Based on the results obtained in the present study, the following conclusions are drawn:
1. A comprehensive kinetic model is developed for metallocene catalyzed ethylene
polymerization which accurately predicts the polymerization rate, polymer
molecular weight distribution, mole fraction of dead polymer chains with terminal
double bond and density of long-chain branches and short-chain branches.
2. A comprehensive kinetic model is developed for propylene polymerization with
metallocene catalysts, which efficaciously predicts the polymerization rate,
polymer molecular weight distribution, the fractions of end groups generated by
various modes of chain transfer i.e. fraction of vinyl-terminated chains, butenyl-
terminated chains, isobutyl-terminated chains and vinylidene-terminated chains,
with different catalysts.
3. A novel 'natural logarithmic differential evolution (NLDE), algorithm of
optimization is proposed and applied successfully for estimating optimum kinetic
parameters.
229
4. Polymerization rates and polymer properties are calculated with population
balance approach and method of moments. Combination of population balance
approach and NLDE optimization approach is found to be extremely effective and
robust for parameter estimation.
5. A large value of initial population (NP) in NLDE should be used to ensure a
globally optimum set of parameters. In this work, NP is taken as high as fifty
times or more the size of parameter vector.
6. For semibatch reactor with constant monomer concentration throughout the course
of polymerization, analytical solution of model equations (to determine pk , dk
and tMk only) is presented and applied in simulations with all catalyst systems.
7. Simple models are solved analytically or numerically to obtain the coarse values
of kinetic parameters which are used in judging the range of parameters those are
required in the estimation of full set of parameters.
In case of gas phase ethylene polymerization with silica-supported
Me2Si[Ind]2ZrCl2 (E1)/MAO:
8. Rate of initiation increases with increase in temperature, ethylene pressure and
initial catalyst amount.
9. Rate of propagation increases with increase in temperature, ethylene pressure and
initial catalyst amount. At high temperatures (40 °C - 60 °C), propagation rate
decreases rapidly after reaching a maximum value. Propagation rate linearly
depends on ethylene pressure (concentration). At low pressures (1-3 bar),
propagation rates are low and maintained, but at high pressures (5-7 bar), high
propagation rates are obtained which decrease after reaching a maximum.
230
10. Rate of spontaneous deactivation increases with increase in temperature and initial
catalyst amount.
11. Rate of chain transfer to monomer increases with increase in temperature and
ethylene pressure. Chain termination dominantly takes place by chain transfer to
monomer.
12. Average molecular weights decrease with increase in temperature. Average
molecular weights increase slightly with increase in ethylene pressure. Increase in
initial amount of catalyst does not change the average molecular weight of
polyethylene.
13. A polydispersity index of 1.9999 is obtained which indicates an ideal molecular
weight distribution. PDI is not affected by change in temperature, ethylene
pressure and initial catalyst amount.
In case of solution phase ethylene polymerization with in-situ-silica supported
Et[Ind]2ZrCl2(E2)/MAO:
14. Rate of initiation increases with increase in temperature, ethylene pressure and
initial catalyst amount.
15. Propagation rate increases with increase in temperature. At 40 °C and 60
°C,
propagation rates are very low, but at 80 °C, 100 °C and 120 °C, rate increases
significantly. Propagation rate linearly increases with increase in ethylene pressure
and initial catalyst amount. Till 80 °C, maximum polymerization rate is
maintained with in-situ-supported catalyst system. Propagation rate also increases
with increase in cocatalyst to catalyst mole ratio. For a ratio of more than 500, rate
decreases after reaching a maximum.
16. Rate of spontaneous deactivation increases rapidly with increase in temperature
from 80 °C to 120 °C. Unlike E1/MAO, a change in initial catalyst amount and
231
cocatalyst to catalyst mole ratio do not affect the rate of spontaneous deactivation
of catalyst due to in-situ arrangement.
17. Rate of chain transfer to monomer increases with increase in temperature. Chain
termination dominantly takes place by chain transfer to monomer.
18. β-hydride elimination increases with increase in temperature. Frequency of β-
hydride elimination increases aggressively beyond 100 °C. Increase in initial
catalyst amount and cocatalyst to catalyst mole ratio do not affect the rate of β-
hydride elimination.
19. Rate of chain transfer to cocatalyst decreases with increase in temperature and
increases with increase in cocatalyst to catalyst mole ratio.
20. Average molecular weights decrease with increase in temperature and slightly
increase with increase in ethylene pressure. Increase in initial amount of catalyst
and cocatalyst to catalyst mole ration do not appreciably change the average
molecular weights of polyethylene.
21. Molecular weight distribution is close to ideal with PDIs ranging in between 1.814
and 2.0 for different polymerization conditions.
22. Fraction of dead chains with terminal double bond is very high (0.781 - 0.992),
which indicates that most chains are terminated via chain transfer to monomer, β-
hydride elimination and chain transfer to cocatalyst.
23. Polyethylene produced with E2/MAO catalysts, comprises of linear chains and
posseses high density since long chain branching frequency is found to be
negligibly low.
24. Gas phase ethylene polymerization with E1/MAO catalysts produces polyethylene
of high molecular weights as compared to that in solution phase polymerization
with in-situ-supported E2/MAO system.
232
In cases of solution phase propylene polymerization with various catalyst systems
(P1 through P7):
25. Rate of initiation increases with increase in temperature, propylene pressure
(concentration), initial catalyst concentration and cocatalyst to catalyst mole ratio.
26. Rate of propagation increases with increase in temperature, propylene pressure,
initial catalyst concentration and cocatalyst to catalyst mole ratio. At high
temperatures (75 °C in P1 and P2, 80 °C in P3 and P4, 70 °C in P7), propagation
rate decreases after reaching a maximum with increase in pressure and initial
catalyst concentration. Under similar reaction conditions, catalyst P1offers highest
propagation rate, followed by P2 through P7.
27. Rate of spontaneous deactivation increases with increase in temperature in all
cases of propylene polymerization.
28. Rate of chain transfer to monomer increases with increase in temperature and
propylene pressure. Highest rate of chain transfer (ktM = 10.2576 M-1
.s-1
at 75 °C)
is observed with P1/MAO, whereas lowest rate (ktM = 5.3048×10-4
M-1
.s-1
at
80 °C) is found with P4/MAO system.
29. Frequency of β-hydride elimination increases with increase in temperature and
initial catalyst concentration. The magnitude of specific rate constant for the
systems studied lies in the order of 10-6
s-1
to 10-3
s-1
. specific rate of reactivation
after β-hydride elimination also increases with increase in temperature.
30. All the catalyst systems studied, yield highly isotactic polypropylene as the rates
of secondary (2, 1) insertion obtained are very less. The rates are extremely low
(ks ~10-5
- 10-9
) with P1-P4/MAO system. However, the rates of secondary
insertion, propagation after secondary insertion and chain transfer after secondary
insertion increase, with increase in temperature.
233
31. Chain transfer to cocatalyst is sensitive to temperature and the rate decreases with
increase in temperature. The inverse trend is possibly due to change in structure of
MAO from simple (linear or cyclic) at low temperatures to a congested one
(ladder or cage) at high temperatures. Rate of reactivation after transfer to
cocatalyst is very high as compared to rate of initiation in each case and increases
with increase in temperature.
32. Average molecular weights of polypropylene decrease with increase in
temperature. With increase in propylene pressure, average molecular weights
increase slightly with zirconium based catalyst systems and increase significantly
with hafnium based systems. With increase in catalyst concentration, molecular
weights do not change with zirconium based catalyst systems but decrease
significantly with hafnium based catalyst systems. P4/MAO gives highest
molecular weight polypropylene (of the order of 108) followed by P3/MAO (of the
order of 105), all other catalysts yield polypropylene with relatively lower
molecular weights (of the order of 104).
33. The model predicts Schulz-Flory distribution with a polydispersity index around
2.0 with all the catalyst systems, which evidences an existence of single site types
in catalyst.
34. Fraction of dead chains with vinyl end group (fv) increases with increase in
temperature but decreases with increase in cocatalyst to catalyst mole ratio.
Fraction of dead chains with butenyl end group (fb) decreases with increase in
temperature and cocatalyst to catalyst mole ratio. Fraction of dead chains with
isobutyl end group (fi) decreases with increase in temperature but increases with
increase in cocatalyst to catalyst mole ratio.
234
5.3 Major contributions
1. A comprehensive kinetic model is developed for metallocene catalyzed ethylene
polymerization capable of predicting polymerization rate, polymer molecular
weight distribution, mole fraction of dead polymer chains with terminal double
bond and density of long-chain branches and short-chain branches.
2. Developed ethylene polymerization model is validated with experimental data
available in open literature, for two different catalyst systems with dissimilar
reaction phase and conditions.
3. A comprehensive kinetic model is developed for metallocene catalyzed propylene
polymerization capable of predicting polymerization rate, polymer molecular
weight distribution, the fractions of end groups generated by various modes of
chain transfer i.e. fraction of vinyl-terminated chains, butenyl-terminated chains,
isobutyl-terminated chains and vinylidene-terminated chains.
4. Developed propylene polymerization model is validated with experimental data
available in open literature, for seven different catalyst systems with dissimilar
reaction conditions in solution phase.
5. The natural logarithmic initialization of normalized population and mutation
operations are included in original differential evolution algorithm (an
evolutionary search based algorithm) and natural logarithmic differential
evolution is proposed.
6. Natural logarithmic differential evolution approach of optimization, which is
capable of handling multiple objective functions simultaneously, is used to
estimate the kinetic model parameters for each catalytic polymerization process
studied.
7. Parametric study is carried out for all the polymerization systems in order to
examine the effect of variation in monomer pressure (concentration),
235
polymerization temperature, initial amount of catalyst and cocatalyst to catalyst
mole ratio on polymerization kinetics and macro- and microstructural properties
of polymer.
8. A systematic approach for, translating mechanistic insight of olefin
polymerization with metallocene catalyst systems into kinetic model and
estimation of model parameters is developed and presented, which is well
applicable to the polymerization of any terminal alkene.
5.4 Future scope of research
The future scope of this work is enumerated below:
1. The models developed are extendable to different metallocene catalyst systems.
Simulation studies can be carried out with various catalyst systems used for
polymerization of ethylene and propylene.
2. Kinetic models for metallocene catalyzed homo- and copolymerization of other
terminal alkenes may be developed and simulated by utilizing the systematic and
comprehensive approach developed in this work.
3. Kinetic models developed in this work may be integrated with transport models to
study the behaviour of metallocene polymerization at macroscopic level
effectively.
236
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251
LIST OF PUBLICATIONS
International Journals (Referred)
1. Nikhil Prakash and Arvind Kumar Sharma, Modeling of Ethylene
Polymerization with Zirconocene Catalyst and Estimation of Kinetic Parameters
using Differential Evolution Algorithm, International Journal of Chemical
Modeling, 2012, 4 (4), 499-514.
2. Nikhil Prakash and Arvind Kumar Sharma, Kinetic Modeling of Gas Phase
Ethylene Polymerization with Silica Supported Metallocene Catalyst and
Determination of Kinetic Parameters using Differential Evolution Algorithm,
International Journal of Polymers and Technologies, 2011, 3 (2), 109-115.
3. Nikhil Prakash, Sushil Kumar and Arvind Kumar Sharma, Ethylene
Polymerization with Supported Metallocene Catalyst: Modeling and Simulation
with Natural Logarithmic Differential Evolution, International Journal of
Chemical Kinetics, (Communicated).
4. Nikhil Prakash, Sushil Kumar and Arvind Kumar Sharma, Modeling and
Simulation of isospecific propylene polymerization catalyzed by rac-
dimethylsilylbis(2,4,6-trimethyl-1-indenyl)zirconium-dichloride/MAO (To be
communicated).
5. Nikhil Prakash, Sushil Kumar and Arvind Kumar Sharma, Modeling of steric
effects of substituents on the microstructure of polypropylene prepared by with
bis(2,4,6-trimethylindenyl)zirconium dichloride and bis(2,4,7-
trimethylindenyl)zirconium dichloride (To be communicated).
International Conference Proceedings (in India)
1. Nikhil Prakash, Sushil Kumar and Arvind Kumar Sharma, Metallocene
Catalyzed Propylene Polymerization: Modeling, simulation and parameter
estimation using differential evolution approach, International Symposium & 66th
Annual Session of IIChE in association with International Partners (CHEMCON-
2013), Depertment of Chemical Engineering, Institute of Chemical Technology,
Mumbai, India, December 27-30, 2013. (Accepted).
