Mathematicai Modelling and Parameter Estimation in

Living Isobutylene Polymerization

Qian Liu

A thesis submitted to the Department of Chernical Engineering

in conformity with the requirernents for the degree of

Master of Science (Engineering)

Queen's University

Kingston, Ontario, Canada

March, 2001

Copyright O Qian Liu, 2001

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Abstract

The synthesis of polyisobutylene (PB) based thennoplastic elastomer (TPE) is of great interest to the polymer industry due to its unique chernical and physical properties. Although PIB with high molecular weight and narrow rnolecular weight distribution (MWD) can be synthesized through living cationic polyrnerization, the kinetics of this polymerization are still not clear, and the validity of a newly developed cornprehensive rnechanism needs to be assessed.

A mathematical mode1 was developed in this work to fit the experimental data, The responses include the dynamic evolutions of monomer concentration, initiator concentration, number average and weight average molecular weights. Individual rate constants of the polyrnerization reaction were estimated using the parameter estimation software package, GREG, which is capable of accomrnodating multiple response cases. The parameter estimation procedure involves an iterative process, which consists of solving stiff differential and algebraic equations (DAE), calculating sensitivity coefficients, and determining the optimal values of parameters.

Estimation results showed that when there are more data used in the estimation, smaller correlation between parameter estimates can be obtained. The simulations with estimated parameters showed good fitting to the experiments for the responses such as monomer concentration, initiator concentration, and number average degree of polyrnerization, although discrepancies remained between simulated and observed weight average degree of polymerization.

Acknowledgements

I wish to first thank my supervisors, Dr. Kim B. McAuley and Dr. Michael F. Cunningham for their guidance, insight, and support throughout this Master's degree.

Secondly, I wish to express thanks to Dr. Judit Puskas and Haihong Peng at University of Western Ontario for giving me access to the valuable experimental data that became the foundation of this work and for our enlightening conversations at the onset of my project.

Lastly, 1 thank Jin for her love and support.

Financial provision from Queen's University, and Materials and Manufacturing Ontario is gratefully acknowledged.

TABLE OF CONTENTS

Chapter 1 : lntroduction

Chapter 2: Literature Review 2.1 Introduction 2.2 Early Work in Living IB Polymerization 2.3 Kinetics of Living IB Polymerization

2.3.1 Initiation 2.3.2 Solvent 2.3.3 Propagation and Reversible Termination 2.3.4 Temperature Effect 2.3.5 Determination of Rate Constant 2.3.6 Reaction Order With Respect To Reactants

2.4 The Kinetic Models 2.4.1 Developrnent of Kinetic Mechanism 2.4.2 Development of Comprehensive Mechanistic Model 2.4.3 Simulation of Simplified Mechanism

2.5 Summary

Chapter 3: Development of Cornprehensive Model Equations 3.1 Introduction 3.2 Material Balance Equations 3.3 Molecular Weight Distributions 3.4 Moment Balance Equations

Chapter 4: Model Simulations 4.1 lntroduction 4.2 Model Simulation with ScientistTM 4.3 Model Simulation with DDASAC

Chapter 5: Parameter Estimation 5.1 lntroduction to Parameter Estimation Software 5.2 Parameter Estimation Package, GREG

5.2.1 Objective Function for Multiresponse Parameter Estimation 5.2.2 Sensitivity Coefficients

5.3 Parameter Estimation using GREG 5.3.1 Estimation Algorithm for Multiresponse Model 5.3.2 Calculation of Response Variance 5.3.3 Parameter Constraints 5.3.4 Parameter Transformation

5.4 Results and Discussion

Chapter 6: Parameter Estimation with Additional Replicate Data 6.1 Additional Replicate Run 6.2 Parameter Estimation with AI1 Replicate Runs

Chapter 7: Conclusions and Recomrnendations for Future Research 7.1 Summary and Conclusions 7.2 Recommendations for Future Research

References

Ap pendix A: Simulation of Original Parameters Appendix B: Simulation of Polydispersity lndex (PDI) of PIB with Original Values of Parameters Appendix C: Effect of Fonward and Backward Rate Constants on Responses Appendix D: Replicate Runs for the Calculation of Response Variances Appendix E: Simulation with Estimated Values of Parameters Appendix F: Experirnental Data Used for Parameter Estimation Appendix G: Simulation of Polydispersity lndex (PDI) of PIB with Estimated Parameters Appendix H: Simulation with New Estimated Paramaters Appendix 1: FORTRAN Code for Pararneter Estimation Vita

List of Tables

Table 2.1 Rate constants for cationic IB polyrnerization in solution (Plesch, 1993)

Table 2.2 Experirnentally obtained composition rate constants

Table 2.3 Simple reaction mechanism and rate constants for sirnulating living IB polymerization using PREDlCl

Table 2.4 Experimental conditions of living IB polyrnerization

Table 2.5 Parameters deterrnined by Puskas and Peng (1999)

Table 3.1 Reaction mechanisrns for comprehensive mode1

Table 3.2 Material balance equations for cornprehensive mode1

Table 3.3 Zeroth, first and second moments of polymer species

Table 3.4 Moment balance equations

Table 4.1 Initial settings for different experiments

Table 5.1 Bartlett's test of constant variance throughout time

Table 5.2 Results of parameter estimation

Table 5.3 Corresponding correlation matrix

Table 5.4 Estimation results at different values of ko

Table 6.1 Pooled variance determined from al[ replicate runs

Table 6.2 Results of parameter estimation

Table 6.3 Corresponding correlation matrix

List of Figures

Scheme 2.1 First order reaction mechanisrn with respect to T U 4

Scheme 2.2 Comprehensive mechanisrn for living IB polymerization (Puskas and Lanzedofer, 1998)

Figure 4.1. Monomer concentration for polyrnerization of IB at - 80°C in 60140 Hx/MeCI. [IB]=2 molll, [DtBP]=0.0007 molll, [TiC14]=0.04 mol/l, F M PCI]=0.004 molll

Figure 4.2. lnitiator concentration for polymerization of IB at -80°C in 60140 Hx/MeCI. [l B]=2 molll, [DtB P]=0.0007 molll, piC14]=0.04 molll, [TMPCI]=0.004 mol/l

Figure 4.3. Nurnber average degree of polymerization for polymerization of IB at -80°C in 60/40 HxIMeCI. [1B]=2 molll, [DtBP]=0.0007 mol/l, [TiCI4]=0.04 molll. F M PCI]=0.004 molll

Figure 4.4. Weight average degree of polyrnerization for polymerization of IB at -80°C in 60/40 HxIMeCI. [l B]=2 mol/l, [DtBP]=0.0007 molll. [TiCI4]=0.04 rnolll. [TM PCI]=0.004 molll

Figure 6.1 Replicate data of rnonomer concentration in experiment 21 7-1 O

Figure 6.2 Replicate data of initiator concentration in experiment 21 7-1 O

Figure 6.3 Replicate data of number average degree of polyrnerization in experiment 21 7-1 0

Figure 6.4 Replicate data of weight average degree of polymerization in experiment 2 1 7-1 0

Figure 6.5 Simulation of monomer concentration in experiment 21 7-1 O

Figure 6.6 Simulation of initiator concentration in experiment 21 7- I O

Figure 6.7 Simulation of number average degree of polymerization in experiment 21 7-1 0

Figure 6.8 Simulation of weight average degree of polymerization in ex~eriment 21 7-1 0

Nomenclature

BartlettYs Statistic Value

Covariance matrix of multireponse mode1

Number average degree of polymerization

Weight average degree of polymerization

Expectation of random variable x

Expected responses matrix

I nitiator (TM PCI)

I nitiator

Measurement of initiator,

[rln,,urei = [ I l I [I'LA] + [I'LA-] + [ I ' L K ]

Intermediate complex initiating species; also referred to as I'LA

lnterrnediate cornplex initiating species; also referred to as 1'

Active initiator with monornei-ic counterion; also referred to as

Active initiator with monomeric counterion; also referred to as I'LA-

Active initiator with dimeric counterion; also referred to as

Active initiator with dirneric counterion; also referred to as I'LR-

Run length at [M]=[MIo

Lewis acid;

Objective function of parameter estimation

Apparent rate constant

vii

Composite rate constant ; kz = k p K , = KokpK1 = Kokpkl / k-,

Composite rate constant; kt = Kokl

Composite rate constant; k: = K, kl [LA],

Composite rate constant; k, = K , k , K , = (k, /k-, ) - k , - (k2 /k-, )

Composite rate constant; k: = k , ~ : = k,K,K, [LA],

Equilibrium rate constant; K, = k, l k,

lonization rate constant for simplified model, mol-'.sec-' )

De-ionization rate constant for simplified model, (sec-' )

lonization rate constant at pMPCl]o>>piC1~o for sirnplified mode[, ( L.rnoZ-'.sec-' )

Deionization rate constant at [TMPCl]o>>~iC14]o for sirnplified model, (sec-' )

Overall reaction rate constant; ki = KI k,

Activation rate constant of cornplex intermediate, (~.rno(-'.sec-' )

Deactivation rate constant of complex intermediate. (sec-' )

Activation rate constant of active sites with monomeric counterions, ( L . ~ O Z - ' .sec-' )

Deactivation rate constant of active sites with monomeric counterions, (sec-' )

Activation rate constant of active sites with dimeric counterions, ( mol-'.sec-' )

Deactivation rate constant of active sites with dimeric counterions, (sec-' )

Propagation rate constant, mol-'.sec-' )

Molecular weight of repeat units in a polyrner sample

a,, MW

Mi

Monomer (IsobutyIene)

Nurnber of response variables

Number average rnolecular weight

Weight average rnolecular weight

Molecular weight of polymer with chain length of I

Maximum measured value of the ith response

Symbol of BCZ, or Tic4

Total number of observations for each response

Number of molecules with chain length of 1

Number chain length distribution, (Nc/CNj)

Dormant polymer chain of length n

Active growing chain of length n with monomeric counterion; also referred to as fnTiC[,-

Active growing chain of length n with monomeric counterion; Pn+ LA-

Intermediate complex of length i

Chlorinated dormant polymers; also referred to as P,

Chlorinated dormant polymers; also referred to as <CI

Active growing chain of length n with dimeric counterion; also referred to as cTi,CI,-

Active growing chain of length n with dirneric counterion; also referred to as P,'LA-

lnstantaneously active growing chains

Polymerization rate

Slope

Sample variance of the ith response

Pooled sample variance

Monomeric counterion

Dimeric counterion

kt'' moment of chain length distribution

Zeroth moment of intermediate complex

First moment of intermediate complex

Second moment of intermediate complex

Zeroth moment of polymer chain with dimeric counterions

First moment of polymer chain with dimeric counterions

Second moment of polymer chain with dimeric counterions

Zeroth moment of polymer chain with monomeric counterions

First moment of polymer chain with rnonomeric counterions

Second moment of polymer chah with monomeric counterions

Zeroth moment of dormant polymer chains

First moment of dormant polymer chains

Second moment of dormant polymer chains

Sum of individual zeroth moments of al1 species

Sum of individual first moment of al1 species

Sum of individual second moment of al1 species

Diagonal entries of Box and Draper matrix

Wi Weight of molecules of length 1

Wi Weight chain length distribution, (WJCWi)

w(t) Sensitivity coefficients - x The average of a set of sarnples

Observed responses matrix

Residual matrix

Greek Letters E Normally distributed noise

h Average number of monorner units added to each active center

UV Entry of covariance matrix

P Parameter matrix to be estimated

P Number of experimental runs

z Number of measurement for each response

7

X i .v Chi-square distribution with significance level cr and degree of freedom of v

Subscripts O Indicates an initial value

Acronyms

BDF

CSTR

DAE

DMA

DMP

DMSO

DtBP

ED

EtOAc

GPC

Hx

IB

MeCHx

MeCl

MWD

NaN

ODE

PD1

PFR

PIB

SQP

SSH

Backward Differential Formula

Continuous Stirred Tank Reactor

Differential and Algebraic Equation

Dimethylacetarnide

2,4-dimethylpyridine

Dimethylsulfoxide

Di-tert-butyl pyridine

Electron Donor

Ethyl acetate

Gel Perrneation Chromotography

Hexane

lsobutyiene

Methyl cyclohexane

Methyl chloride

Molecular Weight Distribution

Not a Number

Ordinary Differential Equation

Polydispersity Index ( PD1 = DP, / D e )

Plug Flow Reactor

Polyisobutylene

Sequential Quadratic Programrning

Stationary State Hypothesis

xii

TCC Tricumyl chloride

TMPCl 2-chloro-2,4,4-trimeth yl-pentane

TPE Thermoplastic Elastomer

... X l l l

Chapter 1 : Introduction

The primary objective of this study is to develop a mathematic model, which will

represent the kinetics of living cationic lsobutylene (IB) polymerization. The

model that fits the data will be helpful in predicting the dynamic evolution of

monomer concentration, initiator concentraticn, and number average and weight

average molecular weight under different reaction conditions. The model is being

developed to assist in the development of improved processes for rnaking

polyisobutylene (PIB) based thermoplatic elastomers (TPE).

PIB-based TPE is a block copolymer developed two decades ago. This material

behaves like vulcanized N bber at room temperature and like thermoplastics at

higher temperature. The superior oxidative stability, vibration-absorbing

properties and other outstanding chernical and physicai properties make PIB-

based TPE a desirable material to the polymer industry (Puskas and Kaszas,

1996).

The PIB-based TPEs are synthesized by block copolymerization. First, the PIB

nibbery segment with high molecular weig ht and narrow molecular weight

distribution (MWD) is made by living cationic polymerization; then a second

monomer, such as styrene, is added to make the glassy outer segment (Puskas

and KaszasJ 996). Although the synthesis of PIB through living cationic

polymerization is a well developed process, the fundamental kinetics of the living

IB polymerization are still not clear.

Chapter 2 gives a review of IB polymerization kinetics. The comprehensive

model developed by Puskas and Lanzendorfer (1 998), and the simulation results

based on simplified models are also presented.

In Chapter 3, the developrnent of the model equations is described. The method

of moments is applied to transform an infinite number of equations to a finite set.

Al1 the assumptions used to develop the model equations are described as well.

The kinetic model equations are solved in Chapter 4. The failure of the

ScientistTM simulation package indicated that the modified EPISODE (Byrne and

Hindmarsh,1976) algorithm is not suitable for the problem due to the stiffness of

the ordinary differential equations (ODEs) in the model. DDASAC, a stiff ODE

solver (Stewart et al., 1994), was then chosen to simulate the model for two

reasons. First, it has been used widely in the simulation of complex reaction

kinetic models, and is very effective in solving stiff ODEs systems. Second, it can

be integrated with parameter estimation software, such as GREG (Stewart,I 995),

by providing sensitivity coefficients of parameters with respect to measured

responses.

In Chapter 5, several parameter estimation software packages are introduced.

Among them, GREG (Generalized Regression Analysis) is used to estimate

parameters in a four-response model. The responses include monomer

concentration, initiator concentration, number average degree of polymerization,

and weight average degree of polymerization. The diagonal entries of the

covariance matrix are approximated by the response sample variances that are

determined from the duplicate runs. Among seven rate constants, the

propagation rate constant, k,, and one backward rate constant, ka, are kept

constant; other pararneters are adjusted according to the constraints defined

from the expenmental analysis done by Puskas and Peng (1999). Simulations of

molecular weight distribution (MWD) are also addressed.

Parameter estimation is performed with additional replicate runs taken ai two

different experimental settings in Chapter 6. The response variances are re-

evaluated. The simulation results are compared with experirnental data.

A summary and conclusions to the work investigated in this thesis are given in

Chapter 7. Recornmendations for future consideration are also presented in this

final chapter.

Chapter 2: Literature Review

2.1 Introduction

PIB-based TPEs are probably the rnost important commercial polymers produced

by living cationic polymerization. As PIB is only synthesized through cationic

polyrnerization, the kinetic study of living IB polymerization is critical to the

irnprovement of production of this novel material (Puskas and Kaszas, 1996).

In this chapter, a cornparison of ideal living polymerization and quasiliving

polyrnerization are presented in the first section. Studies that contrïbuted to the

development of the kinetics and mechanism are introduced. The review also

describes some simplified rnodels and their simulation results.

2.2 Early Work in Living Il3 Polymerization

Living polymerization was first successfuliy applied to anionic polymerization of

styrene and butadiene using sodium naphthalide (Smarc,1956). According to

Szwarc, the term "living polyrnerization" is defined as a chain-growth

polyrnerization system that rnay resurne growth when fresh monomers are

supplied. Later, Flory (1953) sumrnarized living polyrnerization with the following

criteria: O

(1 ) Monomer is added to a terminal group of active growing polymer;

(2) Termination and chah transfer reactions are absent;

(3) The concentration of active sites remains rigorously constant from the

beginning of polymerization.

Under these stringent conditions, al1 active centers are generated

sirnultaneously. When conversion is high, a Poisson distribution is obtained for

the nurnber chain length distribution:

n = 12-' /(i - 1) ! ] exp (-A) (2-1

where h. is the average number of monomer units added to each active center.

The corresponding weight distribution is give by:

wi = n,[i/(A+l)]

Due to the lack of termination and transfer reactions, the growing polyrners do

not die but stay active. The polydispersity (PDI) is close to unity when the

molecular weight is high. If a second rnonorner is added after the first monomer

is consumed, a biock copolymer will be produced.

