MECHANISTIC MODELING AND MODEL-BASED STUDIES

IN SPONTANEOUS SOLUTION POLYMERIZATION

OF ALKYL ACRYLATE MONOMERS

A Thesis

Submitted to the Faculty

of

Drexel University

by

Felix Suryaman Rantow

in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

August 2006

ii

Copyright © 2006 by Felix Suryaman Rantow

All Rights Reserved

iii

ACKNOWLEDGEMENTS

I thank my co-advisors Prof. Masoud Soroush and Dr. Michael Grady for the

opportunity, the guidance and the invaluable advice given to me throughout my

graduate studies. I was fortunate to have been given the chance to engage in an

applied and interactive research work with people at Drexel Chemical

Engineering Department and Dupont Marshall Laboratory Process Engineering

Group. Undoubtedly the experience has positively influenced my personal and

professional growth. Prof. Soroush and Dr. Grady have always emphasized the

importance of ingenuity, originality, passion and effective communication in

making meaningful scientific contributions.

At Drexel, I was fortunate to have been given the chance to interact with fellow

graduate students. They have played roles in shaping how I perceive a

stimulating work environment should be. I thank students in Prof. Soroush’s

group, Nasir Mehranbod, Congling Quan, Chanin Panjapornpon, Swa Swarup

Metta and Sriraj Srinivasan for contributing positively to the work environment.

We as a group have shared work space with Prof. Nily Dan’s and Prof. Cameron

Abrams’ groups. I thank Veena Pata, Kaveh Komaee, Kevin Towles, Ehsan

Jabbarzadeh and Yelena Sliozberg for the stimulating discussions on formal and

informal matters. I also thank other students in the Chemical Engineering

Department who I have been given the chance to interact with daily, and through

many exciting graduate student activities. I believe these are also important to

the creation of a healthy work environment. There are too many names to

mention. I thank them nonetheless.

At Dupont Marshall Laboratory, I was very fortunate to have been given the

chance to interact with the process operators, the process engineers and the

scientists. I was able to absorb the work culture and the professionalism shown

by the people I interact with. I would like to thank Dr. George Kalfas for the

invaluable insights on the modeling and optimization studies. I would also like to

iv

thank Dr. Thomas Scholz and Mark Karalis for help with the polymer

characterization work. I thank Chris Bruni and the Drexel Co-op students Amutha

Jeyarajasingam and Erik Amrine for the help with scheduling experimental runs.

And I also thank Peter Markos, Mike Darczuk and Mike Rullo for running and

tending the experiments.

I would also like to express my gratitude to my committee members Prof.

Masoud Soroush, Prof. Charles Weinberger, Prof. Giuseppe Palmese of Drexel,

and Dr. Michael Grady, Dr. George Kalfas and Dr. Joan Hansen of DuPont. They

have given me their valuable time and advice in the process of my thesis writing

and defense.

I would also like to acknowledge my undergraduate Professor, Prof.

Raghunathan Rengaswamy, currently at Clarkson University, whose advice led

me to pursue work under the guidance of Prof. Masoud Soroush.

On a more personal note, I would not have been able to enjoy and truly

experience, or absorb, each day without the unconditional support given by my

parents, brother, and a special person in my life, Ervina Widjaja. I dedicate my

graduate work, and the way I enjoyed it, to my family; my parents, Joannes and

Jeanny Widodo Rantow, and my brother, Giovanni Darmawan Rantow.

v

Dedicated to my parents,

Joannes and Jeanny Widodo Rantow,

And my brother,

Giovanni Darmawan Rantow

vi

High-temperature (above 120°C) free-radical polymerization of alkyl acrylate

monomers is studied to gain a better quantitative understanding of the governing

reaction network behind the observed process measurement dynamics. At

higher reaction temperatures, mechanistic models developed based on lower-

temperature data fail to quantitatively predict the dynamics of polymer property

indices such as monomer conversion and average molecular weights. In recent

studies, this failure has been attributed to the occurrence of secondary chain

reactions that become increasingly significant with temperature increase.

Experimental data obtained from high-temperature batch polymerization of n-

butyl acrylate (nBA) in xylene points to the occurrence of short chain branch

formation, as opposed to long chain branch formation. This observation

substantiates the backbiting (intra-molecular chain transfer) mechanism

postulated in earlier studies on alkyl acrylate polymerization.

Abstract Mechanistic Modeling and Model-Based Studies in Spontaneous Solution

Polymerization of Alkyl Acrylate Monomers

Felix Suryaman Rantow

Masoud Soroush, Ph.D. and Michael Charles Grady, Sc.D.

vii

In this study the activation energy is estimated through multivariate data fitting of

several measurements. The estimated value of the activation energy of the

postulated self-initiation reaction points to actual occurrence of self-initiation. The

measurements include monomer conversion, number- and weight-average

molecular weights and microstructure quantities (number-average of terminal

solvent group, terminal double bond and chain branch per polymer chain). This

provides an explanation to the observed spontaneous polymerization in alkyl

acrylate polymerization at higher temperatures.

A free-radical polymerization mechanistic model for the spontaneous solution

high-temperature polymerization of n-butyl acrylate in xylene is developed. This

effort involves characterizing polymer solutions, postulating reaction networks,

screening the networks by performing global parameter identifiability analyses,

reducing the complexity of the developed mechanistic model, estimating

unknown kinetic parameters, and validating the model. A method of global

parameter identifiability analysis is developed for models that are affine in

parameters. The model reduction techniques used in this study include the

method of moments modeling method, the tendency modeling method, and the

quasi-steady-state assumptions on the concentrations of free radicals. The

model is used to (a) develop a model-based optimal control policy for a

spontaneously-initiated semi-batch polymerization reactor and (b) design of a

multi-rate state observer to estimate polymer-chain microstructure properties

from currently-available on-line and off-line measurements.

viii

ix

NOTATION

A, [A] Concentration of species A in reactor, A = M, TDB, etc.

[A]B Concentration of species A in feed flowrate of B, B = M, S, I

BP Chain-branching = SCB + LCB (Chapter 3)

BPH Chain-branching per 100 monomer unit = SCBH + LCBH (Chapter 3)

D 'Dead' polymer chain

FA Volumetric flow rate of species A, A = M, S, I

kbb Rate constant of backbiting reactions

kii Rate constant of thermal initiator decomposition reactions (Chapter 3)

kp Rate constant of propagation reactions

kpb Rate constant of chain-branch formation reactions (Chapter 3)

ktc Rate constant of termination by combination reactions

ktd Rate constant of termination by disproportionation reactions

ktdb Rate constant of propagation on terminal double bond reaction

kti Rate constant of self-initiation reactions

ktm Rate constant of chain-transfer to monomer reactions

ktp Rate constant of chain-transfer to polymer reactions

kts Rate constant of chain-transfer to solvent reactions

kβ Rate constant of β-scission reactions

LCB Long chain-branching

LCBH Long chain-branching per 100 monomer unit

mI Total mass of initiator used in reaction

x

M Monomer

Mn Number-average molecular weight

M Number of discretized values for different feed flow rates

N Number of equal batch time intervals (Chapter 3)

rA Rate of reaction of species A, A = M, TDB, etc.

P ‘Live' polymer chain, mid-chain tertiary radical species

Q ‘Live' polymer chain, near-chain-end tertiary radical species

R ‘Live' polymer chain, secondary radical species

R1 Secondary propagating radical (Chapter 3)

R2 Tertiary mid-chain propagating radical (Chapter 3)

R3 Tertiary near-chain-end propagating radical (Chapter 3)

R4 Unsaturated propagating radical (Chapter 3)

R Propagating radical = R1 + R2 + R3 + R4 (Chapter 3)

S Solvent

SCB Short chain-branching

SCBH Short chain-branching per 100 monomer unit

T Reaction time

TC Tertiary carbon in a polymer chain

TDB Terminal double bond

TDBH Terminal double bond(s) per 100 monomer unit

V Volume of reactor

MWm Molecular weight of monomer

xi

xm Monomer conversion

ωi Weighting factor of an objective in the performance index

xii

TABLE OF CONTENTS

1 INTRODUCTION AND BACKGROUND ....................................................... 1

1.1 SCOPE OF RECENT STUDIES IN MECHANISTIC MODELING

OF SOLUTION AND BULK FREE-RADICAL HOMO-POLYMERIZATION... 3

1.2 KINETICS OF FREE-RADICAL SOLUTION HOMO-

POLYMERIZATION ...................................................................................... 5

1.2.1 High-Temperature free-radical solution homo-polymerization

with depropagation involving styrenic/acrylic monomers ....................... 6

1.2.2 High-Temperature free-radical solution homo-polymerization

with branching and/or scission involving styrenic/acrylic monomers ..... 7

1.2.3 Spontaneous thermal initiation in high-temperature free-radical

polymerization involving styrenic/acrylic monomers ............................ 10

1.3 MODELING METHODS............................................................... 12

2 MECHANISTIC MODELING ....................................................................... 17

2.1 EXPERIMENTAL AND ANALYTICAL PROCEDURES................ 20

2.2 POSTULATION OF REACTION NETWORK ............................... 22

2.3 PARAMETER ESTIMATION........................................................ 27

2.4 DISCUSSIONS AND ANALYSES................................................ 33

2.5 CONCLUSIONS........................................................................... 51

3 MODEL-BASED OPTIMAL CONTROL OF A SEMI-BATCH HIGH-

TEMPERATURE POLYMERIZATION REACTOR ............................................. 52

3.1 EXPERIMENTAL FRAMEWORK................................................. 55

3.2 MODEL DEVELOPMENT ............................................................ 55

3.2.1 Kinetic Scheme and Rate Laws ............................................. 55

3.2.2 Process Model ....................................................................... 58

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3.3 PARAMETER ESTIMATION AND MODEL VALIDATION ........... 60

3.4 OPTIMIZATION AND OPTIMIZATION VALIDATION .................. 62

3.4.1 Performance Index................................................................. 62

3.4.2 Optimization Constraints........................................................ 63

3.4.3 Optimization ........................................................................... 64

3.5 CONCLUSIONS........................................................................... 70

4 REDUCED-ORDER MODEL FOR MONITORING SPECTROSCOPIC AND

CHROMATOGRAPHIC PROPERTIES .............................................................. 71

4.1 EXPERIMENTAL PROCEDURE ................................................. 73

4.1.1 Off-Line and On-Line Measurements..................................... 75

4.2 REDUCED-ORDER MECHANISTIC POLYMERIZATION MODEL .

..................................................................................................... 78

4.3 REDUCED-ORDER MULTI-RATE STATE OBSERVER DESIGN...

..................................................................................................... 84

4.4 STATE OBSERVER PERFORMANCE........................................ 86

4.5 CONCLUDING REMARKS .......................................................... 90

5 GLOBAL IDENTIFIABILITY OF REACTION RATE PARAMETERS IN

MECHANISTIC MODELS FOR THE CHAIN HOMOPOLYMERIZATION .......... 91

5.1 SCOPE AND PRELIMINARIES ................................................... 94

5.1.1 Local and Global Identifiability of Model Parameter and Model

Structure.............................................................................................. 94

5.1.2 Distinguishability of Model Structures .................................... 96

5.1.3 Scope..................................................................................... 96

5.2 CHALLENGES IN FREE-RADICAL POLYMERIZATION............. 97

5.2.1 Example 1 .............................................................................. 99

5.2.2 Example 2 ............................................................................ 101

5.3 GLOBAL PARAMETRIC IDENTIFIABILITY OF CHAIN

HOMOPOLYMERIZATION SYSTEMS ..................................................... 103

5.3.1 Example 3 ............................................................................ 104

5.3.2 Example 4 ............................................................................ 106

5.4 CONCLUSIONS......................................................................... 109

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6 CONCLUSIONS AND FUTURE DIRECTIONS......................................... 110

6.1 PRELIMINARY OBSERVATIONS IN HIGH-TEMPERATURE

SPONTANEOUS SOLUTION HOMO-POLYMERIZATION OF SEVERAL

ALKYL ACRYLATE MONOMERS ............................................................ 111

6.2 RECENT STUDIES IN SOLUTION POLYMERIZATION OF 2-

ETHYLHEXYL ACRYLATE AND N-OCTYL ACRYLATE.......................... 114

6.3 MODELING AND PARAMETER ESTIMATION ......................... 115

APPENDIX A.................................................................................................... 118

BIBLIOGRAPHY............................................................................................... 125

VITA ................................................................................................................. 144

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LIST OF TABLES

Table 1. Different sets of reaction mechanisms used in free-radical co- and

homo-polymerization kinetic studies. IM = initiation by monomer, IO = initiation by

peroxides, oxygen, IC = initiation by chain transfer agents, P = propagation, CTM

= chain-transfer to monomer, CTA = chain-transfer to modifier, CTP = chain-

transfer to polymer, CTS = chain-transfer to solvent, CTC = chain-transfer to

Macromonomer, TC = termination by combination, TD = termination by

disproportionation, D = depropagation, BS = beta-scission, BB = backbiting, TDB

= propagation on a terminal double bond, S = Styrene, E = Ethylene. ................. 4

Table 2. Postulated polymerization reaction network for batch polymerization of

nBA in xylene at 140-180°C with a monomer content of 20-40 wt.%. The

subscripts represent polymer chain length. ........................................................ 26

Table 3. Estimated rate constant values based on measurements of X, Mn, Mw

and CBC at 160 and 180°C, at an initial monomer content of 40wt.%, and the

corresponding Arrhenius parameters. ................................................................ 31

Table 4. Reported and estimated rate constant values for high-temperature nBA

polymerization. aThe self initiation rate constant was estimated while assuming

that initiation by impurities did not occur bInter-molecular chain transfer to

polymer was not considered in this study ........................................................... 34

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Table 5. Local sensitivity of parameter-measurement with respect to nominal

parameter and measurement values at 160°C, 200 min reaction time and initial

monomer content of 40wt.%............................................................................... 36

Table 6. Rate constant values for decomposition of oxygen and peroxides

[Cerinski et al., 2002], used in simulations presented in Figure 8....................... 47

Table 7. Activation energies of self-initiating monomers. aInitial monomer content

at 170°C, ca. 20 min reaction time, 30% monomer conversion. bBased on the

postulated participation of molecular oxygen in the self-initiation mechanism.... 50

Table 8. Values of the estimated parameters ..................................................... 60

Table 9. Optimization Settings............................................................................ 65

Table 10. Optimal feed and temperature profiles................................................ 66

Table 11. Key 1H-NMR and 13C-NMR peak assignments................................. 76

Table 12. Simplified (using the small-molecule representation) reaction network

for polymerization of nBA in xylene, with a monomer content of 20-40 wt%, and

at 140-180°C. ..................................................................................................... 79

Table 13. Simplified (using the small-molecule representation) reaction network

for polymerization of nBA in xylene, with a monomer content of 20-40 wt%, and

at 140-180°C. ..................................................................................................... 85

Table 14. Polymerization reaction networks for initiation-propagation-termination

scheme............................................................................................................... 99

xvii

Table 15. Polymerization reaction networks for initiation-propagation-transfer to

polymer-termination scheme ............................................................................ 104

Table 16. Preliminary observation of alkyl substituent size effect on resulting

molecular weight and monomer conversion. Experiments were carried out in

batch, in four hours, in methyl amyl ketone in the absence of thermal initiators

[Grady, Quan and Soroush, 2003].................................................................... 113

Table 17. Summary of known reaction mechanisms in nBA, nOA and 2-EHA

polymerization (low to high temperature).......................................................... 116

xviii

LIST OF FIGURES

Figure 1. Quality of fit for parameter estimation based on measurements of X, Mn,

Mw and CBC. Solid circle and square markers represent experimental data at 160

and 180°C, with initial monomer content of 40 wt.%, respectively. Solid and

dashed lines represent model predictions at 160 and 180°C, with initial monomer

content of 40 wt.%, respectively. ........................................................................ 30

Figure 2. Surface of the objective function (sum of squared error between model

predictions and measurements of monomer conversion, number-average

molecular weight, weight-average molecular weight and number-average chain

branches per chain) of the parameter estimation as functions of kti and kbb at

160°C, initial monomer content of 40wt.%.......................................................... 32

Figure 3. Model predictions vs. experimental data for X and Mw, predictions are

based on estimated Arrhenius parameters (Table 1). Solid triangle and circle

markers represent experimental data with initial monomer content of 20 and 40

wt.%, at 160°C, respectively. Solid and dashed lines represent model predictions

at initial monomer content of 20 and 40 wt.%, at 160°C, respectively. ............... 38

Figure 4. Model predictions vs. experimental data for X and Mw, predictions are

based on estimated Arrhenius parameters (Table 2). Solid triangle and circle

markers represent experimental data at 140 and 160°C, respectively, with initial

monomer content of 40 wt.%. Solid and dashed represent model predictions at

140 and 160°C, respectively, with initial monomer content of 40 wt.%............... 39

xix

Figure 5. Model predictions vs. experimental data for TDBC and CBC vs.

temperature. predictions are based on estimated Arrhenius parameters (Table 2).

Solid markers represent experimental data. Experimental data at 160, 170 and

180°C have initial monomer contents of 40, 33, and 40 wt.%, respectively

(batch), at reaction time of ca. 40-50 minutes. The experimental data at 140°C

has a final monomer content of ca. 50 wt.% (starved-feed, 90 minute feed time),

at reaction time of 90 minutes. The experimental data at 140°C is recalculated

from [Peck et al., 2004]. Empty markers represent batch model predictions at ca.

60 minute reaction time, and the solid lines are plotted only to connect the model

predictions. ......................................................................................................... 40

Figure 6. (Top) Model predictions vs. experimental data for CBC, predictions are

based on estimated Arrhenius parameters (Table 2). Solid square and circle

markers represent experimental data at 160 and 180°C , with initial monomer

content of 40 wt.%, respectively. Solid and dashed lines represent model

predictions at 160 and 180°C, with initial monomer content of 40 wt.%,

respectively. (Bottom) Experimental trends of CBC. The empty circles are

reproduced from [Peck et al., 2004], based on 140°C starved-feed nBA-xylene

experiments. Solid circles are experimental data from this study, based on 160-

180°C batch polymerization experiments. .......................................................... 42

Figure 7. Experimental data for TDBC and TSGC. The empty circles are

reproduced from [Peck et al., 2004], based on 140°C starved-feed nBA-xylene

experiments. Empty triangles are experimental data from this study, based on

170°C batch polymerization experiments, with initial monomer content of 33

wt.%.................................................................................................................... 45

Figure 8. (Top) Model predictions vs. experimental data for monomer conversion,

X, predictions based on oxygen-initiated polymerization (no self-initiation), with

rate constants given in Table 4. Solid triangle markers represent experimental

xx

data at 160°C, with initial monomer content of 40 wt.%. Solid, dashed, dash-

dotted and dotted lines represent model predictions for oxygen-initiated

polymerization with initial oxygen amount of 800, 600, 400 and 200 ppm, at

160°C and initial monomer content of 40 wt.%, respectively. (Bottom) Model

predictions vs. experimental data for X, predictions based on impurity-initiated

polymerization (no self-initiation), with rate constants given in Table 6. Solid

triangle markers represent experimental data at 160°C , with initial monomer

content of 40 wt.%. Solid, dashed, dash-dotted and dotted lines represent model

predictions for oxygen-initiated, DTBP-initiated, DCP-initiated and TBPC-initiated

polymerization, with initial impurity amount of 800 ppm, at 160°C and initial

monomer content of 40 wt.%. ............................................................................. 49

Figure 9. Measurements and model predictions of monomer conversion (xm),

number-average molecular weight (Mn), number of terminal double bonds per

100 monomer units (TDBH) and number of branching points per 100 monomer

units (BPH) when the reactor is operate isothermally. The dashed and solid lines

represent model predictions at 160 and 180°C respectively, and the triangles and

circles represent measurements at 160 and 180°C respectively. ....................... 61

Figure 10. Optimization validation results for 300-minute optimal control recipe.

Triangles and lines represent measurements and model predictions, respectively.