2. Saket Anil Ingle, and Nikhil Prakash, Constrained Geometry Catalysts for Olefin
Polymerization, International Conference on Recent Advances in Chemical
Sciences (ICRACS -2013), Department of Chemistry, Arya P. G. College, Panipat,
India, February 24-26, 2013.
3. Saket Anil Ingle, and Nikhil Prakash, Modeling and Simulation of Slurry
Ethylene Polymerization with Supported Unbridged Zirconocene Catalyst using
Logarithmic Differential Evolution Approach, International Conference on
Polymers on the frontiers of Science and Technology, (APA-2013), University
252
Institute of Chemical Engineering and Technology, Punjab University,
Chandigarh, India, Feb 21-23, 2013.
4. Krunal Amin, Purva Goel and Nikhil Prakash, Kinetic Modeling and Simulation
of Ethylene Polymerization with Cp2ZrCl2/MAO Catalyst using Genetic
Algorithm Approach, International Conference on Polymers on the frontiers of
Science and Technology, (APA-2013), University Institute of Chemical
Engineering and Technology, Punjab University, Chandigarh, India, Feb 21-23,
2013.
5. Nikhil Prakash, Sushil Kumar and Arvind Kumar Sharma, Modeling and
Simulation of Metallocene Catalyzed α-Olefin Polymerization: A Logarithmic
Differential Evolution Approach, Proceedings of International Symposium & 65th
Annual Session of IIChE in association with International Partners (CHEMCON-
2012), Depertment of Chemical Engineering, National Institute of Technology,
Jalandhar, Punjab, India, December 27-30, 2012.
6. Anshu Yadav, Nikhil Prakash and Arvind Kumar Sharma, Kinetic Modeling of
Propene Polymerization with Silica Supported Zirconocene Catalyst and
Estimation of Kinetic Parameters using Differential Evolution Algorithm,
Proceedings of APA International Congress (HEALTH CARE INDIA 2012) on
Advances in Human Healthcare Systems, India Habitat Centre, New Delhi, India,
February 20-23, 2012.
7. Nikhil Prakash and Arvind Kumar Sharma, Modeling and Kinetic Parameter
Estimation of Ethylene Polymerization with Cp2ZrCl2–MAO Catalytic System
using Differential Evolution, Proceedings of International Symposium & 64th
Annual Session of IIChE in association with International Partners (CHEMCON-
2011), Department of Chemical Engineering, M. S. Ramaiah Institute of
Technology, Bangalore, India, December 27-29, 2011.
8. Nikhil Prakash and Arvind Kumar Sharma, Modeling and simulation of
constrained geometry titanocene catalyzed ethylene/norbornene copolymerization,
Proceedings of International Symposium and 62nd
Annual Session of IIChE in
association with International Partners (CHEMCON-2009), Department of
Chemical Engineering, Andhra University, Visakhapatnam, India, December 27-
30, 2009.
9. Nikhil Prakash and Arvind Kumar Sharma, Modeling and Simulation of High
Performance Polyolefin Production: Applications of homogeneous and supported
(nBuCp)2ZrCl2 in ethylene polymerization, Proceedings of APA International
Conference (Poly–2009) on Advances in Polymer Science & Technology: Vision
& Scenario, India Habitat Centre, New Delhi, India, December 17-20, 2009.
10. Nikhil Prakash and Arvind Kumar Sharma, Modeling and Simulation of Metal
Catalyzed Gas Phase Propylene Polymerization using Aspen Polymer Plus,
Proceedings of International Symposium and 61st Annual Session of IIChE in
association with International Partners (CHEMCON-2008), University Institute
of Chemical Engineering and Technology, Punjab University, Chandigarh, India,
December 27-30, 2008.
253
11. Nikhil Prakash and Arvind Kumar Sharma, Metallocene Catalysts in Olefin
Polymerization: Present State of the Art, Proceedings of APA International
Conference (Poly–2008) on Advances in Polymer Science & Technology, India
Habitat Centre, New Delhi, India, January 28 – 31 2008.
12. Nikhil Prakash and Arvind Kumar Sharma, Mathematical Modeling in
Metallocene Catalyzed Olefin Polymerization, Proceedings of APA International
Conference (Poly–2008) on Advances in Polymer Science & Technology, India
Habitat Centre, New Delhi, India, January 28 – 31 2008.
International Conference Proceedings (Abroad)
1. Nikhil Prakash, Sushil Kumar and Arvind Kumar Sharma, Modeling and Kinetic
Parameter Estimation of Ethylene Polymerization with Silica Supported
dimethylsilylene bis(η5 –inden-1–ylidene)zirconium dichloride Catalyst using
Differential Evolution approach, Proceedings of AIChE Annual meeting 2012,
Pittsburgh Convention Center, Pittsburgh, PA, USA, October 28-November 2,
2012.
National Conference Proceedings
1. Utsav Bhargav, J. Nitin and Nikhil Prakash, Advances in Conducting Polymers,
Proceedings of 8th
Annual Session of Students’ Chemical Engineering Congress
(SCHEMCON-2012), Department of Chemical Engineering, Birla Institute of
Technology and Science, Pilani, Rajasthan, September 21-22, 2012.
2. Nikhil Prakash, Sushil Kumar and Arvind Kumar Sharma, Effect of Reaction
Parameters on Propylene Polymerization with the Me2Si(2-Me-Ind)2ZrCl2
Catalyst: An Artificial Neural Network Approach, Proceedings of Conference on
Technological Advancements in Chemical and Environmental Engineering
(TACEE – 2012), Department of Chemical Engineering, Birla Institute of
Technology and Science, Pilani, Rajasthan, March 23-24, 2012.
3. Poornima Narayanan, Nikhil Prakash and Arvind Kumar Sharma, Reaction
Mechanisms and Kinetics of Olefin Polymerization catalyzed by Metallocenes: A
Review, Proceedings of Conference on Technological Advancements in Chemical
and Environmental Engineering (TACEE – 2012), Department of Chemical
Engineering, Birla Institute of Technology and Science, Pilani, Rajasthan, March
23-24, 2012.
4. Saket Anil Ingle, Nikhil Prakash and Arvind Kumar Sharma, Effects of
Operating Conditions on Ethylene-Norbornene Co-polymerization Catalyzed by
cyclopentadienyl-phenoxytitanium Catalysts: An Artificial Neural Network
Approach, Proceedings of Conference on Technological Advancements in
Chemical and Environmental Engineering (TACEE – 2012), Department of
Chemical Engineering, Birla Institute of Technology and Science, Pilani,
Rajasthan, March 23-24, 2012.
254
5. Nikhil Prakash and Arvind Kumar Sharma, Studies on Ethylene Polymerization
with Metallocene/Methylaluminoxane Catalysts: Mechanism and effects of
reaction conditions, Proceedings of APA National Conference on Advances in
Polymer Science & Engineering (PSE – 2010): Emerging Dimensions, University
Institute of Chemical Engineering and Technology, Punjab University,
Chandigarh, India, November 26-27, 2010.
6. Nikhil Prakash and Arvind Kumar Sharma, Recent Advances in Olefin
Polymerization: Applications of Constrained Geometry Single Site Catalysts,
Proceedings of National Seminar on Recent Advances in Chemical Engineering
Operations and Process in Chemical and Allied Industries, Institute of
Technology, Guru Ghasidas University, Bilaspur, India, February 5 - 6, 2008.
7. Nikhil Prakash and Arvind Kumar Sharma, Modeling and Simulation Trends in
Polymer Processing”, Proceedings of National Seminar on Recent Advances in
Chemical Engineering Operations and Process in Chemical and Allied Industries,
Institute of Technology, Guru Ghasidas University, Bilaspur, India, February 5 -
6, 2008.
Book Chapters
1. Nikhil Prakash, Commodity Thermoplastics with Bespoken Properties using
Metallocene Catalyst Systems in Responsive Materials and Methods: State-of-the-
Art Stimuli-Responsive Materials and Their Applications, Edited by: Ashutosh
Tiwari and Hisatoshi Kobayashi, WILEY-Scrivener Publishing LLC, USA, ISBN:
978-1-118-68622-5, (October 2013).
2. Sushil Kumar, Nikhil Prakash and Dipaloy Datta, Biopolymers Based-on
Carboxylic Acids Derived from Renewable Resources in Biopolymers:
Biomedical and Environmental Applications, Edited by: Susheel Kalia & Luc
Averous, Wiley-Scrivener Imprint, USA, ISBN: 978-0-470-63923-8, (October
2011).
255
APPENDIX - I
Code in MATLAB to Estimate the Kinetic Parameters in Ethylene
Polymerization with Me2Si[Ind]2ZrCl2 (E1)/MAO
------------------------------------------------------------------------------------------------------------
Main Program
------------------------------------------------------------------------------------------------------------
% Gas phase polymerization of ethylene with a silica supported
% metallocene catalyst: influence of temperature on deactivation
% Macromol. Rapid Commun. 18,319-324 (1997) by ROOS et al.
% Author: Nikhil Prakash / Dated 17/July/2012
clear
clc
%%%% ALL NEW THETA %%%%%%%%%
% Theta Set of optimization using kin, kp, kd, ktM values
%%%%%%%% P = 5 bar, V = 1L
%%%%%%%% T = 50 oC
% PARA corresponds to lower and upper bounds to 'k' values, named as pass globally
% Natural lagarithmic values will be used in searching 'k' values, which will be properly
corrected in the function with DEqs.
format long g
% PARA = [kin kp kd ktM]
PARAmin = log([1e-5 1e+1 1e-4 1e-4]); % log (k)min
PARAmax = log([1e-2 1e+5 1e-1 1e+1]); % log (k)max
F = 0.7;
CR = 0.9;
refresh = 1;
VTR = 0;
function_eval_limit = Inf;
Iteration_limit = 31;
% P = 4.9346; % atm
% V = 1; % L
% T = 273.16 + 50; % K
% R = 0.0821;
% Molecular weight of catalyst = 448.57
% weight of catalyst = 0.2 g
% Zr = 0.2/448.57
global MPE
MPE = 0.18599;
Zr = 4.4586e-4; % [Zr] <mol/L>
256
D = length (PARAmin);
NP = 500 * D;
fid = fopen('ALL_NEW_ROOS_50.txt', 'a+');
fprintf (fid,'\n\n\n\n\t\t ALL NEW THETA RUN number = [01]\n Volume 1 L\n\n');
% spacing out data output in file
fclose(fid);
fid = fopen('ALL_NEW_ROOS_50.txt', 'a+');
fprintf (fid,datestr(now));
fclose(fid);
fid = fopen('ALL_NEW_ROOS_50.txt', 'a+');
fprintf (fid, '\n PARA = [ki kp kd ktM] \n\n PARAmin = [%d %d %d %d] \t and \n
PARAmax = [%d %d %d %d]\n\n F = [%f]\t CR = [%f]\t NP = [%f]\n\n',
exp(PARAmin), exp(PARAmax), F, CR, NP);
fclose(fid);
if (length(PARAmin) ~= length(PARAmax))
error('Length of upper and lower bounds does not match.')
end
if (NP < 5)
error('Populationulation size NP must be bigger than 5.')
end
if ((F <= 0) || (F > 2))
error('Difference Factor F out of range (0,2].')
end
if ((CR < 0) || (CR > 1))
error('CR value out of range [0,1].')
end
if (Iteration_limit <= 0)
error('Iteration_limit must be positive.')
end
if (function_eval_limit <= 0)
error('function_eval_limit must be positive.')