The living cationic polymerization of IB was not realized until Kennedy and Kelen

(1 982-83) introduced the quasiliving system. In this systern, rapidly reversible

chain transfer and termination reactions were present, allowing for reinitiation of

growing chains. The only requirement that needed to be satisfied in this "living"

system is the constant concentration of active polymer during the reactions.

The first living IB polymerization was achieved using tert-alkyl and tert-aryl esters

(Faust and Kennedy, 1987) and ethers (Mishra and Kennedy. 1987) in

conjunction with BC13 or TiCI4 as the initiation systern. It was found that the MWD

of polymers in this systern was much broader than a Poisson distribution, and it

was attributed to rapid propagation rate with respect to initiation. In order to

produce PIB with a narrower MWD, Kaszas et al. (1989) introduced extemal

electron-pair donors (EDs) for tertiary chloriderriCl4 initiation systems and tertiary

chloride/BCI3 systems. The use of EDs further advanced the understanding of

the kinetics of living 18 polymerization.

2.3 Kinetics of Living IB Polymerization

2.3.1 lnitiation

lnitiation is the first step in any chain growth polymerization; it generates the

active species for propagation. lnitiation of cationic polymerization is a cornplex

process involving ionization and activation. If the rate of initiation is relatively slow

with respect to propagation. some growing chains will propagate with monomer

during the extended initiation period. thus giving a broader MWD.

A two-component initiating system is needed for the living polymerization of IB to

control the MWD and overall polymerization rate. This two-component initiating

system consists of a protic acid, which provides the initiating species, and a

Lewis acid, which determines the properties of the counteranion:

lnitiation:

1 - CI + TiCI, + I'TiCZ5- (2-3)

where !-CI indicates the initiator, 2-chloro-2,4,4-trirnethyl-pentane (TMPCI), Tic14

is the Lewis acid, and I'TiCI; represents the active initiator. It is generally

accepted that the protic acidlLewis acid combinations are the most efficient

initiating systerns for cationic polymerization (Kennedy and Ivan, 1992).

Tertiary al kyl esters (Faust and Kennedy, 1987) and ether (Mishra and

Kennedy,1987), in conjunction with Lewis acids such as BCla or TiC14, were the

first reported initiating systems for living 1B polyrnerization. In the presence of

EDs, Kaszas et al. (1 989) reported that PIB with narrow MWD could b e achieved

with TMPCIILewis acid (TiC14)/ dimethylsulfoxide (DMSO) or dimethylacetamide

(DMA) systems. Later, the bifunctional dicurnyl chloride or trifunctional tricumyl

chloridefliCWpyridine initiating systems were also found to yield living IB

polymerization (Storey and Lee,1 992).

Kaszas et al. (1989) observed that some deliberately added electron pair donors

had a profound effect on living IB polymerization initiated by cumyl chloride/Lewis

acid systerns. In the presence of EDs, such systems elicited a narrow MWD and

a significant decrease in the rate of polymerization. Kaszas attributed this to the

stabilization effect of EDs on growing species. However, this concept was

challenged by Gyor et al. (1 992). who suggested that EDs, such as d i-tert-butyl

pyridine (DtBP) were acting as proton traps in initiation and were responsible for

the living characteristics and narrow MWD. Recently, Storey and Choate (1 997)

studied living IB polyrnerization using 2.4-dirnethylpyridine (DMP) as electron

donor. They found that excess of EDs over the minimum needed could retard the

rate of polymerization. Therefore, it is more desirable to have a slight excess of

EDs over proton irnpurities than it is to have large excess. ln their kinetic study,

the EDs were observed to eliminate the proton impurities, which apparently

validated the proton trapping theory of Gyor et al. (1 992).

2.3.2 Solvents

The rnost commonly used solvents in living IB polymerization are mixtures of

hexane and methyl chloride with different volume ratios. Faust and Kennedy

(1990) did experiments to determine the effects of solvent polarity on the stability

of growing chain ends, and observed that the rate of polymerization increased

with increasing solvent polarity. The explanation of this observation was based

on the equilibrium theory. When the polarity of solvents increased, the

equilibrium would shift to the active sites. Storey el at. (1 995) also reported

similar results with different initiating systems.

2.3.3 Propagation and Reversible Termination

It is now widely agreed that in living IB polymerization, the overwhelming majority

of growing PIB chains are chlorinated dormant polymers, P, -CI, which are in

equilibriurn with a çrnaller concentration of active ion pairs, P,' -Tic4 (Storey et

al., 1995; Storey and Choate, 1 997):

Scheme 2.1 First order reaction mechanism with respect to TiCI4

where ki and kd represent the rate constants of ionizatian and de-ionization,

respectively; k, is the rate constant of propagation; and M is the monomer.

The theory was first proposed for quasiliving polymerization with fast reversible

termination reactions. This dynamic equilibrium theory was later supported by the

observation of tert-alkyl chloride end-capped PIB with the CurnCI/TiC14 initiating

system (Puskas et al., 1991). Puskas et al. suggested that the active centers in

living IB polymerization in the presence of EDs were ion pairs. Free ions and

covalent species were proven not to participate in the equilibrium with dormant

species. If free ions were involved in the polymerization, the concentration of free

ions would be on the order of 10" moVL, and one would expect a high number

average molecular weight initially. This was opposite to the experimental

observations. The covalent species were also found to be inactive in living IB

polyrnerization, thus indicating they were not the active centers either.

2.3.4 Temperature Effect

Storey et al. (1 995) studied the apparent propagation rate constant ka,, at

different temperatures. It was shown that kapp increased with decreasing

temperature and that living polymerization can be observed at the ternperatures

of -80°C to -90°C. Furthermore, the presence of irreversible termination at -40°C

proved that living IB polymerization was possible only at temperatures less than -

40°C (Fodor et a1.J 998). In this study, Fodor also suggested that higher

ionization of polymer chain ends at lower temperatures could result in negative

activation energy. From the equilibrium perspective, at lower temperatures, the

dynamic equilibrium shifts in favor of active sites, which probably is another

reason for the increase in polymerization rate.

2.3.5 Determination of Rate Constants

The simulation of living IB polymerization requires the determination of rate

constants. The measurement of apparent rate constants, which are combinations

of several rate constants, is quite straightfonivard. Generally, their values can be

determined from the dynamic evolution of monomer concentration and initiator

concentration. However, since the concentration of intermediate complexes is not

tracked, the individual rate constants in the equilibrium reactions concerning

these species can not be determined directly from experiments.

The apparent reaction rate constant, ka,,, was determined from a plot of

In([M]o/[M]) versus time. The linearity of the kinetic plot dernonstrated the

expected first-order dependence on the monomer concentration and indicated a

constant number of growing chains (Le. no irreversible termination). Storey and

Choate (1997) proposed an equation based on Scheme 2.1 :

where [ R @ ] is the concentration of instantaneously active growing chains, which

remains constant. In their report, plots of k a p p ~ e r ~ ~ ~ the concentration of

different species were utilized to determine their effects on the rate of

polymerization. For example, the linear relationship between kapp and [Il,

suggested that propagation occurred primarily through ion-paired species and

was proportional to the concentration of active centers.

However, for mathematical modeling and kinetic simulation, knowledge of the

propagation rate constant, k,,, is more critical. Plesch (1993) calculated two sets

of propagation rate constants, k,, of cationic IB polymerization using different

solvents (Table 2.1 ).

Table 2.1 Rate constants for cationic IB polymerization in solution (Plesch, 1993)

Roth and Mayr (1996) evaluated the propagation rate constant k, from

oligomerization in dichloromethane. Using a diffusion clock method, Roth and

Source

Magagnini et al. (1 977)

Ueno et al. (1 966)

k, ( L . rnoT' . )

1 XI o4 9.1x103

Solvent

MeCl

CH2C12

Temperature (OC)

-50

-80

Mayr determined the value of the propagation rate constant as (6+2)x1CI8 L mol-'

s-'; however, this value is four orders of magnitude greater than the values in

Table 2.7. Roth and Mayr explained this discrepancy by two possibilities. Either

the low molecular weight mode1 was not applicable to high molecular weight

polymers, or the assurned oversimplified kinetic scheme used in Plesch's

calculations deterrnined an apparent rate constant, rather than the true one.

2.3.6 Reaction Order With Respect To Reactants

Reaction orders with respect to the reactants provide important information for

kinetic studies. The reaction rate of living IB polyrnerization was observed to be

first order in monomer concentration (Faust, 1990; Zsuga et al., 1992; Storey et

al.. 1995). The first-order relationship was demonstrated from the straight line of

In[M]J [Ml vs. time plots. Zeroth order monorner dependence was also reported

(Kaszas et al., 1994; Roth et al., 1997), and was attributed to the reduced

ionization constant with an increase in monomer concentration. The reaction rate

with respect to initiator was found to be first order under several initiating

systems (Kaszas et al, 1994; Storey et al., 1995; Roth et al., 1997).

The reaction order with respect to cointiator Lewis acid is still controversial.

Storey et al. (1 995) reported a second order reaction rate with respect to Tic14

under the condition of [I],<[LAJo. Gyor et al. (1 992) suggested that the second

order dependency on TiCI4 was a result of the predominant participation of

dimeric counteranions Ti2C1& in the propagation reaction. Furtherrnore, Storey

and Choate (1997) investigated the kinetics and mechanism of living IB

polymerization using the dicumyl chloridefriCWpyridine initiating systern in

hexane/methyl chloride solvents. Storey suggested that the dirneric conteranions

Ti2CI9- were formed by reaction of additional Tic14 with monomeric couteranions,

rather than direct ionkation of chains by neutral dimeric Ti2CI8.

2TiCI, +CC1 - ECI, + C+T~CI,- , ' cf T~JI,- &d O 4 2

Kaszas and Puskas (1 994) observed a first-order dependency on Tic14 at

[I],>[LA],, which cannot be explained by the formation theory of dirneric

counterions. Recently, Puskas and Lanzendorfer (1 998) proposed a new

mechanism, in which the initiation and propagation were first order with respect

to TiCI4 at [I],>[LA], and were second order at [I]o<[LA]o (Scheme 2.2). This

mechanisrn is discussed in detail in section 2.4.2.

2.4 The Kinetic Models

2.4.1 Development of Kinetic Mechanism

The synthesis of PIB by living cationic polymerization is a well-developed

process in industry. However, improvements in the production of PIB-based

copolymers require further investigation and irnproved kinetic models for living IB

polymerization. At the same tirne, models are usefui for polymer production by

providing quantitative predictions of product properties at different reactor

operating conditions.

The first mechanistic mode1 for living IB polymerization was proposed for a

system initiated by tertiary ester/ MtCI, (MtCI,=BCI3 or TiCI4) with ethyl acetate

(EtOAc) as ED (Kaszas et al., 1990). According to this mechanism. the tertiary

ester and MtCI, formed ionized complexes, which then undenvent propagation or

were transformed to tertiary chioride, depending on the reaction conditions. For

the first time, this rnechanisrn dernonstrated the existence of reversible

termination reactions in living IB polymerization. Storey and Lee (1 992) extended

this mechanism rnodel to the tricumyl chloride (TCC)/TiCI4/pyridine system. In

this system, Storey verified the dynamic equilibrium between active ionized

species and dormant species.

Later, Kaszas and Puskas ( A 994) proposed a more detailed mechanism, which

described the living IB polymerization initiated by TMPCI n iCl J di-t-butylpyridine

(DtBP) in a rnethyl chloride (MeCl)/ methyl cyclohexane (MeCHx) (40/60 vlv)

solvent mixture at -90°C and using excess initiator over coinitiator

(vMPCl]02[TiC14]o). Because the chlorinated dimer of IB, TMPCI, was used as

the initiator, the propagation rate constant and the equilibrium rate constants at

the initiator equilibrium were assumed independent of chain length. The

proposed mechanism is given as follows:

1-Cl tTiCI, , k,, , kd 1

ï+TiCIs-

The propagation rate was given by the following equations:

Two assumptions were made to sirnplify equation 2.9. The concentration of ionic

species is assumed negligible compared with the surn of initial concentration of

initiator and Lewis acid, The initiai concentration of initiator and Lewis acid are

relatively low ([l]o+[LA]ocl 0" mollL), and the ionization equilibrium was assumed

to shift toward the dormant species (K1=kilIkdl~<l). Therefore, when

[I]o2[TiCl4]0, the following equation was derived:

where k,' is the overall rate constant (kP1=kpK1). After integrating equation 2.1 0,

the analytical solution to monomer concentration is:

An equation to determine the initiator concentration was also derived for the

condition of [I]o~~iC14]o. The dynamic material balance was developed for the

active initiator (I'TiCZ;) in equation 2.12. According to the assumption that the

equilibrium reaction was shifted toward the dormant species, the Stationary State

Hypothesis (SSH) was made for the active initiator:

d [I +TiCZ; = ki, [d [LA] - k,, [I'TiCl;] - kp [MI [I'TICI; ] = O

dt

Frorn reaction 2.6, the consumption of initiator is given as follows:

--= k,, [Il [LA] - k,, [I'TiCI; 1 dt

Combining equations 2.12 and 2.14, Kaszas and Puskas obtained a simplified

equation with the assumption kp - [ M ] l k , , >> 1 and [LA]=[LAIo:

--- d'II - ki [Il [LA], dt

The analytical integration of equation 2.14 was given as:

(2.1 5)

It is noted that the assumption that the concentration of Lewis acid was

approximated by its initial value, Le. [LA]=[LAlO, is not proper for the condition of

[I]o~~iC14]o, because Lewis acid was used up fast if there is excess initiator.

Kaszas and Puskas demonstrated a large scatter in the data of the initiator

consurnption plot.

Given differential equations 2.10 and 2.14, Kaszas and Puskas simulated

rnonorner conversion and initiator consumption, using a Runge-Kutta 5M order

numerical integration program with adaptive step s ize control. The simulations

and experirnents showed that the polymerization rate, under the condition of [IlO

2[LAIo, was first order with respect to monomer, and proportional to the initial

concentration of Lewis acid TiCI4 and initiator TMPCl f. The apparent propagation

rate constant was determined from the monomer consumption plot, and k;, was

calculated from the first order initiator consumption. T h e simulation of rnonomer

concentration fits the experiments very well, and can be used to represent the

polymerization in certain initiating systems. However, the mode1 of initiator

concentration was oversimplified; moreover, the polymerization under the

condition of [LAIo 2 [Ilo was not considered in the moodel.

Storey and Choate (1 997) investigated the kinetics o7.f IB polyrnerization using 5-

tert-butyl dicurnyl chloride/TiCI4 with 2,4-dimethylpyridine (DMP) as electron

donor in HxIMeCI cosolvents. Based on the experimental results, Storey and

Choate proposed another complex mechanisrn, which considered the second

order dependency on Lewis acid at [LAIo > [Ilo, and szuggested that this second

order dependency would occur when propagation w a s carried predominantly by

chains of dimeric counterions.

Puskas and Lanzendorfer (l998) repeated the experiments at [LAIo 2 [Ilo in

solvents MeCl and MeCHx (40160 vfv) as suggested by Kaszas and Puskas

(1 994). and extended their experiments to a differenU cosolvent systern, MeCl

and Hexane (Hx) (40160 vlv), a less polar solvent mixture. It was observed that

the reaction order with respect to TiCI4 using Hx as nonpolar cosolvent was

higher than one. and the rates of initiation and propagation were considerably

faster than those obtained in MeCHx (kP1=3.4 vs. 0.54 ~~1rno1~s. respectively)

even at [LNo è [Ilo. The results revealed that the reactions were more complex

than expected.

At the same time, Puskas and Lanzendorfer (1 998) conducted experiments at

[LA], > Li], , and observed the occurrence of living polymerization under this

condition. By measuring the monomer consumption on-line with real-time

fiberoptic IR, Puskas and Lanzendorfer showed that the order of reaction rate

with respect to piC14] was 1.76. Based on this observation, model equation

(2.1 1) propoçed by Kaszas and Puskas (1994) was modified accordingly:

where ki is a composite rate constant, the product of the rate constant for

propagation kp and one or more equilibrium constants. Drawing the plots of

In(d[M]/dt) versus In[LA],, the values of k,' and k , were calculated using the

slope, S. as:

ki = S /([Il, [TiCZ4 1, )

ki = S / ( [ I I , [ T ~ c I , ] ~ )

2.4.2 Development of Comprehensive Mechanistic Model

Puskas and Lanzendofer (1998) performed experiments under a different ratio of

initiator to Lewis acid to study the effects of this ratio on the rate of

polymerization. The polymerization was carried out in a Mbraun LabMaster 130

glove box under dry nitrogen at -80%. TMPCl was added to a 500 ml round

bottom fiask, which was charged with Hx and MeCl (60/40,v/v) mixtures.

Appropriate amounts of DtBP and IB were added to the reactor afterward. The

polymerization was started by the addition of chilled TiCI4 in Hx. Samples were

taken out at specific times, and then quenched by chilled methanol to stop the

reaction. The polymers were purified and dried for analysis.

The concentrations of monomer and electron donor (DtBP) were set at 2 mol/L

and 0.007 mol/L, respectively, while the concentration of Tic14 and TMPCl were

varied to study their effects on the reaction.

Monomer conversions were deterrnined gravimetrically. Nurnber average and

weight average molecular weights (Mn and MW) were measured using Size

Exclusion Chromatography (SEC), and initiator concentration was calculated

su bsequently.

The experiments revealed that at different ratios of initiator and Lewis acid,

polymerization rates are in different orders with respect to TiC14. Consequently, a

comprehensive mechanism (Scheme 2.2) was proposed to describe the

simultaneous and cornpetitive reactions.