........................................................................................................................... 68

Figure 11. Corresponding polymer microstructure profiles for the 300-minute

optimal control recipe. Triangles and lines represent measurements and model

predictions, respectively. .................................................................................... 69

Figure 12. Profiles of the reactor temperature and the flow rates of the monomer

(n-butyl acrylate), the solvent (xylene) and the initiator (t-butyl peroxy acetate)

solution feed streams. ........................................................................................ 74

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Figure 13. Molecular structure of the synthesized poly(n-butyl acrylate).

Molecular structure shows the key atoms used to infer polymer microstructure

concentrations. ................................................................................................... 77

Figure 14. Measurements of polydispersity index at different temperatures

(constant initial monomer weight percent of 40%) and different initial monomer

weight percents (constant temperature of 160°C). ............................................. 82

Figure 15. Flow of information between the polymerization reactor and the

nonlinear reduced-order multi-rate state observer.............................................. 86

Figure 16. Estimated vs. measured values of the number- and weight-average

molecular weights (dashed line = estimate without using feed back

measurement, solid line = estimate using feed back measurement, circles =

measurement). ................................................................................................... 87

Figure 17. Estimated vs. measured values of the concentrations of the solvent

and terminal double bond (dashed line = estimate without using feed back

measurement, solid line = estimate using feed back measurement, circles =

measurement). Bottom figure shows evolution of the 1H-NMR peaks. ............... 88

Figure 18. Estimated vs. measured values of the concentrations of the short

chain branches (dashed line = estimate without using feed back measurement,

solid line = estimate using feed back measurement, circles = measurement).

Bottom figure shows evolution of the 1H-NMR peaks. ........................................ 89

Figure 19. Different sizes of substituent groups in (I) n-butyl acrylate, (II) 2-

ethylhexyl acrylate, and (III) n-octyl acrylate..................................................... 112

1

1 INTRODUCTION AND BACKGROUND

It is estimated that polymer-based materials (e.g. plastics, synthetic fibers,

synthetic rubber, paints and coatings) made via free-radical polymerization

comprise over half of the total production in the United States [Cunningham et

al., 2002]. Particularly in the organic coatings industry, acrylic- and methacrylic-

based resins are conventionally produced through free-radical solution

polymerization. These solvent-borne resins are commonly used as the primary

binder in coatings formulation, due to their photostability and resistance to

hydrolysis [Wicks et al., 1999]. However, as environmental restriction on volatile

organic content (VOC) in coatings increases, paint and coatings manufacturers

are trying to replace the production of high-molecular-weight low-percent-solids

acrylic resins with low-molecular-weight high-percent-solids acrylic resins.

Recently, the utilization of high-temperature free-radical solution polymerization,

in the absence of chain transfer agents, has been recognized as an attractive

method for the production low molecular weight, highly functionalized acrylic-

based resins [Grady et al., 2002]. However, as a consequence of the high

temperature requirement, reactions that were deemed insignificant at low

temperatures now have magnified effects at elevated temperatures, causing the

2

inapplicability of simple kinetic schemes previously used for predicting polymer

qualities in low-temperature polymerization processes. A fundamental

understanding of how the various side reactions affect the resulting polymer

properties is therefore essential for the successful implementation of high-

temperature polymerization processes.

Several high-temperature polymerization processes have received considerable

attention, namely, high-temperature polymerization of styrene and ethylene.

Styrene is known for its ability to polymerize spontaneously at elevated

temperatures, while ethylene is known for forming branched polymer chains at

higher temperatures. Similar, but not all of these, reactions have been observed

in acrylic systems, for example, in high-temperature polymerization of methyl

methacrylate (MMA). However, our knowledge on high-temperature acrylic

polymerization is, in general, still very limited. Recently, chain-transfer reactions

leading to polymer chain-branching and spontaneous thermal initiation in high-

temperature solution homo-polymerization of n-butyl acrylates (nBA), and

depropagation reaction in high-temperature solution homo-polymerization of n-

Butyl Methacrylate (nBMA) were, for the first time, carefully investigated [Grady

et al., 2002; Ahmad et al., 1998; Quan et al., 2006]. These reactions are of

interest, since the depropagation and scission reactions are naturally occurring

reactions that can be utilized as molecular weight regulators [Hirano et al., 2003].

Thermal initiation by monomer allows for the generation of radicals in the

absence of initiators. The absence of initiators, combined with lower solvent

3

requirements at high reaction temperatures, leads to a decrease of the cost of

the coatings formulation and elimination of unwanted moieties in the final

product. A detailed knowledge on such side reactions could therefore improve

control of molecular weight and molecular structure.

1.1 SCOPE OF RECENT STUDIES IN MECHANISTIC MODELING OF SOLUTION AND BULK FREE-RADICAL HOMO-POLYMERIZATION

The ultimate aim of mechanistic modeling studies is to identify the underlying

mechanistic pathways in a given reaction system. In the process of identification

one needs to conform to the chemistry of free-radical polymerization and the

dynamics of the process measurements. For free-radical homo-polymerization

systems in general, different monomer systems often have similar underlying

reaction networks. A thorough examination of the different reaction sets or

networks reported in the literature for free-radical polymerization of different

monomer systems is given in the following table.

4

Table 1. Different sets of reaction mechanisms used in free-radical co- and homo-polymerization kinetic studies. IM = initiation by

monomer, IO = initiation by peroxides, oxygen, IC = initiation by chain transfer agents, P = propagation, CTM = chain-transfer to

monomer, CTA = chain-transfer to modifier, CTP = chain-transfer to polymer, CTS = chain-transfer to solvent, CTC = chain-transfer

to Macromonomer, TC = termination by combination, TD = termination by disproportionation, D = depropagation, BS = beta-scission,

BB = backbiting, TDB = propagation on a terminal double bond, S = Styrene, E = Ethylene.

Reference System IM IO IC P CTM CTA CTP CTS CTC TC TD D BS BB TDB

Kotoulas et al., 2003 S X X X X X

Campbell et al., 2003 S X X X X X X

Villermaux et al., 1984 S X X X X

Qin et al., 2002 E X X X X X X X X X

Hutchinson et al., 2001 E X X X X X

Pladis et al., 1998 E X X X X X X X X X X X

Becker et al., 1998 E X X X X X X

Villermaux et al., 1984 E X X X X X

Fenouillot et al., 2001 MMA X X X X X X X

Fenouillot et al., 1998 MMA X X X X X X X X X

Hoppe et al., 1998 MMA X X X X X X X X

Nagasubramanian et al., 1970 MAce X X X X X

Villermaux et al., 1984 MAce X X X X X X

5

1.2 KINETICS OF FREE-RADICAL SOLUTION HOMO-POLYMERIZATION

One of the simplest kinetic schemes in free-radical solution homo-polymerization

is a chain polymerization system which consists of initiation, propagation and

termination reactions. In deriving the rate of polymerization assumptions such as

the long chain hypothesis (LCH), quasi-steady-state assumption (QSSA), and

irreversible reactions are often invoked. Consequently, mathematical derivation

of the polymerization rate is greatly simplified by making these assumptions.

Through the long chain assumption, we are assuming that the reactivity of all

propagating polymer chains is only dependent on the end-group monomer. And

through the irreversible reaction assumption we are in assuming that de-

polymerization reactions are insignificant. The QSSA further simplifies the

derivation by assuming a relatively long reaction time period, thus neglecting the

early dynamics of the radical concentration.

The inadequacy of simple kinetic schemes in describing high-temperature

polymerization has prompted the investigation of other significant reactions

occurring at higher temperatures. Deviations have been attributed to the

occurrences of chain-transfer reactions, which lead to, among others, backbiting

and scission reactions, thus ultimately forming branched polymer chains.

Furthermore, the classical polymerization kinetics does not take into account the

6

presence of depropagation and spontaneous thermal initiation reactions, which

will have significant effects at elevated reaction temperatures. Recent studies on

branching, scission, depropagation and spontaneous thermal initiation reactions

in several free-radical polymerization systems are reviewed in the following sub-

sections.

1.2.1 High-Temperature free-radical solution homo-polymerization with depropagation involving styrenic/acrylic monomers

The depropagation effect, or in other words, the occurrence of the de-

polymerization reaction, is more conveniently defined through the concept of the

ceiling temperature (Tc) [Sawada, 1976]. By definition, at Tc, the rate of

polymerization is exactly equal to the rate of de-polymerization. The Tc is defined

for systems with an initial monomer concentration of 1 mol/L. Alternatively,

analogous to Tc, the onset of depropagation can also be defined by the

equilibrium (i.e. limiting) monomer conversion, with the system held at a constant

reaction temperature.

Depropagation in homo-polymerization has been observed and studied recently,

in several high temperature polymerization systems, namely, in the solution

polymerization of nBMA at ca. 140 ° C [Grady et al., 2002], in the bulk

polymerization of Methyl Acrylate Trimer (MAT) at ca. 130 ° C [Hirano et al.,

2003], and in the bulk and solution polymerization of Methyl Methacrylate (MMA)

7

at 130-165 ° C [Peck et al., 2002].

In the high temperature polymerization of MAT, the depropagation effect was

attributed to the acceleration of the fragmentation reaction at higher temperatures

[Hirano et al., 2003]. Alternatively, the depropagation effect was attributed to the

limiting monomer conversion, apparent in the conversion profile in the high

temperature polymerization of MMA [Hoppe et al., 1998]. In the high temperature

polymerization of nBMA, simulation results generated by Predici® showed that

both the polymer concentration and the temperature have significant effects on

the rate of depropagation [Grady et al., 2002].

1.2.2 High-Temperature free-radical solution homo-polymerization with branching and/or scission involving styrenic/acrylic monomers

At high reaction temperatures, chain-transfer reactions become significant.

Chain-transfer to polymer, or inter-molecular chain-transfer, leads to the

existence of active mid-chain propagating radical center (structure A), which

upon further propagation would create long-chain branching points (structure C),

or, upon scission reactions would create terminal double bonds (structure B).

8

Similarly, at high reaction temperatures, intra-molecular chain-transfer, i.e.

backbiting, becomes significant. Backbiting reactions (structure D) lead to the

formation of near-chain-end propagating radical (structure E), which could

propagate further with other monomer to form short-chain branching points

(structure F) or undergo scission reactions to form terminal double bonds

(structure G). A classical example of branching and scission reactions is that

observed in the high-temperature high-pressure polymerization of poly-ethylene.

Polymer chains with terminal double bonds created by both inter- and intra-

molecular chain-transfer (structure B and structure G) could undergo further

propagation with a radical center, which could lead, again, to the creation of

terminal double bonds, or long branching points.

9

Through the identification of these molecular structures, i.e. terminal double

bonds (TDB), short-chain branching points (SCB) and long-chain branching

points (LCB), the occurrence of various chain fragmentation reactions were

recently investigated in several acrylic polymerization systems, namely, solution

polymerization of nBA at ca. 140 ° C [Grady et al., 2002; Quan et al., 2005; Peck

et al., 2002], bulk polymerization of MAT at ca. 70-130 ° C [Hirano et al., 2003],

solution polymerization of Di-n-Butyl Itaconate (DBI) at 60-100 ° C [Hirano et al.,

2002] and bulk polymerization of Styrene (ST) at ca. 250-350 ° C [Campbell et

al., 2001; Campbell et al., 2003; Campbell et al., 2003].

Branching and scission reactions were characterized in the starved-feed semi-

batch solution polymerization of n-Butyl Acrylate (nBA) at ca. 140 ° C, through the

branching frequency terminal double bond characterization [Grady et al., 2002;

Quan et al., 2005; Peck et al., 2002]. Fragmentation reactions occurring in the

bulk polymerization of methyl acrylate trimer (MAT) at 70-130° C was observed,

and was characterized by using 1H- and 13C-NMR spectroscopy techniques,

resulting in the formation of a propagating polymer chain with an unsaturated end

group (TDB) [Hirano et al., 2003]. In the solution polymerization of Di-n-Butyl

10

Itaconate (DBI) at 60-100 ° C, unexpected C resonances were observed in the

13C-NMR spectra of the carbonyl groups at higher reaction temperatures. Further

investigation reveals the dependence of the intensity of the C resonance on the

monomer feed level, suggesting the occurrence of intra-molecular chain transfer

reaction [Hirano et al., 2004]. Analogous to that observed in BA polymerization,

backbiting, followed by scission reactions were also observed in the

polymerization of Styrene (ST) at elevated temperatures, ca. 250-350 ° C, carried

out in a continuous stirred tank reactor, in the absence of initiators. Through the

identification of oligomers by the use of the 13C-NMR spectra, 1:3 and 1:5

backbiting/scission reactions were characterized and the mid-chain branching

was observed to be non-existent, thus yielding only linear polymer chains with

0,1 or 2 TDB. In addition, the measurement of the TDB distribution by the use of

the MALDI-TOF MS indicates the low level of chain transfer to polymer, relative

to backbiting, yielding only chains with 1 or 2 TDB, showing the dominance of the

β-scission reactions in the production of polymer chains [Campbell et al., 2001;

Campbell et al., 2003; Campbell et al., 2003].

1.2.3 Spontaneous thermal initiation in high-temperature free-radical polymerization involving styrenic/acrylic monomers

Spontaneous thermal initiation, or thermal initiation by monomer, is best

described by the ability to polymerize spontaneously in the absence of other

11

initiating species or in the absence of irradiation [North, 1966]. As have been long

observed and studied in the polymerization of ST [Odian et al., 1990; Barner-

Kowollik et al., 2002; Campbell et al., 2001; Campbell et al., 2003; Campbell et

al., 2003], spontaneous thermal initiation does occur at appreciable rates at

higher reaction temperatures. Self thermal initiation has also been observed in

the high-temperature polymerization of MMA [ Odian et al., 1990; Barner-

Kowollik et al., 2002; Hoppe et al., 1998; Fenouillot et al., 1998; Fenouillot et al.,

2001]. However, the exact nature of the initiation mechanism of MMA is still not

well understood. Furthermore, due to the difficulty in obtaining reproducible data,

the spontaneous polymerization observed in MMA was attributed to the presence

of adventitious peroxides, whose existence is difficult to exclude by the usual

purification techniques [Lehrle et al., 1988].

The thermal initiation kinetics of the high-temperature polymerization of ST is well

established, and is characterized by a third-order monomer dependence on the

rate of initiation. ST monomers form a Diels-Alder adduct, which can undergo

molecule induced homolysis to produce two radicals, each capable of initiating

polymerization. Recently, spontaneous thermal initiation, with a significant rate of

initiation, was observed in the high-temperature polymerization of nBA [Quan et

al., 2005].

12

1.3 MODELING METHODS

The purpose of constructing a detailed kinetic model is to be able to correlate the

reaction conditions (e.g. temperature, initiator concentration, feed time) with the

polymer quality (e.g. MWD). The multi-scale modeling aspect of polymerization

reactors has been discussed in [Ray, 1986].

Depropagation and branching are commonly observed phenomena, and their

existence in free-radical solution polymerization have been discussed in previous

sections. Gel effect, on the other hand, takes place only when viscosity effects

become significant (e.g. >50 wt% in monomer in MMA polymerization [Odian et

al., 1990]). The meso-scale phenomena are only of importance in heterogeneous

(e.g. emulsion) polymerization. Thus, in a homogeneous (i.e. single phase)

polymerization they are usually not considered. The macro-scale phenomena are

readily modeled through the application of mass and energy balance to the

polymerization reactor content.

From the species balance equations, one could solve or integrate the differential

equations directly to obtain concentrations of polymer chains with different length

13

as a function of time. A more common approach is by integrating a reduced set

of balance equations, which are reduced by defining important moments of the

MWD [Bamford et al., 1953]. Another popular approach, which does not involve

solving differential balance equations is through the use of statistical methods,

which view chain growth as a stochastic process having possible states resulting

from the kinetic mechanism and state transition probabilities dependent on the

kinetic parameters [Ray, 1972]. In any case, all methods depend on an accurate

knowledge of the set of kinetic mechanism. Methods that have found wide

acceptance and those that were recently proposed, are reviewed in the following

paragraphs.

Instantaneous Property Method (IPM) [Bamford et al., 1958]. The

Instantaneous property method employs the long chain hypothesis (LCH) and the

quasi steady-state assumption (QSSA) in deriving analytical expressions for the

rate of polymerization and molecular weight distribution (MWD). However, the

IPM is strictly valid for linear co- and homo-polymers. Hence, scission reactions

such as that observed in the high-temperature polymerization of nBA [Quan et

al., 2005] cannot be accounted for by the IPM.

Method of Moments (MM) [Bamford et al., 1953; Ray 1972; Bamford et al.,

1954]. The method of moments transforms the original high-dimensional systems

of differential equations into a low-order system of equations by introducing the

leading moments of the distributions of interest. The major limitation of models

14

based on the method of moments is that they only track average quantities.

While adequate for most situations, the MM cannot examine, for example, the

combined effects of chain-scission and long-chain branching on polymer

architecture, or to incorporate chain-length–dependent termination kinetics into

the kinetic scheme [Konstadinidis 1992; Achilias et al., 1992].

Property Moments Method (PMM) [Villermaux et al., 1983, 1984; Blavier et

al., 1984; Lorenzini et al., 1992, 1992]. The property moments method is also

known as the tendency modeling method. The PMM does not consider balances

on the individual species present in the reaction mixture. Instead, mass balance

equations are derived for a number of selected quantities that characterize the

polymer quality, namely, the average concentration of ‘live’ radicals, the leading

moments of the polymer number chain-length distribution (NCLD), and the

concentration of structural characters such as short-chain branching (SCB) and

long-chain branching (LCB) and terminal double bonds (TDB) irrespective of the

macromolecules carrying them. The PMM, originally developed for modeling

free-radical homo-polymerization, has also been extended to the description of

free-radical co-polymerization reactions [Konstadinidis 1992; Achilias et al.,

1992].

Discrete Galerkin method [Wulkow, 1996]. The Discrete Galerkin method is

more commonly used through the software package Predici®. Predici® allows one

to analyze polymer reactions with an arbitrary number of species and chain-

15

length distributions and arbitrary number of chain-length dependent reaction

steps, with no restrictions on the form of MWD and no necessity of model

reduction (e.g. by invoking QSSA). The package has also recently been

extended to follow branch-point concentrations as a function of chain length

[Iedema et al., 2000].

Three Stage Polymerization Model [Qin et al., 2000, 2002, 2002, 2002, 2003].

Recently, an empirical kinetic model called the Three Stage Polymerization

Model (TSPM) was proposed. This particular model was proposed in order to

provide a general framework for modeling the bulk free radical polymerization.

The name ‘three stage polymerization’ represents the three prominent stages

observed in bulk free polymerization; the low conversion stage, the gel effect

stage and the glass effect stage. The TSPM satisfactorily depicts the course of

polymerization for the bulk thermal polymerization of ST at 100-200 ° C, and as

expected for polymerization above the Tg of the polymer, only the low conversion

and the gel effect stages were observed. An extension of TSPM to co-

polymerization systems was proven to be effective for modeling the co-

polymerization ST/MMA, MMA/Ethylene Glycol Di-methacrylate (EGDMA), and

AN/AMS, even up to elevated reaction temperatures (ca. 60-120 ° C).

Numerical Fractionation (NF) method [Teymour et al., 1994; Papavasiliou et

al., 2002]. The NF method was specifically developed to address the problem of

gel formation in polymer systems. The method is based on the definition of

16

different generations of branched polymers, where each generation differs in the

number of branches per polymer chain. The population of chains in each

generation is then represented through the method of moments. This model has

also been extended further in order to calculate LCB number as a function of

chain length.

Monte Carlo techniques [Tobita, 1993, 2001, 2003, 2003]. Monte Carlo

techniques have been repeatedly used in modeling MWD development in non-

linear free radical polymerization. Statistical simulation studies were focused on

the study of the effects of branching, scission, and crosslinking reactions on

MWD profile.

17

2 MECHANISTIC MODELING

High-temperature (ca. above 100°C) solution polymerization processes have

been considered as a viable alternative to produce low solvent-level, high solids-

level, environmentally-friendly, acrylic resins [Grady et al., 2002]. Market

competition in industrial resin production has motivated optimal operation of

current commercial processes. The optimal operation requires understanding of

how common process variables such as monomer content and reaction

temperature influence the final product quality indices.