end
refresh = floor(abs(refresh));
population = zeros(NP,D);
for i = 1:NP
population(i,:) = PARAmin + rand (1,D).* (PARAmax - PARAmin);
end
w = rand (NP,1);
wi = w;
population_old = zeros (size (population));
value = zeros (1, NP);
best_member = zeros (1, D);
best_member_iteration = zeros (1 ,D);
number_function_evlauation = 0;
257
ibest = 1;
global pass
pass = population(ibest,:) ;
time = 60*[2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52
54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 102 104 106
108 110 112 114 116 118 120 122 124 126 128 130 132 134 136 138 140 142 144 146
148 150 152 154 156 158 160 162 164 166 168 170 172 174 176 178 180 182 184 186
188 190 192 194 196 198 200]; % in seconds
expRp = [3.2 2.5 2.3 2.4 3 3.2 3.3 3.2 3.2 3.3 3.4 3.4 3.3 3.3 3.4 3.3 3.3 3.3 3.3
3.3 3.3 3.3 3.3 3.3 3.2 3.3 3.2 3.2 3.2 3.2 3.2 3.2 3.1 3.2 3.1 3 3 3.1 3.1 3 3 3
3 3 3.1 3 3 2.9 2.9 2.9 2.9 2.9 2.8 2.8 2.8 2.8 2.7 2.8 2.7 2.6 2.6 2.6 2.6 2.6
2.5 2.5 2.5 2.5 2.4 2.4 2.4 2.4 2.4 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.2 2.2 2.2
2.2 2.2 2.2 2.1 2.2 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2 2 2 2]; % in kg/g/h
tspan= 0:time(length(time)); % <sec>
y0 = [1e-78 1e-77 1e-86 1e-94 1e-73 1e-86 Zr];
[t, y] = ode15s(@FUNroos,tspan,y0);
last = tspan(length(tspan));
LIVE_MOM_0 = y(:,1);
LIVE_MOM_1 = y(:,2);
LIVE_MOM_2 = y(:,3);
DEAD_MOM_0 = y(:,4);
DEAD_MOM_1 = y(:,5);
DEAD_MOM_2 = y(:,6);
Rp = (28.05e-3*3600/0.2)*exp(pass(2))*MPE*LIVE_MOM_0; % kg (ethylene)/h/gZr,
Reactor volume is 1 L
modRp = [Rp(121) Rp(241) Rp(361) Rp(481) Rp(601) Rp(721) Rp(841) Rp(961)
Rp(1081) Rp(1201) Rp(1321) Rp(1441) Rp(1561) Rp(1681) Rp(1801) Rp(1921)
Rp(2041) Rp(2161) Rp(2281) Rp(2401) Rp(2521) Rp(2641) Rp(2761) Rp(2881)
Rp(3001) Rp(3121) Rp(3241) Rp(3361) Rp(3481) Rp(3601) Rp(3721) Rp(3841)
Rp(3961) Rp(4081) Rp(4201) Rp(4321) Rp(4441) Rp(4561) Rp(4681) Rp(4801)
Rp(4921) Rp(5041) Rp(5161) Rp(5281) Rp(5401) Rp(5521) Rp(5641) Rp(5761)
Rp(5881) Rp(6001) Rp(6121) Rp(6241) Rp(6361) Rp(6481) Rp(6601) Rp(6721)
Rp(6841) Rp(6961) Rp(7081) Rp(7201) Rp(7321) Rp(7441) Rp(7561) Rp(7681)
Rp(7801) Rp(7921) Rp(8041) Rp(8161) Rp(8281) Rp(8401) Rp(8521) Rp(8641)
Rp(8761) Rp(8881) Rp(9001) Rp(9121) Rp(9241) Rp(9361) Rp(9481) Rp(9601)
Rp(9721) Rp(9841) Rp(9961) Rp(10081) Rp(10201) Rp(10321) Rp(10441) Rp(10561)
Rp(10681) Rp(10801) Rp(10921) Rp(11041) Rp(11161) Rp(11281) Rp(11401)
Rp(11521) Rp(11641) Rp(11761) Rp(11881) Rp(12001)];
M1 = (1 - (modRp./expRp));
%-------------------------
zero = LIVE_MOM_0 + DEAD_MOM_0;
first = LIVE_MOM_1 + DEAD_MOM_1;
258
second = LIVE_MOM_2 + DEAD_MOM_2;
% molecular weights
Mnbar = 28.05*(first./zero);
Mwbar = 28.05*(second./first);
PDI = Mwbar ./ Mnbar;
modPDI = PDI(last);
expPDI = 2;
M2 = (1 - (modPDI/expPDI));
%---------------------------------
value(1) = sumsqr([M1 M2]);
value_best = value(1);
weight_best = w(1);
number_function_evlauation = number_function_evlauation + 1;
for i = 2 : NP
pass = population(i,:);
tspan= 0:time(length(time)); % <sec>
y0 = [1e-78 1e-77 1e-86 1e-94 1e-73 1e-86 Zr];
[t, y] = ode15s(@FUNroos,tspan,y0);
last = tspan(length(tspan));
LIVE_MOM_0 = y(:,1);
LIVE_MOM_1 = y(:,2);
LIVE_MOM_2 = y(:,3);
DEAD_MOM_0 = y(:,4);
DEAD_MOM_1 = y(:,5);
DEAD_MOM_2 = y(:,6);
Rp = (28.05e-3*3600/0.2)*exp(pass(2))*MPE*LIVE_MOM_0; % kg (ethylene)/h/gZr,
Reactor volume is 1 L
modRp = [Rp(121) Rp(241) Rp(361) Rp(481) Rp(601) Rp(721) Rp(841) Rp(961)
Rp(1081) Rp(1201) Rp(1321) Rp(1441) Rp(1561) Rp(1681) Rp(1801) Rp(1921)
Rp(2041) Rp(2161) Rp(2281) Rp(2401) Rp(2521) Rp(2641) Rp(2761) Rp(2881)
Rp(3001) Rp(3121) Rp(3241) Rp(3361) Rp(3481) Rp(3601) Rp(3721) Rp(3841)
Rp(3961) Rp(4081) Rp(4201) Rp(4321) Rp(4441) Rp(4561) Rp(4681) Rp(4801)
Rp(4921) Rp(5041) Rp(5161) Rp(5281) Rp(5401) Rp(5521) Rp(5641) Rp(5761)
Rp(5881) Rp(6001) Rp(6121) Rp(6241) Rp(6361) Rp(6481) Rp(6601) Rp(6721)
Rp(6841) Rp(6961) Rp(7081) Rp(7201) Rp(7321) Rp(7441) Rp(7561) Rp(7681)
Rp(7801) Rp(7921) Rp(8041) Rp(8161) Rp(8281) Rp(8401) Rp(8521) Rp(8641)
Rp(8761) Rp(8881) Rp(9001) Rp(9121) Rp(9241) Rp(9361) Rp(9481) Rp(9601)
Rp(9721) Rp(9841) Rp(9961) Rp(10081) Rp(10201) Rp(10321) Rp(10441) Rp(10561)
Rp(10681) Rp(10801) Rp(10921) Rp(11041) Rp(11161) Rp(11281) Rp(11401)
Rp(11521) Rp(11641) Rp(11761) Rp(11881) Rp(12001)];
M1 = (1 - (modRp./expRp));
259
%-------------------------
zero = LIVE_MOM_0 + DEAD_MOM_0;
first = LIVE_MOM_1 + DEAD_MOM_1;
second = LIVE_MOM_2 + DEAD_MOM_2;
% molecular weights
Mnbar = 28.05*(first./zero);
Mwbar = 28.05*(second./first);
PDI = Mwbar ./ Mnbar;
modPDI = PDI(last);
expPDI = 2;
M2 = (1 - (modPDI/expPDI));
value(i) = sumsqr([M1 M2]);
number_function_evlauation = number_function_evlauation + 1;
if (value(i) < value_best)
ibest = i;
value_best = value(i);
weight_best = w(i);
end
end
best_member_iteration = population(ibest,:);
value_bestit = value_best;
best_member = best_member_iteration;
pm1 = zeros (NP, D);
pm2 = zeros (NP, D);
pm3 = zeros (NP, D);
pm4 = zeros (NP, D);
pm5 = zeros (NP, D);
bm = zeros (NP, D);
ui = zeros (NP, D);
mui = zeros (NP, D);
mpo = zeros (NP, D);
rot = 0:1:NP-1;
rotd= 0:1:D-1;
rt = zeros (NP);
rtd = zeros (D);
a1 = zeros (NP);
a2 = zeros (NP);
a3 = zeros (NP);
a4 = zeros (NP);
a5 = zeros (NP);
ind = zeros (4);
iter = 1;
while (iter < Iteration_limit) && (value_best > VTR) % &&
(number_function_evlauation < function_eval_limit)
260
fprintf('\n\n\n Iteration No.[%d] \t of \t [%d]\n Going Well', iter, Iteration_limit);
population_old = population;
wold = w;
ind = randperm (4);
a1 = randperm (NP);
rt = rem (rot + ind(1), NP);
a2 = a1(rt+1);
rt = rem (rot + ind(2), NP);
a3 = a2(rt+1);
rt = rem (rot +ind(3), NP);
a4 = a3(rt+1);
rt = rem (rot + ind(4), NP);
a5 = a4(rt+1);
pm1 = population_old(a1,:);
pm2 = population_old(a2,:);
pm3 = population_old(a3,:);
pm4 = population_old(a4,:);
pm5 = population_old(a5,:);
w1 = wold(a1);
w2 = wold(a2);
bm = repmat (best_member_iteration, NP, 1);
bw = repmat(weight_best, NP, 1);
mui = rand (NP, D) < CR;
mui = sort (mui');
for i = 1:NP
n = floor (rand * D);
if n > 0
rtd = rem (rotd + n, D);
mui(:,i) = mui(rtd+1,i);
end
end
mui = mui';
mpo = mui < 0.5;
ui = pm3 + F*(pm1 - pm2);
ui = population_old.*mpo + ui.*mui;
for i = 1:NP
ui(i,:) = max (ui(i,:), PARAmin);
ui(i,:) = min (ui(i,:), PARAmax);
end
for i = 1:NP
261
pass = ui(i,:);
tspan= 0:time(length(time)); % <sec>
y0 = [1e-78 1e-77 1e-86 1e-94 1e-73 1e-86 Zr];
[t, y] = ode15s(@FUNroos,tspan,y0);
last = tspan(length(tspan));
LIVE_MOM_0 = y(:,1);
LIVE_MOM_1 = y(:,2);
LIVE_MOM_2 = y(:,3);
DEAD_MOM_0 = y(:,4);
DEAD_MOM_1 = y(:,5);
DEAD_MOM_2 = y(:,6);
Rp = (28.05e-3*3600/0.2)*exp(pass(2))*MPE*LIVE_MOM_0; % kg (ethylene)/h/gZr,
Reactor volume is 1 L
modRp = [Rp(121) Rp(241) Rp(361) Rp(481) Rp(601) Rp(721) Rp(841) Rp(961)
Rp(1081) Rp(1201) Rp(1321) Rp(1441) Rp(1561) Rp(1681) Rp(1801) Rp(1921)
Rp(2041) Rp(2161) Rp(2281) Rp(2401) Rp(2521) Rp(2641) Rp(2761) Rp(2881)
Rp(3001) Rp(3121) Rp(3241) Rp(3361) Rp(3481) Rp(3601) Rp(3721) Rp(3841)
Rp(3961) Rp(4081) Rp(4201) Rp(4321) Rp(4441) Rp(4561) Rp(4681) Rp(4801)
Rp(4921) Rp(5041) Rp(5161) Rp(5281) Rp(5401) Rp(5521) Rp(5641) Rp(5761)
Rp(5881) Rp(6001) Rp(6121) Rp(6241) Rp(6361) Rp(6481) Rp(6601) Rp(6721)
Rp(6841) Rp(6961) Rp(7081) Rp(7201) Rp(7321) Rp(7441) Rp(7561) Rp(7681)
Rp(7801) Rp(7921) Rp(8041) Rp(8161) Rp(8281) Rp(8401) Rp(8521) Rp(8641)
Rp(8761) Rp(8881) Rp(9001) Rp(9121) Rp(9241) Rp(9361) Rp(9481) Rp(9601)
Rp(9721) Rp(9841) Rp(9961) Rp(10081) Rp(10201) Rp(10321) Rp(10441) Rp(10561)
Rp(10681) Rp(10801) Rp(10921) Rp(11041) Rp(11161) Rp(11281) Rp(11401)
Rp(11521) Rp(11641) Rp(11761) Rp(11881) Rp(12001)];
M1 = (1 - (modRp./expRp));
%-------------------------
zero = LIVE_MOM_0 + DEAD_MOM_0;
first = LIVE_MOM_1 + DEAD_MOM_1;
second = LIVE_MOM_2 + DEAD_MOM_2;
% molecular weights
Mnbar = 28.05*(first./zero);
Mwbar = 28.05*(second./first);
PDI = Mwbar ./ Mnbar;
modPDI = PDI(last);
expPDI = 2;
M2 = (1 - (modPDI/expPDI));
value_temporary = sumsqr([M1 M2]);
if (value_temporary <= value(i))
population(i,:) = ui(i,:);
262
value(i) = value_temporary;
w(i) = wi(i);
if (value_temporary < value_best)
value_best = value_temporary;
best_member = ui(i,:);
weight_best = w(i);
end
end
end
number_function_evlauation = number_function_evlauation + NP;
best_member_iteration = best_member;
if (refresh > 0)
if (rem (iter, refresh) == 0)
fid = fopen('ALL_NEW_ROOS_50.txt', 'a+');
fprintf (fid, 'Iteration: %d, Best: %8.4e, Worst: %8.4e\n', iter, value_best,
max(value));
fclose(fid);
for n = 1:D
fid = fopen('ALL_NEW_ROOS_50.txt', 'a+');
fprintf(fid, '\n k(%d) = %e\n', n, exp(best_member(n)));
fprintf('\n k(%d) = %e\n', n, exp(best_member(n)));
fclose(fid);
end
end
end
iter = iter + 1;
end
fid = fopen('ALL_NEW_ROOS_50.txt', 'a+');