According to Scheme 2.2, initiation starts with ion generation, during which

initiator, in conjunction with Lewis acid TiCb, is in equilibriurn with a complex

species (I*LA). The complex species can undergo two reactions shown in Path A

and Path B. The active centers possess two different kinds of counteranions,

TiCI; and Ti2CI9-; the latter is supposed to be created by reaction between the

intermediate species, [I'LA] , andTiCI4, rather than direct ionization of neutral

Ti2C18, which is similar to the assumption of Storey and Choate (1997).

Path A

.1

Path B

4

Scheme 2.2: Cornprehensive rnechanism for living IB polymerization

(Puskas and Lanzendorfer, 1998)

Following the ion generation reactions, active centers propagate and grow into

active growing chains. The monomeric active growing chains are transformed

into polymer/Lewis acid complexes through an equilibrium reaction. Dimeric

active growing chains could also undergo an equilibrium reaction into

polyrner/Lewis acid complexes with a Lewis acid released. The polymer/Lewis

acid complexes exist in equilibrium with dormant polymer chains and Lewis acid.

Because TMPCI has similar structure to PIB, it is assumed that the initiation has

the same rate constant as that of propagation, i.e. kici=kp. Moreover. TMPCI

provides functional head groups for PIB produced in the homogenous

polyrnerization. This chlorinated PIB can yield block copolymers by addition of a

second monomer.

2.4.3 Simulation of Simplified Mechanism

By modifying equations 2.1 1 and 2.1 5, Puskas and Peng (1 999) extended the

models to fit the data a i [I]oc[LA]o. The kinetic equations were proposed to

describe the reactions along two different paths. When the initial initiator

concentration was greater than that of Lewis acid. i.e. [I]02[LA]o, the propagation

via the species with monomeric counterions was assumed to dominate in

polymerization, and the kinetic equations were described as follows:

VI h o = K,k,[LA],& =kt [LA],! VI

The composite rate constants were defined as :

k,A = k p K 4 = K 0 k p K 1 = K0kpkl / k - l

k t = K,k,

The composite rate constant k z was determined from the first-order monomer

consumption plots and was found to be 3.4 ~~rnol-~sec- ' . The composite rate

constant k: was determined from first-order initiator consumption plots and was

found to be 0.22 L-mol" sec-'. k, was calculated independently (Roth and Mayr,

1 W6), sol the value of k-, was calculated as 3.9 x 1 o7 sec-'.

When the initial initiator concentration was srnaller than that of Lewis acid, Le.

[I]o<[LA],, propagation via dimeric counterion was assumed to dominate the

polymerization. The corresponding kinetic equations would be modified as:

The composite rate constants were defined as:

k,B = Ko k2 [LAI,

k , = K0k,K2 = (k,, p-, ) k, - (k2 /kz )

kp = k, K: = k, K., Kz [LA],

Similar to the case of [I]o<[LA]o, kp was determined experimentally from the first

order monomer consumption. By normalizing the propagation rate by

[I],[T~CI,]'-", the apparent propagation rate constant, k,", was also determined

from monomer consumption plots, and was found to be 52 L' - m o l - -sec-' for all

initial conditions.

It is obvious that if initiator consumption was also tracked for the system, the rate

constants k: would be determined experimentally from the first order initiator

consumption plot. However, the initiation was almost instantaneous at [LAIo 2 [Il,,

so it was very difficult to track the initiator consumption. and klB needed to be

determined in other ways.

Puskas and Lanzendorfer (1 998) observed that in different solvents Le.

hexanelmethyl chloridelmethyl chloride and cyclomethyl hexane/methyl chloride,

the value of kl differed by one order of magnitude, while kl was constant. This

suggested that the backward reaction rate constants in al1 the reversible

equilibrium reactions were constant. Therefore, Puskas and Peng (1 999)

assumed that two reverse rate constants could have the same values,

k-2 = kdl =3.9x107 sec-'. From equations 2.25 and 2.27, the composite rate

constant k: can be calculated from following equation:

k;B = K0k2 [LA], = k: /(k,-)

The composite rate constants were calculated with various initial concentrations

of Lewis acid (Puskas and Peng, 1 999) as listed in Table 2.2. It was found that at

[I],<[LA],, the composite rate constant, k, , was constant, while k;B, kiB and &qB

varied with different ~iC14],.

Puskas and Peng (1 999) performed the simulations of simplified rnodels based

Table 2.2. Experirnentally Obtained Composition Rate Constants

on the rnechanisrn in Table 2.3 using the newly developed simulation software

PREDICI, Polyreaction Distributions by Countable System Integration,

kiB

(~rnol-' . sec-' )

0.25

O .27

0.36

O .43

0.70

(Wulkow,1996). PREDICI converts a reaction scherne to a set of ordinary

k q B

6.3 10‘~

7.0 IO-^

9.3 IO-^ 1.1 x IO-^

1.8 x 1 0 ' ~

Pcblo ( O - L )

32

37.5

40

64

128

differential equations, and then converts this system of differential equations to

discrete partial differential equations. The equations are solved using the

k,' B

( - - 1 3.8

4.2

5.6

6.4

10.6

discrete h-p-method with an adaptive treatment of reaction steps and automatic

k,"

( ~'rno~- ' sec-' )

52

50

54

52

51

error control mechanisms.

Table 2.3 Simple reaction mechanism and rate constants for simulating living IB

polymerization using PREDICI.

I Reaction Mechanism at [1],2[L~, I

Simulation results confirmed that, under the condition of [I],2[LA],, the

concentration of initiator,[l], active initiator, [I*], active polymer chains, e* . and

dormant polymer chains, ZP,, are changing with time, but, the summation of

these species remaineci equal to the initial concentration of initiator:

[II+[I'I+CC* +Ce =[II, (2.34)

The sirnulated polydispersity was found to be close to 2.0, reflecting

simultaneous initiation and propagation reactions.

At [I],<[LA],, Puskas and Peng (1999) simulated polymerization using PREDICI

in a similar way. Under these conditions, the Polydispersity Index (PDI) was

found to approach 1 . A . The run length Io, the average number of units

incorporating during a productive ionization period, was used to describe the

molecular weig ht behaviours.

where Io is the run length at [M]=[M],. The MWD gets narrower with decreasing 1,.

When 1, =1, a Poisson distribution Ss achieved. The simulation results also

revealed that PD1 was greater than tvvo above the critical value of 2, = 250.

2.5 Summary

The studies on kinetics of living IB polymerization have led to a comprehensive

reaction mechanisrn, which appears to be applicable for a wide range of reaction

conditions. Therefore, it is necessary to assess this complex mechanism by

cornparing model simulation results with experimental data.

The experimental condition under investigation is listed in Table 2.4. Note that

the initial concentration of Lewis acid is rnuch higher than that of ED (DtBP),

which was added to kill the protonic impurities in the system.

Initial pararneter estimates, which were determined by Puskas and Peng (1 999)

through experiments and sirnulations of simplified models in Table 2.3, are listed

in Table 2.5. As discussed in the previous sections, al1 reverse rate constants

(ko, ki. k2) are assumed to be equal and were determined from experiments by

Puskas's group. k, was deterrnined by Roth and Mayr (1996). The forward rate

constants listed in Table 2.5 were selected by Puskas and Peng (1 999) to fit the

experirnental data using a trial-and-error rnethod. In this thesis, parameters are

adjusted in Chapter 5 and 6 to fit a mode1 derived frorn Puskas's cornprehensive

mechanisrn using the experimental conditions listed in Appendix F.

Table2. 4 Experimental conditions of living IB polymerization

1 Temperature

1 Unit

I Total Volume

Value

1 Solvent Ratio (HxIMeCI)

1

Table 2.5 Parameters determined by Puskas and Peng (1 999)

Concentration of 1 B

Concentration of DtBP

Rate constant Unit

rnol/L

moVL

Value I

2

0.007

L.rno1-' . sec-'

k o

k.~

L.nzol-' . sec-'

k-I

k2

k-2

sec-' 3.9E+7

sec-'

L.mo1-'. sec-'

sec-'

3.9E+7

8.45E-1

3.9E+7

L.rnol-' sec-' 5.5E-2

Chapter 3: Development of Comprehensive

Model Equations

3.1 Introduction

In the previous chapter, a sirnplified mathematical model of living IB

polymerization (Table 2.3) was discussed, based on the assumption that

polymerization proceeded primarily by one of two pathways, depending on the

experimental conditions. However, the two reaction paths shown in the

comprehensive mechanism (Scheme 2.2) exist simultaneously during the

polymerization. As a result, it is desirable to develop a model that can simulate

the rnechanisrn at al1 experirnental conditions. In this chapter, dynamic material

balance equations are developed for al1 species in the polymerization. In order to

simulate the number average and weight average degree of polymerization, the

rnethod of moments is employed to convert the infinite number of ordinary

differential equations into a finite number.

3.2 Material Balance Equations

The properties of a polyrnerization process are detenined by a number of

fundamental characteristics, among which the monomer concentration, initiator

concentration and rnolecular weight distributions are of particular importance for

understanding the polymerization kinetics. Generally, mathematical tools are

available to engineers to sirnulate these characteristics. First, the comprehensive

mechanism (Scheme 2.2) was broken up into elementary reactions shown in

Table 3.1.

Table 3.1. Reaction rnechanisms for cornprehensive mode1

Reaction Type

Activation and

deactivation of

initiator site

Initiation

Propagation

Activation and

deactivation of

poIymers

Description 1 Reaction Mechanism

Formation of active initiator with

monomeric counterion

Formation of interrnediate complex I+LA-I'LA k-0

I dirneric counterion

Formation of active initiator with I*LA +LA *' V + L 4 - t?

monomeric counterion I lnitiation through active sites with

lnitiation through active sites with 1y4- +M kp >CL&- I - -

dimeric counterion

I+LA- +M kp ,<+,y-

I

Equilibrium between intermediate 1 e+LA-, " 4'LA km

Propagation through active sites

with monomeric counterion I

complex and active sites with

monomeric counterion

<+LA- + M kp , LA- i =3,4,..-CO

Propagation through active sites

with dirneric counterion

~quilibrlurn between intermediate 1 <+,TA- k-7 LA + LA - \ kz

LA- - + M kp , f)-- -

i =3,4,---a

complex and active sites with I i =3,4,~.-co

dimeric counterion l I

Equilibrium between intermediate / IN LA,^ L y e+LA complex and dormant polymers

The mode1 equations are composed of a set of dynamic material balance

equations for each individual species in a batch reactor (Table 3.2). The following

two assumptions are made to sirnplify the mode1 equations:

The effect of temperature on the kinetic rate constants will be ignored,

as the reaction temperature is kept constant at 4 0 OC by the liquid

nitrogen.

The initiator, TMPCI, which has similar structure to that of PIB, is

assumed to have same rate constant for reaction with monomer as

propagating chains do, i.e. ki=kp (Puskas, and Lanzendorfer,l998).

The rate of polymerization is calculated as:

where r, is the rate of propagation, and the symbols within [ ] are molar

concentrations. The evolution of concentration of active centers, intermediates

and dormant species are described by the ordinary differential equations in Table

Table 3.2. Material balance equations for comprehensive rnodel

Low Molecular Weight Species:

d[ I* LA] dt =koCrICLAI-k,[LAIII*LA]-(k-o +k,)[I*U]+k-,[I'LA-]+k-,[I+LR-] (3.3)

d[I+ LA-] = k, [I * LA] - (k-, + k, [ M ] ) [ I + L A - ]

dt

d[i 'LA,] - = k, [I* LA][LA] - (k-2 + k , [M] ) [ I+LA; ] dt

-- dl"1 - k-, ([1 * LA] + g [ e : L ~ l ) - ko[LA]([I] + 2 P i ) dt i=3 î=3

High Molecular Weight Species:

d[P," LA-] rlt

= k,[M]([I'LA-1-[P,'LAd]) -k-,[P,'LA-]+k,[P;LA] (3-8)

i=4 ..... CO (3.1 3)

-= k-, [P; LA] - k, [Pj ] [LA] (3.14) dt

I= ] k,[e*LA] - k, [el [LA] i=4 ... CO (3.15) dl

3.3 Molecular Weight Distributions

Molecular weights are important sources of kinetic information. Molecular weights

represent absolute, readily deterrnined quantities that reflect those elementary

reactions leading to the polymer molecule. Analysis of molecular weights and

molecular weight distributions are often used in forming empirical or indirect

estirnates of polymer characteristics (e.g. rheological behavior).

The molecular weights can be presented in different forms. In practice, it is

comrnon to use two average molecular weights, the nurnber average molecular

weight, p,, , and weight average molecular weight, a, as defined by the

following equations.

where Ni is the number of molecules per volume with exactly i repeat units, and

Mi is its rnolecutar weight.

Number average and weight average molecular weights can be deterrnined by

rneasuring the number average and weight average degree of polymerization,

Le. DPn and DPw, from Gel Permeation Chromotography (GPC). Number

degree of polymerization, is defined as the number of repeat units in a polymer

molecule. For a polymer sample, it is more convenient to use average degree of

polymerization (De ), which represents the average number of repeat units:

DP, =i@,,/rn (3.1 8)

where m is the molecular weight of repeat unit, which is a monomer in a

homopolyrnerization.

The corresponding weight average degree of polymerization is:

DP, = M , / m

Polydispersity index (PDI), which is defined by the ratio of two average molecular

weights, is often quoted as a convenient measurement of the breadth of the

molecular weight distribution. It ranges from values close to 1 .O for the nearly

monodiperse polyrners produced by living polymerization up to values of above

100 for branched polymers.

- - PDi=MwlMn (3.20)

For ideal living polymerization, when al1 of the initiation takes place

simultaneously, the molecular weig ht distribution is a Poisson distribution, for

which the number average and weight average molecular weights become

identical when the average molecular weights increase. However, this perfect

controlled molecular weight distribution is very difficult to realize in practice.

3.4 Moment Balance Equations

The rnethod of moments is a simple tool for calculating number average and

weight average molecular weights by transforming the infinite number of balance

equations to a few moment balance equations. Other methods like Laplace

transforms, z-transforms and statistical methods have also been used to

compute the molecular weight distribution. These methods are described in rnany

review articles (e.g., Ray, 1972; Ray and Laurence, 19777, and wilI not be

discussed here.

For a homopolyrner, the kh moment of chah length distribution is defined as

(Ray, 1972):

Generally, the first three moments provide enough information to approxirnate

unimodal polymer molecular weight distributions. The zeroth, first and second

moments are shown in Table 3.3 for al1 species such as intermediate complexes,

growing chains, and dormant polymers. By replacing material balance equations

with moment equations (Table 3.4). the mode1 becomes a finite set of ODES that

is relatively easy to solve using a numerical algorithm.

Table 3.3 zeroth, first and second moments of polymer species

Zeroth Moment

First Moment

Second Moment

Zeroth Moment

First Moment

Second Moment

Intermediate complexes

Polymer chains with dimeric counterions

Polymer chains with monomeric counterions

oa

1- = [I+ LA-] + C [yu-] i=3

Dormant polymer chains

Table 3.4. Moment Balance Equations

dW1 -= -kp [Ml (um, + ado ) dt

From the moment definitions, the total of these zeroth moments

(uto = u0 +UC, +um, +do) is the total concentration of polyrner chains and other

initiator derived species present, which equals the initial

The total of the first moments ( zrt, = u, + ucl + zm, +udl ) is

concentration of initiator.

the total moles of

monomer units in the polymer chains and initiating species, which equals the

moles of monomer units initially in the initiator, plus the monomer consumed:

ut, = CIIo (3-32)

ut, = 2[1], + [Ml , -[Ml (3.33)

Notice that balances on the second moments of chain Iength distribution are not

included in Table 3.4, as it is unnecessary to calculate second moments for the

various types of polyrneric species. Instead, the total of al[ of the second

moments can be determined from zeroth and first moments:

dut, A= k, [M 1 (2um, + umo + 2ud, + ud, ) dt

The number average and weight average degree of polymerization can be

expressed by three leading moments. As the polymer chains are a combination

of four types of polymer species, sums of the individual moments are required for

the calculation of degree of polymerization:

ub - DP,=-- ut2

ut, zim, + ud, + rtc, + u,

The material balance equations, moment equations, and equations for degree of

polymerization, required to simulate the polymerization process are equations

3.2-3.5, and 3.22-3.36.

Chapter 4: Model Simulations

4.1 Introduction

The comprehensive model described in the last chapter contains four response

variables: rnonomer concentration [Ml, initiator concentration [Il, and number

average and weight average degrees of polyrnerization (DP, and DP,,

respectively). Simulation of this model requires solving 12 ordinary differential

equations (equations 3.2-3-5, 3.22-3.31 ), and four algebraic (equations 3.33-

3.36) simultaneously. In this chapter, two software packages (Scientist and

DDASAC) are used to simulate the polymerization for the condition of [I]o<[LA]o

with parameter values shown in Table 2.5. Due to the stiffness of the ordinary

differential equations, appropriate tolerances were chosen to ensure reliable

simulation results. The results are cornpared with experimental data. The

DDASAC simulation code is rnodified to perform the simulations and sensitivity

coefficients calculations for parameter estimation in Chapter 5.