Recent high-temperature n-butyl acrylate (nBA) solution homo-polymerization

studies have been motivated by the inability of 'classical' chain-polymerization

kinetic models (e.g., based on initiation-propagation-termination) in describing

nBA homo-polymerization at elevated temperatures. This inability has been

attributed to significant effects from secondary reactions, such as inter- or intra-

molecular chain-transfer reactions, which can lead to branching or scission

reactions [Peck et al., 2004]. Hence recent studies on nBA homo-polymerization

have focused on (i) the identification of dominant polymerization reactions

through spectroscopic methods (i.e., polymer characterization studies) [Ahmad et

al., 1998; Chiefari et al., 1999; McCord et al., 1997; Peck et al., 2002], (ii)

18

calculation of reaction rate orders [Fernandez-Garcia et al., 2000], (iii) individual

estimation of reaction rate constants through pulsed-laser initiated polymerization

(PLP) experiments [Asua et al., 2004; Nikitin et al., 2002; Buback et al., 2004;

Beuermann et al., 2002], and (iv) simultaneous estimation of a set of reaction

rate constants based on advanced (e.g., spectroscopic measurement of the

chain branch concentration) and conventional (e.g., gravimetric measurement of

polymeric solids content and chromatographic measurement of molecular weight

distribution) measurements [Peck et al., 2004; Busch et al., 2004].

At first glance, the approach of simultaneously estimating a set of parameters

against a set of measurements is the most practical and direct approach toward

obtaining a process model. Such approach, however, is often hindered by; (i)

non-convexity of the least-squared-error objective function and ill-conditioning of

the minimization problem (i.e., more than one set of rate constants can give

comparable model predictions), (ii) over- or under-postulation of polymerization

reaction networks (i.e., knowledge of dominant reaction pathways is still

evolving), (iii) limited availability in number and type of measurements.

Consequently, recent research directions in this area have included (i)

identification of novel 'experimental sensors' (measurements) to be included in

the parameter estimation [Busch et al., 2004], (ii) identification of optimal

parameter-measurement pairs through sensitivity analyses in order to reduce the

complexity of the parameter estimation problem [Polic et al., 2004; Li et al., 2004]

and (iii) use of global optimization approaches to obtain a unique set of rate

19

constants (e.g., implementation of the simulated annealing algorithm in modeling

package PREDICI® [Wulkow et al., 1996]).

Hence an alternative to reducing complexities in the model building and the

parameter estimation problem is the inclusion of spectroscopic measurements.

The inclusion should limit the number of different combinations of rate constants

that can give (i) comparable model predictions and (ii) satisfactory predictions for

all types of measurements. The availability of these measurements depends on

the observability of signature peaks that can be used as a measure of a certain

chain microstructure quantity. Particularly in polymer characterization problems,

of critical importance is the observance of peaks that are distinguishable from

others arising from polymeric backbone atoms. The complexity of the

simultaneous parameter estimation problem is also reduced by the more reliable

knowledge of the dominant reaction rate constants (e.g., linear propagation and

termination) obtained from PLP experiments.

This study is motivated by the observation of near-complete conversion

polymerization experiments that were carried out in the absence of thermal

initiators [Grady et al., 2005]. Spectroscopic evidence (ESI-FTMS) of the

absence of initiator moieties in the polymer chains that are produced from the

spontaneous nBA polymerization experiments [Quan et al., 2005] further

supports this observation. In addition to the spontaneous polymerization

observed at high temperatures (ca. above 140°C), secondary reactions occurring

20

at these temperatures are also studied. The aims of this study are (i) to gain

insight into initiation mechanism and to probe the significance of the initiation

reaction, (ii) to quantitatively explain the effects of process variables on polymer

quality indices (conversion, molecular weight, chain microstructures). The

understanding of the initiation and secondary mechanisms can potentially lead to

initiator-free commercial polymerization processes and can help process

engineers in the safety assessment of high-temperature polymerization

processes.

Section 2 describes the experimental and analytical procedures, and section 3

describes the postulation of the polymerization reaction network. The parameter

estimation approach used in this study is introduced in section 4. Discussions

and analyses of results are presented in section 5, followed by concluding

remarks in section 6. Derivation of the process model is presented in the

Appendix A.

2.1 EXPERIMENTAL AND ANALYTICAL PROCEDURES

n-Butyl acrylate (BASF, 99.5%, with boiling point of 146.8°C) was passed

through an inhibitor removal column (DHR-4, Scientific Polymer Products, Inc.)

prior to use. Xylene (ExxonMobil Chemical Co., an 80/20 mix of xylene isomers

and ethyl benzene with a boiling point range of 137 - 143°C) and 4-

21

methoxyphenol (99%, ACROS, with boiling point of 243°C) were used as

received. Polymerizations were carried out in a 1-liter glass RC1 calorimeter-

reactor (Mettler-Toledo GmbH, Schwerzenbach, Switzerland).

Each experiment started with loading the xylene solvent into the reactor. Nitrogen

gas was subsequently purged through the solvent and reactor for 15 to 30

minutes. The monomer was then pumped into the reactor and heated under

nitrogen blanket to the desired reaction temperature. Reaction temperature was

maintained throughout the course of the polymerization by adjustment of the

reactor jacket temperature. Samples drawn from the reaction mass at specified

time intervals were diluted (50/50 v/v) in a cold inhibitor solution (1000 ppm 4-

methoxy phenol in xylene) to prevent further reaction from taking place.

Polymer and residual monomer concentrations were determined by gravimetric

and chromatographic (GC) analyses, respectively. GC analyses were carried out

by using Hewlett Packard (HP) 6890 GC-FID system with a 30 m DB-5

separation column. The FID (flame ionization detector) temperature was set to

250°C, with oven temperature ramp at 10°C/min. Measurement of residual

monomer by weight was obtained from the integration of monomer peak area,

adjusted with a dilution factor when necessary.

Molecular weight distributions were determined by using an HP 1090 HPLC (high

performance liquid chromatography) system, equipped with an HP 1047A RI

22

(refractive index) detector and a four-column set configuration (105, 104, 103, 102

Å 30 cm x 7.8mm id microstyragel columns). Tetrahydrofuran was used as the

mobile phase at flow rate of 1mL/min at 40°C. Data from the RI detector was

collected and processed by using a Waters Millennium system. Narrow molecular

weight polystyrene standards were used to calibrate the column set.

Polymeric solid samples for NMR analyses were isolated through high vacuum

and evaporation at 65°C. Quantitative carbon and proton NMR spectra were

recorded at 34°C on a Bruker DRX-400 MHz spectrometer (1H frequency of

400.13 MHz and 13C frequency of 100.62 MHz). Deuterated chloroform (CDCl3)

was used as solvent in all analyses. 13C-NMR spectra were obtained by using a

10 mm broadband probe equipped with 1H decoupler, acquisition time of 1.36 s,

spectral width of 24 kHz, relaxation time of 15 s, number of scans between 1600

and 3200, and a 90° pulse width of 13.9 μs.1H-NMR spectra were obtained by

using a 4 s acquisition time, a 10 kHz spectral width, an 8 s relaxation time,

number of scans of 16 and an 11.5 μs 90° pulse width. Peak assignments

obtained through analyses of DEPT spectra reported by [Quan et al., 2005] is

used in this study.

2.2 POSTULATION OF REACTION NETWORK

23

Chain-initiation. Studies in spontaneous polymerization of methyl mecthacrylate

(MMA) have cited both self-initiation [Stickler et al., 1978; Brandolin et al., 1980;

Lingnau et al., 1980; Stickler et al., 1981; Lingnau et al., 1983; Lingnau et al.,

1984; Lingnau et al., 1984] and decomposition of impurities, i.e. oxygen or

peroxides in general [Clouet et al., 1993; Lehrle et al., 1988], as the underlying

mechanism for the generation of initiating radicals. In a recent study on

thermally-initiated MMA polymerization, it is reported that MMA could also

undergo reaction with air to form macromolecular peroxides [Nising et al., 2005].

Both peroxide decomposition [McManus et al., 2002] and self-initiation [Cao et

al., 2004] have been assumed to be responsible for the generation of initiating

radicals in nBA polymerization. In this study, mechanisms of thermal self initiation

proposed for methyl methacrylate [Lingnau et al., 1983] and the decomposition of

peroxide [Cerinski et al., 2002] are adopted for nBA. The initiation step dictates

the amount of live radicals (initially) present, and will eventually dictate the

amount of dead polymer chains formed. The initiating radicals formed by the

peroxide decomposition is represented by live radical chains of length zero (P0).

While in the self-initiation reaction, the initiating radicals are assumed to be the

monomer itself, thus represented as live (secondary) radical chains of length one

(P1).

24

Chain-propagation, chain-transfer to polymer and chain-fragmentation.

Intra-molecular chain-transfer to polymer or backbiting has been postulated to

dominate at lower monomer content (ca. below 50 wt.%) [Peck et al., 2004], and

is known to occur even at lower temperatures [Asua et al., 2004; Willemse et al.,

2005]. The backbiting event allows for the creation of short chain-branches, upon

propagation of the tertiary (mid-chain) radical near the end of the chain. This

tertiary radical can alternatively undergo β-scission, to create a dead polymer

chain with a terminal double bond (TDB) and a live polymer chain. As only

averages are considered in this study, only the scission reactions which create

long dead chains instead of dead trimer chains with TDBs are included in the

overall kinetic scheme. The propagation, chain transfer to polymer, branching

and scission reactions are presented in unison, since these reactions

simultaneously affect the average molecular weights, the number of TDBs and

the number of branching points. Dead polymer chains are created through the

scission reaction, and are governed by the amount of tertiary radicals (Qn)

present. The scission reaction leads to the creation of a dead polymer chain (Dn-

2) with a TDB in one step. At the same time the tertiary radicals can also form

chain branches through further propagation. SCB denotes a short chain-branch

point formed through such a reaction.

Chain-transfer to monomer and solvent. Chain transfer to monomer allows for

the creation of a dead chain with a TDB and a live chain. The rate of chain

transfer to monomer is influenced by the amount of monomer and live

25

(propagating) chains present at a given time. The concentration of monomer is

much larger compared to the concentration of live chains. The significance of

chain transfer to monomer, however, is often discounted due to the low reaction

rate constant relative to the scission reaction. A recent estimation of the rate

constant for chain transfer to monomer reactions in high-pressure high-

temperature polymerization of nBA indicated that transfer to monomer might

have a more significant contribution towards the overall kinetic scheme [Busch et

al., 2004]. In this study, since temperature effects on the number of terminal

double bonds are of interest, β-scission and chain transfer to monomer reactions

are considered in the model development. Along with chain transfer to monomer

[Maeder et al., 1998], nBA has also been reported to undergo chain transfer to

solvent [Peck et al., 2002; Raghuram et al., 1969]. Upon chain-transfer to

monomer, a live secondary radical of length one or the monomer itself (P1TDB) is

created. When this radical propagates further, a TDB is created for the resulting

chain. The reactive solvent species (i.e., in this study, xylol radical) is

represented as a live chain of length zero (P0), and is assumed to propagate at

the same rate as a secondary radical.

Chain-termination. While termination for acrylates usually take place almost

exclusively by combination, at high temperatures, the extent of termination by

disproportionation can increase [Odian et al., 1991; Vana et al., 2002]. Chain

termination by a combination of both modes is assumed, and the estimates

26

reported by [Peck et al., 2004] on the relative extent of each termination mode

are used in this study to describe the various termination reactions.

The postulated polymerization reaction network is presented in Table 2. Details

of the derivation of the batch polymerization process model are presented in the

Appendix A.

Table 2. Postulated polymerization reaction network for batch polymerization of

nBA in xylene at 140-180°C with a monomer content of 20-40 wt.%. The

subscripts represent polymer chain length.

( )

( )

( )

ti

O2

p

bb

pq

β

tm

p

ts

sec-sectc

sec-sectd

sec-terttc

k1

k1 2 0

kn n+1

kn n

kn n+1

kn n-2 2

k TDBn n 1

kTDB1 2

kn n 0

kn m n+m

kn m n m

kn m

2M 2P

R OOR 2P

P +M P

P Q

Q +M P +SCB

Q D +P +TDB

P +M D +P

P +M P +TDB

P +S D +P

P +P D

P +P D +D

P +Q

⎯⎯→

⎯⎯→

⎯⎯→

⎯⎯→

⎯⎯→

⎯⎯→

⎯⎯→

⎯⎯→

⎯⎯→

⎯⎯⎯→

⎯⎯⎯→

⎯sec-terttc

tert-terttc

tert-terttc

n+m

kn m n m

kn m n+m

kn m n m

D

P +Q D +D

Q +Q D

Q +Q D +D

⎯⎯→

⎯⎯⎯→

⎯⎯⎯→

⎯⎯⎯→

27

2.3 PARAMETER ESTIMATION

The objective of the parameter estimation is to obtain reliable estimates for

spontaneous initiation and secondary reaction rate constants. The rate constants

are simultaneously estimated from a set of measurements. Two different sets of

measurements at two different temperatures are considered in the parameter

estimation.

A simultaneous parameter estimation problem with a sum of squared errors

(between model predictions and measurements) performance index is

formulated. The simplex search method (fminsearch) of the Optimization Toolbox

of the commercial software MATLAB version 6, is used in the parameter

estimation. The parameter non-negativity constraint is implicitly imposed in the

simplex method by estimating the exponents of the parameters to be estimated

(i.e., estimate θ , where θ = 10 θ , instead of directly estimating θ, where θ is the

rate constant actually being estimated. Note that by estimating the exponents

instead of the actual parameters, the algorithm converges faster to a minimum.

To minimize the probability of obtaining a local minimum, different initial

conditions for the parameter estimates were used in the parameter estimation.

28

Rate constants for the primary reactions available in the literature are not re-

estimated and are used in the model. The minimization problem is of the form:

( ) ( )( )

i

2N4

i j i j

θ i=1 j=1 i j

ˆy t -y tmin J=

y t

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

∑∑

where y1,…,y4 represent the measurements of monomer conversion (X), number-

average molecular weight (Mn), weight-average molecular weight (Mw), and

number of (short) chain branches per chain (CBC), respectively. Ni is the number

of measurements of the measured variable yi, and the parameters being

estimated θ1, …, θ4 are the rate constants corresponding to the second-order

self-initiation (kti), propagation of tertiary radicals (kpq), secondary radical

backbiting (kbb), and tertiary radical β-scission (kβ). The measurements were

obtained at temperatures of 160 and 180°C, each at an initial monomer content

of 40 wt.%. The parameters kp, ktm, kts, kt were extrapolated from references

[Asua et al., 2004; Maeder et al., 1998; Raghuram et al., 1969; Beuermann et al.,

2002], respectively. It was assumed that the combination of measurements X,

Mn, Mw, CBC, contains adequate information for the estimation of the rate

constants kti, kpq, kbb, kβ. The relationship between the rate constants and the set

of measurements can be seen by inspection of the analytical model structure

(Appendix A): the measurement of X is directly influenced by kti (mass balance

equation on monomer), CBC is directly influenced by kpq (mass balance equation

on chain branches), and the average molecular weights are indirectly influenced

by all the parameters being estimated. Experimental trends to be captured in the

parameter estimation are: increasing monomer conversion, decreasing average

29

molecular weights and CBC, with increasing temperature. Monomer conversion

and molecular weight measurements were obtained through gravimetric and

chromatographic measurements, respectively.

The measurements of number average of chain branches per chain (CBC) were

obtained by using quantitative nBA 13C-NMR peak area at 47-49 ppm as a

measure of number of branch points (quaternary carbon atom [Ahmad et al.,

1998]) and the peak area at 165-180 ppm (carbon atom on C=O [Quan et al.,

2005]) or the peak area at 65 ppm (carbon atom on C-O [Quan et al., 2005]) as a

measure of number of repeat units. One way to report number of branches is to

use the ratio of number of branches to number of repeat units. When this ratio is

multiplied by 100, the number of branches is expressed as (average) number of

branch points per 100 repeat units (CBH). The effect of temperature on chain

branches was most evident as the quantity CBC is defined as:

100 nSCBHCBC DP= ×

where DPn is the number-average degree of polymerization. The quality of fit at

the two different temperatures is presented in Figure 1. Estimates at each

temperature, and the corresponding estimates of the Arrhenius parameters are

presented in Table 2. To gauge the performance of the simplex search algorithm,

the surface of the objective function is presented as functions of kti and kbb, at

160°C and initial monomer content of 40wt.% in Figure 2. The parameters kti and

kbb represent the two underlying pathways that directly influence all of the

measurements (terms in the rate laws that include kpq and kβ depend on amount

30

of tertiary radicals, which is directly governed by the rate of backbiting).

Assuming that the intervals considered for kti and kbb are sufficiently large, it is

shown that the global minimum was obtained by using the optimization approach

(kti = 1.25 x 10-6 L mol-1 s-1, kbb = 1.92 x 104 s-1).

0

0.2

0.4

0.6

0.8

1

X

0

1

2

3

CB

C

0 20 40 60 80 1000

2

4

6

8

10

Time, min

Mn x

10−

3

0 20 40 60 80 1000

5

10

15

20

Time, min

Mw

x 1

0−3

Figure 1. Quality of fit for parameter estimation based on measurements of X,

Mn, Mw and CBC. Solid circle and square markers represent experimental data at

160 and 180°C, with initial monomer content of 40 wt.%, respectively. Solid and

dashed lines represent model predictions at 160 and 180°C, with initial monomer

content of 40 wt.%, respectively.

31

Table 3. Estimated rate constant values based on measurements of X, Mn, Mw

and CBC at 160 and 180°C, at an initial monomer content of 40wt.%, and the

corresponding Arrhenius parameters.

Rate Constant 160°C 180°C A E (J.mol-1)

kti (L.mol-1s-1) 1.25E-06 3.05E-06 8.27E+02 73,152

kpq (L.mol-1s-1) 5.16E+01 5.20E+01 5.99E+01 534

kbb (s-1) 1.95E+04 2.42E+04 3.87E+06 19,114

kβ (s-1) 7.40E-01 4.42E+00 2.88E+17 145,853

32

Figure 2. Surface of the objective function (sum of squared error between model

predictions and measurements of monomer conversion, number-average

molecular weight, weight-average molecular weight and number-average chain

branches per chain) of the parameter estimation as functions of kti and kbb at

160°C, initial monomer content of 40wt.%.

33

2.4 DISCUSSIONS AND ANALYSES

Parameter estimation results and model validation. From Figure 1, it is

shown that a satisfactory fit was obtained. Based on the parameter estimates at

the two different temperatures, Arrhenius parameters were estimated and are

presented in Table 2. Calculated secondary reaction parameter values (based on

the estimated Arrhenius parameters) were then compared with literature values

at temperatures 140 and 170°C. The comparison is presented in Table 3. Note

that the nominal values of kpq and kbb are in agreement with reported values.

Several different values of kβ have been reported in the literature, particularly the

one reported by [Busch et al., 2004]. It should be noted that the simultaneous

parameter estimation study in [Busch et al., 2004] considers different sets of

measurements, parameters and reaction paths (e.g., formation of long chain

branches preceded by inter-molecular chain transfer is considered). As will be

discussed further in the following subsection, the observed trend of chain

branches vs. number average degree of polymerization suggests the formation of

short chain branches, as opposed to long ones. The estimate for kti is the first

report on the postulated second-order self-initiation reaction. The apparently high

energy activation value for kβ is currently understood as a high temperature

34

phenomenon, as scission is not observed to be prevalent at lower reaction

temperatures.