fprintf (fid, 'Mnbar: %d,\t Mwbar: %d,\t PDI: %d\n\n', Mnbar(last), Mwbar(last),
PDI(last));
fclose(fid);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
pass = best_member;
tspan= 0:time(length(time)); % <sec>
y0 = [1e-78 1e-77 1e-86 1e-94 1e-73 1e-86 Zr];
[t, y] = ode15s(@FUNroos,tspan,y0);
last = tspan(length(tspan));
LIVE_MOM_0 = y(:,1);
Rp = (28.05e-3*3600/0.2)*exp(pass(2))*MPE*LIVE_MOM_0; % kg (ethylene)/h/gZr,
Reactor volume is 1 L; y(1) = Lo
plot(time, expRp, 'k^', tspan, Rp, '-k');
if (iter >= Iteration_limit)
263
warning('max. number of iterations reached (Iteration_limit)') %#ok<WNTAG>
end
if (number_function_evlauation >= function_eval_limit)
warning('max. number of function evaluations reached (function_eval_limit)')
%#ok<WNTAG>
end
if (value_best < VTR)
warning('best value has been obtained') %#ok<WNTAG>
end
------------------------------------------------------------------------------------------------------------
Function
------------------------------------------------------------------------------------------------------------
function dy = FUNroos(t,y)
global MPE
global pass
kin = exp(pass(1));
kp = exp(pass(2));
kd = exp(pass(3));
ktM = exp(pass(4));
%%%%%%% Variables used in DEs corresponds to
%y(1): LIVE_MOM_0 Zeroth Moment of Living Polymer Chain Length Distribution
%y(2): LIVE_MOM_1 First Moment of Living Polymer Chain Length Distribution
%y(3): LIVE_MOM_2 Second Moment of Living Polymer Chain Length Distribution
%y(4): DEAD_MOM_0 Zeroth Moment of Dead Polymer Chain Length Distribution
%y(5): DEAD_MOM_1 First Moment of Dead Polymer Chain Length Distribution
%y(6): DEAD_MOM_2 Second Moment of Dead Polymer Chain Length Distribution
%y(7): Cstar Catalyst activated complex concentration
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% LISTED HERE ARE THE ODEs TO BE SOLVED
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
dy(1) = kin*MPE*y(7)-(kd)*y(1);
dy(2) = kin*MPE*y(7)+kp*MPE*y(1)-kd*y(2)+ktM*MPE*(y(1)-y(2));
dy(3) = kin*MPE*y(7)+kp*MPE*(y(1)+2*y(2))-kd*y(3)+ktM*MPE*(y(1)-y(3));
dy(4) = (kd+(ktM*MPE))*y(1);
dy(5) = (kd+(ktM*MPE))*y(2);
dy(6) = (kd+(ktM*MPE))*y(3);
dy(7) = -kin*MPE*y(7);
dy = dy';
264
APPENDIX - II
Code in MATLAB to Estimate the Kinetic Parameters in Ethylene
Polymerization in-situ-supported Et[Ind]2ZrCl2 (E2)/MAO
------------------------------------------------------------------------------------------------------------
Main Program
------------------------------------------------------------------------------------------------------------
% Modeling and Simulation of
%% "Effect of Experimental Conditions on Ethylene Polymerization with In-Situ-
Supported Metallocene Catalyst"
%%%using Logarithmic Differential Evolution Approach
%%%%%%%%%%%%% TEMP = 60 oC
%%%%%%%%% Experimental values for 60 oC and 80 psig pressure
%%%%%%%%%%% Reactor volume = 500 ml; Solvent hexane = 300 ml
% Author: Nikhil Prakash / Dated 19/June/2013
clear
clc
% optimization using kin, kp, kd, ktM, ktCo, kbeta, klcb values
%%%%%%%%%% MAO/Zr = 500 %%%%%%%%%%%
% PARA corresponds to lower and upper bounds to 'k' values, named as pass globally
% Lagarithmic values will be used in searching 'k' values, which will be properly
corrected in the function with DEqs.
format long g
% PARA = [kin kp kd ktM ktCo kbeta klcb]
PARAmin = log([1e-5 1e+1 1e-5 1e-5 1e-6 1e-6 1e-6]); % log (k)min
PARAmax = log([1e-1 1e+6 1e-1 1e-1 1e+3 1e+3 1e+3]); % log (k)max
F = 0.7;
CR = 0.9;
refresh = 1;
VTR = 0;
function_eval_limit = Inf;
Iteration_limit =201;
global MEE
MEE = 0.5755; %% <mol/L>
molZr = 3e-6; % moles in 300 ml solvent, hexane
Zr = molZr*1000/300; % [Zr] <mol/L>
MAO = 500*Zr;
D = length (PARAmin);
NP = 120*D;
fid = fopen('ET_90_60_T.txt', 'a+');
fprintf (fid,'\n\n\n\n\t\t RUN number = [01]\n\n\n'); % spacing out data output in file
fclose(fid);
265
fid = fopen('ET_90_60_T.txt', 'a+');
fprintf (fid,datestr(now));
fclose(fid);
fid = fopen('ET_90_60_T.txt', 'a+');
fprintf (fid, '\n PARA = [kin kp kd ktM ktCo kbeta klcb ] \n\n PARAmin = [%d %d
%d %d %d %d %d] \t and \n PARAmax = [%d %d %d %d %d %d %d ]\n\n F = [%f]\t
CR = [%f]\t NP = [%f]\n\n', exp(PARAmin), exp(PARAmax), F, CR, NP);
fclose(fid);
if (length(PARAmin) ~= length(PARAmax))
error('Length of upper and lower bounds does not match.')
end
if (NP < 5)
error('Populationulation size NP must be bigger than 5.')
end
if ((F <= 0) || (F > 2))
error('Difference Factor F out of range (0,2].')
end
if ((CR < 0) || (CR > 1))
error('CR value out of range [0,1].')
end
if (Iteration_limit <= 0)
error('Iteration_limit must be positive.')
end
if (function_eval_limit <= 0)
error('function_eval_limit must be positive.')
end
refresh = floor(abs(refresh));
population = zeros(NP,D); % initialize population
for i = 1:NP
population(i,:) = PARAmin + rand (1,D).* (PARAmax - PARAmin);
end
w = rand (NP,1);
wi = w;
population_old = zeros (size (population));
value = zeros (1, NP);
best_member = zeros (1, D);
best_member_iteration = zeros (1 ,D);
number_function_evlauation = 0;
ibest = 1;
global pass
pass = population(ibest,:) ;
266
time = 60*[0 1 2 4 5 7 8 9 11 12 14 15 17 18 20 21 23 25 26 28 29 31
32 34 35 36 37 38 39 40 41 42 43 44 46 47 48 49 50 51 52 54 55 57 58 59
60];
actexpRp=[10.9372 27.2702 43.676 54.3952 65.1143 70.3649 75.6154 80.866 75.0704
74.7797 74.489 74.2347 73.944 73.6896 73.3989 73.1446 72.8539 72.5632 77.7774
77.4868 82.701 65.8955 76.651 70.8554 70.6011 70.4194 70.3104 70.0197 69.8743
69.729 63.9697 69.3293 63.679 68.9296 68.6752 73.8895 73.7441 68.0939 67.8759
67.7668 78.5223 72.7267 72.436 72.1453 71.891 71.7093 71.6003]; % in cc/min
expRp = (3.42728e-6)*actexpRp; % in moles / s
tspan= 0:time(length(time)); % <sec>
y0 = [1e-89 1e-77 1e-94 1e-82 1e-73 1e-75 1e-85 1e-95 1e-65 Zr MAO];
[t, y] = ode15s(@FUNE2,tspan,y0);
last = tspan(length(tspan));
LIVE_MOM_0 = y(:,1);
LIVE_MOM_1 = y(:,2);
LIVE_MOM_2 = y(:,3);
DEAD_MOM_0 = y(:,4);
DEAD_MOM_1 = y(:,5);
DEAD_MOM_2 = y(:,6);
LIVE_MOM_0 = y(:,7); % Zeroth Moment of Dead Polymer Chain Length Distribution
with terimnal '='
LIVE_MOM_1 = y(:,8);% First Moment of Dead Polymer Chain Length Distribution
with terimnal '='
LIVE_MOM_2 = y(:,9);% Second Moment of Dead Polymer Chain Length Distribution
with terimnal '='
P0 = y(:,10); % P(0) Catalyst activated complex concentration
CCocat = y(:,11); % Cocat Cocatalyst concentration
Rp = exp(pass(2))*MEE*LIVE_MOM_0; % mol(ethylene)/s/L, y(1) = Lo
modRp = [Rp(1) Rp(61) Rp(121) Rp(241) Rp(301) Rp(421) Rp(481) Rp(541) Rp(661)
Rp(721) Rp(841) Rp(901) Rp(1021) Rp(1081) Rp(1201) Rp(1261) Rp(1381) Rp(1501)
Rp(1561) Rp(1681) Rp(1741) Rp(1861) Rp(1921) Rp(2041) Rp(2101) Rp(2161)
Rp(2221) Rp(2281) Rp(2341) Rp(2401) Rp(2461) Rp(2521) Rp(2581) Rp(2641)
Rp(2761) Rp(2821) Rp(2881) Rp(2941) Rp(3001) Rp(3061) Rp(3121) Rp(3241)
Rp(3301) Rp(3421) Rp(3481) Rp(3541) Rp(3601)];
M1 = (1 - (modRp./expRp));
%-------------------------
zero = LIVE_MOM_0 + DEAD_MOM_0 + LIVE_MOM_0;
first = LIVE_MOM_1 + DEAD_MOM_1 + LIVE_MOM_1;
second = LIVE_MOM_2 + DEAD_MOM_2 + LIVE_MOM_2;
% molecular weights
Mnbar = 28.08*(first./zero);
Mwbar = 28.08*(second./first);
modMnbar = Mnbar(last);
267
modMwbar = Mwbar(last);
PDI = Mwbar ./ Mnbar;
modPDI = PDI(last);
expPDI = 2;
M2 = (1 - (modPDI/expPDI));
value(1) = sumsqr([M1 M2]);
value_best = value(1);
weight_best = w(1);
number_function_evlauation = number_function_evlauation + 1;
for i = 2 : NP
pass = population(i,:);
y0 = [1e-89 1e-77 1e-94 1e-82 1e-73 1e-75 1e-85 1e-95 1e-65 Zr MAO];
[t, y] = ode15s(@FUNE2,tspan,y0);
last = tspan(length(tspan));
LIVE_MOM_0 = y(:,1);
LIVE_MOM_1 = y(:,2);
LIVE_MOM_2 = y(:,3);
DEAD_MOM_0 = y(:,4);
DEAD_MOM_1 = y(:,5);
DEAD_MOM_2 = y(:,6);
LIVE_MOM_0 = y(:,7);
LIVE_MOM_1 = y(:,8);
LIVE_MOM_2 = y(:,9);
P0 = y(:,10);
CCocat = y(:,11);
Rp = exp(pass(2))*MEE*LIVE_MOM_0; % mol(ethylene)/s/L, y(1) = Lo
modRp = [Rp(1) Rp(61) Rp(121) Rp(241) Rp(301) Rp(421) Rp(481) Rp(541) Rp(661)
Rp(721) Rp(841) Rp(901) Rp(1021) Rp(1081) Rp(1201) Rp(1261) Rp(1381) Rp(1501)
Rp(1561) Rp(1681) Rp(1741) Rp(1861) Rp(1921) Rp(2041) Rp(2101) Rp(2161)
Rp(2221) Rp(2281) Rp(2341) Rp(2401) Rp(2461) Rp(2521) Rp(2581) Rp(2641)
Rp(2761) Rp(2821) Rp(2881) Rp(2941) Rp(3001) Rp(3061) Rp(3121) Rp(3241)
Rp(3301) Rp(3421) Rp(3481) Rp(3541) Rp(3601)];
M1 = (1 - (modRp./expRp));
zero = LIVE_MOM_0 + DEAD_MOM_0 + LIVE_MOM_0;
first = LIVE_MOM_1 + DEAD_MOM_1 + LIVE_MOM_1;
second = LIVE_MOM_2 + DEAD_MOM_2 + LIVE_MOM_2;
% molecular weights
Mnbar = 28.08*(first./zero);
Mwbar = 28.08*(second./first);
modMnbar = Mnbar(last);
modMwbar = Mwbar(last);
268
PDI = Mwbar ./ Mnbar;
modPDI = PDI(last);
expPDI = 2;
M2 = (1 - (modPDI/expPDI));
value(i) = sumsqr([M1 M2]);
number_function_evlauation = number_function_evlauation + 1;
if (value(i) < value_best)
ibest = i;
value_best = value(i);