4.2 Model Simulation with ScientistTM

ScientistTM, an experimental data fitting software package by MicromathB, is

designed for parameter estimation in models containing Ordinary Differential and

Algebraic Equations (DAEs). The calculation options consist of model simulation

and parameter optimization. There are four standard ODE solvers in ScientistiM,

including Euler's rnethod, a Runge-Kutta fourth order rnethod, an error controlled

Runge-Kutta method and a Bulirsch-Stoer method. Stiff ODES are solved by

EPISODE (Byrne and Hindmarsh, 1976), which uses a Backward Differention

Formula (BDF) rnethod, which is superior for stiff systems.

For ODE systems, stiffness problems arise when the ratio of the largest

eigenvalue of the Jacobian to the smallest eigenvalue is large. This ratio is

referred to as the stiffness ratio (Luyben, 1990). Physically, if an ODE systern

represents a mixture of fast dynamics and slow dynamics, the solution of state

variables with large eigenvalues will change rapidly, while the solution of state

variables with small eigenvalues will change slowly, producing a stiff system.

Stiffness often exists in kinetic models of polymerization because the rate

constants for various reactions may Vary by many orders of magnitude.

The mode1 equations were solved for one experimental condition (Exp 217-5)

shown in Table 4.1. The rate constants used are listed in Table 2.5. The total

reaction time simulated was 7620 seconds. Results are presented in Figures 4.1-

4.4. Figure 4.1 shows good agreement between simulated and measured

monomer concentration. Figure 4.2 shows that there is a discrepancy between

sirnulated and measured initiator concentration.

It is noted that the measurement of initiator concentration contained ail initiator

complexes, Le. [Il ,,,, = [Il + [l'LA] +[l'LA-] + [l'LA;] , the simulated initiator

concentration shown in this thesis includes al1 these different species. As

discussed in chapter 2, initiator, TMPCI, is consumed nearly instantaneously at

the condition of [I]o<[LA]o.

Table 4.1. Initial settings for different experiments

Expt. ID 1 VMPCI] (mollL) 1 F i C u (mollL) 1 Comments

In Figures 4.3 and 4.4, simulation of number average degree of

polymerization, D e , fits the experimental data quite well, while the simulated

weight average degree of polymerization, DP,, is lower than the corresponding

experimental data.

In Figure 4.1, no simulation results for monomer concentration are shown for

times greater than 6000 seconds. An unexpected error message appeared for

monomer concentration after a long time of reaction. This NaN (Not a Number)

error message occurs when computer is attempting to calculate ''CO - CO "or

" O x cg ". One possible reason for this problem may be the stiffness of the ODE

system. When rate constants were changed to make the ODE system stiffer,

NaN problems became more severe.

The occurrence of the "NaN" error message and inability to simulate this and

other experimental data sets gave great concern about the feasibility of using

ScientistTM. An example of chemical kinetics, which was used to demonstrate the

application of the FORTRAN version of EPISODE in initial value problems (Byrne

and Hindrnarsh, l987), was used to test Scientistm, which uses a modified

version of the EPISODE code. The example involves the following three

nonlinear differential equations with initial values yi(0)=1 ,y2(0)=y3(O)=0 at t=O:

y,'= - 0 . 0 4 ~ ~ +104 (4-3)

y,'= O.O4y, IO^^,^, - 3 - 10' (4.4)

Y31=3-107 y: (4-5)

where ' (single apostrophe) indicates the time derivative of a variable.

This example was originally solved using the EPISODE package in FORTRAN77

on a CDC 7600 machine at Lawrence Livermore Laboratory. With relative error

tolerance of 1 E-6, a simulation was performed using ScientistTM to a final

reaction time 4E+5 seconds. The results agreed with the original solutions.

However, when the tirne limit was increased to 4E+7 seconds, NaNs appeared

and could not be eliminated, whereas the original version of EPISODE had no

difficulty. Moreover, Scientistm only has an overalI tolerance for state variables.

Rather than continuing to work with ScientistTM. and reducing stiffness by making

the SSH to simplify some our mode1 equations, we elected to abandon

ScientistTM in favor of the FORTRAN solver DDASAC.

8 W experiment - simulated i

Figure 4.1. Monorner concentration for polymerization of IB at -80°C in 60140

HxIMeCI. [1B]=2 rnollL, [DtBP]=0.0007 mollL, piCld=0.04 mollL, ~MPCI]=0.004

molIL

1 4 ex~eriment - simulated i

Figure 4.2. lnitiator concentration for polymerization of IB at -8OoC in 60/40

Hx/MeCI. [IB]=2 rnollL, [DtBP]=0.0007 molIL. FiCl4]=0.04 mollL, ~MPCI]=0.004

moI/L

: + experiment - simulated i

Figure 4.3. Number average degree of polymerization for polymerization of IB at

-80°C in 60140 HxIMeCI. [1B]=2 mollL, [DtBP]=0.0007 mollL, ~iCI4]=0.04 molR,

[TM PCl]=0.004 mol/L

i + ex~erirnent - simulated

Figure 4.4. Weight average degree of polymerization for polymerization of IB at - 80°C in 60/40 Hx/MeCI. [1B]=2 moliL, [DtBP]=0.0007 moliL, viCl4]=0.04 moliL,

[TMPCI]=0.004 mollL

4.3. Model Simulation with DDASAC

DDASAC (Double Precision Differential-Algebraic Sensitivity Analysis Code) is

an extension of the irnplicit integrator DDASSL, modified (Stewart et al., 1994) to

handle sensitivity analyses and solve nonlinear initial value problems containing

DAEs. Using a variable-order, variable-step predictor-corrector approach (Gear,

1971), the integrator can handle stiff, coupled systems of DAEs. Pararneter

sensitivity coefficients are calculated a i the end of each solution step by solving

the sensitivity equations numerically. These sensitivity coefficients provide

information to the optirnizer about the effects of parameters on the state and

response variables.

The system of DAEs was solved with DDASAC to simulate the polymerization

experiments under the experimental conditions in Table 4.1, using the rate

constants listed in Table 2.5. The simulation routine required approximately thirty

seconds on a Pentium II 266 MHz PC for a two-hour reaction. Individual

tolerances were specified for each state variable, and the Jacobian matrix was

calculated numerically in DDASAC. The Tolerance values and Jacobian

subroutine are given in the code in Appendix 1.

Simulation results for al1 of the experimental conditions in Table 4.1 are plotted in

Appendix A, and the experimental data used are tabulated in Appendix F. The

simulation results are better than the ones using ScientistTM. The simulated

monomer concentrations tend to fit the data quite well, except for the shape of

the curve for experiment 21 7-1 0. The number average and weight average

degree of polymerization agree with the experimental values at low tirnes.

However, at longer reaction times, the degrees of polymerization show

discrepancies with the experimental values. Although the simulated results tend

to be lower for DPn than the experimental data, these results tend to agree better

with the experiments than DPw does.

The simulations of initiator concentration fit the data quite well, especially for the

experimental conditions of [i]o<[LA]o (21 7-8, 21 7-9, 21 7-1 O), at which the initiator

was consurned much slower than that at [I]or[LA]o.

The simulations were also performed using the SSH on the zeroth moments

(umo, uco. udo, and uo) and Lewis acid. In this case, the derivative terms in the

ordinary differential equations for these state variables were set to zero, and

these variables were calculated through algebraic equations. The simulation

results using the SSH are alrnost identical with the ones obtained without the

SSH in Appendix A. Although the SSH can be applied for the polymerization

under the experimental conditions and parameter values concerned so far, we

were not sure that it could be used accurately for a wide range of conditions and

parameter values, and so the full set of dynamic equations was used for

subsequent parameter estimation and simulation studies.

The Polydispersity Index (PDI) is plotted in Appendix B, in which PD1 = DP,/DP,.

At [I]oz[LA]o, PD1 approaches 1 .O at high molecular weight, and the simulated PD1

is lower than that of measured data. At [I]o<[LA]o, there are great discrepancies

between the PD1 determined from simulation and these observed at the

beginning of each experiment. At longer reaction tirnes, the çimulated PDIs are

close to 2.0, while the observed PDIs are almost constant during the

polymerization, and are much lower than that the predicted PDI.

Chapter 5: Parameter Estimation

The development of a mechanistic rnodel is a helpful tool for understanding and

predicting the kinetics of polyrnerization. However, the model equations often

contain many unknown kinetic parameters, which need to be deterrnined to

ensure good rnodel predictions. In this chapter, we estimaie the unknown

parameters in Our dynamic model using the experimental data. Parameter

estimation for dynamic models requires specialized parameter estimation

software that calls an ODE solver subroutine to perform simulations and

calculate sensitivity information.

5.1 Introduction to Parameter Estimation Software

Several software programs are available for parameter estimation of nonlinear

ODE models. In the following section, a few parameter estimation programs are

briefly discussed.

ScientistTM (MicroMath Scientific soffware) is a Microsoft Window-based

simulation and parameter estimation package. The minirnization algorithms in

ScientistTM include nonlinear Sirnplex and Least Squares methods. The primary

minimization method in ScientistTM is the Least Squares method. This method

uses a Powell variant of the Levenberg-Marquardt approach (Powe11,1970) to find

the minimum of the sum of squares of residuals. The algorithm is a hybrid of

Gauss-Newton and steepest descent methods. When a good initial

approximation of parameters is not available, the region of the minimum is

located by a noniinear sirnplex algorithm. with which the initial values are

improved prior to the irnplementation of Least Squares minimization. Once the

change of the sum of squares of residuals is less than the tolerance.

convergence is considered achieved.

Scientistm is easy to use and is one of the sirnplest packages of pararneter

estimation for batch reactors. However, because of its inability to solve our stiff

system of ODEs reliably, we did not use it for the parameter estimation.

PREDlCl (Polyreaction Distributions by Countable System Integration).

developed by Wulkow (1 996), is a comprehensive simulation package for kinetic

models of polymerization reactions. In addition to its simulation function,

PREDlCl is also able to estimate parameters in kinetic models. The parameter

estimation is based on a damped Gauss-Newton rnethod, and a penalty function

is used to weight the differences between measured data and sirnulated values.

This new functionality in PREDlCl was not available when we began Our

pararneter estimation work.

MULTI (Guay,1997) is a powerful parameter estimation program written in

FORTRAN. In MULTI, the values of ODEs and the associated sensitivity

coefficients are calculated sirnultaneously by ODESSA (Leis and Kramer, 1988).

ODESSA is a rnodified LSODE package (Hindmarsh. 1980). It solves nonstiff

and stiff ODES with Adam's method and Gear's backward differentiation formulas

(BDFs), respectiveiy. At each integration step. ODESSA calculates the

parametric sensitivity coefficients, and solves the corresponding sensitivity

equations. Parametric sensitivity coefficients are the derivatives of state variables

with respect to the parameters. These sensitivity coefficients are used by an

optimizer in selecting improved values for parameters. The calculation of

sensitivity coefficients is discussed in section 5.2.2.

GREG, a FORTRAN subroutine for Generalized Regression, is designed to solve

multiple response nonlinear parameter estimation problems (Stewart,I 995). This

program was used successfully by a previous student (Tremblay, 1999) to

estimate the kinetic parameters with a known diagonal covariance matrix in a

three-response mode1 of Ziegler-Natta ethylene homopolymerization with a

missing data structure. In the present research, this program is used to estimate

the individual rate constants in a living IB polymerization with a full data structure.

The parameter estimation algorithm as well as the stiff DAE solver, DDASAC,

used in GREG will be discussed in the following section. GREG, rather than

MULTI, was chosen for parameter estimation in this thesis project because it is

available commercially, and has been used successfully by researchers at a

number of different universities and companies.

5.2 Parameter Estimation Package, GREG

5.2.1 Objective Function for Multiresponse Parameter Estimation

In parameter estimation, parameter values in a model are adjusted to obtain the

best fit to experimental data. GREG is a package designed to solve parameter

estimation problems for single or multiple types of responses. Assume that a

dynamic multiresponse model has M observed response variables, and that

r different measurements of each response are made during p different

experimental runs, given a total of N = .r - p observations for each response. The

predictions of these responses will depend on experimental conditions and the

vector o f parameters, p:

~ n m = f m ( x n P I + E,, n =l,2 ... IV ; m = 42 ..., M (5-1 )

where y,, is the observed value of the mth response for the nth observation,

fm (x, , f i ) is the model function for the mth response, specified by experimental

settings, x, and parameters p.

The noise term, 8, , is assumed to be normally distributed, and have the

following properties:

E(&d =O (5 -2)

for n3cr

for n=r

where L i s a MxM covariance matrix for responses, and 1% represents the

element in the mth row and sth column of 2'. Equations 5.3 and 5.4 are based

on the assumption that there is no interaction among responses for different

observations but that different responses made at the sarne time may be

correlated with each other.

In matrix form, the measured responses are represented by a NxM matrix, y, the

expected response matrix is represented by H(x,, p) ,

and the residual matrix is generaied as:

By studying the conditional maximum Iikelihood and Bayesian arguments, Box

and Draper (1 965) showed that the maximum likelihood estimates of P are those

that minirnize the determinant of 1 ~ ~ ~ 1 .

In GREG, minimization of the determinant criterion is detenined using

Sequential Quadratic Programming (SQP), a variant of the general Gauss-

Newton method. At each iteration, the objective function is approximated by a

quadratic function at some point Bo by taking a second order Taylor series

expansion:

WB) =P(B) +Y~($,)(P-P.) + l U P -Bo)'r(Bo)@ - B o ) (5.8)

where is the gradient and T($,) is the Hessian matrix at pointe,.

Note that in GREG, each parameter is bounded by the values stored in CHMAX,

which indicates the biggest magnitude change allowed for the parameter during

the computation. The bounds are generated by computer automatically, and are

not determined from the physical meaning of the parameters.

In GREG, two convergence criteria are available. One tolerance criterion,

RPTOL, is based on step size at iteration i:

If ~ A S ' 1 5 max(RPT0L - ' l,l .e - 8) , (5.9)

then optirnization stops.

RPTOL is a user-selected parameter, and p'is the parameter vector at Rh

iteration. Note that if the elements of p differ by orders of magnitude, it would be

appropriate to scale the parameters in the model so that they are of similar sizes.

Othewise, this criterion will focus on the changes only in the largest parameters.

Another criterion, RSTOL, is based on the size of the change in the objective

function at iteration i:

then optimization stops.

where ASk is the predicted change in the objective function using the quadratic

approximation. The convergence criteria are checked at the end of each

iteration. If none of them are satisfied, another iteration will be perforrned with a

smaller step size.

5.2.2 Sensitivity Coefficients

The sensitivity coefficients needed for parameter estimation are solved in

DDASAC. Sensitivity coefficients can be calculated using two different

algorithms. The simpiest uses a finite-difference approximation. in which the

changes in response variables are calculated with respect to a small perturbation

of parameters.

where ~,û, is a srnall perturbation in the Rh element of P . However, since the

responses need to be calculated for each parameter at every iteration. this is

very costly in cornputaiion time.

Rather than using a finite-difference approximation method. we opted to have

DDASAC determine sensitivity coefficients using user-provided Jacobian

information. Because the mode1 equations and sensitivity equations share the

same Jacobian matrix, the sensitivity coefficients can be calculated directly along

with the solution of model equations. This idea was applied to a general ODE

solver DASSL (Petzold, 1982) by Caracotsios and Stewart (1 985), who extended

DDASSL to DDASAC in order to handie the sensitivity analysis.

5.3 Parameter Estimation using GREG

5.3.1 Estimation algorithm for multi-response model

GREG provides different objective functions for parameter estimation depending

on the structure of the experimental data and corresponding covariance matrk.

Users can choose different Levels according to their specific problem. For the

kinetic model of living IB polymerization, we chose Level 22, at which GREG

provides a sirnplified objective function, J , for multi-response pararneter

estimation, with known diagonal covariance matrix:

where m is the total number of responses, oii are the diagonal entries of

covariance matrix, and Vii are the diagonal entries of Box and Draper matrix of

T z z.

By definition, the diagonal entries in the covariance matrix are the variances of

corresponding variables, and can be approximated from replicate runs using the

sample variance s: for the corresponding response variable. The non-diagonal

entries are the correlations between response variables at the same and different

observations. Due to the lack of information, it is assumed that in our model, the

correlations between the responses at the same sampling tirne are negligible,

and there is no interaction for responses at different observations. The non-

diagonal entries therefore are assurned to be zero for the estimation.

5.3.2 Calculation of Response Variance

The response variances required for the weights of the Least Squares method

are approximated with sample variances. The sarnple variances can be

calculated from the replicate runs, and are denoted as:

where n is the number of samples.

When the samples are taken at different times du ring the reaction. the overall

variances can be determined by pooling the variances at different sample times.

assurning that the variances are constant at al1 sample times. One commonly

used test for constant variances hypothesis is Bartlett's test (Maçon. et a1.,1989).

in which the hypothesis will be rejected if the Barttett's statistic value B exceeds

x:." 1 a chi-squared statistic with significance leve1 a and degree of freedom v=k-

where S: is pooled sample variance:

Here, at each of k sampling intervais, the number of repeated experiments is

denoted by ni and sample variance by s: .

Appendix D lists the replicate runs at experimental setting 217-5, which were

provided by the researchers at University of Western Ontario. The Bartlett's test

statistic value, B, was calcuiated and listed in Table 5.1 as well as chi-squared

statistics XLv with significance level a=l% and degree of freedom v=5. The

results show that the Bartlett's statistics for monomer concentration, initiator

concentration, and number average degree of polymerization are not statistically

significant, thus the variances could be considered homogenous throughout time.