Table 4. Reported and estimated rate constant values for high-temperature nBA

polymerization. aThe self initiation rate constant was estimated while assuming

that initiation by impurities did not occur bInter-molecular chain transfer to

polymer was not considered in this study

Rate Constant A E (J.mol-1) 140°C 170°C Ref.

kti (L.mol-1.s-1) 8.27E+02 73,152 4.66E-07 1.97E-06 This Studya

kp (L.mol-1.s-1) 2.21E+07 17,900 1.21E+05 1.91E+05 Asua et al., 2004

-- -- -- 1.67E+05 Busch et al., 2004

kpq (L.mol-1.s-1) kpx1.20E-03 -- 1.45E+02 -- Peck et al., 2004

5.99E+01 534 5.13E+01 5.18E+01 This Study

ktm (L.mol-1.s-1) kpx1.60E-02 15,200 2.31E+01 4.43E+01 Maeder et al., 1998

-- -- -- 1.81E+05 Busch et al., 2004

kts (L.mol-1.s-1) kpx1.95E+01 32,175 2.01E+02 5.39E+02 Raghuram et al., 1969

kbb (s-1) 3.50E+07 29,300 6.91E+03 1.23E+04 Plessis et al., 2003

-- -- -- 2.13E+04 Busch et al., 2004

-- -- 4.00E+03 -- Peck et al., 2004

3.87E+06 19,114 1.48E+04 2.16E+04 This Study

ktp (L.mol-1.s-1) -- -- -- 5.00E+03 Busch et al., 2004b

kβ (s-1) -- -- 6.00E+00 -- Peck et al., 2004

-- -- -- 6.28E+04 Busch et al., 2004

2.88E+17 145,853 1.04E-01 1.85E+00 This Study

kt (L.mol-1.s-1) 2.57E+08 5,590 5.05E+07 5.64E+07 Beuermann et al., 2002

-- -- -- 1.10E+07 Busch et al., 2004

35

To quantify the reliability or variability of the reported estimates, we consider the

local sensitivity of the parameters with respect to the measurements (we assume

that the parameters are linearly related to the measurements within one order of

magnitude of the nominal values). The parameter-measurement sensitivity, with

respect to measurements of monomer conversion and weight-average molecular

weight, is presented in Table 4. The variability of the estimates can then be

calculated as follows:

exp experimental erimentaltheoretical

yyθθ

⎛ ⎞Δ ⎟⎜ ⎟Δ = ×Δ⎜ ⎟⎜ ⎟⎜Δ⎝ ⎠

hence a conservative approach towards quantifying the variability of the reported

estimates is by using measurement with the largest variability (e.g. monomer

conversion, +/- 10% off the average) as basis for the above calculation. By using

such approach, the variability with respect to the nominal values at 160°C are

approximately: kti x 5 ≥ kti ≥ kti / 3, kpq x 5 ≥ kpq ≥ kpq / 2, kbb x 3 ≥ kbb ≥ kbb /

2 and kβ x 60 ≥ kβ ≥ kβ / 60.

36

Table 5. Local sensitivity of parameter-measurement with respect to nominal

parameter and measurement values at 160°C, 200 min reaction time and initial

monomer content of 40wt.%

Parameter (θ) Δθ ΔX (%) ΔMw (%)

kti X10 +17.75 -46.54

/10 -26.49 +96.48

kpq X10 +19.50 -123.89

/10 -19.72 +53.79

kbb X10 -34.90 -81.95

/10 +19.57 +129.49

kβ X10 +1.48 -11.44

/10 -0.15 +1.40

The model was then validated by comparing model predictions with experimental

data at extrapolated conditions (at experimental conditions outside the range

used in the parameter estimation). In Figure 3, the model predictions are

compared with experimental data at 160°C, and at initial monomer content of

20wt.% (the model was based on subset of data obtained at 160-180°C, initial

monomer content of 40wt.%). Based on separate triplicate experiments, the

variability of the monomer conversion data can be captured by using constant

error bars on the average value. The constant error bars represent +/- 10% off

the average for the monomer conversion data and represent +/- 5% off the

37

average for the weight-average molecular weight data. The comparison shows

that the model was able to capture the monomer conversion trend quantitatively

and the weight-average molecular weight trend qualitatively (slight over-

prediction). Similarly, the model predictions are compared with experimental data

at 140°C, at initial monomer content of 40wt.%. The comparisons are presented

in Figure 4. The model was over-predicting the extent of monomer conversion at

140°C, and slightly under-predicting the weight-average molecular weight profile.

To understand the cause of the discrepancies, the model was then validated

against molecular structure quantities as function of temperature. This validation

is presented in Figure 5. From the figure, it is shown that the model is able to

capture the temperature-chain branch trend, however, is not able to capture the

temperature-terminal double bond trend. This suggests that the model has not

fully captured the interplay between the chain branch and terminal double bond

formation. This also explains the high variability of the estimate for kβ. It should

also be mentioned that many efforts have been focused on understanding the

effects of chain length on the termination rate coefficient [Buback et al., 2002]. In

this study, it is assumed that the rate of chain-termination does not depend on

chain length at low initial monomer content (shorter polymer chains throughout

reaction time). The rate constant for chain-termination is therefore not re-

estimated, instead, the chain termination parameters reported in [Peck et al.,

2004] is adopted.

38

0

0.2

0.4

0.6

0.8

1

X

0 50 100 150 200

0

0.5

1

1.5

2x 10

4

Time, min

Mw

Figure 3. Model predictions vs. experimental data for X and Mw, predictions are

based on estimated Arrhenius parameters (Table 1). Solid triangle and circle

markers represent experimental data with initial monomer content of 20 and 40

wt.%, at 160°C, respectively. Solid and dashed lines represent model predictions

at initial monomer content of 20 and 40 wt.%, at 160°C, respectively.

39

0

0.2

0.4

0.6

0.8

1

X

0 50 100 150 2000

1

2

3x 10

4

Time, min

Mw

Figure 4. Model predictions vs. experimental data for X and Mw, predictions are

based on estimated Arrhenius parameters (Table 2). Solid triangle and circle

markers represent experimental data at 140 and 160°C, respectively, with initial

monomer content of 40 wt.%. Solid and dashed represent model predictions at

140 and 160°C, respectively, with initial monomer content of 40 wt.%

40

0

1

2

3

4

5

CB

C

120 140 160 180 2000

0.2

0.4

0.6

0.8

1

Temperature, oC

TD

BC

Figure 5. Model predictions vs. experimental data for TDBC and CBC vs.

temperature. predictions are based on estimated Arrhenius parameters (Table 2).

Solid markers represent experimental data. Experimental data at 160, 170 and

180°C have initial monomer contents of 40, 33, and 40 wt.%, respectively

(batch), at reaction time of ca. 40-50 minutes. The experimental data at 140°C

41

has a final monomer content of ca. 50 wt.% (starved-feed, 90 minute feed time),

at reaction time of 90 minutes. The experimental data at 140°C is recalculated

from [Peck et al., 2004]. Empty markers represent batch model predictions at ca.

60 minute reaction time, and the solid lines are plotted only to connect the model

predictions.

Microstructural evolution and dominant modes of chain-initiation and

chain-termination. In this study, temperature was found to have the largest

effect on microstructures, and this effect was most evident when microstructural

quantities are represented as quantities per chain. The linear increase of average

number of chain branches per chain as a function of DPn (Figure 6) suggests the

formation of short chain branches and confirms the reaction network postulated

in [Peck et al., 2004]. This trend suggests a constant 'rate of formation' of chain

branches (i.e., if there is a shift from intra- to inter-molecular chain transfer, the

'rate of formation' should not be constant as DPn increases). This also supports

studies that attribute the (short) chain branch formation to backbiting. Data from

[Peck et al., 2004], recalculated and presented in Figure 6, also support this

observation.

42

0

1

2

3

CB

C

0 20 40 60 80 1000

2

4

6

8

DPn

CB

C

Figure 6. (Top) Model predictions vs. experimental data for CBC, predictions are

based on estimated Arrhenius parameters (Table 2). Solid square and circle

markers represent experimental data at 160 and 180°C , with initial monomer

content of 40 wt.%, respectively. Solid and dashed lines represent model

predictions at 160 and 180°C, with initial monomer content of 40 wt.%,

43

respectively. (Bottom) Experimental trends of CBC. The empty circles are

reproduced from [Peck et al., 2004], based on 140°C starved-feed nBA-xylene

experiments. Solid circles are experimental data from this study, based on 160-

180°C batch polymerization experiments.

Figure 7 presents the microstructural data of number-average terminal double

bonds per chain (TDBC) and number-average terminal solvent groups per chain

(TSGC). TDBC and TSGC measurements were obtained from quantitative nBA

1H-NMR analyses. The 1H-NMR peak area at 5.5 and 6.2 ppm (2 hydrogen

atoms on the terminal double bond [Chiefari et al., 1999]) was used as a

measure of the concentration of terminal double bonds, and the peak area at 7.1

ppm (4 hydrogen atoms on the terminal solvent group [Quan et al., 2005]) was

used as a measure of the concentration of terminal solvent groups. Number of

repeat units was measured by using the peak area at 4.1 ppm (2 hydrogen

atoms on CH2 next to COO [Quan et al., 2005]). Number of terminal double

bonds per 100 repeat unit (TDBH) was calculated by dividing the peak area at

either 5.5 or 6.2 ppm by half of the peak area at 4.1 ppm, multiplied by 100; and

the number of terminal solvent groups per 100 repeat unit (TSGH) by dividing

one-fourth of the peak area at 7.1 ppm by half of the peak area at 4.1 ppm,

multiplied by 100. Similarly, the number of terminal double bonds per chain

(TDBC) and the number of terminal solvent groups per chain (TSGC) are given

by:

44

100

100

n

n

TDBHTDBC DP

TSGHTSGC DP

⎛ ⎞⎟⎜= ×⎟⎜ ⎟⎜⎝ ⎠

⎛ ⎞⎟⎜= ×⎟⎜ ⎟⎜⎝ ⎠

At the moment there are not enough data to draw conclusions on the evolution of

terminal solvent groups per chain, however, the values of TSGC agree with

previous studies. FTMS studies by Grady et al. [Grady et al., 2002], Peck and

Hutchinson [Peck et al., 2002] and Quan et al. [Quan et al., 2005] reported that at

elevated temperatures most of chains are xylol-initiated or β-scission

initiated/terminated. Experimental trends show that approximately 3 out of 10

chains are xylol-initiated, and 7 out of 10 chains are β -scission terminated at

170°C and at initial monomer content of 33wt.% (Figure 7). A more thorough

study on the evolution of chain microstructures for broader experimental ranges

of temperature and monomer content is currently being carried out.

45

0

0.2

0.4

0.6

0.8

1

TD

BC

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

DPn

TS

GC

Figure 7. Experimental data for TDBC and TSGC. The empty circles are

reproduced from [Peck et al., 2004], based on 140°C starved-feed nBA-xylene

experiments. Empty triangles are experimental data from this study, based on

170°C batch polymerization experiments, with initial monomer content of 33

wt.%.

46

Spontaneous initiation mechanism. While this study reports the estimated rate

constant for self-initiation, direct evidence for self-initiation is still absent. Thus we

also consider the possibility of initiation by decomposition of impurities (e.g.

oxygen and peroxides). The Arrhenius expressions reported by [Cerinski et al.,

2002] are used to represent rate of decomposition of common peroxides (Table

5). In Figure 8, two cases are considered: (i) in the case that an unknown amount

of impurities initiate the polymerization and the decomposition rate is

approximately equal to the decomposition rate of molecular oxygen, and (ii) when

the amount of impurities is known, however the rate at which they decompose is

not known. The first plot (top plot, Figure 8) shows that trace amount of

impurities, ca. 800 ppm, is able to initiate and sustain a considerable rate of

polymerization, with a slow rate of impurity decomposition. It should be noted,

however, that solubility of oxygen in organic medium is well below 800 ppm.

Hence the hypothetical scenario of oxygen-initiated polymerization (in

conjunction with the reported secondary constants) is deemed unrealistic. The

second plot (bottom plot, Figure 8) shows that faster impurity decomposition

rates lead to lower final monomer conversion. Thus much higher amounts (>>

800ppm), of impurities with faster decomposition rates, need to be present to

initiate and sustain observed monomer conversion profiles.

47

Table 6. Rate constant values for decomposition of oxygen and peroxides

[Cerinski et al., 2002], used in simulations presented in Figure 8.

Type of impurity A E (J.mol-1) Value at 160°C (s-1)

Oxygen (O2) 6.00E+09 116,000 6.15E-05

di-t-butyl peroxide

(DTBP) 6.64E+15 156,000 1.02E-03

di-cumyl peroxide

(DCP) 1.12E18 170,000 3.53E-03

t-butyl perisopropyl

carbonate (TBPC) 1.42E+15 141,000 1.41E-02

From the reported estimates, we are also able to analyze the activation energy of

the postulated second-order self-initiation, by comparing the estimated activation

energy for n-butyl acrylate with other self-initiating monomers. The comparison is

presented in Table 6. Note that the amount of initial monomer required to achieve

30% monomer conversion in ca. 20 minute reaction time at 170°C, increases

with increasing energy of activation. In other words, the lower the initial monomer

content required to achieve a certain degree of monomer conversion (at a given

temperature and reaction time) the lower the activation energy associated with

the self-initiation reaction. Thus the observed trend of the self-initiation activation

energy and the high amount of impurities required to obtain observed conversion

profiles suggest the occurrence of self-initiation. However, at the moment there is

48

still no direct evidence for self-initiation. It should also be noted that removal of

oxygen/peroxide-type impurities through nitrogen bubbling of solvent prior to

monomer feed, might not be adequate [Albertin et al., 2005]. A more direct

approach toward identifying the initiating molecules or quantifying amount of

impurities will be presented in a future study.

49

0

0.2

0.4

0.6

0.8

1

X

0 40 80 120 160 2000

0.2

0.4

0.6

0.8

1

Time, min

X

Figure 8. (Top) Model predictions vs. experimental data for monomer

conversion, X, predictions based on oxygen-initiated polymerization (no self-

initiation), with rate constants given in Table 4. Solid triangle markers represent

50

experimental data at 160°C, with initial monomer content of 40 wt.%. Solid,

dashed, dash-dotted and dotted lines represent model predictions for oxygen-

initiated polymerization with initial oxygen amount of 800, 600, 400 and 200 ppm,

at 160°C and initial monomer content of 40 wt.%, respectively. (Bottom) Model

predictions vs. experimental data for X, predictions based on impurity-initiated

polymerization (no self-initiation), with rate constants given in Table 6. Solid

triangle markers represent experimental data at 160°C , with initial monomer

content of 40 wt.%. Solid, dashed, dash-dotted and dotted lines represent model

predictions for oxygen-initiated, DTBP-initiated, DCP-initiated and TBPC-initiated

polymerization, with initial impurity amount of 800 ppm, at 160°C and initial

monomer content of 40 wt.%.

Table 7. Activation energies of self-initiating monomers. aInitial monomer content

at 170°C, ca. 20 min reaction time, 30% monomer conversion. bBased on the

postulated participation of molecular oxygen in the self-initiation mechanism

Monomer E (J.mol-1) Monomer contenta (wt.%) Ref.

nBA 73,152 33 This study

MMA 85,600b 50 Nising et al., 2005

Styrene 108,500 100 Hui et al., 1972

51

2.5 CONCLUSIONS

A mechanistic model for high-temperature batch polymerization of nBA was

developed and presented. The model was able to capture the process dynamics

between 140 and 180°C and between 20 and 40 wt.%. Quality of estimated rate

constants, along with the predictive capabilities of the developed model, is

presented and discussed. New experimental observations that support previous

studies on high-temperature solution of alkyl acrylates are also presented. This

study sheds light on the nature of the spontaneous initiation, backbiting, β-

scission and branch formation reaction mechanisms. It should also be noted that

there are limitations to the extents by which extrapolations can be made based

on the reported parameters. Future research directions in this field include; (i)

identification of novel 'experimental sensors' or measurements for parameter

estimation studies, (ii) design of experiments that provide direct evidence of self

initiation (e.g. better impurity removal techniques), (iii) development of efficient

parameter estimation approaches and (iv) development of iterative and data-

driven approaches towards identification of polymerization reaction networks and

the corresponding rate constants.

52

3 MODEL-BASED OPTIMAL CONTROL OF A SEMI-BATCH HIGH-TEMPERATURE POLYMERIZATION REACTOR

Batch reactors are widely used in the commercial production of polymer solutions

(resins), especially in low-volume production of high-quality resins [Srinivasan et

al., 2003; Grady et al., 2002]. While traditional batch processes are equipped

with minimum instrumentation [Srinivasan et al., 2003], efforts have been made

to use advanced non-conventional analytical methods such as near-infra-red

spectroscopy to measure monomer conversion and polymer average molecular

weights in batch solution polymerization of styrene [Fontoura et al., 2003].

Optimal control of batch polymerization reactors has received considerable

attention over the past several decades. Choices of monomer(s), solvent(s),

initiator(s) and operating conditions dictate the underlying polymerization reaction

mechanism and the complexity of the dynamics of the polymerization system

(e.g. homo- vs. co-polymerization, solution vs. emulsion polymerization, absence

vs. presence of gel effect, etc.). Recent optimal control studies on batch

polymerization systems include: solution and emulsion polymerization of styrene

[Karagoz et al., 2000; Gentric et al., 1997], solution and bulk polymerization of

53

methyl methacrylate [Zhang, 2004; Garg et al., 1999], and semi-batch emulsion

copolymerization of vinyl acetate/butyl acrylate [Immanuel et al., 2002].

Optimal control problems in batch polymerization reactors are inherently multi-

objective. Optimizing variables often have competing effects on an objective

function. The effect of an optimizing variable on one objective function can be in

reverse of the effect of the same optimizing variable on another objective

function. While the optimization problem can be formulated and solved in a

number of ways, choices are limited by computational expense [Srinivasan et al.,

2003]. Constraints of the optimization problem are feasible range of optimizing

variables, balance equations, product quality limits, instrument limitations, and

safety limits. Advances have been made to improve numerical algorithm (e.g.,

genetic algorithm [Garg et al., 1999; Immanuel et al., 2002; Tian et al, 2001] and

to develop alternative optimization approaches (e.g., extended iterative dynamic

programming [Wang et al., 1997] and measurement inclusion [Srinivasan et al.,

2002]).

Most optimization studies are based on first-principle models, derived from

knowledge of a predominant set of polymerization reaction mechanism

(mechanistic models). The need for reliable process models in the presence of

process-model mismatch and disturbances has motivated the use of hybrid

models such as a combination of neural networks and a first-principles model

[Zhang 2004].

54

This paper presents a study on optimal control of high-temperature (ca. above

140°C) semi-batch solution polymerization of alkyl acrylates. Challenges in the

high-temperature polymerization include the inadequacy of low-temperature

mechanistic models for the high temperatures and the limited industrial

experience in high-temperature batch operation. The inadequacy of the low-

temperature mechanistic models is a consequence of the profound effect of

secondary reactions that are negligible at low temperatures on the reactor

dynamics at the high temperatures. Motivated by these, in this work, a

mechanistic semi-batch reactor model based on a proposed reaction mechanism

for n-butyl acrylate is derived. The model parameters (reaction rate constants)

are estimated from off-line measurements of conversion, average molecular

weight, and number of terminal double bonds and branching points per 100

monomer units. The model is validated against measurements made when the

reactor is operated in regions different from those in which parameter estimation

was based on. By using the model, optimal solvent, monomer solution, and

initiator solution feed and reactor temperature profiles that minimize a multi-

objective performance index, are calculated. The reactor is operated in real-time

according to the optimal recipe and measurements are made to validate the

optimality of the recipe.

55

3.1 EXPERIMENTAL FRAMEWORK

The polymerization system being considered is the thermal solution

polymerization of n-butyl acrylate in xylene. Polymerizations were carried out at

160 and 180°C, low monomer concentration (20-40 monomer wt.%), in the

absence of thermal initiators, in a 2-Liter calorimeter. Details of the experimental

setup and spectroscopic analyses can be found in [Peck et al., 2004].

Experiments were carried out at DuPont Marshall Laboratory, Philadelphia, PA.

3.2 MODEL DEVELOPMENT

3.2.1 Kinetic Scheme and Rate Laws

Reaction mechanism postulated for the polymerization of n-Butyl Acrylate is

based on the characterization results on different polymer micro-structures. The

approach introduced in tendency modeling method [Quan et al., 2005] is used in

this work. The approach assumes that rate of reactions are only dependent upon

concentrations of micro-structural entities that are involved in the reaction.