weight_best = w(i);
end
end
best_member_iteration = population(ibest,:);
value_bestit = value_best;
best_member = best_member_iteration;
pm1 = zeros (NP, D);
pm2 = zeros (NP, D);
pm3 = zeros (NP, D);
pm4 = zeros (NP, D);
pm5 = zeros (NP, D);
bm = zeros (NP, D);
ui = zeros (NP, D);
mui = zeros (NP, D);
mpo = zeros (NP, D);
rot = 0:1:NP-1;
rotd= 0:1:D-1;
rt = zeros (NP);
rtd = zeros (D);
a1 = zeros (NP);
a2 = zeros (NP);
a3 = zeros (NP);
a4 = zeros (NP);
a5 = zeros (NP);
ind = zeros (4);
fprintf('going well');
iter = 1;
while (iter < Iteration_limit) && (value_best > VTR) && (number_function_evlauation
< function_eval_limit)
fprintf('\n\n\n Iteration No.[%d] \t of \t [%d]\n Going Well', iter, Iteration_limit);
population_old = population;
269
wold = w;
ind = randperm (4);
a1 = randperm (NP);
rt = rem (rot + ind(1), NP);
a2 = a1(rt+1);
rt = rem (rot + ind(2), NP);
a3 = a2(rt+1);
rt = rem (rot +ind(3), NP);
a4 = a3(rt+1);
rt = rem (rot + ind(4), NP);
a5 = a4(rt+1);
pm1 = population_old(a1,:);
pm2 = population_old(a2,:);
pm3 = population_old(a3,:);
pm4 = population_old(a4,:);
pm5 = population_old(a5,:);
w1 = wold(a1);
w2 = wold(a2);
bm = repmat (best_member_iteration, NP, 1);
bw = repmat(weight_best, NP, 1);
mui = rand (NP, D) < CR;
mui = sort (mui');
for i = 1:NP
n = floor (rand * D);
if n > 0
rtd = rem (rotd + n, D);
mui(:,i) = mui(rtd+1,i);
end
end
mui = mui';
mpo = mui < 0.5;
ui = pm3 + F*(pm1 - pm2);
ui = population_old.*mpo + ui.*mui;
for i = 1:NP
ui(i,:) = max (ui(i,:), PARAmin);
ui(i,:) = min (ui(i,:), PARAmax);
end
for i = 1:NP
pass = ui(i,:);
y0 = [1e-89 1e-77 1e-94 1e-82 1e-73 1e-75 1e-85 1e-95 1e-65 Zr MAO];
[t, y] = ode15s(@FUNE2,tspan,y0);
last = tspan(length(tspan));
270
LIVE_MOM_0 = y(:,1);
LIVE_MOM_1 = y(:,2);
LIVE_MOM_2 = y(:,3);
DEAD_MOM_0 = y(:,4);
DEAD_MOM_1 = y(:,5);
DEAD_MOM_2 = y(:,6);
LIVE_MOM_0 = y(:,7); % Zeroth Moment of Dead Polymer Chain Length Distribution
with terimnal '='
LIVE_MOM_1 = y(:,8);% First Moment of Dead Polymer Chain Length Distribution
with terimnal '='
LIVE_MOM_2 = y(:,9);% Second Moment of Dead Polymer Chain Length Distribution
with terimnal '='
P0 = y(:,10); % P(0) Catalyst activated complex concentration
CCocat = y(:,11); % Cocat Cocatalyst concentration
Rp = exp(pass(2))*MEE*LIVE_MOM_0; % mol(ethylene)/s/L, y(1) = Lo
modRp = [Rp(1) Rp(61) Rp(121) Rp(241) Rp(301) Rp(421) Rp(481) Rp(541) Rp(661)
Rp(721) Rp(841) Rp(901) Rp(1021) Rp(1081) Rp(1201) Rp(1261) Rp(1381) Rp(1501)
Rp(1561) Rp(1681) Rp(1741) Rp(1861) Rp(1921) Rp(2041) Rp(2101) Rp(2161)
Rp(2221) Rp(2281) Rp(2341) Rp(2401) Rp(2461) Rp(2521) Rp(2581) Rp(2641)
Rp(2761) Rp(2821) Rp(2881) Rp(2941) Rp(3001) Rp(3061) Rp(3121) Rp(3241)
Rp(3301) Rp(3421) Rp(3481) Rp(3541) Rp(3601)];
M1 = (1 - (modRp./expRp));
zero = LIVE_MOM_0 + DEAD_MOM_0 + LIVE_MOM_0;
first = LIVE_MOM_1 + DEAD_MOM_1 + LIVE_MOM_1;
second = LIVE_MOM_2 + DEAD_MOM_2 + LIVE_MOM_2;
% molecular weights
Mnbar = 28.08*(first./zero);
Mwbar = 28.08*(second./first);
modMnbar = Mnbar(last);
modMwbar = Mwbar(last);
PDI = Mwbar ./ Mnbar;
modPDI = PDI(last);
expPDI = 2;
M2 = (1 - (modPDI/expPDI));
value_temporary = sumsqr([M1 M2]);
% value_temporary = sumsqr([M1 M2 M3]);
if (value_temporary <= value(i))
population(i,:) = ui(i,:);
value(i) = value_temporary;
w(i) = wi(i);
if (value_temporary < value_best)
value_best = value_temporary;
271
best_member = ui(i,:);
weight_best = w(i);
end
end
end
number_function_evlauation = number_function_evlauation + NP;
best_member_iteration = best_member;
if (refresh > 0)
if (rem (iter, refresh) == 0)
fid = fopen('ET_90_60_T.txt', 'a+');
fprintf (fid, 'Iteration: %d, Best: %8.4e, Worst: %8.4e\n', iter, value_best,
max(value));
fclose(fid);
for n = 1:D
fid = fopen('ET_90_60_T.txt', 'a+');
fprintf(fid, '\n k(%d) = %e\n', n, exp(best_member(n)));
fprintf('\n k(%d) = %e\n', n, exp(best_member(n)));
fclose(fid);
end
end
end
iter = iter + 1;
end
fid = fopen('ET_90_60_T.txt', 'a+');
fprintf (fid, 'Mnbar: %d,\t Mwbar: %d,\t PDI: %d, \n\n', Mnbar(last), Mwbar(last),
PDI(last));
fclose(fid);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
pass = best_member;
tspan= 0:time(length(time)); % <sec>
y0 = [1e-89 1e-77 1e-94 1e-82 1e-73 1e-75 1e-85 1e-95 1e-65 Zr MAO];
[t, y] = ode15s(@FUNE2,tspan,y0);
last = tspan(length(tspan));
LIVE_MOM_0 = y(:,1);
LIVE_MOM_1 = y(:,2);
LIVE_MOM_2 = y(:,3);
DEAD_MOM_0 = y(:,4);
DEAD_MOM_1 = y(:,5);
DEAD_MOM_2 = y(:,6);
LIVE_MOM_0 = y(:,7);
272
LIVE_MOM_1 = y(:,8);
LIVE_MOM_2 = y(:,9);
P0 = y(:,10);
CCocat = y(:,11);
Rp = exp(pass(2))*MEE*LIVE_MOM_0; % mol(ethylene)/s/L, y(1) = Lo
modRp = [Rp(1) Rp(61) Rp(121) Rp(241) Rp(301) Rp(421) Rp(481) Rp(541) Rp(661)
Rp(721) Rp(841) Rp(901) Rp(1021) Rp(1081) Rp(1201) Rp(1261) Rp(1381) Rp(1501)
Rp(1561) Rp(1681) Rp(1741) Rp(1861) Rp(1921) Rp(2041) Rp(2101) Rp(2161)
Rp(2221) Rp(2281) Rp(2341) Rp(2401) Rp(2461) Rp(2521) Rp(2581) Rp(2641)
Rp(2761) Rp(2821) Rp(2881) Rp(2941) Rp(3001) Rp(3061) Rp(3121) Rp(3241)
Rp(3301) Rp(3421) Rp(3481) Rp(3541) Rp(3601)];
%-------------------------
zero = LIVE_MOM_0 + DEAD_MOM_0 + LIVE_MOM_0;
first = LIVE_MOM_1 + DEAD_MOM_1 + LIVE_MOM_1;
second = LIVE_MOM_2 + DEAD_MOM_2 + LIVE_MOM_2;
% molecular weights
Mnbar = 28.08*(first./zero);
Mwbar = 28.08*(second./first);
modMnbar = Mnbar(last);
modMwbar = Mwbar(last);
PDI = Mwbar ./ Mnbar;
modPDI = PDI(last);
expPDI = 2;
fid = fopen('ET_90_60_T.txt', 'a+');
fprintf (fid, '\n\n');
fclose(fid);
fid = fopen('ET_90_60_T.txt', 'a+');
fprintf (fid, 'P0= %d,\n\n', P0);
fclose(fid);
fid = fopen('ET_90_60_T.txt', 'a+');
fprintf (fid, '\n\n');
fclose(fid);
fid = fopen('ET_90_60_T.txt', 'a+');
fprintf (fid, 'Cocat= %d,\n\n', CCocat);
fclose(fid);
fid = fopen('ET_90_60_T.txt', 'a+');
fprintf (fid, '\n\n');
fclose(fid);
fid = fopen('ET_90_60_T.txt', 'a+');
273
fprintf (fid, 'fraction with tdb f(=) = %d,\n\n',
LIVE_MOM_0/(LIVE_MOM_0+LIVE_MOM_0));
fclose(fid);
lcb = exp(pass(7))*LIVE_MOM_0*LIVE_MOM_0;
lcb = lcb(last);
lcb1000C = (500*lcd)/first;
fid = fopen('ET_90_60_T.txt', 'a+');
fprintf (fid, 'lcb/1000C = %d,\n\n', lcb1000C);
fclose(fid);
plot(time, expRp, 'k^', tspan, Rp, '-k', time, modRp, '>g');
if (iter >= Iteration_limit)
warning('max. number of iterations reached (Iteration_limit)') %#ok<WNTAG>
end
if (number_function_evlauation >= function_eval_limit)
warning('max. number of function evaluations reached (function_eval_limit)')
%#ok<WNTAG>
end
if (value_best < VTR)
warning('best value has been obtained') %#ok<WNTAG>
end
------------------------------------------------------------------------------------------------------------
Function
------------------------------------------------------------------------------------------------------------
function dy = FUNE2(t,y) global MEE pass % TEMP = 60 oC and Pressure = 80 psig % kin = exp(pass(1)); kp = exp(pass(2)); kd = exp(pass(3)); ktM = exp(pass(4)); ktCo = exp(pass(5)); kbeta = exp(pass(6)); klcb = exp(pass(7)); %%%%%%% Variables used in DEs corresponds to %y(1): LIVE_MOM_0 Zeroth Moment of Living Polymer Chain Length Distribution %y(2): LIVE_MOM_1 First Moment of Living Polymer Chain Length Distribution %y(3): LIVE_MOM_2 Second Moment of Living Polymer Chain Length Distribution %y(4): DEAD_MOM_0 Zeroth Moment of Dead Polymer Chain Length Distribution %y(5): DEAD_MOM_1 First Moment of Dead Polymer Chain Length Distribution %y(6): DEAD_MOM_2 Second Moment of Dead Polymer Chain Length Distribution %y(7): LIVE_MOM_0 Zeroth Moment of Dead Polymer Chain Length Distribution
with terimnal '='
274
%y(8): LIVE_MOM_1 First Moment of Dead Polymer Chain Length Distribution with
terimnal '=' %y(9): LIVE_MOM_2 Second Moment of Dead Polymer Chain Length Distribution
with terimnal '=' %y(10): P(0) Catalyst activated complex concentration %y(11): Cocat Cocatalyst concentration dy(1) = kin*MEE*y(10)-(kd+kbeta)*y(1); dy(2) = kin*MEE*y(10)-
(ktM*MEE+kd+ktCo*y(11)+kbeta)*y(2)+(ktM*MEE+ktCo*y(11)+kp*MEE+klcb*y(8))
*y(1); dy(3) =
kin*MEE*y(10)+(ktM*MEE+ktCo*y(11)+kp*MEE+klcb*y(9))*y(1)+2*(kp*MEE+klcb
*y(8))*y(2)-(ktM*MEE+kd+ktCo*y(11)+kbeta)*y(3); dy(4) = kd*y(1); dy(5) = kd*y(2); dy(6) = kd*y(3); dy(7) = (ktM*MEE+ktCo*y(11)+kbeta-klcb*y(7))*y(1); dy(8) = (ktM*MEE+ktCo*y(11)+kbeta)*y(2)-klcb*y(8)*y(1); dy(9) = (ktM*MEE+ktCo*y(11)+kbeta)*y(3)-klcb*y(9)*y(1); dy(10) = -kin*MEE*y(10)+ktCo*y(11)*y(1)+kbeta*y(1); dy(11) = -ktCo*y(11)*y(1); dy = dy';
275
APPENDIX - III
Code in MATLAB to Estimate the Kinetic Parameters in Propylene
Polymerization with Me2Si[Ind]2ZrCl2 (P1)/MAO
------------------------------------------------------------------------------------------------------------
Main Program
This script is representative one, and is used (with modifications) for kinetic parameter
estimation in propylene polymerization with all catalysts studied in this work.