The BartleWs statistics for weight average degree of polyrnerization is larger than

the chi-squared statistics, Iargely because of a single measurement that was very

different from the other values. This rneasurement may have been an outlier.

Therefore, we assumed that al1 variances are constant through al1 sample tirnes.

Table 5.1 Bartlett's Test of constant variance throughout time

5.3.3 Parameter Constraints

As discussed in chapter 2, Puskas and Peng (1999) proposed two different

sirnplified kinetic equations for initiator consumption along paths A and B. At

C ~ l o ~ [ ~ ~ l o

The composite rate constants Kokl at [I],>[LA], were determined frorn the first

S:

B

order initiation reaction:

(k, 1 k,) k, = 0.22

DPw 7. 2e2

16.33

(5.1 8)

Because of the fast initiation at [I],>[LAl,, it is not possible to track initiator

[Il (moI/L) 2.1 6e-7

3.16

xh., (a=1 %.v=5)

[Ml (rnol/L) 3.0e-2

7.1 7

consumption precisely. An alternative equation for the monorner consumption

rate was employed to derive the relationship among the rate constants:

1 5.09

DPn 7.09e2

7.66

where k, is the composite rate constant, k p " = K,k, K 2 = (k, /k-O ) k p - (kL /k-?) .

Frorn the plots of In(d[M]/dt) versus tirne, the values of ki were calculated as the

SIII],[TiCl~o '-", where S is the slope of the plot. If the value of propagation rate

constant k, is fixed at 6E8 (1-mol-'.sec-'), the second constraint becomes:

(k, 1 k - , ) - (k2 Ik-,)=8.67E-8

ln constraints 5.1 8 and 5.20, there are five un known parameters, and two

constraining equations, so only the values of three parameters must be

estimated, and the other two can be expressed as the functions of these three.

From the experimental results (Puskas and Lanzendorfer. 1998) diçcussed in

Chapter 2, it is known that the reverse initiation rate constant ki can be derived

from the experîmentai data. and values of the reverse rate constants k o and k 2

were assumed equal to kl. Therefore, hnro forward rate constants kl and k2 were

derived from equations 5.18 and 5.20 as:

k, = 0.22 /(ko / k-, ) (5.2 1 )

k, = 8.67e - 8 - k-? /(ko / k-, ) (5.22)

Therefore, if values for ko, ko, kll and k2 can be determined from the

experirnental data, then the other two rate constants, kl and k2 can be

subsequently calculated.

5.3.4 Parameter Transformation

The parameters to be estimated are individual rate constants in the

comprehensive mechanism of living IB polymerization. To be physically relevant,

the values of rate constants must be positive numbers. The pararneters were

transformed to their logarithmic forms to ensure this constraint. A second benefit

of the logarithmic transformation is that it scales the parameters to similar sizes,

eliminating the dominating effects on the tolerance test caused by changes of

parameters with large size.

It is noted that not every individual rate constant could be estimated for the

model. According to the comprehensive mechanisrn (Scheme 2 3 , when the

reactions are very fast, the overall rates of reaction may not be significantly

affected by increasing the forward rate constants and backward rate constants of

equilibrium reactions by the same factors (Appendix C). Because the effects of

forward rate constants on reaction rate could be cancelled by a corresponding

change of backward rate constants, it is impossible to estimate both fonvard and

backward rate constants independently. Therefore, the value of ko is kept

constant at the nominal value in Table 2.5, and only the forward rate constant ko

is estimated. Taking a logarithmic transformation, the parameters to be estimated

in GREG are:

5.4 Results and Discussion

Given parameter constraints and experimental data for the responses of

monomer concentration, initiator concentration, and number average and weight

average degrees of polyrnerization, the estimations were performed on a

Pentium Il 266 PC in less than one minute. The estimation results are listed in

Table 5.2, with the corresponding correlation matrix in Table 5.3. The strong

correlations between parameters indicate that these rate constants are difficult to

estimate independently from current data. The simulation results using the

estimated pararneters are given in Appendix E. The experirnental data used in

the estimation are shown in Appendix F.

Table 5.2 Resuits of Pararneter Estimation

Para rneter

kl (sec-')

Table 5.3 Corresponding Correlation matrix

Estirnated Value

The measured data points with error bar are plotted along the simulation in

Appendix E as well. The measured error of monorner concentration was

deterrnined frorn sample variance at Table 5.1 , Le. s = S, = 0.173 (mollL) (s , is the

95% Confidence

pooled sarnple standard deviation). The total length of the error bar above and

below the measured points takes the value of 2s. The error bars for [Il, DPn and

DPw plots were deterrnined in a sirnilar way.

At [I]02[LA]o, the simulations of rnonomer concentration and number average

degree of polymerization for 21 7-5, 21 7-5-Klara, 21 7-6, and 21 7-7 are in good

agreement with experimental data, while simulations of monomer concentration

and number average degree of polymerization for 21 7-5-2 are lower than

experimental data. The initiator is consumed very fast throughout this range, and

the simulations of initiator concentration fit the data quite well. The simulations of

weight average degree of polymerization show that, as the molecular weight

increases, the discrepancies become larger between simulated values and

measurements.

At [I]oc[LA]o, three sets of experiments were sirnulated (Le. 21 7-8, 21 7-9, 21 7-

10). The simulations of monomer concentration and DPn for 21 7-8 and 21 7-9 fit

the data well, however, the simulated monomer concentration for 21 7-1 0 does

not show the S-shape of the experimental data. The simulated DPws are higher

than the experimental data. It is also observed that the simulations of initiator

concentration show good agreement with the experiments.

The ratios of number average and weight average degree of polymerization (PDI)

are shown in Appendix G. The simulations of PD1 are found to converge to one at

[I]oZ[LA]~, and converge to two at [I]o<[LA]o. The discrepancies between

simulated and measured values may be attributable to some complex reactions,

which were not recognized in our model.

Parameter estimations were repeated when was given the values on the order

of 1 09,1 08- 1 06,1 os, and 1 04. When was increased up to 3.9 x 10' sec-', or was

decreased down to 3 . 9 ~ 10'' sec-', the estimation failed due to computation

difiiculties. The results at other values are shown in Table 5.4. When ko was

large enough, i.e. k, = 3.9 x 1070r3.9x 10' sec-', the estimate of ko changed with

ko, but the ratios of these two rate constants were kept constant, indicating that

the ratio has bigger effect on the responses than the individual rate constants.

The estimates of kl and k2 converged to the same values when

k, = 3 . 9 ~ 1 0 ~ sec-' or ko=3.9x108 sec-', indicating that ki and k2 were

independent of ka when k - ~ changes within a certain range. When ko is relatively

srnall, like 3.9 x 106 sec-' or 3 . 9 ~ IO* sec-' , the ratios of ko and ko are still kept

constant, but are not equal to the ratios when is large.

Table 5.4 Estimation Results at Different Values of ko

1 Initial Value

ko

(sec-')

3.9e8

3.9e7

3.9e6

Estimated Value

ko

(~.rnol".sec-l)

1.56e8

1.56e8

1.56e8

k-1

(sec-')

3.9e7

3.9e7

3.9e7

k-2

(sec-')

3.9e7

3.9e7

3.9e7

ko

mo mol-'.sec-') 2.08e9

2.09e8

9.87e6

kl

(sec-')

2.63e7

2.13e7

3.64e7

k-2

(sec-')

3.85e7

3.87e7

1.37e7

Chapter 6: Parameter Estimation with Additional Replicate

Data

To improve the fitting of weight average degree of polymerization, we considered

adding a catalytic deactivation reaction to the mechanism to account for the

effects of poisons on the molecular weight distribution. Unfortunately, adding a

deactivation term in the model would Iead to make the predicted PD1 higher,

rather than lower, so we abandoned this idea.

ln this chapter, the parameter estimation was done with al1 replicate data sets.

With one additional replicate run at the experiment, 21 7-1 0, the response

variances were re-evaluated, and then the simulation results with new pararneter

estimates are discussed.

6.1 Additional replicate Run

A replicate run was performed at University of Western Ontario to verify the

S-shape of the monomer concentration curve observed in experiment 21 7-1 O

(Appendix F). The replicate runs at this setting are shown in Figures 6.1-6.4. In

Figure 6.1, it can be observed that there is no obvious S-shape for monomer

concentration in the new data set, suggesting that the S-shape of the original

monorner concentration data is not reproducible.

2-5 I

! - 2 2 i 1.5

4

g 1 C* /*new i l l

Y

0.5 - d + ; i original i i i + I i

i Z O I 1 I i -

1

1 -0.5 500 3000 1 500

Figure 6.1 Replicate data of monomer concentration in experimental217-10.

Initial Conditions: [IB]=2 mollL. [DtBP]=0.0007 mol/L, FMPCI]=O.O5 rnollL,

[TiC14]=0.0 125 rnol/L.

: + new i original . ;

Figure 6.2 Replicate data of initiator concentration in experirnent 21 7-1 0, Initial

Conditions: [I B]=2 rnol/L, [DtBP]=0.0007 mol/L, F M PCI]=0.05 mol/L,

[TiC14]=0.0 125 moi/L.

I

60 l I

8 *i i ;+new I I

e t original , :

Figure 6.3 Replicate data of nurnber average degree of polymerization in

experiment 21 7-1 0, lnitial Conditions: [1B]=2 rnollL, [DtBP]=0.0007 mollL,

[TMPCI]=0.05 mollL, DiCl4]=0.01 25 mollL.

1 + new , i e P 40 : . original , ;

20 1

Figure 6.4 Replicate data of weight average degree of polymerization in

experiment 21 7-1 0, Initial Conditions: [IB]=2 rnoilL, [DtBP]=0.0007 mollL,

[TMPCI]=0.05 mol/L, [TiCI4]=0.01 25 molli.

6.2 Parameter Estimation with Ali Replicate Runs

As there are replicate nins at two different experimental settings, i.e. 21 7-5 and

21 7-1 0 (Appendix F), the overall sarnple variances can be determined by pooling

the variances deterrnined at different sarnpling times cf replicate nins. The

pooled sample variances are listed at Table 6.1.

Table 6.1. Pooied variance determined from a11 replicate runs

Compared with the pooled variances in Table 5.1. the new sample variance of

rnonorner concentration does not change much. The estimated variance of the

initiator concentration is larger than the original one, and the variances of number

average and weight average degrees of polyrnerization are smaller with

additional replicate runs at 21 7-1 0.

Parameter estimation was perforrned with additional data from replicate runs

using the pooled sarnple variances in Table 6.1. The estimated results are listed

in Tables 6.2 and 6.3. The parameter estimates shifted to new values. Because

the correlations between the estimates decreased for the estimation with

additional data. these new estirnated parameters are more reliable than the ones

determined in Chapter 5.

[Il (mol/L)

1.41 E-5

DPw

563.9 s$

[Ml (mol/L)

0.0388

DPn

542.1

The simulation results with estimated pararneters at experimental setting 21 7-1 0

are shown in Figure 6.5-6.8. The simulations fit the original data as well as or

better than the new data. The figures of simulations with new estimates at other

experimental settings are shown in Appendix H. The Figures show that although

the simulations of monomer concentration, initiator concentration, and number

average degree of polyrnerization fit the data quite well, the simulations of weight

average degree of polyrnerization still have discrepancies with the experiments,

and the discrepancies between simulations and experiments increase as reaction

time increases.

Table 6.2 Results of Parameter Estimation

I I I interval

95% Confidence

Table 6.3 Corresponding Correlation matrix

Estimated Value Parameter

r2

ko (Lmol-'.sec-')

kl (sec-')

k2(se c" )

Initial Value

1 7.48

1.56E8

3.9E7

3.9E7

16-36

7.31 E7

2.13E7

1.27E7

16.14

4.31 E7

2.00E7

1.02E7

16.57

1.23E8

2.27E7

1.57E7

21 7-1 0 i + original data ; l 4-

1 simulation 1 ! A newdata ,

Figure 6.5 Simulation of monomer concentration in experiment 21 7-10, Initial

Conditions: [IB]=2 molIL, [DtBP]=0.0007 rnollL, rMPCI]=0.05 rnollL,

viCl4]=0.0 125 mollL.

21 7-1 O j + original data , i-

4

simulation !

-- - -- -

Figure 6.6 Simulation of initiator concentration in experiment 21 7-1 0, Initial

Conditions: [IB]=2 rnolIL, [DtBPj=0.0007 rnoVL, [TMPCI]=0.05 rnol/L,

~iC14]=0.01 25 mollL.

21 7-1 O T '

1 + original data I , simulation '

j 6.00E+01 I I :

j A newdata

! ! b = 400E+01 1 Fu I i

i 2.00E+OI

Figure 6.7 Simulation of number average degree of polymerization in experiment

21 7-1 0, Initial Conditions: [1B]=2 mol/L, [DtBP]=0.0007 niollL. rMPCI]=0.05

mol/L, riCf4]=0.0 1 25 mol/L.

21 7-1 O + original data ;

,- simulation 1 ' A newdata ;

0.00E+00 Y ! , O 500 1 O00 1500

Figure 6.8 Simulation of weight average degree of polymerization in experiment

21 7-1 0, Initial Conditions: [IB]=2 rnollL, [DtBP]=0.0007 mollL, FMPCI]=0.05

mol/L, viCl4]=O.O 1 25 mol/L.

Chapter 7: Conclusions and Recommendations for Future

Research

The objectives of this thesis were to develop and solve a set of mode: equations

based on the mechanism of living IB polymerization proposed by Puskas and

Lanzendorfer (1 998); to estimate the values of parameters by fitting the model to

experimental data sets; and to assess the validity of the comprehensive model

through model simulations.

7.1 Summary and Conclusions

Because of severe stiffness of the ODEs, special algorithms were employed to

solve the ODEs. In Chapter 4, two software packages (Scientistm and

DDASAC), which were designed especially for stiff systerns, were used and

compared. The simulations were performed using the rate constants provided by

Puskas and Peng (1 999) for the experimental conditions specified in Table 4.2.

The Stationary State Hypothesis was verified for the zeroth moments of ail

species and Lewis acid. ScientistTM was abandoned because it gave unreliable

results for some simulations and gave numerical error messages during some

simulations. DDASAC was chosen to simulate the dynamic model due to its

capability and reliability for handling stiff ODE systems.

The focus of Chapter 5 was on parameter estimation using GREG, a FORTRAP

subroutine designed to solve multiple response parameter estimation problems.

The first key result stemmed from the successful estimation of three parameters

(16. ki, k2) using responses of monomer concentration, initiator concentration,

and number average and weight average degrees of polyrnerization. The

estimation was performed using Level 22 in GREG, in which the objective

function is a weighted Least Squares function with the weight approximated by

the sample variances determined from the replicate runs provided by Dr. Puskas

Two parameter constraints, which were derived from the experimental analysis

done by Puskas and Peng (1 999), were applied to the ODE system. kl, the

activation rate constant for active sites with monomeric counterion, and k2, the

activation rate constant for active sites with dirneric counterion, were expressed

as functions of other rate constants, so that they would not have to be estimated.

The deactivation rate constants kl and k2 were estirnated along with ko, the

activation rate constant for the interrnediate complexes. A natural logarithmic

transformation was applied to each parameter to ensure positive rate constants

and to scale the parameters that would be estimated to the same order.

The associated simulation results in Appendix E show good agreement with the

experimentai data for monomer concentration, number average degree of

polymerization, and initiator concentration. The simulations of weight average

degree of polymerization are lower than the data ai [I]o>[LA]o, and are higher than

the data at [I]O<[LA]~. The results also showed that, when assumed value of k o

(the deactivation rate constant for the complex intermediate) was changed over a

certain range, the ratio of estimated value of to k - ~ remain essentially constant.

This observation demonstrated that the equilibrium constants (the ratio of rate

constants) have a greater effect on the responses concemed than the individual

rate constants. The simulations of PD1 show that, for long reaction times. the PD1

is close to 1 .O at [Ilo 2 [LAIo, but is much [arger (approxirnately equal to 2.0) at

CKio<[LAlo-

In Chapter 6, a replicate run was investigated for the experimental setting, 217-

10. for which the original data set showed an S-shape in the plot of monomer

concentration vs. time. The replicate results showed that the S-shape was not

reproduced for the new data. Six additionai data sets from replicate runs were

used in parameter estimation in Chapter 6. Although there are still discrepancies

between simulations and experiments. the correlations between the estimates

became smaller, thus giving more reliable parameter estimates.

In conclusion, the monomer concentration, initiator concentration and nurnber

average molecular weight of living IB polyrnerization with TMPCITTiC14 initiating

system c m be well simulated using the proposed comprehensive model in

Scheme 2.2. However, the mode1 simulations have probtems fitting the weight

average degree of polymerization at some conditions. and the simulated PD1 is

larger than those determined from the experiments at [Ilo < [m.

7.2 Recommendations for Future Research

There are a number of potential topics for future research that arise from this

thesis. They are summarized as follows:

The simulation package, PREDICI, should be used to predict the entire

molecular weight distribution of the polymer, if this type of information is

desired. Comparisons could then be made between the predicted and

measured molecular weight distributions to discern any difference in

shape. PREDlCl simulations would provide additional verification that the

DDASAC simulation resuks are reliable.

More experiments should be conducted. The parameter estirnates will

have smaller confidence intervals when the number of data is large. Also,

better estimates of the variances of the responses would be obtained,

leading to better ability to make firm conclusions about whether or not the

mode1 can fit the data.