Reactivity of propagating radicals is also assumed to be independent of chain

length.

56

Thermal self-initiation and propagation. n-Butyl Acrylate has been observed to

undergo polymerization in the absence of thermal initiators. The self-initiation

reaction is currently assumed to be second-order with respect to monomer

concentration [Rantow et al., 2006]:

iik1I 2R⎯⎯→ Initiation by initiator

tik12M 2R⎯⎯→ Initiation by monomer

pk1 1R +M R⎯⎯→ Propagation

Inter- and Intra-molecular chain transfer. Backbiting and chain-transfer to

polymer (CTP) have been observed to be dominant at low and high initial

monomer concentrations, respectively [Rantow et al., 2006].

bbk1 3R R⎯⎯→ Backbiting

tpk1 2R +D+TC R +D⎯⎯→ CTP

Chain-transfer to small molecules. Assuming that amount of impurities in

solution polymerization is kept to a minimum, the only chain-transfer reactions to

be considered are chain-transfer to monomer (CTM) and solvent (CTS). A

terminal double bond (TDB) is produced every time chain-transfer to monomer

occurs.

tsk1R+S R +D⎯⎯→ CTS

tmk4R+M R +D⎯⎯→ CTM

Long and short chain-branching formation. Formation of long chain-branching

(LCB) and short chain-branching (SCB) have been observed in solution

polymerization of n-Butyl Acrylate at 140°C [Grady et al., 2002].

57

pbk2 1R +M R +LCB⎯⎯→ LCB Formation

pbk3 1R +M R +SCB⎯⎯→ SCB Formation

Terminal double bond formation. Scission reactions are in competition with the

branching-formation reactions [Grady et al., 2002]. Notice that TDB is also

formed in CTD.

pk4 1R +M R +TDB⎯⎯→ TDB Formation

βk2 1R R +D+TDB⎯⎯→ Beta-scission

βk3 1R R +D+TDB⎯⎯→ Beta-scission

TDB propagation. Existence of TDBs allows further polymerization to take

place. TDB propagation is in competition with the scission reactions [Grady et al.,

2002].

tdbk1 2R +TDB+D R⎯⎯→ TDB Propagation

Chain termination. Both known modes of chain-termination is assumed to occur

in this work; termination by combination (CTC) and by disproportionation (CTD).

tdk2R 2D+TDB⎯⎯→ CTD tck2R D⎯⎯→ CTC

Based on the postulated reaction mechanism, the following rate laws are derived:

58

( )

( ) { }

2M ti p 1 pb 2 3 tm p 4

I ii

S ts

2TDB p 4 β 2 3 tdb 1 td 1 2 2 3 3

SCB pb 3

LCB pb 2

D tm t

r =-2k [M] -k [M][R ]-k [M] [R ]+[R ] -k [M][R]-k [M][R ]

r =-k [I]

r =-k [S][R]

r =k [M][R ]+k [R ]+[R ] -2k [TDB][R ]+k [R ][R]+[R ]([R ]+[R ])+[R ]

r =k [M][R ]

r =k [M][R ]

r =k [M][R]+k

{ }s β 2 3

2tc td 1 2 2 3 3 tdb 1

[S][R]+k ([R ]+[R ])

+(k +2k ) [R ][R]+[R ]([R ]+[R ])+[R ] -2k [TDB][R ]

where by invoking the quasi-steady-state-assumption on all the radical

concentrations, we end up with:

( )( ) ( )

( )( ) ( )

0.52

ti ii

tc td

1

2 1

3 1

tm4

tp

tp tdb

pb tm ts β tc td

bb tdb

pb tm ts β tc td

f

k [M] +2k [I][R]=k +k

[R][R ]=1+α+β

[R ]=α[R ]

[R ]=β[R ]

k[R ]= [R]k

k [TC]+k [TDB]α=

k +k [M]+k [S]+k + k +k [R]

k +k [TDB]β=

k +k [M]+k [S]+k + k +k [R]

[TC]=[M] -[M]-

⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

[TDB]-[SCB]-[LCB]

3.2.2 Process Model

59

An isothermal semi-batch solution polymerization process model can then be

derived:

( )

( )

( )

( )

( )

( )

( )

T

M M M T

I I I T

S S I I S T

TDB T

SCB T

LCB T

D T

dV =Fdtd[M] 1= F [M] +r V-[M]F

dt Vd[I] 1= F [M] +r V-[I]Fdt V

d[S] 1= F [S] +F [S] +r V-[S]Fdt V

d[TDB] 1= r V-[TDB]Fdt V

d[SCB] 1= r V-[SCB]Fdt V

d[LCB] 1= r V-[LCB]Fdt V

d[D] 1= r V-[D]Fdt V

where FT = FM + FS + FI. The measured variables are given by:

fm

f

t

f M M0

fn m

f

f

[M] -[M]x =[M]

1[M] = F [M] dtV

[M] -[M]M = ×MW[D]

[TDB]TDBH= ×100[M] -[M]

[SCB]+[LCB]BPH= ×100[M] -[M]

∫

60

3.3 PARAMETER ESTIMATION AND MODEL VALIDATION

Reaction rate constants are then estimated from the four types of measurements

shown in Figure 9. The parameter values are given in Table 8. The rate of

thermal initiator decomposition that is used in this study is that of t-butyl peroxide;

kii = 3.15 x 1014 exp(-1.38 x 105/ RT) s-1. The quality of the model predictions is

shown in Figure 9.

Table 8. Values of the estimated parameters

Parameter Unit 160°C 180°C kti L.mol-1.s-1 1.33E-10 1.33E-09 kp L.mol-1.s-1 1.73E+02 9.25E+02 ktp L.mol-1.s-1 1.57E-01 1.69E+00 kbb s-1 2.15E+01 1.20E+02 ktm L.mol-1.s-1 3.38E-04 9.05E-03 kts L.mol-1.s-1 1.18E-01 1.47E+00 kpb L.mol-1.s-1 8.47E+00 1.14E+01 kβ s-1 1.84E+01 3.06E+01 ktdb L.mol-1.s-1 3.97E+02 1.63E+03 ktc L.mol-1.s-1 5.76E+00 2.02E+02 ktd L.mol-1.s-1 1.76E+01 7.96E+01

61

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

time, min

x m

0 50 100 150 200

0

2000

4000

6000

8000

10000

time, min

Mn

0 20 40 60 80 100 120

0

1

2

3

4

5

6

7

8

time, min

TD

BH

0 20 40 60 80 100 120

0

1

2

3

4

5

6

7

8

time, minB

PH

Figure 9. Measurements and model predictions of monomer conversion (xm),

number-average molecular weight (Mn), number of terminal double bonds per

100 monomer units (TDBH) and number of branching points per 100 monomer

units (BPH) when the reactor is operate isothermally. The dashed and solid lines

represent model predictions at 160 and 180°C respectively, and the triangles and

circles represent measurements at 160 and 180°C respectively.

62

3.4 OPTIMIZATION AND OPTIMIZATION VALIDATION

3.4.1 Performance Index

The performance index to be minimized is formulated by considering typical resin

recipes and general knowledge of industrial polymerization processes. In

particular, it is desirable to achieve the following objectives:

• Monomer conversion. A monomer conversion of 100% is desirable,

since unreacted monomers can undergo further (unwanted)

polymerization.

• Number-average molecular weight. Low viscosity resins are desired for

automotive coatings applications. This generally translates to lower

number-average molecular weight resins. In a typical batch production of

resins, a certain number-average molecular weight is desired.

• Number of branching points. Effect of branching on polymer properties

in general is still not well understood. Thus, currently production is based

on experience or knowledge of the 'optimum' recipe. The number of

branching points is kept constant in each recipe.

• Number of terminal double bonds. The existence of terminal double

bonds creates opportunities for macromonomer production. Manufacturers

63

would ultimately like to have control over number of terminal double bonds

in the end product.

• Initiator. Thermal initiators are the most expensive component in the resin

production recipe. Thus, it is desirable to minimize the amount of initiator

used in the resin production.

Based on the above considerations, the following performance index is

formulated:

( )2 22 2

2 n I1 m 2 3 4 5

n,des des des I,t

M mBPH TDBHJ=ω 1-x +ω -1 +ω -1 +ω -1 +ωM BPH TDBH m

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

where ω1, … , ω5 are the weights on the individual objectives. Their values

depend on the goal of optimization.

3.4.2 Optimization Constraints

The above performance index is minimized subject to:

• Optimizing Variable Constraints. These constraints represent the

feasible ranges of the optimization variables.

• Balance equations. These are the conservation equations that govern

the process.

64

• Hardware and safety limits. Safety concerns and instrumentation

capabilities are represented as limits on how steep change in

temperatures can be.

the optimization constraints are as follows:

( )

N

i,j j1

N

i,I I1

i,j j

i i+1 des

i

f =f , j=M,S

f f

0 f f , j=M,S,I i=1,…,N

T -T ΔT i=1,…,N

393.2 T 573.2 i=1,…,N

≤

≤ ≤

≤

≤ ≤

∑

∑

3.4.3 Optimization

Different batch times are selected and evaluated. Each batch time is divided into

N equal time intervals, and each feed flowrate and temperature profile is

discretized into M equally-spaced constant values. Thus, the feed flowrate and

temperature profiles are piece-wise constant. Since we have 3 different flow

rates (monomer, solvent and initiator), we end up with 4 x M x N optimizing

variables. When choosing N and M one should take into account the

65

computational cost of the optimization. In other words, N and M should be large

enough such that the performance index is minimized adequately.

In this study, the performance index minimum is obtained by simulating all

possible profiles subject to the defined constraints (simulation), and

subsequently, searching through the list of the resulting performance indices for

a minimum (optimization). An optimal reactor feed policy subject to an isothermal

condition is first calculated. Subsequently, based on the optimal feed policy,

different temperature profiles are simulated to obtain a better optimum.

Polymerization recipe used in the optimization and optimization condition is given

in Table 9. Optimization results are shown in Table 10.

Table 9. Optimization Settings

Monomer (n-butyl acrylate) = 300 g

Solvent (xylene) = 600 g

Initiator (t-butyl peroxide) = 0.01% x Monomer

ω1 = ω2 = ω3 = ω4 = ω5 = 1

N = M = 4

Mn,des = 2000

TDBHdes = 5

BPHdes = 10

(ΔT)des = 5°C

66

Table 10. Optimal feed and temperature profiles

Time(min) 100 200 300 400

Fraction of monomer 0.5 0.5 0.75 0.75 Added 0.5 0.5 0.25 0.25 0 0 0 0 0 0 0 0 Fraction of solvent 1 1 1 0.5 Added 0 0 0 0.5 0 0 0 0 0 0 0 0 Fraction of initiator 0 0 0 0 Added 0.5 0 0.5 0.5 0 0.5 0 0 0 0 0 0 Temperature (°C) 170 170 170 170 165 175 175 175 160 175 180 180 155 180 185 175 xm 100% 100% 100% 100% Mn 2253 2053 2035 1978 TDBH 5.4 5.1 4.6 4.8 BPH 7.8 8.1 9.7 10.2 J 0.8089 0.7708 0.7455 0.7387

Optimization results were validated experimentally. The optimization validation

results are shown in the Figure 10. The validation is a comparison between

experimental data and model predictions from the method of moments model

presented in the mechanistic modeling section. Optimal feed policy from

tendency model and method of moments models yield the same results. The

microstructures are presented in terms of quantities per chain, calculated as

follows:

67

n

n

TDBHTDBC= ×DP100

BPHBPH= ×DP100

Challenges in experimental validation include; (i) limited process operator

experience in carrying out high-temperature experiments, (ii) possible existence

of process-model mismatch, (iii) sub-optimal control policies and (iv) limited

experience in utilization of modern polymer characterization techniques in

optimal control studies.

68

0

0.2

0.4

0.6

0.8

1

Con

vers

ion

0

2E3

4E3

6E3

Mn

0 100 200 3001

1.5

2

2.5

3

Time, minutes

Pol

ydis

pers

ity

Figure 10. Optimization validation results for 300-minute optimal control recipe.

Triangles and lines represent measurements and model predictions, respectively.

69

0

0.05

0.1

0.15

0.2

[SC

B],

mol

/L

0 100 200 3000

0.02

0.04

0.06

0.08

0.1

Time, minutes

[TD

B],

mol

/L

Figure 11. Corresponding polymer microstructure profiles for the 300-minute

optimal control recipe. Triangles and lines represent measurements and model

predictions, respectively.

70

3.5 CONCLUSIONS

In this study, a class of batch polymerization reactor was investigated. A

mechanistic model for an isothermal semi-batch reactor was developed based on

a proposed set of reaction mechanism of high-temperature solution

polymerization of alkyl acrylates. Optimal control policies were calculated based

on minimization of a proposed performance index. The performance index was

formulated based on desired micro-structural properties such as number of

terminal double bonds and branching points, and more conventional objectives

such as maximization of monomer conversion, minimization of thermal initiator

and obtainment of a pre-determined number-average molecular weight.

Optimization results revealed non-intuitive control policies which allow for the

optimization of end-product properties and the minimization of production cost.

71

4 REDUCED-ORDER MODEL FOR MONITORING SPECTROSCOPIC AND CHROMATOGRAPHIC PROPERTIES

With increasing accessibility to advanced measurement techniques, recent

studies focus on improving process modeling and monitoring approaches in a

variety of applications. The unifying theme of recent studies in kinetic modeling is

the integration of chemical information obtained from a variety of measurements.

The goal of these studies has been to elucidate the dynamics governing the

behavior of a process. For example, a combination of mass action kinetic model

and artificial neural network has recently been applied to monitor different levels

of pesticide contents in fruit and vegetable samples, whose levels were obtained

by using spectrophotometry techniques [Nising et al., 2004]. Similar studies

focused on (a) identifying and estimating a unique set of rate coefficients by

simultaneously fitting a set of measurements [Bezemer et al., 2001; Zogg et al.,

2004; Rantow et al., 2006], (b) understanding how changes in measurements

affect model parameter values and initial rate of reactions [Carlson et al., 1993],

(c) decreasing parameter search space by including parametric non-negativity

constraints in the parameter estimation [Jaumot et al., 2005], and (d)

understanding the effect of the initial guesses on an unconstrained optimization

approach to parameter estimation [Zhu et al., 2002]. These studies emphasize

the need for a systematic approach to using different measurements in modeling

and monitoring strategies.

Mechanistic models often provide inaccurate extrapolating predictions in complex

polymerization systems. This can be attributed to phenomena that occur at the

extrapolated conditions but are not accounted for in the model. To address this

72

problem, one usually resorts to continual improvement of the mechanistic models

or development of process monitoring strategies that use off-line and on-line

measurements to correct the predictions. Research directions in this area include

multivariate monitoring or data processing strategies by using statistical tools

[Kourti et al., 1995; Negiz et al, 1997; Neogi et al., 1998], model-based

monitoring strategies [Soroush et al., 1997; Tatiraju et al., 1999] and

characterization of batch reactions with in situ spectroscopic measurements,

calorimetry and dynamic modeling [Ma et al., 2003]. Particularly in process

monitoring of classical batch polymerization reactors, with the advancements

made in `hardware' sensors [Kammona et al., 1999], the current challenge is in

integrating data from emerging measuring devices into existing monitoring

systems (e.g., monitoring of polymer chain end groups by using nuclear magnetic

resonance spectroscopy [Lutz et al., 2005]).

This study was motivated by the observation of complex phenomena [Grady et

al., 2002] in high-temperature free-radical polymerization of acrylates. One such

phenomenon is spontaneous thermal polymerization of n-butyl acrylate, i.e.

polymerization without the addition of ‘known’ free-radical initiators.

Reproducible, sustained conversion of monomer to polymer in a few hours was

observed [Quan et al., 2005]. Other phenomena such as back-biting, beta-

scission and long chain branching also occur and complicate the underlying

polymerization kinetics. The method-of-moments model presented in [Rantow et

al., 2006] is able to successfully predict monomer conversion and number-

average molecular weight, but it over-predicts the polymer polydispersity index

and under-predicts the concentration of terminal double bonds (subsection

3.4.3). In view of these predictive deficiencies, this study aims to (i) develop a

simplified reduced-order polymerization reactor model for high-temperature

polymerization of n-butyl acrylate, (ii) develop a multi-rate state estimator using

the method described in [Zambare et al., 2003] and the reduced order model,

and (iii) evaluate the performance of the state observer in predicting number-

and weight-average molecular weights (chromatographic properties) and polymer

73

chain microstructures (spectroscopic properties). The approach of modeling

macromolecular kinetics as small-molecular kinetics [Villermaux et al., 1984] is

adopted as a model-reduction technique. The polymerization reaction network in

study contains 15 irreversible reaction pathways and 10 rate coefficients. The

reduced-order reactor model is of order 7 (has 7 state variables), compared to

the 20 state variables of the original model. The measurements of the reactor

temperature and feed flow rates of monomer, solvent and initiator are the

frequent measurements, and the measurements of molecular weight averages

and microstructural quantities are the infrequent measurements. The multi-rate

state estimator estimates from these measurements.

4.1 EXPERIMENTAL PROCEDURE

The information of the raw material suppliers and preparation of raw materials

used in the experiments are reported in [Rantow et al., 2006]. The polymerization

is carried out in a 1-liter Mettler-Toledo RC1 Reactor-Calorimeter, which is a

jacketed semi-batch glass reactor. The feed and temperature profiles are the

optimal profiles already calculated in [Rantow et al., 2005] to achieve certain end-

product properties.

Initially, 600 g of solvent (xylene) is loaded to the reactor at room temperature.

The reactor temperature is then increased to 170°C. Subsequently after the

temperature is reached, 300 g of monomer (n-butyl acrylate) is fed to the reactor.

At this time, the cooling system embedded in the reactor-calorimeter

automatically decreases the jacket temperature (by adjusting the flow rate of

cooling fluid to the jacket), to balance the heat generated by the polymerization

reaction, thus maintaining the reactor temperature at 170°C. This initial

temperature drop can be seen in Figure 12.

74

Figure 12. Profiles of the reactor temperature and the flow rates of the monomer

(n-butyl acrylate), the solvent (xylene) and the initiator (t-butyl peroxy acetate)

solution feed streams.

After 75 minutes into the reaction, the reactor temperature is then increased to

175°C. Once the temperature is reached, 100 g of monomer is then fed to the

reactor, along with approximately 40 mg of initiator solution (10 wt.% of t-butyl

peroxy acetate in xylene). This results in a milder temperature drop (Figure 12).

Then 75 minutes later, the temperature is again increased to 180°C, and 75

minutes after that, the temperature is again increased to 185°C. The reaction

temperature is then maintained at about 185°C for another 75 minutes before the

reactor temperature is dropped back to room temperature. It should be noted that

in the absence of the thermal initiation phenomenon of n-butyl acrylate this could

lead to a monomer pooling (accumulation of unreacted monomer in the reactor)

75

event, although in this particular study the total monomer content is reasonably

low (40% by weight). Thus care needs to be taken in terms of knowing whether

polymerization is occurring or has occurred prior to the addition of the free-radical

initiators.