------------------------------------------------------------------------------------------------------------
% Estudo Comparativo de "Polimerização de Propileno com Diferentes Catalisadores
Metalocênicos
% Através de um Planejamento de Experimentos"
% by Maria et al.
% % % % % % Polímeros: Ciência e Tecnologia, vol. 12, nº 1, p. 48-59, 2002.
%%%%%%%%%%%%% TEMP = 75 oC TEMP = 75 oC TEMP = 75 oC
%%%%%%%%%%%%%%
% Author: Nikhil Prakash / Dated 20/Aug/2012
clear all
clear
clc
%%%% ALL NEW THETA %%%%%%%%%
% Theta Set of optimization using kin, kp, kd, ktM, kH, krH, ks, ksp, ksM, kAl, krAl
values
%%%%%%%%%% MAO/Zr = 500 %%%%%%%%%%%
% PARA corresponds to lower and upper bounds to 'k' values, named as pass globally
% Lagarithmic values will be used in searching 'k' values, which will be properly
corrected in the function with DEqs.
format long g
% PARA = [kin kp kd ktM kH krH ks ksp ksM kAl krAl]
PARAmin = log([1e-4 1e+4 1e-5 1e-2 1e-6 1e-2 1e-7 1e-4 1e-3 1e-3 1e+1]); % log
(k)min
PARAmax = log([1e-2 1e+6 1e-3 1e+0 1e-3 1e+3 1e-2 1e+0 1e+3 1e+3 1e+5]); % log
(k)max
F = 0.7;
CR = 0.9;
refresh = 1;
VTR = 0;
function_eval_limit = Inf;
Iteration_limit = 81;
global MPP
% P = 2; % <bar>
276
% x = 0.0002*P^6 - 0.0015*P^5 + 0.0048*P^4 - 0.0069*P^3 + 0.007*P^2 + 0.1529*P -
0.0012; %% P-x relation at T = 0.0 0C
% moltol = 100*0.8669/92.1381;
% molpro = (x/(1-x))*moltol;
T = 25+273;
MPP = 0.354-(6.75e-3)*T+(3.66e-5)*T^2; %<mol/l>
% MPP = molpro*10; %per liter
Zr = 10e-6; % [Zr] <mol/L>
MAO = 500*Zr;
D = length (PARAmin);
NP = 200*D;
fid = fopen('P3C1_75_500.txt', 'a+');
fprintf (fid,'\n\n\n\n\t\t RUN number = [00002]\n\n\n');
fclose(fid);
fid = fopen('P3C1_75_500.txt', 'a+');
fprintf (fid,datestr(now)); % writing date and time in file
fclose(fid);
fid = fopen('P3C1_75_500.txt', 'a+');
fprintf (fid, '\n PARA = [kin kp kd ktM kH krH ks ksp ksM kAl krAl] \n\n PARAmin
= [%d %d %d %d %d %d %d %d %d %d %d] \t and \n PARAmax = [%d %d %d %d
%d %d %d %d %d %d %d]\n\n F = [%f]\t CR = [%f]\t NP = [%f]\n\n', exp(PARAmin),
exp(PARAmax), F, CR, NP);
fclose(fid);
if (length(PARAmin) ~= length(PARAmax))
error('Length of upper and lower bounds does not match.')
end
if (NP < 5)
error('Populationulation size NP must be bigger than 5.')
end
if ((F <= 0) || (F > 2))
error('Difference Factor F out of range (0,2].')
end
if ((CR < 0) || (CR > 1))
error('CR value out of range [0,1].')
end
if (Iteration_limit <= 0)
error('Iteration_limit must be positive.')
end
if (function_eval_limit <= 0)
error('function_eval_limit must be positive.')
end
refresh = floor(abs(refresh));
population = zeros(NP,D); % initialize population
277
for i = 1:NP
population(i,:) = PARAmin + rand (1,D).* (PARAmax - PARAmin);
end
w = rand (NP,1);
wi = w;
population_old = zeros (size (population));
value = zeros (1, NP);
best_member = zeros (1, D);
best_member_iteration = zeros (1 ,D);
number_function_evlauation = 0;
ibest = 1;
global pass
pass = population(ibest,:) ;
time = [117 175 233 292 358 417 483 533 600 650 700 775 842 892 958 1008 1075 1142
1192 1258 1317 1383 1442 1492 1558 1617 1675 1733 1800 1858 1917 1967 2033 2100
2150 2217 2267 2333 2400 2450 2525 2583 2633 2700 2758 2817 2883 2933 2992 3050
3108 3175 3225 3300 3358 3425 3492 3533 3600]; %% (seconds)
expRp = [3.48168 4.94764 5.16754 5.46073 5.31414 5.38743 5.35079 5.24084 5.27749
5.16754 5.09424 5.05759 4.94764 4.80105 4.72775 4.6911 4.6178 4.5445 4.43455
4.43455 4.32461 4.21466 4.25131 4.10471 3.99476 3.95812 3.92147 3.81152 3.77487
3.73822 3.66492 3.62827 3.55497 3.40838 3.48168 3.33508 3.33508 3.33508 3.29843
3.15183 3.11518 2.96859 2.89529 2.82199 2.74869 2.71204 2.71204 2.63874 2.5288
2.4555 2.4555 2.41885 2.34555 2.34555 2.34555 2.1623 2.2356 2.12565 2.08901];
tspan= 0:time(length(time)); % <sec>
y0 = [1e-50 1e-52 1e-55 1e-49 1e-48 1e-52 1e-49 1e-66 1e-59 1e-56 Zr 1e-57 MAO 1e-
77 1e-87];
[t, y] = ode15s(@FUNPROP,tspan,y0);
last = tspan(length(tspan));
LIVE_MOM_0 = y(:,1);
LIVE_MOM_1 = y(:,2);
LIVE_MOM_2 = y(:,3);
MIS_MOM_0 = y(:,4);
MIS_MOM_1 = y(:,5);
MIS_MOM_2 = y(:,6);
DEAD_MOM_0v = y(:,7);
DEAD_MOM_0B = y(:,8);
DEAD_MOM_1 = y(:,9);
DEAD_MOM_2 = y(:,10);
DEAD_MOM_0I = y(:,15);
Rp = exp(pass(2))*MPP*LIVE_MOM_0;
modRp = [Rp(118) Rp(176) Rp(234) Rp(293) Rp(359) Rp(418) Rp(484) Rp(534)
Rp(601) Rp(651) Rp(701) Rp(776) Rp(843) Rp(893) Rp(959) Rp(1009) Rp(1076)
278
Rp(1143) Rp(1193) Rp(1259) Rp(1318) Rp(1384) Rp(1443) Rp(1493) Rp(1559)
Rp(1618) Rp(1676) Rp(1734) Rp(1801) Rp(1859) Rp(1918) Rp(1968) Rp(2034)
Rp(2101) Rp(2151) Rp(2218) Rp(2268) Rp(2334) Rp(2401) Rp(2451) Rp(2526)
Rp(2584) Rp(2634) Rp(2701) Rp(2759) Rp(2818) Rp(2884) Rp(2934) Rp(2993)
Rp(3051) Rp(3109) Rp(3176) Rp(3226) Rp(3301) Rp(3359) Rp(3426) Rp(3493)
Rp(3534) Rp(3601)];
M1 = (1 - (modRp./expRp));
zero = LIVE_MOM_0 + MIS_MOM_0 + DEAD_MOM_0v + DEAD_MOM_0B +
DEAD_MOM_0I;
first = LIVE_MOM_1 + MIS_MOM_1 + DEAD_MOM_1;
second = LIVE_MOM_2 + MIS_MOM_2 + DEAD_MOM_2;
% molecular weights
Mnbar = 42.08*(first./zero);
Mwbar = 42.08*(second./first);
modMnbar = Mnbar(last);
modMwbar = Mwbar(last);
% expMnbar = ;
% expMwbar = ;
% M2 = (1 - (modMnbar/expMnbar));
% M3 = (1 - (modMwbar/expMwbar));
PDI = Mwbar ./ Mnbar;
modPDI = PDI(last);
% expPDI = ;
% M2 = (1 - (modPDI/expPDI));
%---------------------------------
modB =
(DEAD_MOM_0B(last)*100)/(DEAD_MOM_0B(last)+DEAD_MOM_0v(last)+DEAD_
MOM_0I(last));
% expB = ;
% M3 = (1 - (modB/expB));
%----------------------------------
modI =
(DEAD_MOM_0I(last)*100)/(DEAD_MOM_0B(last)+DEAD_MOM_0v(last)+DEAD_
MOM_0I(last));
% expI = ;
% M4 = (1 - (modI/expI));
%-----------------------------
value(1) = sumsqr([M1]);
% value(1) = sumsqr([M1 M2 M3 M4]);
value_best = value(1);
weight_best = w(1);
number_function_evlauation = number_function_evlauation + 1;
279
for i = 2 : NP
pass = population(i,:);
tspan= 0:time(length(time)); % <sec>
y0 = [1e-50 1e-52 1e-55 1e-49 1e-48 1e-52 1e-49 1e-66 1e-59 1e-56 Zr 1e-57 MAO 1e-
77 1e-87];
[t, y] = ode15s(@FUNPROP,tspan,y0);
last = tspan(length(tspan));
LIVE_MOM_0 = y(:,1);
LIVE_MOM_1 = y(:,2);
LIVE_MOM_2 = y(:,3);
MIS_MOM_0 = y(:,4);
MIS_MOM_1 = y(:,5);
MIS_MOM_2 = y(:,6);
DEAD_MOM_0v = y(:,7);
DEAD_MOM_0B = y(:,8);
DEAD_MOM_1 = y(:,9);
DEAD_MOM_2 = y(:,10);
DEAD_MOM_0I = y(:,15);
Rp = exp(pass(2))*MPP*LIVE_MOM_0;
modRp = [Rp(118) Rp(176) Rp(234) Rp(293) Rp(359) Rp(418) Rp(484) Rp(534)
Rp(601) Rp(651) Rp(701) Rp(776) Rp(843) Rp(893) Rp(959) Rp(1009) Rp(1076)
Rp(1143) Rp(1193) Rp(1259) Rp(1318) Rp(1384) Rp(1443) Rp(1493) Rp(1559)
Rp(1618) Rp(1676) Rp(1734) Rp(1801) Rp(1859) Rp(1918) Rp(1968) Rp(2034)
Rp(2101) Rp(2151) Rp(2218) Rp(2268) Rp(2334) Rp(2401) Rp(2451) Rp(2526)
Rp(2584) Rp(2634) Rp(2701) Rp(2759) Rp(2818) Rp(2884) Rp(2934) Rp(2993)
Rp(3051) Rp(3109) Rp(3176) Rp(3226) Rp(3301) Rp(3359) Rp(3426) Rp(3493)
Rp(3534) Rp(3601)];
M1 = (1 - (modRp./expRp));
zero = LIVE_MOM_0 + MIS_MOM_0 + DEAD_MOM_0v + DEAD_MOM_0B +
DEAD_MOM_0I;
first = LIVE_MOM_1 + MIS_MOM_1 + DEAD_MOM_1;
second = LIVE_MOM_2 + MIS_MOM_2 + DEAD_MOM_2;
% molecular weights