The parameter estimation that was done using GREG assumed a

diagonal covariance matrix for the responses. However, this assumption is

likely not valid. For example, we anticipate that there is correlation

between the nurnber and weight average degrees of polymerization, and

also between the initiator concentration and the monomer concentration. If

more replicate runs were perforrned, then GREG Level40 or 41 could be

used to estimate the off-diagonal entries of the covariance matrix, leading

to more reliable parameter estimates.

Experiments should be perforrned at conditions that are different from

those used to fit the rnodel, in order to validate the mode1 by testing its

prediction ability.

References:

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Faust, R. and J.P. Kennedy (1 990). Living Carbocationic Polymerization XXI. Kinetic and Mechanistic Studies of Isobutylene Polymerization lnitiated by Trimethylpentyl Esters of Different Acids, J. Macromol. Sci-Chem., A27(6), 649- 667.

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Appendix A

Simulation with Original Parameters

Experirnent setting: -80°C in 60/40 HxIMeCI

Initial Conditions: [IB]=2 mol/L, [DtBP]=0.0007 rnol/L

The parameters used in the simulation (Table 2.5)

The experimental data are tabulated in Appendix F.

A l : Simulation of monomer concentration

A2: Simulation of number average degree of polymerization

A3: Simulation of weight average degree of polyrnerization

A4: Simulation of initiator concentration

Unit

mol-' sec-'

sec-'

Rate constant

ko

k o

Lmol-' . sec-'

sec -'

~.rnol-'. sec-'

sec-'

L.rnol-'. sec-'

k.i

k-1

k2

k-2

kp

Value

1.56E+8

3.9E+7

1

5.5E-2

3.9E+7

8.45E-1

3.9E+7

6E+8

I l P217-5-2 P217-5-KIara

2.5 - 2.5 - observed + observed

2 4 - sirnulated 2 4 - sirnulated î î g1.5 - ' 51.5 - E 5C

E Y

- 1 - ! cl 1 - 5 3

0.5 - a 0.5 - l

' l

O , 1 , O 1 1 1

O 40 O 6000 8000 1 1 O 2000 40 O 6000 8000 2000 time ?sec) ! , time ?sec)

observed ! . '" 1

+ observed - sirnulated

1 0.5 - i i l I

' O -r a 1 1 1

O 40 O 6000 8000 ! 2000 time ?sec) l

1

O 40 O 6000 8000 '

2000 time Psec)

P217-8 l

I , 1

4 observed :

P217-7

+ observed - sirnufated

I

A + A - , I 1

O 40 O 6000 8000 1 2000 time ?sec)

1 + observed !

I

j

+ observed - simufated

a 1 1 v i

500 1 O00 l

1500 / time (sec) l

1 O00 'O0tirne (sec)

Al . Simulation of monomer concentration

5.00E+02 - I 1 5.00E+02 - A 1 I i

4.00E+02 - 1 4.00E+02 - f !

observed 1 : $3.00~+02 - 1

- + observed i simulated , l - simulated i

I

, 1 i I 1 I

I I

4000 6000 8000 1 O 2000 time (sac)

O 2000 - 4000 6000 8000 1 l trme (sec)

l 1

L I

j y- + observed

if - simulated

1.00E+02

+ observed - simulated

1 .OOE+O2

P217-7

+ observed - simulated

+ observed - simulated

O 2000 40 O 6000 8000 , ! O 504 üme &es)

1000 time (sec)

1500 ,

1 O00 ''fime (sec)

1500 1

P217-9 P217-I O

A 2 Simulation of number average degree of polymerization

6.00E+02 - 5.00E+02 -

, 4.00E+02 - g3.00~+02 -

j 2.00E+02 - i 1.00E+02 -

6.00€+02 - ; 5.00€+02 -

0 observed , . 4.00E+02 - -

simulated 1 g3.00E+02 - ; I ; : 2.00E+02 - ; 1 1.00E+02 1

+ observed - simulated

1 I

P217-5-2 7.00E+02 - 1

+ : 6.00E+02 - I

: 5.00E+02 - ~ ' 34.00E+02 -

! + observed 1 - simulated 1

1 .OOE+O2 - ! I

1 1

O 40 O 6000 8000 ' 'Oo0 time ?sec)

+ observed i - simulated f

i 2.00E+02 !

1 40 O 6000 8000 : O 2000 time ?sec)

7.00E+02 - + 7.00E+02 - + . 6.00E+02 - 6.00E+02 - .. I

5.00E+02 - 5.00E+02 - l

I I 4

' 34.00E+02 - + Observed 1 83-00E+02 + observed , - simulated I 2-00E+02 - - ' 2.00E+02 - simulated 1

1 1.00€+02 - I 1.00E+02 a t i r A O.OOE+OO 1 t i

O 2000 40 O 6000 8000 O 2000 40 O 6000 8000 1 time &ec> tirne ?sec)

I

I ~4.00E+02 + observed . - simulated ,

g3.00~+02 6 observed - simulated

O 2000 40 O 6000 8000 i 1 O 508me (sec! 1 O0 1500 ; time ?sec) i l ,

P217-9 i 7.00E+02 -

L

A3: Simulation of weight average degree of polymerization

a 6.00€+02 - 1 5.00E+02 -

B4-OOE+O2 - : &3.00~+02 -

2,00E+02 -

I i I I

: l . l PZ1 7-1 0

. 8 7.00E+02 - obse~ed

- sirnulated

: I

! 1 6.00€+02 - 1

! 1 5.00E+02 - j 1 24.00E+02 - i i o3.00€+02 - I

j j 2.00E+02 - + observed - simulated

O observed - simulated

O 40 O 6000 8000 , 2000 time ?sec)

3.5OE-O3 1 obsewed ,

- simula ted

P217-7

O obsewed , - simulated

I I

, 1

I i

' -5.00E-04 4 40 O 6000 8000 2000 time ?sec)

' 24.00E-02

3.00E-02 ' - : s2.00E-02

l 1.00E-02 i

l

O 500. 1 O00 time (sec)

1500 1 1

+ observed - simula ted

4 obsewed !

2 2.50E-03 - sirnulated 1 E I = 1 SOE-03

1 - Y 1

5.00E-04 - 1 f

A v I , i l

-5.00E-04 Q 40 O 6000 8000 i 2000 time ?sec)

O observed j - simulated

1 O observed 1 - simulated

O 500 1000 1500 ! time (sec) l

A4: Simulation of initiator concentration

Appendix B

Simulation of Polydispersity Index (PDI) of PIB with original

values of parameters

Experiment setting: -80°C in 60/40 Hx/MeCI

Initial Conditions: [1B]=2 rnol/L, [DtBP]=0.0007 mol/L, pMPCl]=0.004 molfL,

[TiCI4]=0 .O4 rnol/L

Values of parameters used in the simulations (Table 2.5)

Rate constant

ko

k o

kl

k-1

k2

k 2

k~

Value

1.56E+8

3.9E+7

5.5E-2

3.9E+7

8.45E-1

Unit

~ m o l - ' . sec-'

sec-'

Lrnol-'. sec-'

sec-'

mol-'. sec-'

3.9E+7 sec-'

6E+8 Lrnol-'. sec-'

i P2i75-2 observed I j P217-5-Klara

2.00E+00 - 2.OOE+OO sirnulated 1 : simulated

: 1-50E+OO ! lSOE+00 I I

9 1 .OOE+OO !

9 1 .OOE+OO I

5.00E-01 1 : 5.00E-01

0.00E+00 i 1 0*00E+00 rn I

O 2000 40 O 6000 8000 i O 2000 40 O 6000 8000 ! time ?sec) , t time ?sec) !

P217-5 0 observed

2.OOE+OO 1. - simulated

P217-6 2,OOE+OO + observed - sirnulated 1.50E+00

g 1 .OOE+OO

I I O 2000 40 O 6000 8000 j i

time ?sec) , I

i ' PZ1 7-7 i P217-8 + observed

2.OOE+OO 1 O observed 6.50€+00 7 - sirnulated b... . - simulated . 5.5OE+OO 1.50E+00 4.50E+00

8 1 .OOE+OO 5 3.50€+00 1 n2.50E+00 -

5.00E-01

1 5.00E-01 ' 0.00E+00

O 2000 40 O 6000 8000 -5.00E-01 O

5 0 h a ( s ~ O O 1500 / time ?sec) O ,

1 l 1

I l j , , i i 5.00E-01 I ; -5.00~-01 O i j -5.00~-01 500 1 000 1500 / 1 'OBme (sekyoo 1500 ' / ! time (sec) I

P2 i 7-9 2 P217-1 O

6.5OE+OO - O observed 6.50Et00 - O observed

B: Simulation of PD1 with original values of parameters

5.50E+00 - 1 4.5OE+OO -

- simulated , 5-50E+00 - simulated

I i 4.50E+00 - : j - 3.5OE+OO - n ' a 2.50E+00 - ' j 1.50E+00-

Appendix C

Effects of Fonivard and Backward Rate Constants on

Responses

Experirnent setting: -80°C in 60140 HxfMeCI

Initial Conditions: [IB]=2 rnollL, [DtBP]=0.0007 mollL, ~MPCI]=0.004 mollL,

Values of parameters used in the simulations (Table 2.5)

Rate constant

L-mol-'. sec-'

ko

ko

kl

k-1

k2

k-2

C l : Simulations with the values of ko and k o up 100%

C2: Simulations with the values of kl and kl up 100%

C3: Simulations with the values of k2 and k 2 up 100%

Value Unit

1.56E+8

3,9E+7

5.5E-2

3.9E+7

8.45E-1

3.9E+7

mol-'. sec-'

sec-'

~.mol-'. sec-'

sec-'

m mol-'. sec-'

sec-'

O 2000 4000 6000 8000 10000

time(s)

- base-case A k0 8k-0 up100%

- base-case A kO&k-OuplOOOh

time(s)

- base-case A kO&k-0up100%

Cl: Simulations with the values of ko and k o up 100%

1 - base-case A kl &k-1 up 100%

- base-case A kl&k-1 up%

- base-case A k1 &k-1 up100%

C2: Simulations with the values of kl and k.l up 100%

- base-case A k2&k-2 up 100%

- base-case A k2&k-2 up 100%

- base-case A k2&k-2 up 100%

C3: Simulations with the values of k2 and k 2 up 100%

Appendix D

Replicate Runs for the Calculation of Response Variances

Experiment ID: 21 7-5 (Appendix F):

Experiment ID FMPCI] (mol/L)

21 7-5-2 0.004

D l : Experimental data for monomer concentration

D2: Experimental data for initiator concentration

D3: Experimental data for number average DP

D4: Experimental data for weight average DP

Note: The values of initiator concentration supplied by J. Puskas were not

measured directly. Instead, they were calculated from monomer concentration,

number average molecular weight and mass of polymer produced according to

the equations:

and

Due to errors in measured values of [Ml. Mn, and mass of polymer. the smaller

initiator concentration were sornetirnes calculated to be negative. Although these

negative values do not make physical sense, they were used directly for

parameter estimation.

D l : Experimental data for monomer concentration

I + 31-May '24-May A Garba-1 XGarba-2 i

02: Experimental data for initiator concentration

D3: Experimental data for number average DP

D4: Experimental data for weight average DP

Appendix E

Simulations with Estimated Values of Parameters

Experiment setting: -80°C in 60/40 HxiMeCI

Initial Conditions: [IB]=2 rnol/L, [DtBP]=0.0007 mol/L

Values of parameters used in the simulations (Corresponding to Table 5.2)

Parameter

ko (L. mol-' .sec-' )

The experirnental data are tabulated in Appendix F.

Value

2.09E8

kl ( mol-'. sec-' )

kl (sec-')

k2( Lrnol-'.sec-' )

kî(s e c-' )

E l : Simulations of monomer concentration

E2: Simulations of number average DP

E3: Simulations of weight average DP

E4: Simulations of initiator concentration

4.1 1 E-2

2.63E7

6.26E-1

3.87E7

observed : : + observed j - simulated

I

I

!

f , I 1

1

O r 1

O 40 O O 6000 8000 , -0.5 - 2000 4000 6000 8000 , 2000 time !sec) time (sec) I

P217-5

+ observed - sirnulated ,

! l

x & l

I ' 2000 4000 6000 8000

1

tirne (sec)

O observed - simulated

1

I 1

2000 4000 6000 8000 '

-0.5 time (sec)

P217-7 2.5

+ observed - sirnulated

time (sec) 500 1000 1500 j

time (sec) , I

1 + observed

h - simulated

1 O00 Motirne (sec)

observed i

- simulated i

1 ; 2.5 7 + observed - simulated

E l : Simulation of monorner concentration with estimated parameters

5-OOE+O2 - r s 5.00E+02 - I =

4.00E+02 - i 4.00E+02 - r 3.00€+02 - ' i r 3.00E+02 - ! I n O observed ;

O obsewed , - sirnulated j simulated ! : 1.00E+02 - ,

I 1 , 1 I

2900 4000 6000 8000 2000 4000 6000 8000 time (sec) time (sec)

P217-5 P217-ô 6.00E+02 - 0 6.00E+02 -

9 i 5.00E102 - ' 5.00€+02 - 1

! 1 4.00€+02 - 4.00E+02 -

E 3.00E+02 - r 3.00E+02 - ! obsewed j 2.00E+02 observed , - simulated -

1.00E+02 simulated 1

1 I i 0.00E+00 I 1 1

I

2000 4000 6000 8000 -1.00€+02 2000 4000 6000 8000 ' time (sec) I time (sec) l

c 3.00E+02 O observed : , a

-simulated i 2.00E+02 observed i

- . 1 .OOE+O2 simulated 1.00E+02

0.00E+00

-1.00E+02 time (sec) tirne (sec)

!

i -1.00€+02 500 1000 1500 / -1.00E+02 ) 500 1 O00 1500 l l time (sec) 1 I time (sec) 1

1 1

P217-9 I P217-1 O 6.00€+02 - i 6-OOE+O2 -

I J ! 1

E2: Simulation of nurnber average degree of polymerization with estirnated parameters

5.00€+02 - 4.00€+02 -

E 3.00€+02 - a

' 0 2.00E+02 - 1.00E+02 -

Z , 0*00€+00 & 4

i 1 1

f ! 5.00€+02 - ! ' 4.00€+02 -

O observed , - r3.00E+02-

O observed - simulated simulated

obsewed

ssirnuIated + observed - simulated

I 1 I 1

2000 4000 6000 8000 time (sec)

h 2000 4000 6000 8000 tirne (sec)

i

0~00E+00 L 1 I i

-1.00E+02 40 O 6000 8000 time ?sec)

2000 4000 6000 8000 ' time (sec)

-1.00E+02 ) 2000 4000 6000 8000 , -1.00~+02 1 500 1 O00 1500 i time (sec) time (sec) ,

P217-7 P217-8

P217-9 1 I P217-10 1

?.OOE+O2 - 1 I

6.00E+02 - i 5.00E+02 - 1

1 , + observed l

4.00€+02 - + obsewed 3 - a 3.00E+02 - - simulated ' b 3.00E+02 simulated - P a

2.00E+02 - ( 2.00€+02 I

7.00€+02 - 6.00E+02 - f ' 6.00E+02 -

5.00E+02 - 4.00€+02 -

+ obsewed L 3 3.00E+02 - - simulated 2.00E+02 -

1 i 1.00~+02 ; 1.00E+02 * l

0.00E+00 I j 0.00E+00 1 !