4.1.1 Off-Line and On-Line Measurements

A sample is taken every 30 minutes. The drawn samples are diluted (50/50 v/v)

in a cold inhibitor solution (1000 ppm 4-methoxy phenol in xylene). Polymer

solids content in the samples is determined by gravimetry, while molecular

weight distributions are determined by using an HP 1090 high performance liquid

chromatography (HPLC). Polymer-chain microstructure properties are quantified

by using proton nuclear magnetic spectroscopy (1H-NMR) and carbon NMR (13C-

NMR). More details on the analytical procedures can be found in [Rantow et al.,

2006]. Time delay associated with each of the infrequent measurements is 30

minutes. The solvent concentration in the reactor is inferred from 1H-NMR

measurement of the hydrogen atoms on the terminal solvent (xylol) groups. The

concentration of the solvent in the reactor is calculated from the number of

solvent groups per 100 repeat units (SH) in the polymer:

S

M

H

Ht

S 00

wt

Mw 0

number of solvent groups in NMR sampleSH = number of repeat units in NMR sample

area of 1H-NMR peak at 7.1 ppm 4= 100area of 1H-NMR peak at 4.1 ppm 2

ρ F dt[ ]

M V= 100

ρ F dt [ ]M V

smS

M

δδ= ÷ ×= ÷

+−

×−

∫

∫

76

where ms0 is the mass of the solvent in the reactor at time t = 0 (right after initial

reactor loading and before the feeding starts). Thus, from the above equation, [S]

can be inferred from:

S M

t t

S s0 Mw w0 0

1 SH ρ[S] = ρ F dt + m F dt [M]M V 100 M V

⎛ ⎞⎛ ⎞ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ − − ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟ ⎟⎝ ⎠ ⎝ ⎠∫ ∫

Note that this is a bulk average quantity. Also note that the other quantities, i.e.

concentrations of terminal double bonds and short chain branches can also be

inferred in the same way. Key peak assignments are given in Table 11 and the

corresponding molecular structure of poly(n-butyl acrylate) is given in Figure 13.

Table 11. Key 1H-NMR and 13C-NMR peak assignments

Hδ , ppm 1H-NMR Peak Assignment Atom(s) in Figure Reference

4.1 H on CH2 of ester linkage next to COO i Quan et al.,

2005

5.5 H on CH2 or terminal double bond b Chiefari et

al., 1999

6.2 H on CH2 of terminal double bond a Chiefari et

al., 1999

7.1 H on terminal xylol group d,e,f,g Quan et al.,

2005

Cδ , ppm 13C-NMR Peak Assignment Atom(s) in Figure Reference

47.2-48.4 Quaternary backbone C c Ahmad et

al., 1998

165-180 C on C=O h Quan et al.,

2005

77

n m

a

b

c

d

e f

g

HH

CH3CH2

CH2

CH3CH2

CH2

CH3

CH2

CH2

CH3CH2

CH2

CH3

CH2CH2

H

HH

H

CH3

H

H

CC

CC

CC

CH2

O OC

C CH2

CH2

O OC

CH2CH2

CH2

O OC

CHCH2CC CH2CH2 CH

C

OC

O

CH2

OOCh

ii

Figure 13. Molecular structure of the synthesized poly(n-butyl acrylate).

Molecular structure shows the key atoms used to infer polymer microstructure

concentrations.

The concentration of solvent can also be readily measured using and on-line or

at-line gas chromatography. The reactor temperature, the volume of monomer

added to the reactor, the volume of the reacting mixture inside the reactor, and

the feed flow rates are measured at a sampling rate of one minute with no time

delay; these are frequent measured inputs.

78

4.2 REDUCED-ORDER MECHANISTIC POLYMERIZATION MODEL

The polymerization reaction network relevant to polymerization reactions of alkyl

acrylates at higher temperatures has been reported in [Peck et al., 2004; Rantow

et al., 2006]. In this study, we simplify the polymerization reaction network given

previously in Table 2 by using a small-molecule representation. The underlying

assumption behind the simplification of the reaction network is in the

classifications of the reacting species. In the reduced-order model we assume

that the polymer chain microstructures are formed as if new small molecules are

formed, and reactive end-group radicals are viewed as regular radical-containing

small molecules. This idea was first put forth by [Villermaux et al., 1984]. This

approximation becomes more accurate as the polymer chain length and the

concentration of polymer decrease. For this particular study, the polymer chains

produced have 30 repeating units on average. The resulting reaction network is

given in Table 12, which leads to a reduced-order kinetic model.

79

Table 12. Simplified (using the small-molecule representation) reaction network

for polymerization of nBA in xylene, with a monomer content of 20-40 wt%, and

at 140-180°C.

sec sec

sec sec

sec

sec

2 2

2

2 2

2

2

2

ti

p

bb

pq

tm

p

ts

tc

td

terttc

terttd

tert terttc

terttd

k

k

k

k

k

k TDB

kTDB

k

k

k

k

k

k

k

M P

P M P

P Q

Q M P SCB

Q D P TDB

P M D P

P M P TDB

P S D P

P D

P D

P Q D

P Q D

Q D

Q

β

−

−

−

−

−

−

⎯⎯→

+ ⎯⎯→⎯⎯→

+ ⎯⎯→ +

⎯⎯→ + ++ ⎯⎯→ +

+ ⎯⎯→ ++ ⎯⎯→ +

⎯⎯⎯→

⎯⎯⎯→

+ ⎯⎯⎯→

+ ⎯⎯⎯→

⎯⎯⎯→

2tert

D⎯⎯⎯→

The reduced-order reactor model is then developed by writing mass and

structural-character balances for the semi-batch reactor. An energy balance is

not needed to be included in the process model, because the measured reactor

temperature is an input to the model (is a measured input). It should be noted

that the density of reacting solution is assumed to be constant. This assumption

is deemed appropriate as polymerization is carried out at the lower end of the

concentration range (less than 40wt.%), the operating temperature range is

narrow, and that the solvent (xylene) and the monomer (n-butyl acrylate) have

similar densities. The resulting ordinary differential equations (ODEs) describing

the dynamics of the reactor are:

80

( )( )( )( )( )

, ,[ ], , ,[ ], ,[ ],[ ], , ,[ ]

, ,[ ],[ ],[ ], , ,[ ], ,[ ],[ ],[ ], , ,[ ], ,[ ],[ ],[ ], , ,[ ]

, ,[ ],[ ],[ ],[ ],[ ]

I I M S

M I M S

S I M S

TDB I M S

SCB I M S

D

f T V I F F FIf T V M I F F FM

f T V S M I F F FSdf T V M TDB I F F FTDBdtf T V M SCB I F F FSCBf T V M S D ID

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ( ), ,I M SF F F

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

where

( ) ( ) ( )

( ) ( )

( )

( ) ( )

( )

1, ,[ ], , , [ ], [ ]

, ,[ ],[ ], , , [ ],[ ],[ ],[ ],

1 [ ]

, ,[ ],[ ],[ ], , , [ ],[ ],

1 [ ]

I

M

II I M S I I M S I

w

TDBM I M S M

MI M S M

w

S I M S S

SI M S

w

f T V I F F F R I T I F F F FV M

f T V M I F F F R M P Q P T

M F F F FV M

f T V S M I F F F R S P T

S F F FV M

ρ

ρ

ρ

⎧ ⎫⎪ ⎪⎪ ⎪= − + + −⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭=

⎧ ⎫⎪ ⎪⎪ ⎪− + + −⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭=

− + + −

( ) ( )

( ) ( )

( ) ( )

, ,[ ],[ ],[ ], , , [ ],[ ],[ ],

[ ]

, ,[ ],[ ],[ ], , , [ ],[ ],

[ ]

, ,[ ],[ ],[ ],[ ], , , [ ],[ ],[ ],[ ],

[ ]

S

S

TDBTDB I M S TDB

I M S

SCB I M S SCB

I M S

D I M S D

I

F

f T V M TDB I F F F R Q M P T

F F FTDBV

f T V M SCB I F F F R Q M TF F FSCB

Vf T V M S D I F F F R Q M S P T

F FD

⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭=

+ +−

=+ +−

=+− M SF

V+

where the state variables, [I], [M], [S], [TDB], [SCB] and [D] are the concentration

of the initiator (t-butyl peroxy acetate), concentration of the monomer (n-butyl

acrylate), concentration of the solvent (xylene), concentration of terminal double

bonds, concentration of short chain branches, and concentration of `dead'

polymer chains, respectively. The measured inputs V, FI, FM, FS, and T are the

volume of the reacting mixture, the reactor temperature, and the volumetric flow

81

rates of the initiator solution, the monomer and the solvent, respectively. The

vector of infrequent measured outputs:

( )( )n

w

M Mn

w M M

H V ,V,[M],[D]MM H V ,V,T,[M],[D]

=[TDB] [TDB][SCB] [SCB]

[S] [S]

⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

where

( )

( )

n

M

w

M

M M mM M

w

M M M MM M 1 2 3

w

t

M M0

ρ V MWH V ,V,[M],[D] = -[M] ×M V [D]

ρ V ρ V -[M]H V ,V,T,[M],[D] = c +c +c TρV VM [D]

V = F dt

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎛ ⎞⎧ ⎫⎛ ⎞⎪ ⎪ ⎟⎜⎪ ⎪⎟⎜ ⎟⎜⎟⎨ ⎬⎜ ⎟⎜⎟⎜ ⎟⎟⎜⎪ ⎪⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

∫

VM is also a frequent measured input.

Note that weight-average molecular weight is calculated by multiplying number-

average molecular weight by polydispersity index, PDI, given by the empirical

correlation:

M M1 2 3

ρ VPDI=c +c +c TρV

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

where the correlation constants c1=2.66031, c2=0.03337 and c3=-0.00929. The

second term on the right hand side of the correlation is c2 times the weight

fraction of the monomer fed to the reactor. The correlation constants are

obtained by fitting the correlation to the measurements shown in Figure 14.

82

0 10 20 30 40 501

1.5

2

2.5

3

Initial Monomer Content, wt.%

Pol

ydis

pers

ity

140oC

160oC

180oC

120 140 160 180 2001

1.5

2

2.5

3

Temperature, oC

Pol

ydis

pers

ity

10wt.%20wt.%30wt.%40wt.%

Figure 14. Measurements of polydispersity index at different temperatures

(constant initial monomer weight percent of 40%) and different initial monomer

weight percents (constant temperature of 160°C).

83

The reaction rate laws are:

( )

( ) ( )( )

I d

2 sec TDB tertM ti p p tm

S tssec TDB

TDB β p

tertSCB p

sec-sec sec-sec 2D β tm ts tc td

sec-tert sec-terttc td

R =-k [I]

R =-2k [M] -k [M] [P]+[P ] -k [M][Q]-k [M][P]

R =-k [S][P]

R =k [Q]+k [M][P ]

R =k [M][Q]

R =k [Q]+ k [M]+k [S] [P]+ k +2k [P]

+ k +2k [P][Q ( )tert-tert tert-tert 2tc td]+ k +2k [Q]

where the concentrations of the free radicals are given by the following algebraic

equations, obtained by applying the quasi-steady-state assumption:

( )

( )

0.522I M S I M S

t ti d

2ti d

bb

tert I M Sp β t

tm

sec I M Sp

TDB

F +F +F F +F +F+ +8k k [M] +2k [I]V V

[R]=4 k [M] +2k [I]

[R][P]=1+α+β

kα= F +F +Fk [M]+k +k [R]+V

k [M]β= F +F +Fk [M]+V

[Q]=α[P][P ]=β[P]

⎡ ⎤⎛ ⎞⎢ ⎥⎟⎜ ⎟⎜⎢ ⎥⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦

84

4.3 REDUCED-ORDER MULTI-RATE STATE OBSERVER DESIGN

On the basis of the developed reduced-order model, a reduced-order multi-rate

state observer is designed using the method described in [Tatiraju et al., 1999;

Zambare et al., 2003]. The estimator is of the form:

( )( )( ) { }( ) { }

*31

*42

ˆ ˆ[ ] , ,[ ], , ,

ˆ ˆ ˆ[ ] , ,[ ],[ ], , ,

ˆ ˆ ˆˆ ˆ[ ] , ,[ ],[ ],[ ], , , [ ] [ ]

ˆ ˆ ˆ ˆ ˆ[ ] , ,[ ],[ ],[ ], , , [ ] [ ]

ˆ ˆ ˆ ˆ[ ] , ,[ ],[ ],[ ], , ,

I I M S

M I M S

S I M S

TDB I M S

SCB I M

I f T V I F F F

M f T V I M F F F

S f T V S M I F F F L S S

TDB f T V TDB M I F F F L TDB TDB

SCB f T V SCB M I F F

=

=

= + −

= + −

=

&

&

&

&

& ( ) { }( ) ( ){ }

*53

*64

ˆ[ ] [ ]

ˆˆ ˆ ˆ ˆ ˆ ˆ[ ] , ,[ ],[ ],[ ],[ ], , , , ,[ ],[ ]n

S

D I M S n M M

F L SCB SCB

D f T V D S M I F F F L M H V V M D

+ −

= + −&

where each term with an asterisk represents the predicted present value of the

variable with the corresponding infrequent measurement. This present value is

obtained by the least-squared-error fit of a line to the three most recent

measurements of the infrequent measured output. The second term in the right

hand side of the estimator equation is simply a corrective feedback term; as long

as there are discrepancies between the predicted present value of a variable

(variable with asterisk) and the corresponding estimate (variable with hat), the

corrective action is made.

The estimator calculates continuous estimates of the concentrations of the

initiator, the solvent, terminal double bonds, short polymer chain branches and

dead polymer chains from:

• The frequent measured inputs: the reactor temperature, the reacting

mixture volume, the flow rates of the monomer, the solvent and the

85

initiator solution feed streams, and the volume of monomer fed to the

reactor; and

• The infrequent measured outputs: the concentrations of the solvent,

terminal double bonds, short polymer chain branches, and the number-

average molecular weight of the polymer.

. An analysis of the local observability of the system (Table 13) led to setting all

of the estimator gain entries except for the entries L31, L42, L53 and L64 to zero.

Table 13. Simplified (using the small-molecule representation) reaction network

for polymerization of nBA in xylene, with a monomer content of 20-40 wt%, and

at 140-180°C.

Measurement [S] [TDB] [SCB] Mn Mw

State Variable

V Yes Yes Yes Yes Yes

[I] Yes Yes Yes Yes Yes

[M] Yes Yes Yes Yes Yes

[S] Yes No No Yes Yes

[TDB] No Yes No No No

[SCB] No No Yes No No

[D] No No No Yes Yes

The non-zero observer gain entries L31 = 1 x 10-1, L42 = 1 x 10-1, L53 = 1 x 10-1,

and L64 = -5 x 10-6, which ensure that all eigenvalues of the Jacobian of the

observer error dynamics lie in the left half of the complex plane. The non-zero

terms in effect indicate which slow measurement is used to correct a state

variable estimate. A more thorough discussion on how the estimation error

(between the estimates and the true values) depends on the estimator gain

matrix entries is given in [Zambare et al., 2003].

86

4.4 STATE OBSERVER PERFORMANCE

The flow of information between the reactor and the multi-rate observer is

depicted in Figure 15. Measurements of the reactor temperature and the feed

flow rates were fed to the nonlinear reduced-order multi-rate state estimator.

The estimator initial conditions were set to that of the actual reactor.

Figure 15. Flow of information between the polymerization reactor and the

nonlinear reduced-order multi-rate state observer.

The performance of the estimator in providing estimates of the number-average

and weight-average molecular weights is shown in Figure 16.

87

0

2E3

4E3

6E3

8E3

Mn

0 100 200 3000

0.5

1

1.5

2x 10

4

Time, minutes

Mw

Figure 16. Estimated vs. measured values of the number- and weight-average

molecular weights (dashed line = estimate without using feed back

measurement, solid line = estimate using feed back measurement, circles =

measurement).

88

In the estimator implementation, the entries L31, L42, L53 and L64 are set to zero

until the first set of infrequent measurements are available. To calculate the

present value of each infrequently measured output, a linear polynomial is fitted

to the most recent three data points (measurements), except for when two sets of

infrequent measurements are available. The latter case, a line is fitted to two

measurements.

0 100 200 3004

4.5

5

5.5

6

Time, minutes

[S],

mol

/L

0 100 200 300

0

0.02

0.04

0.06

0.08

0.1

Time, minutes[T

DB

], m

ol/L

Figure 17. Estimated vs. measured values of the concentrations of the solvent

and terminal double bond (dashed line = estimate without using feed back

measurement, solid line = estimate using feed back measurement, circles =

measurement). Bottom figure shows evolution of the 1H-NMR peaks.

89

The estimates of the polymer microstructure quantities calculated by the multi-

rate state estimator are shown in Figures 17 and 18, which demonstrate that the

state estimator provides accurate continuous estimates of the variables that have

slow measurements. The selection of the non-zero estimator gain entries on the

basis of the local observability analysis is shown to be good enough to ensure

satisfactory estimates.

Figure 18. Estimated vs. measured values of the concentrations of the short

chain branches (dashed line = estimate without using feed back measurement,

solid line = estimate using feed back measurement, circles = measurement).

Bottom figure shows evolution of the 1H-NMR peaks.

90

Selection of the non-zero estimator gain entries based on the local observability

analyses is shown to provide adequate information to the state observer.

Selection of the non-zero entries show that the reactor volume, monomer

concentration and initiator concentration can be estimated solely based on the

fast measurable inputs. And there is a one-to-one pairing between the slow

measurable outputs and the rest of the state variables.

4.5 CONCLUDING REMARKS

A reduced-order high-temperature mechanistic polymerization model was

presented. The model is capable of predicting spectroscopic and

chromatographic polymer properties. On the basis of the model, a reduced-order

multi-rate nonlinear state observer was developed for a semi-batch, high-

temperature, n-butyl acrylate, solution polymerization reactor. Selection of the

non-zero estimator gains based on the local observability analysis reveals

optimal 'pairings' between the observer estimates and measurements of the

chromatographic and spectroscopic variables. Real-time results from the on-line

implementation of the estimator (soft sensor) confirmed the reliability of the

estimator to provide accurate estimates of spectroscopic and chromatographic

polymer properties. The feedback feature of `closed-loop’ observers/estimators

provide process models with much needed robustness to process-model

mismatch.

91

5 GLOBAL IDENTIFIABILITY OF REACTION RATE PARAMETERS IN MECHANISTIC MODELS FOR THE CHAIN HOMOPOLYMERIZATION

Recent efforts in the modeling and monitoring of polymerization reactors have

been focused on understanding the underlying reaction mechanisms in broader

ranges of process operation. For example, such efforts have been motivated by

the increasing average temperature used in industrial polymerization reactors

(e.g., batch chain-polymerization reactor for acrylics [Grady et al., 2002] and

continuous chain-polymerization reactor for styrene [Campbell et al., 2003]).

Broadening the range of process operations often leads to increasing

complexities in the underlying polymerization chemistry (e.g., in alkyl acrylate

polymerization at 25-140°C [Peck et al., 2004]). The added complexity requires

one to revisit/re-evaluate models that were developed in the narrower range of

process operation, and subsequently, obtain a more representative set of

reaction mechanisms (reaction network) and the corresponding parameter (rate

constant) values.

The parameter estimation problem has been formulated as follows. Given a set

of process measurements, postulate a set of reaction mechanisms and estimate

the corresponding set of parameters. While such formulation may seem straight-

forward, the simultaneous parameter estimation is often hindered by (i) limited

92

number and type of available measurements, (ii) limited knowledge of dominant

reaction mechanisms, and (iii) the obtaining of more than one set of parameters

that give comparable model predictions. Hence, what at first seems to be a

simple parameter estimation problem is also a kinetic model identification

problem. The problem is inherently complex, as one needs to simultaneously

consider the nature of the measurements (outputs), the underlying

polymerization chemistry, the modeling techniques, and all the elements of the

parameter estimation problem (e.g., optimization/search algorithms [Maria,

2004], optimization problem formulation [Bindlish, 2003]). A more comprehensive

treatment of the problem has been called design of experiments [Jacquez, 1998],

where the quality (e.g., sampling frequency) of output measurements and the

type of input sequences are also of interest. Often times, estimates are solely

based on available (conventional) measurements (e.g., in polymerization of

styrene [Hui et al., 1972] and methyl methacrylate [Stickler et al., 1981]).

Currently, estimates are based on both advanced and conventional

measurements (e.g., in high-temperature acrylates polymerization [Busch et al.,

2004; Rantow et al., 2006]). Spectroscopic measurements help reveal

characteristic polymer chain structures that are formed by specific reactions.