Mnbar = 42.08*(first./zero);
Mwbar = 42.08*(second./first);
modMnbar = Mnbar(last);
modMwbar = Mwbar(last);
% expMnbar = ;
% expMwbar = ;
% M2 = (1 - (modMnbar/expMnbar));
% M3 = (1 - (modMwbar/expMwbar));
PDI = Mwbar ./ Mnbar;
modPDI = PDI(last);
280
% expPDI = ;
% M2 = (1 - (modPDI/expPDI));
%---------------------------------
modB =
(DEAD_MOM_0B(last)*100)/(DEAD_MOM_0B(last)+DEAD_MOM_0v(last)+DEAD_
MOM_0I(last));
% expB = ;
% M3 = (1 - (modB/expB));
%----------------------------------
modI =
(DEAD_MOM_0I(last)*100)/(DEAD_MOM_0B(last)+DEAD_MOM_0v(last)+DEAD_
MOM_0I(last));
% expI = ;
% M4 = (1 - (modI/expI));
%-----------------------------
value(i) = sumsqr([M1]);
% value(i) = sumsqr([M1 M2 M3 M4]);
number_function_evlauation = number_function_evlauation + 1;
if (value(i) < value_best)
ibest = i;
value_best = value(i);
weight_best = w(i);
end
end
best_member_iteration = population(ibest,:);
value_bestit = value_best;
best_member = best_member_iteration;
pm1 = zeros (NP, D);
pm2 = zeros (NP, D);
pm3 = zeros (NP, D);
pm4 = zeros (NP, D);
pm5 = zeros (NP, D);
bm = zeros (NP, D);
ui = zeros (NP, D);
mui = zeros (NP, D);
mpo = zeros (NP, D);
rot = 0:1:NP-1;
rotd= 0:1:D-1;
rt = zeros (NP);
rtd = zeros (D);
a1 = zeros (NP);
a2 = zeros (NP);
a3 = zeros (NP);
a4 = zeros (NP);
a5 = zeros (NP);
281
ind = zeros (4);
iter = 1;
while (iter < Iteration_limit) && (value_best > VTR) % &&
(number_function_evlauation < function_eval_limit)
fprintf('\n\n\n Iteration No.[%d] \t of \t [%d]\n Going Well', iter, Iteration_limit);
population_old = population;
wold = w;
ind = randperm (4);
a1 = randperm (NP);
rt = rem (rot + ind(1), NP);
a2 = a1(rt+1);
rt = rem (rot + ind(2), NP);
a3 = a2(rt+1);
rt = rem (rot +ind(3), NP);
a4 = a3(rt+1);
rt = rem (rot + ind(4), NP);
a5 = a4(rt+1);
pm1 = population_old(a1,:);
pm2 = population_old(a2,:);
pm3 = population_old(a3,:);
pm4 = population_old(a4,:);
pm5 = population_old(a5,:);
w1 = wold(a1);
w2 = wold(a2);
bm = repmat (best_member_iteration, NP, 1);
bw = repmat(weight_best, NP, 1);
mui = rand (NP, D) < CR;
mui = sort (mui');
for i = 1:NP
n = floor (rand * D);
if n > 0
rtd = rem (rotd + n, D);
mui(:,i) = mui(rtd+1,i);
end
end
mui = mui';
mpo = mui < 0.5;
ui = pm3 + F*(pm1 - pm2);
ui = population_old.*mpo + ui.*mui;
282
for i = 1:NP
ui(i,:) = max (ui(i,:), PARAmin);
ui(i,:) = min (ui(i,:), PARAmax);
end
for i = 1:NP
pass = ui(i,:);
tspan= 0:time(length(time)); % <sec>
y0 = [1e-50 1e-52 1e-55 1e-49 1e-48 1e-52 1e-49 1e-66 1e-59 1e-56 Zr 1e-57 MAO 1e-
77 1e-87];
[t, y] = ode15s(@FUNPROP,tspan,y0);
last = tspan(length(tspan));
LIVE_MOM_0 = y(:,1);
LIVE_MOM_1 = y(:,2);
LIVE_MOM_2 = y(:,3);
MIS_MOM_0 = y(:,4);
MIS_MOM_1 = y(:,5);
MIS_MOM_2 = y(:,6);
DEAD_MOM_0v = y(:,7);
DEAD_MOM_0B = y(:,8);
DEAD_MOM_1 = y(:,9);
DEAD_MOM_2 = y(:,10);
DEAD_MOM_0I = y(:,15);
Rp = exp(pass(2))*MPP*LIVE_MOM_0;
modRp = [Rp(118) Rp(176) Rp(234) Rp(293) Rp(359) Rp(418) Rp(484) Rp(534)
Rp(601) Rp(651) Rp(701) Rp(776) Rp(843) Rp(893) Rp(959) Rp(1009) Rp(1076)
Rp(1143) Rp(1193) Rp(1259) Rp(1318) Rp(1384) Rp(1443) Rp(1493) Rp(1559)
Rp(1618) Rp(1676) Rp(1734) Rp(1801) Rp(1859) Rp(1918) Rp(1968) Rp(2034)
Rp(2101) Rp(2151) Rp(2218) Rp(2268) Rp(2334) Rp(2401) Rp(2451) Rp(2526)
Rp(2584) Rp(2634) Rp(2701) Rp(2759) Rp(2818) Rp(2884) Rp(2934) Rp(2993)
Rp(3051) Rp(3109) Rp(3176) Rp(3226) Rp(3301) Rp(3359) Rp(3426) Rp(3493)
Rp(3534) Rp(3601)];
M1 = (1 - (modRp./expRp));
zero = LIVE_MOM_0 + MIS_MOM_0 + DEAD_MOM_0v + DEAD_MOM_0B +
DEAD_MOM_0I;
first = LIVE_MOM_1 + MIS_MOM_1 + DEAD_MOM_1;
second = LIVE_MOM_2 + MIS_MOM_2 + DEAD_MOM_2;
% molecular weights
Mnbar = 42.08*(first./zero);
Mwbar = 42.08*(second./first);
modMnbar = Mnbar(last);
modMwbar = Mwbar(last);
% expMnbar = ;
283
% expMwbar = ;
% M2 = (1 - (modMnbar/expMnbar));
% M3 = (1 - (modMwbar/expMwbar));
PDI = Mwbar ./ Mnbar;
modPDI = PDI(last);
% expPDI = ;
% M2 = (1 - (modPDI/expPDI));
%---------------------------------
modB =
(DEAD_MOM_0B(last)*100)/(DEAD_MOM_0B(last)+DEAD_MOM_0v(last)+DEAD_
MOM_0I(last));
% expB = ;
% M3 = (1 - (modB/expB));
%----------------------------------
modI =
(DEAD_MOM_0I(last)*100)/(DEAD_MOM_0B(last)+DEAD_MOM_0v(last)+DEAD_
MOM_0I(last));
% expI = ;
% M4 = (1 - (modI/expI));
%-----------------------------
value_temporary = sumsqr([M1 M2]);
% value_temporary = sumsqr([M1 M2 M3 M4]);
if (value_temporary <= value(i))
population(i,:) = ui(i,:);
value(i) = value_temporary;
w(i) = wi(i);
if (value_temporary < value_best)
value_best = value_temporary;
best_member = ui(i,:);
weight_best = w(i);
end
end
end
number_function_evlauation = number_function_evlauation + NP;
best_member_iteration = best_member;
if (refresh > 0)
if (rem (iter, refresh) == 0)
fid = fopen('P3C1_75_500.txt', 'a+');
fprintf (fid, 'Iteration: %d, Best: %8.4e, Worst: %8.4e\n', iter, value_best,
max(value));
fclose(fid);
for n = 1:D
fid = fopen('P3C1_75_500.txt', 'a+');
284
fprintf(fid, '\n k(%d) = %e\n', n, exp(best_member(n)));
fprintf('\n k(%d) = %e\n', n, exp(best_member(n)));
fclose(fid);
end
end
end
iter = iter + 1;
end
fid = fopen('P3C1_75_500.txt', 'a+');
fprintf (fid, 'Mnbar: %d,\t Mwbar: %d,\t PDI: %d, \t B: %d,\t I: %d, \n\n',
Mnbar(last), Mwbar(last), PDI(last), modB, modI);
fclose(fid);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
pass = best_member;
tspan= 0:time(length(time)); % <sec>
y0 = [1e-50 1e-52 1e-55 1e-49 1e-48 1e-52 1e-49 1e-66 1e-59 1e-56 Zr 1e-57 MAO 1e-
77 1e-87];
[t, y] = ode15s(@FUNPROP,tspan,y0);
last = tspan(length(tspan));
LIVE_MOM_0 = y(:,1);
Rp = exp(pass(2))*MPP*LIVE_MOM_0;
modRp = [Rp(118) Rp(176) Rp(234) Rp(293) Rp(359) Rp(418) Rp(484) Rp(534)
Rp(601) Rp(651) Rp(701) Rp(776) Rp(843) Rp(893) Rp(959) Rp(1009) Rp(1076)
Rp(1143) Rp(1193) Rp(1259) Rp(1318) Rp(1384) Rp(1443) Rp(1493) Rp(1559)
Rp(1618) Rp(1676) Rp(1734) Rp(1801) Rp(1859) Rp(1918) Rp(1968) Rp(2034)
Rp(2101) Rp(2151) Rp(2218) Rp(2268) Rp(2334) Rp(2401) Rp(2451) Rp(2526)
Rp(2584) Rp(2634) Rp(2701) Rp(2759) Rp(2818) Rp(2884) Rp(2934) Rp(2993)
Rp(3051) Rp(3109) Rp(3176) Rp(3226) Rp(3301) Rp(3359) Rp(3426) Rp(3493)
Rp(3534) Rp(3601)];
plot(time, expRp, 'k^', tspan, Rp, '-k', time, modRp, '>g');
if (iter >= Iteration_limit)
warning('max. number of iterations reached (Iteration_limit)') %#ok<WNTAG>
end
if (number_function_evlauation >= function_eval_limit)
warning('max. number of function evaluations reached (function_eval_limit)')
%#ok<WNTAG>
end
if (value_best < VTR)
warning('best value has been obtained') %#ok<WNTAG>
end
------------------------------------------------------------------------------------------------------------
285
Function
------------------------------------------------------------------------------------------------------------
function dy = FUNPROP(t,y)
% This function contains all ODEs for Estudo Comparativo de
% "Polimerização de Propileno com Diferentes Catalisadores Metalocênicos
% Através de um Planejamento de Experimentos"
% by Maria etal.
% % % % % % Polímeros: Ciência e Tecnologia, vol. 12, nº 1, p. 48-59, 2002.