1 -1.00€+02 -1.00E+02 1500 / ttme (sec) f

1

I

+ observed - simulated .

~~~~~ -

E3: Simulation of weight average degree of polymerization with estimated parameters

I I 1 1 1 i

0 observed - simulated

T T

tirne (sec)

1

P217-5-Klara 1 I

1

l

1 i

O observed I 1 - simulated ,

T

time (sec)

PZ1 7-5

4-00E-03

3.00E-03 O observed

î - simulated % 2.00E-03

O observed , a 2.00E-03 4 - simulated

Ë Y - E

0~00E+00

~ 1 " ' - 0 3 ~ 3 - & 3 0 -1 .OOE-03 ; ~ ~ $ + j ~ ~ ~ -1 -0OE-O3 time (sec) , .

I tirne (sec) I

-2.50E-03 -1 O observed - simulated 1 + observed . - simulated !

-1.50E-03 - O 500. 1000 1500 ; tirne (sec) time (sec)

c

P217-9 P217-10

5-OOE-02

+ observed - simulated ;

I

O l

l

E4: Simulation of initiator concentration with estimated parameters

Appendix F

Experimental Data for Parameter Estimation

Experiment setting: -80°C in 60140 HxlMeCI

Initial Conditions: [18]=2 rnollL, [DtBP]=0.0007 rnollL

Initial Conditions of lnitiator and Coinitiator ( Table 4.1 )

The experimental data were obtained from the papers published by Puskas and

Lanzendorfer (1 998). and Puskas and Peng, (1 999).

No. of Experiments

2 1 7-5-2

2 1 7-5-KIara

VMPCI] (moVL)

0.004

0.004

viCl4] (rnol1L)

0.032

0 .O375

DPn / DPw time rsl

DPw

Exp217-5-Klara

time [s] O

DPn 2.00

[Ml (mol/L) 2.00000

300

DPw 2.00

1.73500 1 74.67

Dl (mol/L) 0.00400

140.63 0.00047

DPn 2.00

DPw DPn

DPn 2.00

DPw

DPw 2.00

tirne[s] O 120

[M](mol/L) 2.00000 1.82232

DPn 2.00 28.69

DPw 2.00 51.22

[Il (mol/L) 0.05000 0.04382

Appendix G

Simulation of Polydispersity index (PDI) of PIB with

estimated parameters

Experiment setting: -80°C in 60/40 HxJMeCI

Initial Conditions: [1B]=2 mol/L, [DtBP]=0.0007 rnollL, [TMPCI]=0.004 mollL,

piC14]=0.04 mol/L

Values of parameters used in the simulations (Corresponding to Table 5.2)

Parameter

ko (1.mol-' .sec-')

ko(sec-' )

k1 ( L.moZ-'. sec-' )

kl (sec")

k2( L.moZ-'. sec-' )

k2(seë1 )

Value

2-09E8

3.9E+7

4.1 1 E-2

2.63E7

6.26E-1

3.87E7

40 O 6000 8000 : O 2000 time !ses,

I I b

P217-5-2 P217-5-KIara I

I

O 2000 40 O 6000 8000 time ?sec) I

0 observed ; , .

I

, I

+ observed - l

simulated , 1 1

1 a . I !

2.OOE+OO - 2.00E+00 -

2-OOE+OO

1.50E+00

i ~ ~ . O O E + O O

- simulated , ' 5.00E-01

n t rn I . 0.00E+00

~ ~ . O O E + O O - 5.00E-01 -

O 2000 40 O 6000 8000 , time ?sec)

lSOE+OO - 1-=+00-\** ; , -

, : 2 1 .OOE+OO - + observed - simulated , 5.00~-01 -

I

1 + observed

O.OOE+OO , . 1 . 1 i O.OOE+OO -

1 - simulated

O 2000 time 40 ?sec) O 6000 8000 1

- simulated

P217-7 P217-8 6.50E+00 - observed

40 O 6000 8000 i O 2000 time ?sec)

5.50E+00 - 1

, , 4.50€+00 - f

/ ] -3SOE+OO - P

+ observed ! n2-50E+00 -

+ observed - simulated !

- simuiated -

1 ; -5.00E-01 O 1 O00 MPlme (sec) 1500 ! j

y 0 0 ''Pime (sec

1500 ,

G: Simulation of PD1 with estirnates of pararneters

Appendix H

Simulations with New Estimated Parameters

Experiment setting: -80°C in 60140 Hx/MeCI

Initial Conditions: [l B]=2 mol/L, [DtBP]=0.0007 mol/L

Values of Parameters used in the Simulation (Corresponding to Table 6.2)

Parameter

ko mol-' .sec-')

Value

7.31 E7

ko(sec-' )

ki ( mol-'. sec-' )

kl (sec-' )

kZ( mol-' .sec-' )

kz(s e C-' )

H 1 : Simulations of monomer concentration

H2: Simulations of number average DP

H3: Simulations of weight average DP

H4: Simulations of initiator concentration

3.9E7

1.1 7E-1

2.13E7

5.87E-1

1.27E7

k, ( Lmol-'. sec-' ) 6E+8

PZ1 7-5-2

0 observed - simulated ,

i

,

O 2000 4000 6000 8000 time (sec)

P217-5

+ observed '

simula ted

I I

o A 1

2000 4000 6000 8000 - time (sec)

+ observed ! - simulated I

1

1

f , i I

+ observed - simulated

1

1 -0.5 ) 2000 4000 6000 8000 I tirne (sec)

I P217-6

+ observed j - simulated

& a 1

2000 4000 6000 8000: time (sec)

+ observed - simula ted

- time (sec)

+ observed - simulated

l U Z X r 1 1 I

500 1000 1500 tirne (sec)

+ observed ; 0 obsemed ,

- - simulated ,

i I

i

T 1 1 1 1 I

1000 1000 1500 ' I time (sec) time (sec) 1

H i Simualtion of monomer concentration

1

P217-5-2 PZ1 7-5-Klara I

6.00€+02 - ' I

6.00E+02 - I

5,00E+02 - Z L Z I ; 5.00€+02 -

f 9 8

4.00€+02 - 1 4.00€+02 - 1

r 3.00E+02 - ' i c 3.00E102 - observed + observed ' 8 2.00~+02 - - - sirnulated 1

simulated ' j 1-00~+02 - 7

, ' I

1 1 0.00E+00 i

2000 4000 6000 8000 1 -1.00~+02 time (sec)

2000 4000 6000 8000 j tirne (sec)

l 1

c 3,00E+02 a + observed a 2.00E+02 - simulated '

1.00E+02

0.00E+00 L I 1 1

-1 .OOE+O2 2000 4000 6000 8000 time (sec) l

c 3.00E+02 a 0 2.00E+02 O obsetved

1.00E+02 simulated

' 0.00E+00

-1 .OOE+02 time (sec)

6.00E+02

k 5.00E+02

4.00€+02

2 3.00E+02 O observed , Q

2.00E+02 - simulated

O.OOE+OO I I 1 I

-1.00E+02 2000 4000 6000 8000 time (sec)

4.00E+02 4 observed ; - sirnulated ;

l

1.00E+02 I kkoo : 0.00E+00

' -1.00E+02 tirne (sec)

P217-9 P217-10 6.00E+02

I

: 6.00E+02 - 5.00€+02 j ! 5.00E+02 - 4.00E+02 + observed . i 4.00€+02 - observed

1 c 3.00€+02 - simulated ; j 3.00~+02 - - sirnulated

0 2.00E+02 I V n . a 2.00E+02 -

-1 .OOE+02 dl 500 1000 IMo -1.00€+02 1 500 1000 1500 tirne (sec) tirne (sec)

H2. Simualtion of number average degree of polymerization

O observed - simulated

I I I 1

2000 4000 6000 8000 time (sec)

0 observed ' - simulated i

2000 4000 time (sec)

- I + s

0 observed - sirnulated

& 2000 40 O 6000 8000 time bec)

I

0 observed :

- simulated

7.00E+02 - 6.00E+02 - 5.00E+02 - observed ,

4.0OEt02 - - simulated obsefved 2 3.00E+02 - - simulated 0

2.00E1.02 - 2.00€+02 - 1.00E+02 - , . 1.00E+02 - O.OOE+OO , , . I O.OOE+OO

-1.00E+02 2000 4000 6000 8000 1 , -1 .OOE+O2 time (sec) I

Q 1

i -1.OOE+O2 6 500 i 000 1500 i -I.OOE+O~ 6 500 1 000 1500 , time (sec) t i m (sec) 1 1 / I

PZ1 7-9 P217-1 Ii 7.00E+02 - ' 7.00€+02 -

I /

H3. Simualtion of weight average degree of polymerization

+ observed - simula ted

6.00E+02 - 5.00€+02 -

3 4-00E+02 : 3.00E+02 2.00E+02 -

, , 6.00E+02 - + observed 5.00E+02 - - sirnulated , 4-00E*02 -

I 3.00E+02 - ' 1 a . 2.00E+02 -

+ 0bseNed - sirnulated

-6.OOE-03 J tirne (sec)

P217-5 1 -0OE-02

8.00E-03 O obse~ed - simulated ,

: 6.00E-03

time (sec)

! 1 P217-5-Klara !

! 1,OOE-02 1 O observed 1

-6.00E-03 -1 time (sec)

I + observed

simulated

W+LO -6.OOE-03

time (sec)

+ observed 2 4.00E-03 simulated

-2.00E-03

-4.00E-03

-6.00E-03 A time (sec) ,

0 observed - 2.00E-02 - - sirnulated i

a 0.00E+00 - I i 1

: -1.00E-02 !b 500 1000 1500 j tirne (sec) !

!

2 4.00E-02 4 -

O observed , 0 3.00-02 , , ; g 3.00"-02 kbserve; , E simulated - 2.00E-02 simulated ! 1 2-00E-02 = f 1.00E-02 1 .OOE-02

I I ,

l 0.00E+00 0.00E+00 l ! -1.00E-02 500 1000 i 500

time (sec) 1 O 500 1 O00 1MO / 1 time (sec) i

H4. Simulation of initiator concentration

Appendix I

FORTRAN Code for Parameter Estimation

PROGRAM multiple !Fitting multiple responses, [M],[I],DPn and DPw ! Estimate three parameters with constraints !At GREG level=22 !complete data set with a known diagonal covariance matrix !derivatives values is provided by the sensitivity analysis in DDASAC. ! Author: Jack Liu !Date: Aug 22.2000

IMPLlClT DOUBLE PRECISION (A-H.0-Z) PARAMETER(LUN=I O, 1 LUNDAT=2, 2 NPAR=3, ! The parameter to be estimated is k0,k-1 ,k-2

3 NEXP=52, !number of experimental points excluding exp217-10 4 MGLL=4,

5 NRESP=4, 6 L22GRP=NEXP,

7 N08GRP=NRESP*L22GRP1 8 MDSC=3+NPAR*(NPAR+NOBGRP+8)+3*NOBGRPl 9 MISC=3'NPAR+NRESP*(NRESP+.S)+(NRESP+l), 9 NRPAR=5) ! nurnber of total parameters DIMENSION BNDLW(NPAR),BNDUP(NPAR),CHMAX(NPAR), 1 DEL(NPAR),PAR(NPAR),DSC(MDSC), 2 STRACK(0:MGLL-1 ),PIPPAR(NPAR),VTOL(NRESP), 3 VTOLIM(NRESP),VO(NRESP,NRESP) DIMENSION OBS(NRESP,L~~GRP),~OU~(NEXP)~PRE(NRESP,NP) DIMENSION RPAR(NRPAR) DIMENSION ISC(MISC) DIMENSION a1(8),aLA(8),aM(8) EXTERNAL MODEL COMMON/setting/aI,aLA,aM COMMON/time/tout COMMON/para/RPAR

!open file of variables of time and Cm. OPEN(UNIT=LUNDAT,FILE='fulldata.dat',STATUS='UNKNOWN')

!open file of values of initial parameters OPEN(UN IT=LUN DAT+1 ,F ILE='iniparl .dat',STATUS='UN KNOW N') OPEN(UNIT=LUN,FILE='full.out',STATUS='UNKNOWN') LEVEL=22

! read in time intervals and responses DO 5 1=1 ,NEXP READ(LUNDAT,*) t,Crn,DPn,DPw,el tout(l)=t OBS(1 ,I)=Cm OBS(2,1)=DPn OBS(3,1)=DPw OBS(4,1)=el

5 CONTINUE ! read in initial parameters ! RPAR(1 )=kO(I.mol-1 .sec-1 ); ! RPAR(2)=k-O(sec-1) ! RPAR(3)=k-1 (sec-1 ) ! RPAR(4)=k-Z(sec-1) ! RPAR(S)=kp(I.mol-1 .sec-1 ) ! kl=0.2U(kOlk-O)(l.moI-1 .sec-1 ) ! k2=3.38/(kO/k-O)(l.moI-1 .sec-1 )

DO 1 O I=1 ,NRPAR READ(LUNDAT+l ,*) RPAR(I) W RITE(*,f)lRPAR(',II1)=',RPAR(I)

10 CONTINUE ! transformation of parameters

PAR(1 )=dlog(RPAR(i )) ! par(1 )=ln(kO) PAR(2)=dlog(RPAR(3)) ! par(Z)=ln(k-1 ) PAR(3)=DLOG(RPAR(4)) ! par(3)=ln(k-2) write(Iun,*) par

!read in experimental setting. OPEN(UNIT=LUNDAT+2,FILE='fullexp.dat1,STATUS='OLD') do 15 1=1,8 READ(LUNDATt2,') al(l),aLA(l),aM(I)

15 CONTINUE CALL SETUP(IDIF,IDPROB,IRESD,ITMAX,LEVEL,LINSRT,LISTS, 1 LISTSC,LUNSCR,MNLSSlNARMIJINU(HEVINPAR,NPROBINR€SPl 2 NSUPRS,ALPHA,ATOL,EMOD,EPSMCH,RPTOL,RSTOL,BNDLW,BNDUP, 3 CHMAX,DEL,PAR,TYPPAR,VTOL,VO)

! To use DDASSAC, set IDPROB to 2. IDPROB=2

! set convergence tolerance as default. RPTOL=l .D-5 RSTOL4 .D-1

!Set response diagonal covariance matrix. V0(1,1)=4.e-2 V0(Zl2)=3.2e3 VO(3,3)=4.9e3 V0(4,4)=2.15d-7 LISTSC=2 LISTS=4 CALL GREG(IDIF,IDPROB,IRESD,ITMAX,ITNO,LEVEL,LINSRT,LISTS, 1 LISTSC,LUN,LUNDAT,LUNSCR,L22GRP,MDSClMGLL,MISClMNLSS, NARMIJ,NBLOCK,NU(HEV,NU(P,NFUNC,NOB,NPAR,NPAT,NPROB,NRESP, NSUPRS,NTERCO,IBLOCKlIOBSIISC,ALPHA,ATOL,EMODlEPSMCHlRPTOL, RSTOL,BNDLW,BNDUP,CHMAX,DEL,DSC,OBS,PAR,STRACK, TYPPAR,VTOL,VTOLIMIVOIMODEL) write(lun,50) par(l), dexp(par(1 )),par(Z),dexp(par(2)), 1 PAR(3),DEXP(PAR(3)) FORMAT(2(1 PD1 8.9)) STOP END

SUBROUTINE MODEL(IDER,LtMIT,L22GRPlMINFOINOBlNPAR, 1 NRESP,IOBS,DERIV,F,OBS,PAR) IMPLICIT DOUBLE PRECISION(A-H,O-Z) DlMENSlON PAR(NPAR),F(NRESP,52),DERIV(NRESP,L22GRP,*),OBS(NRESP,*) DIMENSION RPAR(S),PRE(NRESP,52)

COMMON/para/RPAR MINFO=O RPAR(1 )=DEXP(PAR(l )) RPAR(3)=D EXP (PAR(2)) RPAR(4)=D EXP(PAR(3)) W RITE(*,*) RPAR(1 ), RPAR(3),RPAR(4)

!cal1 subroutine in which the ODES are solved. !PAR is the storing array of parameters; PRE is storing array of Predicted values; !DERIV is storing array derivatives. ! put prediction into array F.

CALL ODE(RPAR,PRE,DERIV) do 10 i=l ,nresp do 20 j=1,52

! F hotds the residuals. F(i, J)=OBS(l, J)-PRE(1, J)

!df/d(par(l ))=dfldrpar(l )*rpar(l ) D ERIV(I, JI I )=DERIV(I, JI 1 )*RPAR(I ) D ERIV(I, J ,2)=DERIV(I ,J,2)*RPAR(3) DERIV(1, J,3)=DERIV(I ,J,3)*RPAR(4)

20 CONTINUE 10 CONTINUE

RETURN END

SUBROUTINE ODE(RPAR,PRE,DER) IMPLlClT NONE INTEGER NSNAR,NPAR,NSPAR,LUN,LUNl ,IWORK,LRW ,LIW,

a INFO,I,J,K,IOUT,IDID,IPAR,leform,ki INTEGER NRESP,NU(P,NSET,NRUN,NCOUNT