What is not always understood is how such inclusion help alleviate the parameter

estimation or kinetic model identification problem. Therefore, this question brings

to light other questions such as (i) whether a unique set of parameter values can

be inferred from the available measurements (e.g., parametric identifiability

93

[Walter et al., 1996], parametric observability [Vidal et al., 2002; Vecchio et al.,

2003]) and (ii) how sensitive a measurement is to a parameter (e.g., sensitivity

analyses [Polic et al., 2004]). Recent case studies mostly deal with

biological/biochemical systems [Audoly et al., 2001; Vajda et al., 1994], and a

few with catalytic chemical systems [Yao et al., 2003; Ben-Zvi et al., 2004].

Several methods for analyzing identifiability have been put forth; they include:

analyses of explicit solutions through series expansions [Walter et al., 1996],

analyses of the state isomorphism matrix [Walter et al., 1996; Vajda et al., 1994],

and analyses of the linearized structures of the output (or input-output) functions

through differential algebra manipulations [Ljung et al., 1994; Denis-Vidal et al.,

2000].

Motivated by the need to understand the nature of parametric identifiability,

particularly in relevance to kinetic model identification and parameter estimation

in polymerization systems, this study aims to (i) analyze the degree of

identifiability of the individual parameters (reaction rate constants) from state-of-

art measurements [Kammona et al., 1999] and (ii) assess the distinguishability of

process models that arise from different reaction networks. Chain

homopolymerization systems considered in this study are those that represent

the high-temperature polymerization kinetics of alkyl acrylates.

94

5.1 SCOPE AND PRELIMINARIES

This section describes the scope of this study and reviews the notions of local

and global identifiability of model structure and model parameter, and

distinguishability of model structures.

5.1.1 Local and Global Identifiability of Model Parameter and Model Structure

Consider the class of processes that can be represented by a parameterized

non-linear state-space model of the form:

( )( )( )

x=f x,θ,uM θ :

y=h x,θ,u

⎧⎪⎪⎪⎨⎪⎪⎪⎩

&

where x = [x1 … xn]T denotes the vector of model state variables, θ = [θ 1 … θ i]T

denotes the vector of model parameters, u = [u1 … up]T denotes the vector of

model inputs, and y = [y1 … ym]T denotes the vector of model outputs. The vector

of output measurements is represented by ŷ = [ŷ1 … ŷm]T. It is assumed that

continuous measurements of the outputs are available; that is, ŷ = y(t).

Model parameter values are commonly obtained by solving the least squared-

error (non-linear regression) problem:

95

( ) ( )TN 2

i iθi=1 0

ˆmin h x,θ,u -y t⎡ ⎤⎣ ⎦∑∫

Definition 1 [Bellman et al., 1970]: A model structure is locally identifiable

from a given set of output measurements, ŷ1(t) … ŷm(t), generated by the vector

of parameter values θ , if ˆθ=θ is a local minima of the non-linear regression

problem. The model structure is globally identifiable, if ˆθ=θ is a global minima of

the non-linear regression problem. In other words, if the model is not identifiable,

then there are more than one parameter vectors that correspond to exactly the

same input-output behavior, and it is impossible to eliminate them from the data

alone.

Remark 1: The model structure generally refers to the vector function f(x,θ,u)

which represents analytical relationships between model states, model

parameters and model inputs.

Remark 2: The notion of structural identifiability is associated with the nature of

the solution to the parameter estimation problem. Hence in this study the notion

of parametric identifiability is interchangeably used in place of structural

identifiability.

96

5.1.2 Distinguishability of Model Structures

Definition 2 [Walter et al., 1996]: A model structure, ( )M .% , is distinguishable from

a model structure, ( )M . , if and only if for almost any parameter vector value, θ% ,

there is no other parameter vector value, θ , such that

( ) ( )M θ M θ≡%%

where M denotes the model structure, and the above equation reads ( )M θ%% has

the same input-output behavior as ( )M θ . In other words, if ( )M .% is not

distinguishable from ( )M . , then there is no way to discover from experimental

data that the structure M(.) of the model does not correspond to that M(.) of the

process that has generated the data.

5.1.3 Scope

In this study, we limit the scope to the special form of the general nonlinear

model put forth in 3.4.1.1:

( )( )( )

:θ

θ⎧⎪ =⎪⎪⎨⎪ =⎪⎪⎩

x f xM

y h x

&

where θ is a parameter matrix and m=n. It is assumed that the

model outputs are not redundant; that is, the nxn matrix

97

( )∂∂

h xx

is nonsingular locally. Then it is possible to write:

( )=G y Hx

where H is a constant nonsingular nxn matrix, and the states can be expressed

in terms of the outputs:

( )1−=x H G y

The model in terms of the outputs takes the form:

( ) ( )( )1 1θ− −⎡ ⎤ =⎣ ⎦d H G y f H G ydt

Then linear regression can be used to estimate the parameter matrix [Ljung et

al., 1994]. The parameter matrix θ will be unique (model will be globally

identifiable), if the right hand side of the above equation depends on elements of

θ only.

5.2 CHALLENGES IN FREE-RADICAL POLYMERIZATION

Current issues in mechanistic models for free radical polymerization chemistry

include:

98

• Measurements of the concentration of intermediate radicals are not

available, as the intermediate radicals exist in very low concentrations and

have very short life time.

• Batch free-radical models are usually not cast in a linear form (with

respect to the parameter matrix); the optimization problem of the

parameter estimation is non-convex.

• Role of measurements on parametric identifiability is not well understood

yet.

• More than one postulated network is able to capture the dynamics of a

given set of measurements; more than one set of parameters is able to

capture the given set of measurements.

Before carrying out polymerization and collecting data on the reactor, one should

check to see if there is any possibility of selecting the best model structure and

estimating uniquely its parameters from the experiments considered, irrespective

of the quality of the experimental data. To this end, the global identifiability of

mechanistic models with different sets of measurements and the

distinguishability of globally identifiable model structures are assessed. A review

of current state-of-art on-line measurements for polymerization reactors can be

found in [Kammona et al., 1999].

99

Table 14. Polymerization reaction networks for initiation-propagation-termination

scheme

R1 R2

2⎯⎯→dkI R 2⎯⎯→dkI R

+ ⎯⎯→pkR M R + ⎯⎯→pkR M R

2 ⎯⎯→tkR D 2 2⎯⎯→ +tkR D TDB

5.2.1 Example 1

Consider a polymerization system in which only the primary chain polymerization

reactions take place; i.e., chain initiation, chain propagation and chain

termination by combination (reaction network R1 in Table 14). Using species

mass balances, one can develop the chain-polymerization batch-reactor model:

1 1 1

2 2 2 4

23 3 4

24 1 1 3 4

1 1

22

2,0

2,0 23

3

1

θ

θ

θ

θ θ

=−

=−

=−

= −

=

= −

−=

x x

x x x

x x

x x x

y x

xyx

x xy

x

&

&

&

&

1 1,0

2 2,0

3

4

(0)

(0)

(0) 0

(0) 0

=

=

=

=

x x

x x

x

x

where x = [x1 x2 x3 x4]T = [ [I] [M] [D] [R] ]T is the vector of state variables, θ = [θ1 θ2

θ3]T = [kd kp kt]T is the vector of parameters, and y = [y1 y2 y3]T = [ [I] xm DPn ]T is

the vector of measurements. Invoking quasi steady state assumption (QSSA) for

100

the molar concentration of radicals, x4, and writing the states in terms of the

outputs, we obtain:

10.51

112 2 0.5

1 233

1

0

0

0

θ

θθθ

θ

⎡ ⎤−⎢ ⎥⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎛ ⎞ ⎡ ⎤⎢ ⎥⎟⎜⎢ ⎥ ⎢ ⎥⎟= − ⎜⎢ ⎥⎟⎜⎢ ⎥ ⎢ ⎥⎟⎜⎢ ⎥⎝ ⎠ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦

xxd x

x xdtx

( )( )

1 1

2,0 2 2

2 32,0

3

1⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

G y y xx y x

y xxy

The following observations can be made:

• The parameters θ1 and 0.5

12

3

θθθ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

are globally identifiable from the set of

measurements.

• The individual parameters of θ2 and θ3 are not identifiable from the set of

measurements.

• Inclusion of measurement of y3 does not improve the parametric

identifiability of the system.

Considering the above observations, one can experimentally determine the

globally identifiable lumped parameters, and not rely on the measurement of

number-average molecular weight (Mn) on improving the quality of the parameter

estimates. Also note that the measurement of y1 (concentration of initiator) is a

101

measurement where the solution to the globally identifiable parameter θ1 and

lumped parameter 0.5

12

3

θθθ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

are dependent upon.

5.2.2 Example 2

One may also consider a reaction network that is the same as the above, except

for the chain termination mechanism (reaction network R2 in Table 14). Such a

case is considered when one is not certain of the path by which the propagating

chain terminates. To our advantage, terminal double bonds (TDBs) are created

whenever the chain termination by disproportionation occurs. Therefore, the

TDBs can be used as a measure of how often (reaction rate constant) the chain

termination by disproportionation occurs. The structure of TDB is typically

measured by using proton nuclear magnetic spectroscopy (1H-NMR) [Peck et al.,

2004; Quan et al., 2005; Rantow et al., 2006]. By using the same modeling

approach we arrive at the following model:

1 1 1

2 2 2 4

23 3 4

24 1 1 3 4

25 3 4

2

x x

x x x

x x

x x x

x x

θ

θ

θ

θ θ

θ

=−

=−

=

= −

=

&

&

&

&

&

1 1,0

2 2,0

3

4

5

0

0

0

x x

x x

x

x

x

=

=

=

=

=

102

1 1

22

2,0

2,0 23

3

54

2,0 2

1

=

= −

−=

=−

y x

xyx

x xy

x

xyx x

where x = [x1 x2 x3 x4 x5]T = [ [I] [M] [D] [R] [TDB] ]T is the vector of state variables,

θ = [θ1 θ2 θ3]^T = [kd kp kt]T is the vector of parameters, and y = [y1 y2 y3 y4]T = [ [I]

xm DPn TDB]T is the vector of measurements. Making QSSA for the molar

concentration of the intermediate radicals, x4, we obtain:

10.5

1 112

2 3 0.51 2

3 1

5 1

0

0

2 00

θ

θθθ

θθ

⎡ ⎤ ⎡ ⎤−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎛ ⎞⎢ ⎥ ⎢ ⎥⎟⎜ ⎡ ⎤⎟− ⎜⎢ ⎥ ⎢ ⎥⎟⎜ ⎢ ⎥⎟= ⎜⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

xxd x

x xdtx

x

( )

( )1 1

2,0 2 2

2 32,0

3 5

2,0 2 4

1⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

G y y xx y x

y xxy x

x y y

Note that the two models are indistinguishable from the measurements and the

addition of the spectroscopic measurement of TDB does not improve the

parametric identifiability of the system. Such prior knowledge is useful as the

measurement of TDB requires a time-consuming procedure and is not always

obtainable on-line. It should also be noted, however, that the experimental

103

evidence of the existence of TDBs does suggest that the network R2 is the more

accurate description of the system.

5.3 GLOBAL PARAMETRIC IDENTIFIABILITY OF CHAIN HOMOPOLYMERIZATION SYSTEMS

Considered here are models that are postulated for homopolymerization systems

which show polymer chain branch formations. Based on current NMR

techniques, short- and long-chain branches are not always experimentally

distinguishable [Quan et al., 2005]. Two reaction networks are considered. The

first one attributes branching formation to the preceding intra-molecular chain

transfer reaction (reaction network R3 in Table 15), and the second reaction

network attributes the branching formation to the preceding inter-molecular chain

transfer (reaction network R4 in Table 15). In the reaction network R1, short

chain branches (SCB) are formed through the intra-molecular chain transfer

(backbiting) followed by propagation of the tertiary radical (Q). While in the

reaction network R2, the long chain branches (LCB) are formed through the inter-

molecular chain transfer (chain transfer to polymer) followed by the propagation

of the tertiary radical (P). The two reaction networks are typical representations

of how chain branches are formed. From experimental measurements, it is

difficult to distinguish one reaction network from the other. Currently, branch

quantities are determined through carbon nuclear magnetic spectroscopy (13C-

104

NMR) [Peck et al., 2004; Quan et al., 2005; Rantow et al., 2006]. Often times,

both reaction networks can capture the dynamics of the concentration of chain

branches.

Table 15. Polymerization reaction networks for initiation-propagation-transfer to

polymer-termination scheme

R3 R4

2dkI R⎯⎯→ 2dkI R⎯⎯→

pkR M R+ ⎯⎯→ pkR M R+ ⎯⎯→

2 tkR D⎯⎯→ 2 tkR D⎯⎯→ bbkR Q⎯⎯→ tpkR TC D P+ + ⎯⎯→

pqkQ M R SCB+ ⎯⎯→ + ppkP M R TC LCB+ ⎯⎯→ + +

5.3.1 Example 3

Similar to Examples 1 and 2, based on species mass balances, we can develop

an isothermal chain-polymerization batch-reactor model [Rantow et al., 2006]:

1 1 1

2 2 2 4 5 2 5

23 3 4

24 1 1 4 4 5 2 5 3 4

5 4 4 5 2 5

6 5 2 5

2 2

x x

x x x x x

x x

x x x x x x

x x x x

x x x

θ

θ θ

θ

θ θ θ θ

θ θ

θ

=−

=− −

=

= − + −

= −

=

&

&

&

&

&

&

1 1,0

2 2,0

3

4

5

6

0

0

0

0

x x

x x

x

x

x

x

=

=

=

=

=

=

105

1 1

22

2,0

2,0 23

3

64

2,0 2

1

y x

xyx

x xy

x

xyx x

=

= −

−=

=−

where x = [x1 x2 x3 x4 x5 x6]T = [ [I] [M] [D] [R] [Q] [SCB] ]T is the vector of state

variables, θ = [θ1 θ2 θ3 θ4 θ5]^T = [kd kp kt kbb kpq]T is the vector of parameters, and

y = [y1 y2 y3 y4]T = [ [I] xm DPn SCB]T is the vector of measurements. Again, making

QSSA for the concentration of free radicals, we obtain:

10.5 0.5

1 12 4 11

3 3 0.51 22

1 0.53 10.5

16 4

3

0 0

0

0 0

0 0

xxd x xxdt

x xx

θ

θ θθ θθ θ

θ

θθθ

⎡ ⎤−⎢ ⎥⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎟ ⎟⎜ ⎜⎢ ⎥ ⎡ ⎤⎟ ⎟− −⎜ ⎜⎢ ⎥⎟ ⎟⎢ ⎥ ⎜ ⎜ ⎢ ⎥⎟ ⎟⎜ ⎜⎢ ⎥⎝ ⎠ ⎝ ⎠⎢ ⎥ ⎢ ⎥⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥ ⎛ ⎞⎢ ⎥⎟⎜⎢ ⎥ ⎟⎜⎢ ⎥⎣ ⎦ ⎟⎜ ⎟⎜⎢ ⎥⎝ ⎠⎣ ⎦

( )

( )1 1

2,0 2 2

2 32,0

3 6

2,0 2 4

1G y y x

x y xy xxy x

x y y

⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

Observations:

106

• The parameters θ1, 0.5

12

3

θθθ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

and 0.5

14

3

θθθ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

are globally identifiable from the

measurements.

• The parameters θ2, θ3, and θ4 are not identifiable from the measurements.

• The measurements y3 and y4 do not improve the parametric identifiability

of the model.

• The parameter θ5 does not have any effect on the measurements; it is not

identifiable due to the model structure.

As in Example 2, measurement of the number-average molecular weight (y3)

does not improve the global identifiability of the model. The spectroscopic

measurement of the short chain branches (SCB) also does not improve the

identifiability of the model. By analyzing the parameter matrix, we can also see

that the parameter θ5 does not at all affect the set of measurements. This is

contrary to the common intuition which assumes that once we have included the

measurement of branch quantities (y4)that we would obtain a more reliable

measurement of the rate of branch formation (θ5).

5.3.2 Example 4

One can also postulate a reaction network where the branch formation is

attributed to the inter-molecular chain transfer to polymer (reaction network R4 in

107

Table 2). Note that in the reaction network, TC (tertiary carbon) is not a reactive

species, it is only used to represent how the rate of chain transfer to polymer

reaction is governed by the number of tertiary carbons on the polymer chain.

Again species mass balances lead to [Rantow et al., 2006]:

1 1 1

2 2 2 4 5 2 5

23 3 4 4 4 6

24 1 1 4 4 6 5 2 5 3 4

5 4 4 6 5 2 5

6 5 2 5 4 4 6 5 2 5

7 5 2 5

2 2

x x

x x x x x

x x x x

x x x x x x x

x x x x x

x x x x x x x

x x x

θ

θ θ

θ θ

θ θ θ θ

θ θ

θ θ θ

θ

=−

=− −

= −

= − + −

= −

= − +

=

&

&

&

&

&

&

&

1 1,0

2 2,0

3

4

5

6

7

0

0

0

0

0

x x

x x

x

x

x

x

x

=

=

=

=

=

=

=

1 1

22

2,0

2,0 23

3

74

2,0 2

1

y x

xyx

x xy

x

xyx x

=

= −

−=

=−

where x = [x1 x2 x3 x4 x5 x6 x7]T = [ [I] [M] [D] [R] [P] [TC] [LCB]] ]T is the vector of

state variables, θ = [θ1 θ2 θ3 θ4 θ5]^T = [kd kp kt ktp kpp]T is the vector of parameters,

and y = [y1 y2 y3 y4]T = [ [I] xm DPn LCB]T is the vector of measurements. Upon the

rearrangement of the output equations:

108

( )

10.5 0.5

1 12 4

3 3 110.5 0.5

1 22 11 4 0.5

3 1 2 73

0.57

14

3

0 0

0

0

0 0

xxd x xxdt

x x x xx

θ

θ θθ θθ θ

θθ θθ

θθθ

⎡ ⎤−⎢ ⎥⎢ ⎥⎛ ⎞ ⎛ ⎞⎡ ⎤ ⎢ ⎥⎟ ⎟⎜ ⎜⎟ ⎟⎢ ⎥ − ⎜ ⎜⎢ ⎥⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎢ ⎥ ⎡ ⎤⎢ ⎥⎝ ⎠ ⎝ ⎠⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎛ ⎞=⎢ ⎥ ⎢ ⎥⎢ ⎥⎟⎜ ⎟⎢ ⎥ ⎜ ⎢ ⎥⎢ ⎥⎟⎜ ⎟⎜ +⎢ ⎥ ⎝ ⎠ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎛ ⎞⎣ ⎦ ⎟⎢ ⎥⎜ ⎟− ⎜ ⎟⎢ ⎥⎜ ⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦

( )

( )1 1

2,0 2 2

2 32,0

3 7

2,0 2 4

1G y y x

x y xy xxy x

x y y

⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

Observations:

• The parameters θ1, 0.5

12

3

θθθ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

, and 0.5

14

3

θθθ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

are globally identifiable from the

set of measurements.

• The parameters θ2, θ3, and θ4 are not globally identifiable from the set of

measurements.

• The measurements y3 and y4 do not improve the parametric identifiability

of the system.

• The parameter θ5 does not have any effect on the measurements; it is not

identifiable due to the model structure.

109

Note that the dynamics of state x6 can be represented by a combination of states

x2 and x7. The reaction networks R3 and R4 are therefore indistinguishable from

one another from the set of outputs considered.

5.4 CONCLUSIONS

A class of chain-homopolymerization systems has been analyzed in terms of its

global parametric identifiability. This study considers identifiability analyses as an

integral part of parameter estimation efforts in polymerization systems. The

advantages or disadvantages of including measurements were evaluated and

presented. Globally identifiable parameters or lumped parameters were obtained,

and the implications of their global identifiability properties were discussed. The

study can be readily extended to more complex polymerization systems, and can

also consider alternate models that are developed by different modeling

techniques. The importance of the identifiability and distinguishability analyses in

developing complex polymerization models is also re-emphasized, particularly for

parameter estimation and modeling efforts in high-temperature polymerization

systems. Lastly, the study also signifies the need for both non-redundant

measurements and identifiable model structures, to avoid issues associated with

different sets of models and parameters that can explain a given set of

measurements.

110

6 CONCLUSIONS AND FUTURE DIRECTIONS

This study led to the following contributions:

• Estimates of the Arrhenius parameters for the secondary reactions observed

at high reaction temperatures (self-initiation, backbiting, scission and

propagation of tertiary radicals)

• First report on estimate of the activation energy of the self-initiation reaction

• Experimental evidence of short chain branch formation

The study also led to the following open questions:

• How to isolate experimentally the self initiation species?

• How to calculate theoretically the energetically-favorable molecular structure

of the self-initiation species?

Based on the current knowledge on high-temperature spontaneous solution

polymerization of n-butyl acrylates (nBA), one could establish a fundamental and

general understanding of the underlying pathways in high-temperature

spontaneous solution and bulk polymerization of alkyl acrylate monomer family,

111

particularly of alkyl acrylate monomers that contain longer substituent alkyl

groups.

Currently, experimental observations that shed light on the underlying

mechanistic pathways in polymerization of 2-ethylhexyl acrylates (2-EHA) and n-

octyl acrylate (nOA) are still scattered and non-unified.

6.1 PRELIMINARY OBSERVATIONS IN HIGH-TEMPERATURE

SPONTANEOUS SOLUTION HOMO-POLYMERIZATION OF SEVERAL

ALKYL ACRYLATE MONOMERS

When considering different alkyl acrylate monomers, a fundamental question that

comes to mind is how the size of the substituent group affects the rate of

polymerization. The molecular structures of the monomers are depicted in the

figure below.

112

Figure 19. Different sizes of substituent groups in (I) n-butyl acrylate, (II) 2-

ethylhexyl acrylate, and (III) n-octyl acrylate

It has been reported based on pulsed-laser initiated polymerization studies that

rate of chain propagation increases with increasing size of acrylate substituent

(Beuermann, Paquet, McMinn and Hutchinson, 1996). This study was carried out

at low temperatures (ca. 5-30°C). In contrast at higher temperatures, preliminary

experiments carried out at DuPont Marshall Laboratory indicated that resulting

weight-average Molecular weight decreases, and % monomer conversion

increases as size of acrylate substituent group increases (i.e., methyl, propyl,

butyl, octyl, etc.).

113

Table 16. Preliminary observation of alkyl substituent size effect on resulting

molecular weight and monomer conversion. Experiments were carried out in

batch, in four hours, in methyl amyl ketone in the absence of thermal initiators

[Grady, Quan and Soroush, 2003]

Monomer Resulting Mw %Conversion

methyl acrylate 53887 85

ethyl acrylate 25755 94

n-butyl acrylate 9965 98

2-ethylhexyl acrylate 6898 99

While the increase in propagation rate could explain the increasing trend in

%conversion, it does not explain the decreasing trend in molecular weight. One

possible explanation is the increasing tendency for chain transfer events to take

place with increasing alkyl substituent size. In a separate study, the propagation

rate constant of 2-EHA was reported to be lower to that of nBA [Plessis et al.,

2001].

114

6.2 RECENT STUDIES IN SOLUTION POLYMERIZATION OF 2-ETHYLHEXYL ACRYLATE AND N-OCTYL ACRYLATE

Recent studies report scattered observations on occurrences of various reactions

in different reaction conditions and temperature regimes. In a high-temperature

bulk polymerization of n-octyl acrylate, self-initiation was reported at

temperatures above 190 C, with order of dependence of rate of polymerization

with respect to monomer of 0.8 [Katime et al., 1988]. It was also reported that the

dependence of the rate of polymerization on monomer concentration was

observed to be in the order of 1.5 in the solution polymerization of 2-EHA at 50°C

[Kothandaraman et al., 1993]. n-octyl acrylate has also been used in atom

transfer radical polymerization under microwave irradiation [Xu et al., 2003].

Systematic modeling study on polymerization of 2-EHA and nOA has not been

reported in the open literature. It has been observed that n-octyl acrylate (nOA)

polymerizes spontaneously at ca. 190°C. While 2-EHA spontaneously

polymerizes at ca. 150°C. We are thus interested in understanding how initial

monomer content and temperature affects the tendency of the monomers (nOA

and 2-EHA) to polymerize spontaneously.

Based on 1H-NMR spectra of nOA-Styrene copolymer, terminal double bonds

and terminal solvent groups were not observed [Bisht et al., 2003]. On the other

hand, chain transfer to polymer was observed based on 13C-NMR spectra

measurement in the polymerization of 2-EHA in cyclohexane, with branching

115

levels much higher than those reported for nBA [Heatley et al., 2001]. Another

study determined that the chain transfer to polymer event in emulsion

polymerization of 2-EHA is preceded by an intra-molecular chain transfer

reaction (backbiting), based on 13C-NMR analyses [Plessis et al., 2001]. It was

suggested that the reaction scheme proposed for n-BA also applies to 2-EHA.

Chain transfer to solvent (cyclohexane) was also observed in the pulse

radiolysis-initiated polymerization of 2-EHA at room temperature [Takacs et al.,

1999]. In this study, peak assignments of 13C- and 1H-NMR spectra of poly-OA

and poly-2-EHA will be re-confirmed and reported.

6.3 MODELING AND PARAMETER ESTIMATION

The mechanistic modeling approach used in the preliminary study on high-

temperature spontaneous solution polymerization of n-butyl acrylate will be used

to study polymerization of 2-EHA and nOA. The following table summarizes

known reaction mechanisms in nBA, nOA and 2-EHA solution polymerization,

from low (ca. room temperature) to high temperature (ca. 200°C).

116

Table 17. Summary of known reaction mechanisms in nBA, nOA and 2-EHA

polymerization (low to high temperature)

Reaction nBA 2-EHA nOA

Chain Initiation (Spontaneous) X X X

Chain Propagation X X X

Chain Termination X X X

Chain Transfer to Solvent X X

Chain Transfer to Monomer X

Chain Transfer to Polymer (Intra) X X

Chain Transfer to Polymer (Inter)

Chain Fragmentation (Scission) X

It should be re-emphasized that the current knowledge on underlying mechanistic

pathways in high-temperature spontaneous solution polymerization of 2-EHA and

nOA is still very limited.

Hence the following are open research questions that can be pursued in the

future:

• The nature of branching in high-temperature solution and bulk polymerization

of n-butyl acrylate, 2-ethylhexyl acrylate and n-octyl acrylate

117

• The effect of size of alkyl substituent group on primary and secondary

reaction pathways, and on the overall rate of polymerization

• Effect of temperature on the dynamics of polymer chain microstructure

formation

• The rate of reactions of primary and secondary free-radical polymerization

pathways

118

APPENDIX A

DERIVATION OF THE BATCH POLYMERIZATION PROCESS MODEL

A.1 RATE LAWS FOR SMALL MOLECULES.

Rate laws for dimers, oxygen radical, xylol radical, monomer, solvent and micro-

structural properties such as short chain-branching points (SCBs) and terminal

double bonds (TDBs) were derived based on the postulated reaction network

presented in Section 3. They are:

119

1 2 2

0 1 2

20 1 2 1

3

3 3

3

1 2

3

13

1 23

2 [ ] [ ]

[ ] [ ]

[ ]

[ ]

[ ]

[ ]

2 [ ] [ ] [ ]

TDBM ti p n

n

pq n tm nn n

S ts nn

R OOR O

SCB pq nn

TDBTDB n p

n

P R OOR ts n pn

r k M k M P P P P P

k M Q k M P

r k S P

r k R OOR

r k M Q

r k Q k M P

r k R OOR k S P k M P

β

∞

=

∞ ∞

= =

∞

=

∞

=

∞

=

∞

=

⎛ ⎞⎟⎜=− − + + + + ⎟⎜ ⎟⎜ ⎟⎝ ⎠

− −

=−

=−

=

= +

= + −

∑

∑ ∑

∑

∑

∑

∑

1

2

1

0

2 sec sec0 1

1 1 23

13

2 [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ]TDB

L LP ti p p

TDBP n p p p

n

TDBtm n pP

n

r k M k M R k M R

r k Q k M P k M P k M P

r k M P k M P

β

∞

=

∞

=

= + −

= + + −

= −

∑

∑

A.2 RATE LAWS FOR `LIVE' AND `DEAD' POLYMER CHAINS.

Rate laws for live secondary propagating radicals, live tertiary propagating

radicals and dead chains of length n, based on postulated reaction network

presented in Section 3, take the following form:

120

( ) ( )

( ) ( )

( ) ( )

( )

1 1

sec sec secsec sec sec

3 3

3

secsec

3

[ ] [ ] [ ] [ ]

2

[ ] 2

n

n

P p n n pq n bb tm ts n

tertterttc td n m tc td n m

m m

tert terttert tertQ bb n pq n tc td n m

m

tertterttc td n m

m

r k M P P k M Q k k M k S P

k k P P k k P Q

r k P k M k Q k k Q Q

k k Q P

β

− −

∞ ∞− −− −

= =

∞−−

=

−−

=

= − + − + +

− + − +

= − + − +

− +

∑ ∑

∑

( ) sec sec2

3

sec sec

3 3

sec sec

3 3

sec sec

3 3

[ ] [ ]

2

2 2

2

nD tm ts n n tc n m mm

tert terttc n m m tc n m m

m m

tert terttc n m m td n m

m m

tert terttd n m td n m

m m

tert terttd

r k M k S P k Q k P P

k P Q k Q P

k Q Q k P P

k P Q k Q P

k Q

β

∞

∞−

+ −=

∞ ∞− −

− −= =

∞ ∞−−

−= =

∞ ∞− −

= =

−

= + + +

+ +

+ +

+ +

+

∑

∑

∑ ∑

∑ ∑

∑ ∑

3n m

m

Q∞

=∑

A.3 METHOD OF MOMENTS

The kth moment of the live and dead chains are defined as:

3

3

3

kk n

n

kk n

n

kk n

n

n P

n Q

n D

λ

δ

μ

∞

=

∞

=

∞

=

=

=

=

∑

∑

∑

121

When the above moment definitions were applied to the rate laws, the following equations were obtained;

( )

1 2 2

0 1 2

1

2

20 1 2 1 0

0 0

0

1 2

0

0 1

1 2 0 0

2 sec sec0 1

0

2 [ ] [ ]

[ ] [ ]

[ ]

[ ]

[ ]

[ ]

2 [ ] [ ] [ ]

2 [ ] [ ] [ ]

TDBM ti p

pq tm

S ts

R OOR O

SCB pq

TDBTDB p

P R OOR ts p

L LP ti p p

P

r k M k M P P P P

k M k M

r k S

r k R OOR

r k M

r k k M P

r k R OOR k S k M P

r k M k M R k M R

r k

β

β

λ

δ λ

λ

δ

δ

λ

δ

=− − + + + +

− −

=−

=−

=

= +

= + −

= + −

= +

1

1 1 2

0 1

[ ] [ ] [ ]

[ ] [ ]TDB

TDBp p p

TDBtm pP

k M P k M P k M P

r k M k M Pλ

+ −

= −

122

( )

( ) ( )

( ) ( )

( )( )

0

0

0

2 0 0

sec sec sec2sec sec sec0 0 0

20 0 0

secsec0 0

0 0

[ ] [ ] [ ] [ ]

2

[ ] 2

[ ] [ ]

p pq bb tm ts

tertterttc td tc td

tert terttert tertbb pq tc td

tertterttc td

tm ts t

r k M P k M k k M k S

k k k k

r k k M k k k

k k

r k M k S k k

λ

δ β

μ β

δ λ

λ λ δ

λ δ δ

λ δ

λ δ

− −− −

−−

−−

= + − + +

− + − +

= − + − +

− +

= + + +( )

( )

( )( ) ( )

( ) ( )

( )

1

1

sec sec 2sec sec0

secsec0 0

20

2 0 1 1

sec sec secsec sec sec0 1 1 0

1

2

2 4

2

[ ] [ ] [ ] [ ]

2

[ ]

c td

tertterttc td

tert terttert terttc td

p pq bb tm ts

tertterttc td tc td

bb pq

k

k k

k k

r k M P k M k k M k S

k k k k

r k k M k

λ

δ β

λ

λ δ

δ

λ δ λ

λ λ λ δ

λ

−−

−−

−−

− −− −

+

+ +

+ +

= + + − + +

− + − +

= − + ( )

( )( ) ( )

( )( ) ( )( )

1

2

1 1 0

secsec0 1

sec secsec sec1 1 0 1

secsec0 1 1 0 0 1

2 0 1

2

[ ] [ ] 2

2 2 2

[ ] 2

tert terttert terttc td

tertterttc td

tm ts tc td

tert tert terttert tert terttc td tc td

p pq

k k

k k

r k M k S k k k

k k k k

r k M P k

μ β

λ

δ δ δ

λ δ

λ δ λ λ

λ δ λ δ δ δ

λ λ

−−

−−

−−

− −− −

− +

− +

= + + + +

+ + + + +

= + + + ( )

( ) ( )

( ) ( )

( )( )

2

2

2 2

sec sec secsec sec sec0 2 2 0

2 2 2 0

secsec0 2

sec sec2 2 0 2

[ ] [ ] [ ]

2

[ ] 2

[ ] [ ] 2

bb tm ts

tertterttc td tc td

tert terttert tertbb pq tc td

tertterttc td

tm ts tc

M k k M k S

k k k k

r k k M k k k

k k

r k M k S k k

δ β

μ β

δ λ

λ λ λ δ

λ δ δ δ

λ δ

λ δ λ λ

− −− −

−−

−−

−

− + +

− + − +

= − + − +

− +

= + + + ( )

( ) ( )

( )

21

2sec0 2 1 1 2 0 0 2 1

sec sec sec0 2 0 2 2 0 0 2

2 2 2

2 2 2

tert tert terttc tc

tert tert terttd td td

k k

k k k

λ

λ δ λ δ λ δ δ δ δ

λ λ λ δ λ δ δ δ

− −

− − −

+

+ + + + +

+ + + +

123

A.4 BATCH PROCESS MODEL

The rate laws were then incorporated in conservation (balance) equations for a

batch polymerization reactor model. Perfect mixing and constant volume

(negligible volume changes in reaction medium during reaction time) were

assumed. Quasi steady-state assumption on the live chains is not made. The

batch polymerization reactor model is of the form;

( )

( )

( )

( )

( )

( )

( )

( )

1 2

0

1

2

1

0

0

1 21 2 1 2 0

00

11

22

11

[ ] , [ ] 0 [ ]

[ ] , [ ] 0 [ ]

[ ] , [ ] 0 [ ]

[ ] , [ ] 0 0

[ ] , [ ] 0 0

[ ] , [ ] 0 0

[ ] , [ ] 0 0

[ ] , [ ] 0 0

[ ] , [TDB

M

S

R OOR

TDB

SCB

P

P

P

TDBT

P

d M r M Mdt

d S r S Mdt

d R OOR r R OOR R OORdt

d TDB r TDBdt

d SCB r SCBdt

d P r Pdt

d P r Pdt

d P r Pdt

d P r Pdt

= =

= =

= =

= =

= =

= =

= =

= =

= ( )

( )

( )

( )

] 0 0

, 0 0, 0,1,2

, 0 0, 0,1,2

, 0 0, 0,1,2

k

k

k

DB

kk

kk

kk

d r kdt

d r kdt

d r kdt

λ

δ

μ

λ λ

δ δ

μ μ

=

= = =

= = =

= = =

124

The measured variables are related to the state variables of the preceding model

according to:

1

0

2

1

0

1

1

0

1

[ ]1[ ]

[ ] 100

[ ] 100

[ ] [ ] 100

n m

w m

M MW

M MW

MXM

TDBTDBH

SCBSCBH

S STSGH

μμ

μμ

μ

μ

μ

= ×

= ×

= −

= ×

= ×

−= ×

125

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144

VITA

FELIX S. RANTOW

PERSONAL Birth date: 16 July 1980

Birth place: Jakarta, Indonesia

Citizenship: Indonesian

EDUCATION Ph.D. Candidate, Drexel University

Chemical Engineering, September 2002 to September 2006

Thesis: “Mechanistic Modeling and Model-Based Studies in Spontaneous

Solution Polymerization of Alkyl Acrylate Monomers”

Advisors: Prof. Masoud Soroush and Dr. Michael C. Grady

M.S., Drexel University

Chemical Engineering, September 2002 to December 2005

B. S., Purdue University

Chemical Engineering, August 1998 to May 2002

Undergraduate Research Project: “Development of a JAVA-XML based

process scheduling language”

Advisors: Dr. Seza Orcun and Prof. Joseph F. Pekny

145

AWARDS Graduate Student Research Award, Drexel University, 2006

Best Paper Presentation Award, American Control Conference, 2005

EXPERIENCE DuPont Marshall Laboratory, Process Engineering Group, 2004 to 2006

Visiting Research Assistant. Designed and analyzed batch solution

polymerization experiments.

Chemtura Quality Assurance, Summer 2001

Database System Developer. Designed and developed an online

database system for document request.

Purdue University Civil Engineering Department, 2000 to 2002

Undergraduate Research Assistant. Designed and developed a visual

basic-based decision support system for choosing trenchless technology

equipments. Designed and developed an online roadmap application for

the Indiana department of transportation.

Purdue Computer Science Department, Spring 1999

Lab Instructor. Teach introductory JAVA programming language. Present

lab material.

JOURNAL PUBLICATIONS

F. S. Rantow, M. Soroush, M. C. Grady, G. A. Kalfas, “Spontaneous Polymerization and Chain Microstructure Evolution in High-Temperature Polymerization of n-Butyl Acrylate”, 47(4), 1431-1443, Polymer, 2006

F. S. Rantow, M. Soroush, M. C. Grady, “A Reduced-Order Kinetic Model for Process Monitoring of Polymer Microstructure Spectroscopic Data”, to be submitted to Journal of Chemometrics, 2006

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F. S. Rantow, M. Soroush, M. C. Grady, “Global Parameter Identifiability of Affine-In-Parameter Polymerization Mechanistic Models”, in preparation

M. Soroush, F. S. Rantow, Y. Dimitratos, “Control Quality Loss in Analytical Control of Input-Constrained Processes”, Accepted for publication in Industrial and Engineering Chemistry Research, 2006

CONFERENCE PROCEEDINGS

F. S. Rantow, M. Soroush, M. C. Grady, “Global Parametric Identifiability of Mechanistic Models in Chain Polymerization”, Proceedings of American Control Conference, 3068-3073, 2006

F. S. Rantow, M. Soroush, M. C. Grady, “Optimal Control of a High-Temperature Semi-Batch Polymerization Reactor”, Proceedings of American Control Conference, 3102-3107, 2005

F. S. Rantow, M. Soroush, M. C. Grady, “Resin Design and Optimization: High-Temperature (140-200°C) n-Butyl Acrylate Polymerization”, Proceedings of Foundations of Computer-Aided Process Design, 577-580, 2004

PRESENTATIONS

American Control Conference 2006. “Global Parametric Identifiability of Mechanistic Models in Chain Polymerization”

AICHE Annual Meeting 2005. “Reaction Network Identification in High-Temperature Polymerization of n-Butyl Acrylate: A Simultaneous Parameter Estimation Approach”

AICHE Annual Meeting 2005. “Model for Polymer Microstructure Monitoring and Control in Polymerization of Acrylates”

American Control Conference 2005. “Optimal Control of a High-Temperature Semi-Batch Polymerization Reactor”

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AICHE Annual Meeting 2004. “High-Temperature Acrylate Polymerization: Decentralized Parameter Estimation and Multi-Rate State Estimation”

Foundations of Computer-Aided Process Design 2004. “Resin Design and Optimization: High-Temperature (140-200°C) n-Butyl Acrylate Polymerization”

CERTIFICATION

Safety and Chemical Engineering Education (SACHE) Faculty Workshop. “What Chemical Engineering Students Need To Know”, Rohm and Haas Company Engineering Division, Croydon, PA, September 2005

EXTRACURRICULAR AND PROFESSIONAL AFFILIATIONS

President. Drexel University Chemical Engineering Graduate Student Association, 2004 to 2005

Member, American Institute of Chemical Engineers, 2001-present

Cantor. St. Agatha-St. James Church, 2003-2005

Social Events Committee. Indonesian Students Association, 1999-2002

Intramural Soccer Captain. Drexel University, 2004