%%%%%%%%%%%%% TEMP = 75 oC %%%%%%%%%%%%%%
global MPP
global pass
kin = exp(pass(1));
kp = exp(pass(2));
kd = exp(pass(3));
ktM = exp(pass(4));
kH = exp(pass(5));
krH = exp(pass(6));
ks = exp(pass(7));
ksp = exp(pass(8));
ksM = exp(pass(9));
kAl = exp(pass(10));
krAl = exp(pass(11));
%%%%%%% Variables used in DEs corresponds to
%y(1): Lo Zeroth Moment of Living Polymer Chain Length Distribution
%y(2): LIVE_MOM_1 First Moment of Living Polymer Chain Length Distribution
%y(3): LIVE_MOM_2 Second Moment of Living Polymer Chain Length Distribution
%y(4): MIS_MOM_0 Zeroth Moment of (2,1) inserted Polymer Chain Length
Distribution
%y(5): MIS_MOM_1 First Moment of (2,1) inserted Polymer Chain Length
Distribution
%y(6): MIS_MOM_2 Second Moment of (2,1) inserted Polymer Chain Length
Distribution
%y(7): DEAD_MOM_0v Zeroth Moment of Dead Polymer Chain Length Distribution
(vinylidene end group)
%y(8): DEAD_MOM_0B Zeroth Moment of Dead Polymer Chain Length
Distribution(Butentl end group)
%y(9): DEAD_MOM_1 First Moment of Dead Polymer Chain Length Distribution
%y(10): DEAD_MOM_2 Second Moment of Dead Polymer Chain Length Distribution
%y(11): Cstar Catalyst activated complex concentration
%y(12): CHstar Hydride catalyst activated complex concentration
%y(13): MAO Cocatalyst concentration
%y(14): CMestar Methyl catalyst activated complex concentration
%y(15): DEAD_MOM_0I Zeroth Moment of Dead Polymer Chain Length
Distribution(I end group)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
286
% LISTED HERE ARE THE ODEs TO BE SOLVED
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
dy(1) = kin*MPP*y(11)-
(kd+kH+(ks*MPP)+(kAl*y(13)))*y(1)+(ksp+ksM)*MPP*y(4)+krH*MPP*y(12)+krAl*
MPP*y(14);
dy(2) = kin*MPP*y(11)+kp*MPP*y(1)-
(kd+kH+(ks*MPP)+(kAl*y(13)))*y(2)+ktM*MPP*(y(1)-y(2))+ksp*MPP*(y(4)+
y(5))+ksM*MPP*y(4)+krH*MPP*y(12)+krAl*MPP*y(14);
dy(3) = kin*MPP*y(11)+kp*MPP*(y(1)+2*y(2))-
(kd+kH+ks*MPP+kAl*y(13))*y(3)+ktM*MPP*(y(1)-
y(3))+ksp*MPP*(y(4)+2*y(5)+y(6))+ksM*MPP*y(4)+krH*MPP*y(12)+krAl*MPP*y(1
4);
dy(4) = ks*MPP*y(1)-(ksp+ksM)*MPP*y(4);
dy(5) = ks*MPP*(y(1)+y(2))-(ksp+ksM)*MPP*y(5);
dy(6) = ks*MPP*(y(1)+2*y(2)+y(3))-(ksp+ksM)*MPP*y(6);
dy(7) = (kd+kH+(ktM*MPP))*y(1);
dy(8) = ksM*MPP*y(4);
dy(9) = (kd+kH+(ktM*MPP)+kAl*y(13))*y(2)+ksM*MPP*y(5);
dy(10) = (kd+kH+(ktM*MPP)+kAl*y(13))*y(3)+ksM*MPP*y(6);
dy(11) = -kin*MPP*y(11);
dy(12) = kH*y(1)-krH*MPP*y(12);
dy(13) = -kAl*y(13)*y(1);
dy(14) = kAl*y(13)*y(1)-krAl*MPP*y(14);
dy(15) = kAl*y(13)*y(1);
dy = dy';
287
BIOGRAPHIES
Biography of the Candidate
Mr Nikhil Prakash is serving as a Lecturer (since August 2005) in Department of Chemical
Engineering at BITS-Pilani, Pilani Campus, and pursuing his PhD under the supervision of
Dr Arvind Kumar Sharma and Dr Sushil Kumar. He earned his B Tech degree in Chemical
Engineering from Bundelkhand Institute of Engineering & Technology (BIET)-Jhansi in
2001 and M E (Chemical Engg.) from BITS-Pilani in 2003.
Mr Prakash has 10 years of experience in teaching undergraduate and post graduation
students in the field of Chemical Engineering, Polymer Engineering & Petroleum
Engineering and his areas of Research are Polymer Science and Engineering; Process
Engineering; Modeling, Simulation and Optimization; Reaction Engineering; Process
Dynamics and Control; Artificial Neural Networks and Catalysis.
He has guided 1 ME Dissertation, 2 BE Thesis and 260+ Projects (Study oriented
projects; Computer projects; Design projects; Special projects; Professional Practice;
Research Practice and Course projects of ME). He has published 21 research papers in
national & international journals and conferences, 2 book chapters and participated in 25
seminars/conferences/symposia/workshops.
He is the reviewer of 4 International Journals (American Journal of Polymer Science;
International Journal of Materials Engineering; Nanoscience and Nanotechnology; Science
and Technology). He is awarded Minor Research Project by University Grant Commission
(UGC), New Delhi India (2013-15) and Travel Grant by Department of Science &
Technology (DST), New Delhi, India to attend 2012-AIChE Annual Meeting held at
Pittsburgh PA, USA.
He has been a Organizing Committee Member & Resource Faculty for Workshop on
Analytical Instruments for Chemical and Environmental Engineers (WAICEE - 2013), BITS
Pilani, March 22-23, 2013, Organizing Committee Member for 8th Annual Session of
Students' Chemical Engineering Congress (SCHEMCON-2012), BITS Pilani, Sept. 21-22,
2012; Conference on Technological Advancements in Chemical and Environmental
Engineering (TACEE-2012), BITS Pilani, March 23-24, 2012; Conference on Photonic
Polymers: Materials, Devices and Applications (PPMDA-2008), BITS Pilani, April 3-4,
2008; National Conference on Environmental Conservation (NCEC-06), BITS Pilani, Sept.
1-3, 2006 and Organizing Faculty Member for Intensive Teaching Workshop (ITW), BITS
Pilani, April, 2006.
Mr Prakash is the Life associate member of Indian Institute of Chemical Engineers
(IIChE); Life member of Asian Polymer Association (APA), International Association of
Engineers (IAENG), Asia-Pacific Chemical, Biological & Environmental Engineering
Society (APCBEES); Member of AIChE from 2012, and Honorary Treasurer of Indian
Institute of Chemical Engineers (IIChE), Pilani Regional Centre, Pilani.
288
Biography of the Supervisor
Dr Arvind Kumar Sharma graduated (B Tech) from Harcourt Butler Technological
Institute (HBTI) Kanpur in 1986, did Masters (MS) in 1992 and PhD in 2005, both from
Indian Institute of Technology (IIT) Madras.
He worked as a Junior Research Fellow in the Department of Post Harvest Process
and Food Engineering, College of Technology, Govind Ballabh Pant University of
Agriculture and Technology, Pantnagar, during 1986-87. Later at IIT Madras, during 1990-
92, he also worked as a Senior Project Officer for a project on Biogas Generation from
Tannery Effluents funded by the Department of Nonconventional Energy Sources of
Ministry of Energy and as a Research Associate for a project on Colour Removal of
Industrial Effluents using Fluidized Bed funded by Council of Scientific and Industrial
Research (CSIR) during 1997–2002.
Presently he is an Assistant Professor (since June 2006) in the Department of
Chemical Engineering at Birla Institute of Technology and Science (BITS) Pilani – Pilani
Campus, which he joined in December 2004 as a Lecturer.
He also served as Head of the Department (Jan 2007 to August 2012). During this
period, department was granted three major development funds: Departmental Research
Support (DRS) – Special Assistance Programme (SAP) of University Grant Commission
(UGC) [Rs. 48 Lakhs (2011 - 16)], UGC-Infrastructure Support [Rs. 20 Lakhs (2011-12)
and Fund for Improvement of S & T Infrastructure (FIST) of Department of Science and
Technology (DST) [Rs. 80.5 Lakhs (2011-16)]. Five (5) UGC-BSR research fellowships
were also granted to the department. The department also conducted two prominent
conferences: Technological Advancements in Chemical and Environmental Engineering
(TACEE-2012, March 23-24, 2012) and 8th
Annual Session of Students’ Chemical
Engineering Congress (SCHEMCON-2012, September 21-22, 2012), an annual event of
Indian Institute of Chemical Engineers (IIChE).
Dr. Sharma teaches in the field of Chemical Engineering, Environmental Engineering
& Biochemical Engineering and his areas of Research are Environmental Engineering
(Water and Wastewater Treatment), Adsorption, Fluidization, Fluid Dynamics, Biochemical
Engineering (Bioreactor Analysis and Design), Reaction Mechanism & Kinetics and
Modeling & Simulation. He has guided 2 ME Dissertations, around 20 Professional and
Research Practice students, 2 BE Theses and around 50+ BE Projects. He has published
around 30+ research papers in national & international journals and conferences, 5 chapters
in lecture notes and participated in 15+ short term
courses/seminars/conferences/symposiums/workshops/conventions.
He has reviewed research papers submitted for CURIE Journal (Journal of
Cooperation among University, Research and Industrial Enterprises); paper/proposal for 2nd
National Convention on “Energizing Entrepreneurship through Innovation”, Nov. 2-3,
2007, Pilani, India and coordinated the proof reading of 25 review/research papers for the
special issues of Journal of Energy, Heat and Mass Transfer – Dec. 1996 and March 1997
issues, viz., FESTSCHRIFT ISSUES in honour of Prof. Y B G VARMA, on his retirement
from the Department of Chemical Engineering, IIT Madras.
He delivered an Expert Lecture in Workshop on Analytical Instruments for Chemical
and Environmental Engineers (WAICEE 2013), March 22-23, 2013, Pilani, India; Co-
judged a Session in SCHEMCON – 2012, Sept. 21-22, 2012, Pilani, India and Chaired a
Session on Green Chemistry in International Conference on Sustainable Manufacturing :
Issues, Trends and Practices (ICSM 2011), Nov. 10-12, 2011, Pilani, India.
He has been a Member of Organizing Committee for Workshop on Analytical
Instruments for Chemical and Environmental Engineers (WAICEE 2013), March 22-23,
2013, Pilani, India; Member of Local Advisory Committee for National Conference on
Green and Sustainable Chemistry, Feb. 19-21, 2010, Pilani, India; Member of Local
Organizing Committee (Registration) for National Conference on Environmental
289
Conservation (NCEC-06), Sept. 1-3, 2006, Pilani, India; Member of Registration
Committee for Indian Chemical Engineering Congress (CHEMCON) – 2001, Dec. 19–22,
2001, Chennai, India and Student Member of Technical Programme Committee for
International Conference on Advances in Chemical Engineering, ICAChE – 96, Dec. 11–13,
1996, Chennai, India.
He has been an examiner for MTech Project at HBTI Kanpur (April 2013); BTech
(Industry) Projects at Banasthali University, Banasthali (Jan 2013 & Jan 2012); PhD Thesis,
ME Dissertations, Professional and Research Practice, BE Theses, Practice School (PS)
Project Reports and BE Projects at BITS Pilani.
At BITS Pilani, he is/has been a member of Senate, Research Board, Doctoral
Counseling Committee, Doctoral Advisory Committee, Departmental Research Committee,
IIChE - Pilani Regional Centre, Academic Counseling Cell and Library Committee.
He is a Member of Board of Management for Krishna Vidya Niketan, Muradnagar
(Gaziabad), India and Life Member of IIChE. For more details, please visit:
http://universe.bits-pilani.ac.in/pilani/arvinds/profile.
290
Biography of the Co-Supervisor
Dr Sushil Kumar, Assistant Professor, Department of Chemical Engineering at Motilal
Nehru National Institute of Technology (MNNIT), Allahabad has over 10 years of industrial,
teaching, and research experience. Prior to MNNIT, Allahabad, he served as an Assistant
Professor in Department of Chemical Engineering at BITS-Pilani, Pilani Campus. He also
worked with Central Institute of Plastic Engineering and Technology (CIPET), Lucknow for
one and half years as Technical Officer and Graduate Engineer Trainee. He did his B Tech
from Harcourt Butler Technological Institute (HBTI) - Kanpur, M Tech from Indian Institute
of Technology (IIT) - Kanpur and PhD from BITS - Pilani.
His current research interests include Process Intensification, Polymer Science &
Technology, Biochemical Engineering, Green Technology, Chemical Thermodynamics, and
Renewable Energy Sources. He has around 68 research publications (21 refereed journals,
45 conferences and 2 book chapters) to his credit which have been published over the years
in various International and National Journals and Conference Proceedings. Dr Kumar
guided one PhD in the area of Process Intensification (Reactive Extraction) and currently, he
is supervising 3 scholars for their doctoral research. Besides this, he has guided 5 ME
Dissertations and around 20 BE Project students under his supervision.
He is the referee and expert reviewer of 14 International Journals of repute (Journal
of Chemical and Engineering Data, Industrial and Engineering Chemistry Research,
Separation and Purification Technology, Fluid Phase Equilibria, Biotechnology and
Bioprocess Engineering, Desalination etc.). He also reviewed three books of Tata McGraw
Hill publisher. He is awarded Research Project by Department of Science and Technology
(DST), New Delhi, India under Fast Track Scheme for Young Scientists, 2012-2014.
Dr Kumar is the Life member of Indian Institute of Chemical Engineers (IIChE),
Fellow member of International Congress of Chemistry and Environment (ICCE), member
of AIChE for the year - 2010 to 2013, and Executive Committee Member, Lucknow
Regional Centre of IIChE chapter. He organized a national conference on “Technological
Advancements in Chemical and Environmental Engineering (TACEE - 2012)” held at BITS-
Pilani during March 23-24, 2012, and also worked as a Treasurer for SCHEMCON 2012
held during September 27-28, 2012 at BITS-Pilani.