!Declare state variables array Y ! Y(l )=umO(mol/l), zeroth moment of monomeric growing species; ! Y(2)=udO(mol/l), zeroth moment of dimeric growing species; ! Y(3)=ucO(mol/l), zeroth moment of complex intermediates; ! Y(4)=uO(mol/I), zeroth moment of dormant species; ! Y(5)=LA (moI/I), Lewis Acid concentration; ! Y(G)=M(mol/l), monomer concentration; ! Y(7)=uml (rnol/l), first moment of monomeric growing species; ! Y(8)=udl (mol/[), first moment of dimeric growing species; ! Y(S)=ucI (moI/l), first moment of complex intermediates; ! Y(? O)=ul (mol/I), first moment of dormant species; ! Y ( l l )=ut2(mol/l), second moment of al1 polymer species; ! Y(12)= I (molll), initiator concentration; ! Y(13)=lm (mol/l), (I+LA-), monomeric active species concentration; ! Y(I4)=ld(mol/l), (I+LA-2), dimeric active species concentration; ! Y(15)=lc(mol/l), (l'LA), intermediate complex concentration; ! Y(lG)=utl (molll), first moment of al1 polymer species; ! Y(17)=DPn, Number average degree of polymerization; ! Y(18)=DPw Weight average degree of polyherization.

DOUBLE PREClSlON Y, 1 YPRIME

DOUBLE PREClSlON t,tout,leff DOUBLE PREClSlON alO,al, ! (mol/l),initial Initiator concentration;

1 aMO,aM, ! (molIl), initial monomer concentration; 2 aLA0,aLA ! (mol/l), initial Lewis Acid concentration

DOUBLE PREClSlON RPAR DOUBLE PREClSlON PREIDER

DOUBLE PRECISION RWORK,RTOL,ATOL PARAMETER (NSNAR=18,NPAR=S,NSPAR=3,LUN=l O,NRESP=4, 1 NEXP=52,NSET=8) PARAMETER (LRW=l0000,LIW=100) DIMENSION Y(NSNAR,NSPAR+l ),YPRIME(NSNAR,NSPAR+l ), 1 RTOL(NSTVAR,NSPAR+l ),ATOL(NSTVAR,NSPAR+l ), 2 PRE(NRESP,NEXP),DER(NRESP,NEXP,NSPAR) DIMENSION INFO(l8),RWORK(LRW),LWORK(LIW) DIMENSION RPAR(NPAR),tout(nexp),IPAR(NSPAR),Ieff(4) DIMENSION a1(8),aLA(8),aM(8) DIMENSION NCOUNT(NSET) EXTERNAL fsub,Esub,Jac,Bsub COMMON/setting/al,aLA,aM COMMON/ib/alO,aLAO,aMO COMMON/time/tout

! Number of points in each experiment set. NCOUNT(1)=8 !217-5 NCOUNT(2)=8 !217-5K NCOUNT(3)=9 !217-5-2 NCOUNT(4)=7 !217-6 NCOUNT(5)=6 !217-7 NCOUNT(6)=4 !217-8 NCOUNT(7)=5 !217-9 NCOUNT(8)=5 !217-10 NRUN=O DO 20 K=1 ,NSET alO=al(K) a LAO =a LA(K) aMO=aM(K) t=O.ODO

!initial conditions of states variables DO 30 J=l ,NSTVAR Y(J, 1 )=O.do

30 CONTINUE y(l,l)=l.Od-14 !umO=O y(2,1)=1.0d-14 !udO=O y(3,1 )=l.Od-14 !ucO=O y(4,l )=al0 !uO=a10 y(5,l )=aLAO ! LA=aLAO y(6,l )=aMO !M=aMO y(7,1)=1.0d-15 !um1 =O y(8,1)=1.0d-14 !udl=O y(9,l)=l.Od-l3 !UCI =O y(10,1)=2.ODO'a10 !u1=2*alO y(? 1 ,l)=4.DO*alO !ut2=4*alO y(12,I )=al0 ![l]=alO y(13,1)=1 .d-15 ![lm]=O y(14,1)=1.d-15 ![ld]=O y(? 5.1 )=l .d-15 ![lc]=O y(l6,1)=2.DO*alO !ut1 =2'al0 y(17,1)=2.0dO !DPn=2 y(l8,1)=2.0DO !DPw=2

!identify the sensitivity parameters 1 PAR(1)=1 IPAR(2)=3 1 PAR(3)=4

!initial values for the parametric sensitivities of U1 DO 37 J=l ,NSNAR

DO 39 [=2,NSPAR+l Y(J,I)=O.OdO

39 CONTINUE 37 CONTINUE ! initialize and insert Infou) values

do 40 ki=1 , l8 info(ki)=O

40 continue ! set tolerance as scalar infor(2)=0, as array info(2)=1

info(2)=1 ! Evaluate the nonzero elements of Jocobian matrix

info(5)=1 ! Ask DDASAC to evaluate the initial yprime vector:

info(1 I )=1 ! Request sensitivity analysis

info(l2)=NSPAR ! Using diagonal E matrix:

info(l3)=-1 info(l4)=l info(l5)=1

! Using Euclidean norm(root-mean-square value) in testing enor vectors. info(l6)=1

! suppress t-differencing o f f in the initialization; Rwork(44)=O.OdO

! Provide array tolerances in according to info(2) do 45 J=t , NSTVAR DO 50 1=1, NSPAR+l RTOL(J,I)=l .d-4

50 CONTINUE 45 CONTINUE !

ATOL(l,l)=l .Od-17 !the range of urnO is O to 1 .d-12,(mol/l) ATOL(2,1)=1 .Od-17 !the range of udO is O to 1 .d-12,(mol/l) ATOL(3,l )=l .Od-8 !the range of ucO is O to 1 .d-3,(mol/l) ATOL(4,1)=1 .Od-7 !the range of uO is ATOL(S,l)=l.OD-7 !the range of LA is ATOL(G11)=1.0D-8 !the range of M is from 2 to 1 .d-3,(mol/l) ATOL(7,1)=1 .Od-16 !the range of um1 is from O to 1 .d-11 ,(mol/l) ATOL(8,1)=1 .Od-16 !the range of udl is from O to 1 .d-1 1 ,(molIl) ATOL(9,1)=l .Od-7 !the range of ucl is from O to 1 .d-2,(mol/l) ATOL(10,1)=1 .Od-8 !the range of u l is from 4.d-3 to 1 .d-1 ,(molIl) ATOL(11,1)=1.00-7 !the range of ut2 is from 1 .d-2 to 1 .d+3,(mol/l) ATOL(12,1)=1 .D-1 O ! suppose the range of [1] is 1 e-3 to 1 e-13 ATOL(l3,l)=l .D-28 ! suppose lm is 2 .e-23; ATOL(14,1)=1 .d-28 ! suppose Id is 1 .e-24; ATOL(l5,l)=l .D-19 ! suppose Ic is 1 .e-14 ATOL(i6,1)=1 .OD-6 !the range of ut1 is from 8.d-3 to 1 .dO,(mol/l) ATOL(17,1)=1 .OD-4 !the range of DPn is from 2.d0 to 1 .d+2 ATOL(18,1)=1 .OD-4 !the range of DPwis from 2.d0 to 1 .d+2

! ATOL for kO ATOL(1,2)=1 .OD-26 ATOL(2,2)=1 .OD-26 ATOL(3,2)=1 .OD-16 ATOL(4,2)=1 .OD-16

ATOL(5,2)=1 .OD-16 ATOL(6,2)=1 .OD-14 ATOL(7,2)=1 .Od-23 AT0 L(8,2)=l .Od-24 ATOL(9,2)=1 .Od-14 ATOL(10,2)=1 .Od-14 ATOL(11,2)=1.0D-12 ATOL(12,2)=1.0-24 ATOL(13,2)=1 .D-34 ATOL(14,2)=1 .d-34 ATOL(15,2)=1 .D-24 ATOL(16,2)=1 .OD-14 ATOL(17,2)=1 .OD-12 ATOL(l8,2)=1 .OD-12

! ATOL FOR k l ATOL(lr3)=1 .OD-24 ATOL(2,3)=1 .OD-27 ATOL(3,3)=1 .OD-18 ATOL(4,3)=1 .OD-17 ATOL(5,3)=1 .OD-18 ATOL(6,3)=1 .OD-13 ATOL(7,3)=1.Od-21 ATOL(8,3)=1.0d-23 ATOL(9,3)=1 .Od-14 ATOL(1OI3)=1.0d-1 3 ATOL(11,3)=1 .OD-1 O ATOL(l2,3)=l .D-28 ATOL(13,3)=1 .D-32 ATOL(14,3)=l .d-32 ATOL(15,3)=1 .D-28 ATOL(16,3)=1 .OD-13 ATOL(17,3)=1 .OD-11 ATOL(l8,3)=1 .OD-1 O

! ATOL for k-2 ATOL(1,4)=1 .OD-25 ATOL(2,4)=1 .OD-22 ATOL(3,4)=1 .OD-18 ATOL(4,4)=1 .OD-17 ATOL(5,4)=1 .OD-17 ATOL(6,4)=1 .OD-13 ATOL(7,4)=1.0d-23 ATOL(8,4)=1.0d-22 ATOL(9,4)=1 .Od-14 ATOL(10,4)=1 .Od-13 ATOL(11,4)=1 .OD-1 O ATOL(l2,4)=l .D-1 8 ATOL(13,4)=1 .D-28 ATOL(14,4)=1 .d-28 ATOL(15,4)=1 .O-1 8 ATOL(l6,4)=? .OD-13 ATOL(l7,4)=l .CID-1 1 ATOL(I8,4)=1 .OD-10

!Integration loop, cal1 the implicit integrator DDASAC !WARNING!DDASAC is changed at line 2051 to avoid the printout.

do 80 iout=1 ,NCOUNT(K) NRUN=NRUN+l

CALL DDASAC (t.tout(NRUN),NSTVAR,Y,YPRIM El RTOLIATOL, l N FO, 1 RWORK,LRW.IWORK,LIW,RPAR,IPAR,IDID,LUN,Ieform,fsub,Esub, 2 Jac,Bsu b)

! write results regardless ! of the solution status, then stop ! if lDlD is negative.

IF (1DID.LT.O) THEN WRITE(LUN,110) IDlD STOP E NDlF

! Effective [I]=[I]+[I+LA-]+[I+LA-2]+[I*M] leff(1 )=y(12,1 )+y(13,1 )+y(14,l )+y(l5,1)

!put the predictions to Pre, and derivatives to DER PRE(1 ,NRUN)=Y(6,1) PRE(2,NRUN)=Y(1T11) PRE(3,NRUN)=Y(l8,1) PRE(4,NRUN)=leff(I) DO 70 J=2,INF0(12)+1 leff(J)=y(l2, J)+y(13, J)+y(14, J)+y(l5, J) DER(1 ,NRUN,J-1 )=Y(6,J) DER(2,NRUN,J-1 )=Y(17,5) DER(3,NRUN.J-l)=Y(18,5) DER(4,NRUN.J-1 )=leff(J)

70 CONTINUE 80 CONTINUE 20 CONTINUE 1 10 FORMAT(l1 X,'lntegration Failed with fDID='15) 120 FORMAT(ll1 Xl9Number of steps taken so far=',15/

1 1 xI1Nurn ber of function calls =',15/ 2 lx,'Number of Jacobian calls =',15/ 3 1 x,'Number of error test fails =',15/ 4 1 x,'Number of convergence fails =',15/)

return ! end of the ODE subroutine.

END itt+**.ht***tir**~*-*t*i-tL*t****ir*~*******f*****tt*i*

SUBROUTINE fsub(t,NSWAR, U,fval,RPAR,IPAR,leform, Ires) !This subroutine calculates the right-hand side of ODES

IMPLICIT NONE INTEGER IPAR,leform,Ires,NSn/AR DOUBLE PREClSlON fval,

1 U DOUBLE PREClSlON t,RPAR,kl ,k2 DOUBLE PREClSlON aiO,aLAO,aMO ! experimental setting DIMENSION U(NSTVAR),fvaI(NSTVAR),RPAR(*),IPAR(*) COMMON/iblaIO.aLAO,aMO k l =0.22dO/(RPAR(l )IRPAR(2)) k2=5.2dl 'RPAR(4)l(RPAR(S)'RPAR(l )/RPAR(2))

!calculate the function vector fval: ! urnO1=-k-1 umO+kl uc0

fval(1 )=-RPAR(3)*U(l )+kl *U(3) ! udO'=k2IaucO-k-2udO

fval(2)=k2*U(5)*U (3)-RPAR(4)*U(2) ! ucO1=k-1 um0-(kl +k-0+k21a)ucO+k-2ud0+k01au0

fval(3)=RPAR(3)*U(1 )-(kl +RPAR(2)+ 1 k2*U(5))*U(3)+RPAR(4)*U(2)

fvaI(18)=U(18)-U(l1 )/U(16) RETURN END

ft*~*f**irt~******************.H**~*Xir************W*tt

SUBROUTINE Esub(t,NSlVAR, U,Ework,Rpar, Ipar,leform, Ires) !this subroutine calculates the nonzero element of E matrix

IMPLICIT NONE DOUBLE PRECISION t,U(*),Ework(*),Rpar(*) INTEGER NSn/AR,lpar,leform,lres,l do 10 I=i, lS Ework(l)=l

10 continue RETURN END

SUBROUTINE Jac(t,Nstvar,U,Pd,Rpar,lpar,leform,lres) ! The subroutine calculate the analytical Jacobian matrix

IMPLICIT NONE integer Nstvar,lpar(*),Ieform,lres double precision t,U(nstvar),Rpar(*),PD(nstvar,*)

DOUBLE PREClSlON k l ,k2 double precision alO,alAO,aMO COMMON/ib/alO,aLAO,aMO kl=O.22dOI(RPAR(l )/RPAR(2)) k2=5.2d 1 *RPAR(4)/(RPAR(5)'RPAR(l )/RPAR(2)) PD(1,l )=-RPAR(3) PD(? ,3)=kl PD(2,2)=RPAR(4) PD(2,3)=kî*U(5) PD(2,5)=E*U(3) PD(3,l )=RPAR(3) PD(3,2)=RPAR(4) PD(3,3)=-(kl +RPAR(2)+k2*U(5)) PD(3,4)=RPAR(1 )'U(5) PD(3,5)=RPAR(1 )*U(4)-k2*U(3) PD(4,3)=RPAR(2) PD (4,4)=-RPAR(1 )*U (5) PD(4,5)=-RPAR(1 )*U(4) PD(5,2)=RPAR(4) PD(S13)=RPAR(2)-k2*U(5) PD(5,4)=-RPAR(1 )*U(5) PD(5,5)=-RPAR(1 )*U(4)-k2*U(3) PD(6,l )=-RPAR(S)*U(G) PD(6,2)=RPAR(5)*U(6) PD(6,6)=-(RPAR(5)*U(l )+RPAR(S)*U(2)) PD(7,l )=RPAR(S)*U(G) PD(7,6)=RPAR(S)'U(l) PD(7,7)=-RPAR(3) PD(7,9)=kl PD(8,2)=RPAR(5)*U(6) PD(8,5)=k2*U(9) PD(8,6)=RPAR(5)'U(2) PD(8,8)=-RPAR(4) PD(8,9)=kZ*U(5) PD(9,5)=RPAR(I )*U(10)-k2*U(9) PD(9,7)=RPAR(3) PD(9,8)=RPAR(4) PD(9,9)=-(kl +RPAR(2)+k2*U(5)) PD(9,l O)=RPAR(l )*U(5) PD(10,5)=-RPAR(1 )*U(1 O) PD(l O19)=RPAR(2) PD(? O, 1 O)=-RPAR(1 )*U(5) PD(11,l )=RPAR(S)*U(G) PD(? II2)=RPAR(5)*U(6) PD(1l16)=RPAR(5)*(U(1 )+U(2)+2.DO*(U(7)+U(8))) PD(11,7)=2.DO*RPAR(5)'U(6) PD(11,8)=2.DO*RPAR(5)'U(6) PD(12,5)=-RPAR(1 )*U(12) PD(12,12)=-RPAR(1 )*U(5) PD(12,'i 5)=RPAR(2) PD(13,6)=-RPAR(5)*U(13) PD(l3,13)=(RPAR(3)+RPAR(5)*U(6)) PD(13,15)=kl PD(14,5)=k2*U(15) PD(14,6)=-RPAR(5)*U(14) PD(14,14)=-(RPAR(4)+RPAR(5)*U(6))

PD(l4,15)=iQ*U(5) PD(15,5)=RPAR(l )*U(12)-W'U(i.5) PD(15,12)=RPAR(I )*U(5) PD(l5,13)=RPAR(3) PD(? 5,14)=RPAR(4) PD(15,15)=-(RPAR(Z)+kl +kî*U(5)) PD(16,6)=1 PD(l6,l6)=l PD(17,16)=-lia10 PD(l7,17)=1 PD(18,11)=-llU(16) PD(l8,16)=U(l i )/(U(l6)*U(l6)) PD(i8,18)=1 Ires=l RETURN END

SUBROUTINE Bsub(t,NSTVAR,U,dpj,JSPAR,RPAR,IPAR,leform,lres) ! Sensitivity coefficients calculation impIicit none integer NSTVAR,JSPAR,IPAR,leform,lres,i double precision t,U,dpj,RPAR double precision dk1 ,dk2,dk3 dimension U(l8),dpj(18) DIMENSION RPAR(5)

! sensitivity coeff. wrt. kO if (jspar. eq. 'l ) then d kl =-0.22dO*rpar(2)/(rpar(l )*rpar(l )) d k2=-5.2d1 *rpat(4)*rpar(2)/(rpar(5)*rpar(l )*rpar(l )) dpj(1 )=d k l *U(3) dpj(2)=dk2*U(5)*U(3) dpj(3)=U(5)*U(4)-(dkl +dk2*U(5))*U(3) dpj(4)=-U(5)*U(4) dpj(5)=-U(5)'U(4)-d k2*U(5)*U(3) dpj(7)=dkl *U(9) dpj(8)=d k2'U(5)'U(9) dpj(9)=-(dkl +dk2*U(5))*U(9)+U(5)*U(lO) dpj(1 O)=-U(S)*U(l O) dpj(l2)=-U(5)*U(12) dpj(13)=dkl *U(15) dpj(l4)=dk2*U(5)*U(15) dpj(l5)=U(S)*U(12)-(dkl +dk2*U(5))*U(15) else if (jspar.eq.2) then ! sensitivity coefficients wrt. k-1 do 10 i=1,18 dpj(i)=O.OdO

10 continue dpj(1 )=-U(l) dpj(3)=U(I dpj(7)=-U(7) dpj(9)=U(7) dpj(13)=-U(13) dpj(l5)=U(13) else if (jspar.eq.3) then ! sensitivity coefficients wrt. k-2 do20 i=1,18

dpj(i)=O.dO 20 continue

d k3=(5.2d 1 *rpar(2))/(rpar(5)*rp ar(1)) dpj(2)=-U(2)+dk3*~(5)*~(3) dpj(3)=U(2)-dk3*~(5)*~(3) dpj(S)=U(2)-d k3*u(5)*u(3) dpj(8)=-U(8)+dk3*u(s)"u(9) dpj(9)=U(8)-d k3*u(5)*u(9) dpj(l4)=-U(l4)+d k3*u(5)*u(l5) dpj(l5)=U(l4)-dk3*~(5)*~(15) end if Ires=l RETURN END

VlTA

Name: Qian Liu

Place and Year of Birth: Dalian, PRC, 1973

Education: B. Sc. In Chernical Engineering,

Dalian University of Technology, Dalian, P.R.China

1990-1 995

MSc. in Chernical Engineering,

Queen's University, Kingston, Ontario

1998-

Teaching Assistant,

Queen's University, Kingston, Ontario,

1998,1999

Project Engineer,

Dalian Goodyear Tire Co., Dalian, P.R.China,

1995-1 998

Experience: