Kinetic simulation of living carbocationic polymerizations. II. Simulation of living isobutylene polymerization using a mechanistic model

EUROPEAN

European Polymer Journal 41 (2005) 1–14

www.elsevier.com/locate/europolj

POLYMERJOURNAL

Kinetic simulation of living carbocationic polymerizations.II. Simulation of living isobutylene polymerization

using a mechanistic model

Judit E. Puskas a,*, Sohel Shaikh b, Kevin Z. Yao c,Kim B. McAuley c, Gabor Kaszas d

a Macromolecular Engineering Research, Department of Polymer Science, The University of Akron, Akron,

OH 44325-3909, USAb Department of Chemical and Biochemical Engineering, The University of Western Ontario, London, ON, Canada N6A

c Department of Chemical Engineering, Queen’s University, Kingston, ON, Canada K7L 3N6d Lanxess Inc., 1265 Vidal St. South, Sarnia, ON, Canada N7T 7M2

Received 14 July 2004; received in revised form 30 July 2004; accepted 2 August 2004

Available online 25 September 2004

Abstract

This paper discusses the kinetic simulation of TiCl4––coinitiated living carbocationic isobutylene (IB) polymeriza-

tions governed by dormant-active equilibria, using a mechanistic model. Two kinetic models were constructed from

the same underlying mechanism: one using a commercial simulation software package (Predici�), and the other using

the method of moments. Parameter estimation from experimental batch reactor data with Predici yielded a rate con-

stant of propagation kp = 4.64 · 108 ± 2.75 · 108L/mols, with no constraints imposed. This agrees with kp data meas-

ured with diffusion clock and competition methods, but disagrees with kinetically obtained kp values. Estimation of rate

constants with Predici� and the GREG parameter estimation software packages revealed that it was difficult to estimate

the complete set of kinetic parameters, due to correlated effects of the parameters on model predictions. Estimability

analysis confirmed that some of the strongly correlating parameters could not be estimated simultaneously using the

available experimental data. Using kp = 6 · 108 ± 2.75 · 108L/mols measured by Mayr, and using starting estimatesof other rate constants defined by experimentally observed correlations, yielded the set of rate constants required for

the simulations. Both kinetic models yielded good agreement with experimental data, with the exception of Mw values

that slightly diverged from the theoretically predicted �Mw�Mn = constant� relationship. This may indicate the occur-rence of a minor side reaction. However, the kp/k�1 = 17.5L/mol average run length calculated from measured and sim-

ulated MWD data agrees well with earlier literature values.

� 2004 Published by Elsevier Ltd.

Keywords: Living carbocationic polymerization; Isobutylene; Modeling; Kinetics; Mechanism; Run length

0014-3057/$ - see front matter � 2004 Published by Elsevier Ltd.

doi:10.1016/j.eurpolymj.2004.08.006

* Corresponding author. Tel.: +1 330 972 6203; fax: +1 330 972 5290.

E-mail address: [email protected] (J.E. Puskas).

mailto:[email protected]

Nomenclature

I initiator, 2-chloro-2,4,4-trimethyl-pentane

(TMPCl)

IB isobutylene

LA Lewis acid (TiCl4––titanium tetrachloride)

M monomer, isobutylene (IB)

PIB polyisobutylene

MeCl methyl chloride

MeCHx methylcyclohexane

Hx hexane

DtBP di-tert-butylpyridine, proton trap

ED electron pair donor

[ ] concentrations of individual species, mol/L

[ ]0 initial concentrations of individual species,

mol/L

MW molecular weight

Mn number average molecular weight

Mw weight average molecular weight

DPn number average chain length

DPw weight average chain length

MWD Mw=Mn, molecular weight distribution

I * LA initiator/Lewis acid intermediate

I+LA� active initiator with monomeric gegenion

IþLA�2 active initiator with dimeric gegenion

PIB-Cl dormant tertiary chloride capped polyiso-

butylene chain

Pn * LA polymer/Lewis acid intermediate with chain

of n, n P 3

Pn-Cl dormant, tert-chloride capped polymer

chain of length n, where n P 3

Pþn LA

� active growing chain of length n, with mono-

meric Lewis acid gegenions n P 3

Pþn LA

�2 active growing chain of length n, with dimer

Lewis acid gegenions n P 3

kp propagation rate constant of polymeriza-

tion, L/mols

k00p overall propagation rate constant of polym-

erization, L3/mol3 s

k0 rate constant of intermediate formation, L/

mols

k�0 rate constant of intermediate dissociation,

s�1

K0 k0/k�0, equilibrium constant of intermediate

formation, L/mol

k1 ionization rate constant of form active spe-

cies with monomeric gegenion, s�1

k�1 deactivation rate constant of active species

with monomeric gegenion, s�1

K1 k1/k�1, equilibrium constant of the forma-

tion of active species with monomeric

gegenion

k2 ionization rate constant of form active spe-

cies with dimeric gegenion, L/mols

k�2 deactivation rate constant of active species

with dimeric gegenion, s�1

K2 k2/k�2, equilibrium constant of the forma-

tion of active species with dimeric gegenion,

L/mol

u0 zeroth moment of dormant polymer chains

uc0 zeroth moment of intermediate complex

ud0 zeroth moment of polymer chains with

dimeric counterions

um0 zeroth moment of polymer chains with

monomeric counterions

ut0 total zeroth moment (u0 + uc0 + ud0 + um0)

u1 first moment of dormant polymer chains

uc1 first moment of intermediate complex

ud1 first moment of polymer chains with dimeric

counterions

um1 first moment of polymer chains with mono-

meric counterions

ut1 total first moment (u1 + uc1 + ud1 + um1)

u2 second moment of dormant polymer chains

uc2 second moment of intermediate complex

ud2 second moment of polymer chains with

dimeric counterions

um2 second moment of polymer chains with

monomeric counterions

ut2 total second moment (u2 + uc2 + ud2 + um2)

2 J.E. Puskas et al. / European Polymer Journal 41 (2005) 1–14

1. Introduction

Living polymerization is characterized by the ab-

sence of significant chain transfer or irreversible chain

termination reactions. Living conditions were first

achieved in anionic polymerization [1]. Essentially liv-

ing conditions in carbocationic systems had been

achieved experimentally as early as 1974 [2]. Systematic

research in Kennedy�s group in the 1980s led to the suc-cessful production of high molecular weight and nearly

uniform polymers with living/controlled carbocationic

polymerization [3]. By the mid-1990s, living/controlled

radical polymerization was also a reality [4]. Living

polymerization provides control over the molecular

weight (MW) and the architecture of the polymer

produced, and therefore is a preferred method of

macromolecular engineering [5–7]. In both living carbo-

cationic and radical polymerizations there is a reversi-

ble equilibrium between dormant and active chain

ends [7–10].

I + LA

I*LA

I+LA- I+LA2-

Pn+LA- Pn

+LA2-

Pn*LA

Pn + LA

k0k-0

k1

k-1

k-2

k2+ LA

kp kp+ M+ M

k-1k1

k-0k0

kp

+ Mkp

+ M

k2k-2

+ LA

Path A Path B

Scheme 1. Comprehensive mechanism proposed for living IB

polymerization.

I + LA I*Pn + LAPn

*k1

k-1

k-1

k1

kp

+ M

+ M

kp

K1,eq=k1/k-1

Scheme 2. Simplified model of living IB polymerization.

J.E. Puskas et al. / European Polymer Journal 41 (2005) 1–14 3

Pn P�n

In the living carbocationic polymerization of isobutylene

(IB), it has been demonstrated that the active initiating

and propagating centers are paired ions [9,11].

PIB-Clþ LA PIB�==�LA

where PIB-Cl is the dormant PIB chain capped with a

tertiary chloride group, and LA denotes the Lewis acid

coinitiator. The polymerization rate of IB is first-order

in monomer, but under certain conditions apparent zero

order has also been reported [12,13]. The rate is first-

order in initiator for a variety of initiating systems

[14–16]. This implies that the propagation is a simple

bimolecular reaction between active chains and mono-

mer. However, the polymerization rate has shown both

first-order and second-order dependence on Lewis acid

(TiCl4) [15–26]. This close to second-order behavior

was attributed to the fact that TiCl4 may form bimolec-

ular complex gegenions [25,26]. It was also suggested

that the propagation involves both monomeric and

dimeric counterions [27]. The rate of polymerization

decreases with increasing temperature. This phenome-

non manifests itself as an ‘‘apparent negative activation

energy’’ [14,22,28,29]. These facts indicate that the

polymerization rate is a complicated function of all

the factors during a polymerization. When interpreting

the influences of these factors on the polymerization

rate, the equilibrium between active and dormant chains

plays a key role. The shift of the equilibrium is governed

by many factors, including concentrations of initiator,

Lewis acid, monomer, additives, such as proton trap

and electron pair donors (EDs), as well as temperature

and solvent polarity [14–26].

The comprehensive mechanistic model, shown in

Scheme 1, was conceived by Puskas� group [15,25,26]

for 2-chloro-2,4,4-trimethylpentane (TMPCl)/TiCl4 ini-

tiated living IB polymerization. In this model system,

the structure of the TMPCl initiator mimics that of the

growing polymer chains. It has been demonstrated that

if ki � kp is assumed for this system, it greatly simplifies

the kinetic derivations [17]. Subsequently this assump-

tion was verified experimentally; kp = 6 ± 4 · 108L/molswas measured by the diffusion clock method [30] and

with kp/k�1 = 16.5, kp � 3ki [17,23]. In previous papers,

a Predici� simulation of a simplified version of the ki-

netic model, as shown in Scheme 2, was described

[25,26]. The rate constants that were used in the simple

model were combinations of the true kinetic constants

in the proposed comprehensive model. The current pa-

per is particularly aimed at deconvoluting appropriate

values for the true constants, regardless of the different

experimental conditions (e.g., different initiator and

Lewis acid concentrations), and using them in the com-

prehensive mechanistic model to simulate living IB

polymerization.

2. Kinetic background

2.1. The proposed scheme

The proposed comprehensive mechanism [15] in-

volves a series of consecutive and competitive reactions.

Firstly, the initiator and Lewis acid form an intermedi-

ate, I * LA, through an equilibrium involving rate con-

stants k0 and k�0. This intermediate species can

undergo two reactions as shown by the Paths A and

B. In Path A, this intermediate species becomes a mono-

meric active initiator through an equilibrium involving

rate constants k1 and k�1. The monomeric active initia-

tor can polymerize monomers, leading to growth of the

active chain. An active growing chain of any length, n,

can transform back to a polymer/Lewis acid intermedi-

ate through an equilibrium involving k�1 and k1. In Path

B, the initiator/Lewis acid intermediate can incorporate

an additional Lewis acid to form a dimeric active initia-

tor through an equilibrium involving rate constants k2and k�2. The dimeric active initiator can then propagate


by adding monomer. It has been shown that the dimeric

gegenion must form by a two-step process [11,24,31].

Halogenated titanium compounds are known to form

neutral dimers in a variety of crystal structures and

also under cryogenic conditions, but with a very lim-

ited stability range [32,33]. The existence of neutral

dimeric Ti2Cl8 under dilute polymerization conditions

([TiCl4]0 � 10�2–10�1mol/L) has never been demon-

strated experimentally. In Scheme 1, the dimer forma-

tion follows a two-step course as suggested earlier

[11,24,31]. In the first step, the initiator/Lewis acid or

polymer/Lewis acid intermediate dissociate forming

monomeric counteranions, which in turn can react with

additional Lewis acid to form dimeric counteranions.

The species with dimeric counteranion can propagate

with monomer, or can also release Lewis acid and trans-

form back to an initiator/Lewis acid or polymer/Lewis

acid intermediate by an equilibrium reaction involving

k�2 and k2. The polymer/Lewis acid intermediate,

Pn * LA, formed from both Paths A and B can undergo

equilibrium reactions to produce dormant chains, Pn,

and free Lewis acid.

The mechanism in Scheme 1 accounts for a number

of important experimental observations: the depend-

ence of polymerization rate on Lewis acid can be either

first or second order, or in between, depending on the

actual Lewis acid concentration and other experimental

conditions [17–26]; the true propagation rate constant

kp, can be as high as 109L/mols as determined by dif-

fusion clock methods [30,34–36] while the kinetically-

determined rate constant (a combination of K0K1kpand K0K2kp in Scheme 1) can be as low as 104L/mols.

Table 1 lists the high and low values of kp obtained by

various research groups using different experimental

techniques. The inconsistency in reported kp values

has been discussed by Plesch [41,43] and still remains

unresolved [44]. While the mechanistic model shown

in Scheme 1 provides a plausible solution to this appar-

ent discrepancy, as yet there has been no direct experi-

mental confirmation of the existence and the chemical

Table 1

kp values in carbocationic IB polymerizations obtained by various m

kp (L/mols) Initiating system Solvent

6 · 103 AlBr3/TiCl4 Heptane

7.9 · 105 Light/VCl4 In bulk

1.2 · 104 Et2AlCl/Cl2 CH3Cl

9.1 · 103 Ionizing radiation CH2Cl21.5 · 108 Ionizing radiation In bulk

6 · 108 R-Cl/TiCl4 CH2Cl27 · 108 IB n-mer/TiCl4 Hexanes/CH

4.7 · 108 TMP-Cl/TiCl4 Hexanes/CH

1.7 · 109 TMP-Cl/BCl3 Hexanes/CH

a Diffusion clock/competition experiments.

nature of the proposed intermediates, I * LA and

Pn * LA. Several investigators have suggested the exist-

ence of various intermediates and suggested pathways

for their formation. For example, the formation of

polarized (stretched or activated, more-covalent-than-

ionic) dipole intermediates in the reaction of TMPCl

or PIB-Cl with TiCl4 (Winstein spectrum) was sug-

gested by various researchers [17,45]. Plesch suggested

the formation of a monomer-solvated carbocation

intermediate [43] and in this case high monomer con-

centration would result in first-order propagation with

the intermediate. Sigwalt argued that this suggestion

was not convincing since low kp values were also ob-

served at low monomer concentrations [46,47], and in

turn he suggested a two-step propagation with the

formation of solvated carbocationic intermediates

[48]. With this, the apparent second-order rate con-

stant obtained in kinetic experiments would be

kp,app = KSM * kp where KSM is the constant for the

equilibrium between solvated carbocation and mono-

mer-complexed solvated carbocation, this latter pro-

pagating by unimolecular rearrangement. Scheme 1

proposes the formation of polymerization-active carbo-

cations via intermediates involving the initiator and/or

the dormant polymer chain and the Lewis acid. It

may be reasonable to assume that these intermediates

are polarized species. Following the formation of

Pþn ==

�LA or Pþ

n ==�LA2 propagation could proceed

via a collision of these species with monomer as shown

in Scheme 1, or via a two-step reaction as suggested by

Sigwalt: first by monomer solvation, followed by prop-

agation via rearrangement. In this latter case the kp in

Scheme 1 would also represent a composite rate con-

stant involving the monomer solvation equilibrium

constant and the true rate constant of propagation

(KSM * kp). Scheme 1 and Sigwalt�s model show similar-

ities in terms of reasoning that the discrepancy between

kinetically obtained rate constants and those measured

by the diffusion clock method could be due to various

pre-equilibria.

ethods

T/�C Reference

�14 [37]

�20 [38]

�48 [39]

�78 [40,41]

�78 [42]

�78 [30]a

3Cl �80 [34]a

3Cl �80 to �40 [35]a

3Cl �80 to �40 [36]a

Table 2

Reactions

I + LAM I * LA k0, k�0I * LAM I+LA� k1, k�1I � LAþ LA$ IþLA�

2 k2, k�2IþLA� þM! Pþ

n LA� ki � kp

IþLA�2 þM! Pþ

n LA�2 ki � kp

Pþn LA� þM! Pþ

nþ1LA� kp

Pþn LA�2 þM! Pþ

nþ1LA�2 kp

Pþn LA� ! Pn � LA k�1

Pn � LA! Pþn LA

� k1Pþn LA

�2 ! Pn � LAþ LA k�2

Pn � LAþ LA! Pþn LA

�2 k2

Pn * LA! Pn + LA k�0Pn + LA! Pn * LA k0


As mentioned earlier, this paper discusses the decon-

volution of kinetically obtained rate constants using the

mechanistic model of Scheme 1.

2.2. Kinetic modeling

Based on Scheme 1, a kinetic model was developed

using Predici�, an open-ended integrated polymerization

software that is based on the discrete Galerkin h-p

Method [49]. It allows unrestricted input of reaction rate

constants, different types of polymer species, monomers

and reactants, reaction steps and polymerization condi-

tions such as temperature and pressure [50]. During a

typical dynamic simulation, the formation or consump-

tion of any of the individual species can be followed

either with respect to time or conversion. It also has

parameter estimation capabilities; based on a given

kinetic model and sufficient experimental data, rate con-

stants can be estimated. Within the Predici environment,

the series of reactions shown in Table 2, was written.

Independently, a set of ordinary differential equa-

tions (ODEs) were derived as shown in Table 3 [51].

The model equations are composed of a set of dynamic

Table 3

Material balance equations for the comprehensive model

Low molecular weight species:d½I�dt

¼ �k0½I�½LA� þ k�0½I � LA�

d½I � LA�dt

¼ k0½I�½LA� � k2½LA�½I � LA� � ðk�0 þ k1Þ½I � LA� þ k�1½IþL

d½IþLA��dt

¼ k1½I � LA� � ðk�1 þ kp½M�Þ½IþLA��

d½IþLA�2 �

dt¼ k2½I � LA�½LA� � ðk�2 þ kp½M�Þ½IþLA�

2 �

d½M�dt

¼ �kp½M�ð½IþLA�� þX1

i¼3½Pþi LA

�� þ ½IþLA�2 � þ

X1i¼3½P

þi LA

�2

d½LA�dt

¼ k�0ð½I � LA� þX1

i¼3½Pi � LA�Þ � k0½LA�ð½I� þX1

i¼3PiÞ � k2½þP1

i¼3½Pi � LA�Þ þ k�2ð½Iþ LA�2 � þ

P1i¼3½Pþ

i LA�2 �Þ

High molecular weight species:

d½Pþ3 LA

��dt

¼ kp½M�ð½IþLA�� ½Pþ3 LA��Þ � k�1½Pþ

3 LA�� þ k1½P3 � LA�

d½Pþi LA

��dt

¼ kpð½Pþi�1LA

�� ½Pþi LA

��Þ � k�1½Pþi LA

�� þ k1½Pi � LA� i

d½Pþ3 LA

�2 �

dt¼ kp½M�ð½IþLA�

2 � � ½Pþ3 LA�2 �Þ � k�2½Pþ

3 LA�2 � þ k2½P3 � LA�

d½Pþi LA

�2 �

dt¼ kp½M�ð½Pþi�1LA

�2 � � ½Pþ

i LA�2 �Þ � k�2½Pþi LA

�2 � þ k2½P3 � LA

d½P3 � LA�dt

¼ k�1½Pþ3 LA

�� ðk�0 þ k1 þ k2½LA�Þ½P3 � LA� þ k�2½Pþ3 LA

d½Pi � LA�dt

¼ k�1½Pþi LA�� ðk�0 þ k1 þ k2½LA�Þ½Pi � LA� þ k�2½Pþi LA

d½P3�dt

¼ k�0½P3 � LA� � k0½P3�½LA�

d½Pi�dt

¼ k�0½Pi � LA� � k0½Pi�½LA� i ¼ 4 � � �1

material balance equations for each individual species.

Because we have used TMPCl, essentially a dimer PIB,

as the initiator in the experiments, we refer to the first

polymeric species formed by propagation as P3 (Table

3: Eqs. (g), (i) and (m)).

The method of moments was used to determine the

weight and number average molecular weights of the

(a)

A�� þ k�2½IþLA�2 � (b)

(c)

(d)

�Þ (e)

LA�ð½I � LA�(f)

(g)

¼ 4 � � �1 (h)

½LA� (i)

�½LA� i ¼ 4 � � �1 (j)

�2 � þ k0½P3�½LA� (k)

�2 � þ k0½Pi�½LA� i ¼ 4 � � �1 (l)

(m)

(n)

Table 4

Zeroth, first and second moments of polymer species

Intermediate complexes Polymer chains with monomeric counterions

Zeroth moment uc0 ¼ ½I � LA� þP1

i¼3½Pi � LA� um0 ¼ ½IþLA�� þP1

i¼3½Pþi LA

��First moment uc1 ¼ 2½I � LA� þ

P1i¼3i½Pi � LA� um1 ¼ 2½IþLA�� þ

P1i¼3i½Pþi LA

��Second moment uc2 ¼ 22½I � LA� þ

P1i¼3i

2½Pi � LA� um2 ¼ 22½IþLA�� þP1

i¼3i2½Pþ

i LA��

Polymer chains with dimeric counterions Dormant polymer chains

Zeroth moment ud0 ¼ ½IþLA�2 � þ

P1i¼3½Pþ

i LA�2 � u0 ¼ ½I� þ

P1i¼3½Pi�

First moment ud1 ¼ 2½IþLA�2 � þ

P1i¼3i½PþLA�

2 � u1 ¼ 2½I� þP1

i¼3i½Pi�Second moment ud2 ¼ 22½IþLA�

2 � þP1

i¼3i2½Pþ

i LA�2 � u2 ¼ 22½I� þ

P1i¼3i

2½Pi�

Table 5

Moment balance equations

dum0

dt¼ �k�1um0 þ k1uc0 (a)

dud0dt

¼ k2½LA�uc0 � k�2ud0 (b)

duc0dt

¼ k�1um0 � ðk1 þ k�0 þ k2½LA�Þuc0þk�2ud0 þ k0½LA�u0

(c)

du0dt

¼ k�0uc0 � k0½LA�u0 (d)

dum1

dt¼ kpMum0 � k�1um1 þ k1uc1 (e)

dud1dt

¼ kp½M�ud0 � k�2ud1 þ k2½LA�uc1 (f)

duc1dt

¼ k�1um1 � ðk1 þ k�0 þ k2½LA�Þuc1þk�2ud1 þ k0½LA�u1

(g)

du1dt

¼ �k0½LA�u1 þ k�0uc1 (h)

d½M�dt

¼ �kp½M�ðum0 þ ud0Þ (i)

d½LA�dt

¼ k�0uc0 � k0½LA�u0 � k2½LA�uc0 þ k�2ud0 (j)


polymeric species. These moments are defined in Table 4

and the corresponding moment differential equations are

given in Table 5 [51].

From the moment definitions, the total of these zer-

oth moments (ut0 = u0 + uc0 + um0 + ud0) is the total

concentration of polymer chains and other initiator-

derived species present, which equals the initial concen-

tration of initiator. The total of the first moments

(ut1 = u1 + uc1 + um1 + ud1) is the total number of moles

of monomer units in the polymer chains and initiating

species, which equals the moles of monomer units ini-

tially in the initiator, plus the monomer consumed:

ut0 ¼ ½I�0 ð1Þ

ut1 ¼ 2½I�0 þ ½M�0 � ½M� ð2Þ

Notice that balances on the second moments of chain

length distribution are not included in Table 3 as it is

unnecessary to calculate second moments for the various

types of polymeric species. Instead, the total of all of the

second moments can be determined from zeroth and first

moments:

dut2dt

¼ kp½M�ð2um1 þ um0 þ 2ud1 þ ud0Þ ð3Þ

The number average and weight average degree of

polymerization can be expressed by three leading mo-

ments. As the polymer chains are a combination of four

types of polymer species, sums of the individual mo-

ments are required for the calculation of degree of

polymerization:

DPn ¼ut1ut0

¼ um1 þ ud1 þ uc1 þ u1um0 þ ud0 þ uc0 þ u0

¼ 2½I�0 þ ½M�0 � ½M�½I�0

ð4Þ

DPw ¼ ut2ut1

¼ ut2um1 þ ud1 þ uc1 þ u1

ð5Þ

The material balance equations, moment equations, and

equations for degree of polymerization required to sim-

ulate the polymerization process are in Table 3 (Eqs.

(a)–(d)) Table 5 (Eqs. (a)–(i)) and Eqs. (1)–(5). The dy-

namic model equations were solved using DDASAC

[52].

2.3. Determination of initial estimates of rate constants

2.3.1. Experimental kinetic measurements

Based on Scheme 1, selected rate constants can be de-

rived experimentally from initiator and monomer con-

sumption data, to be used as initial estimates for the

parameter estimation. Assuming that experimental

conditions have been selected that the polymerization

proceeds either predominantly by Path A or predomi-

nantly by Path B, simplified expressions were derived

from Scheme 1 to describe the time-varying behavior

of the initiator [I] and monomer [M] concentrations:

the detailed derivations are given in Appendix A (Path

A) and Appendix B (Path B).


Path A

ln½I�0½I� ¼ K0k1½LA�0t ð6Þ

ln½M�0½M� ¼ kpK0K1½I�0½LA�0t ð7Þ

Employing Eqs. (6) and (7), K0k1 = 0.22L/mol s and

K0K1kp = 3.4L2/mol2 s were calculated from previously

published experimental initiator and monomer con-

sumption data in Path A [15].

Path B

ln½M�0½M� ¼ K0K2kp½I�0½LA�

2

0t ð8Þ

In Path B, that is, under conditions normally used in liv-

ing IB polymerizations, initiation is nearly instantaneous

and initiator consumption cannot be followed experi-

mentally. After a very fast initiation period, [Pn] = [I]0.

In classical living polymerization this simplification is

routinely used in formulating rate equations for mono-

mer consumption. Also, in case of instantaneous ini-

tiation the monomer consumed during initiation is

neglected. In contrast, in Path A, initiation and propa-

gation was found to proceed simultaneously, so these

simplifications cannot be considered. In Path B, experi-

mental data have revealed a fractional order of 1.76 in

TiCl4 [15], which is in good agreement with the range

of 1.7–2.2 reported by other researchers [11,16,19,24].

ln½M�0½M� ¼ k00p½I�0½LA�

1:76 ð9Þ

Here k00p is the experimentally measured apparent rate

constant. This indicates that in reality, both Paths A

and B proceed simultaneously, with Path B dominating.

Using Eq. (9), constant k00p ¼ 52L3/mol3 s was obtained

from experimental monomer consumption data [15].

With the interpretation in Eq. (8), K0K2kp = 52L3/

mol3 s. Eqs. (6)–(8) were derived with several assump-

tions and should be viewed as simplified rate expres-

sions, but they provide means to establish initial

estimates for the simulations.

Selected rate constants in IB polymerizations have

been measured or derived independently. For instance,

‘‘diffusion clock’’ and competition experiments methods

yielded kp � 108L/mols [30,35,36]. As discussed earlier,

this high value is contradicting earlier data, presented

in Table 1, with the exception of kp obtained from irra-

diation-initiated IB polymerization. Industrial experi-

ence supports the close to diffusion-limited high value

[53], therefore we accepted this value. Later it will be

shown that parameter estimation also supports a high

kp value. Published capping rate constants, although de-

rived from differing interpretations, were in the range of

3–5 · 107s�1 (Puskas and Peng [26]: k�1 = 3.9 · 107 s�1

(TMPCl); Kim and Faust [55]: k�1 = 5.0 · 107s�1

(TMPCl) and k�1 = 3.4 · 107 s�1 (PIB-36mer). Using

kp = 6 · 108L/mols we get K0K1 = 5.7 · 10�9L/mol andK0K2 = 8.7 · 10�8L2/mol2, which confirms that both

equilibria are dramatically shifted toward the dormant

polymer chains Pn-Cl. From K0k1 = 0.22L/mols and

K0K1 = 5.7 · 10�9L/mol we get k�1 = 3.9 · 107 s�1. Thisagrees very well with published capping rate coefficients,

thus we assumed this value for k�2 and determined

K0k2 = (k0/k�0) Æ k2 = 3.38L2/mol2 s.

As a result, the following correlations have been

established:

C1 : k1K0 ¼ k1 � ðk0=k�0Þ ¼ 0:22L=mols ð10Þ

C2 : k2K0 ¼ k2 � ðk0=k�0Þ ¼ 3:38L2=mol2 s ð11Þ

The correlation between the parameters will influence

the estimability of individual parameters.

2.4. PREDICI parameter estimation

For the PREDICI parameter estimation, the reaction

scheme in Table 2 was used. Table 6 lists initial rate con-

stant values and sets of estimated parameters. The initial

values for k0, k�0, k1 and k2 were established from the

correlations given in (10) and (11) by trial and error, fit-

ting experimental data [54]. The parameter estimation

routine in PREDICI is based on the damped Gauss–

Newton method [56,57] and has the ability to converge

well even with bad starting values. Seven experimental

data sets were used in the simulation [15]. First, all

parameters were estimated simultaneously. The simula-

tion did converge with kp = 6 * 108L/mols as a starting

estimate without any constraints imposed, although with

high error within the 90% confidence intervals; the data

are listed as Set 1 in Table 6. It is interesting to note that

kp converged to a value of 4.64 * 108 ± 2.75 * 10

8L/mol s.

In contrast, the simulation was unable to converge with

kp = 104L/mols as a starting estimate. This further sup-

ports the high kp values obtained in diffusion clock and

competition experiments [30,35,36]. The values for k�1and k�2 converged to 1.77 and 2.75 * 10

7 s�1, close to

the initial values. However, k0, k�0, k1 and k2 carried very

large errors and did not satisfy the correlations estab-

lished in (10) and (11). Next we fixed kp = 6 * 108L/mols

and k�1 = k�2 = 3.9 * 107 s�1, and estimated k0, k1 and

k2. Set 2 shows that the error is still high. In the next step

we enforced the correlations defined in Eqs. (10) and (11)

using interpreter functions in Predici [57]. Set 3 shows

that k0 and k�0 still carry very high error. As was deter-

mined from the estimability analysis, these parameters

were correlated and hence could not be estimated simul-

taneously. Therefore in Set 4, we fixed the value of k�0 to

3.9 * 107s�1. Set 4 shows the estimated k0 with low error

within the 90% confidence interval. The parameters in

Set 4 were used for the simulations with Predici. It should

Table 6

Initial parameter values and parameter estimates used in simulations

Parameter Initial values Set 1 ± CIa (estimating all) Set 2 ± CI (estimating k0, k1, k2)

1(a): Estimates without constraints

k0 (L/mols) 1.56 * 108 2.75 * 10

8 ± 2.79 * 108 7.96 * 107 ± 3.97 * 10

7

k�0 (s�1) 3.90 * 10

7 1.13 * 108 ± 7.62 * 10

8 3.90 * 107

k1 (s�1) 5.50 * 10

�2 1.20 * 10�2 ± 1.23 * 10

�2 1.72 * 10�2 ± 1.23 * 10�2

k�1 (s�1) 3.90 * 10

7 1.71 * 107 ± 1.77 * 107 3.90 * 10

7

k2 (L/mols) 8.45 * 10�1 5.88 ± 3.93 3.50 ± 1.47

k�2 (s�1) 3.90 * 10

7 5.90 * 107 ± 2.75 * 107 3.90 * 10

7

kp (L/mols) 6.00 * 108 4.64 * 10

8 ± 2.75 * 108 6.00 * 108

Set 3 ± CI (estimating k0, k�0) Set 4 ± CI (estimating k0)

1(b): Estimates with constraints

k0 (L/mols) 1.72 * 108 ± 3.84 * 10

8 1.51 * 108 ± 2.03 * 10

7

k�0 (s�1) 4.48 * 10

7 ± 1.16 * 108 3.90 * 10

7

k1b (s�1) 5.73 * 10

�2 5.68 * 10�2

k�1 (s�1) 3.90 * 10

7 3.90 * 107

k2b (L/mols) 8.80 * 10

�1 8.73 * 10�1

k�2 (s�1) 3.90 * 10

7 3.90 * 107

kp (L/mols) 6.00 * 108 6.00 * 10

8

a CI = 90% confidence interval.b The values of k1 and k2 are dictated by correlations defined in Eqs. (10) and (11).

0 200 400 600 8000.00

0.01

0.02

0.03

0.04

0.05 Predici ODE

[TM

PCl],

mol

/L

time (s)

Fig. 1. Comparison of experimental and simulated initiator

consumption plots for Path A: [IB]0 = 2.0mol/L, [TMPCl]0 =

0.05mol/L, [DtBP]0 = 0.007mol/L. Hx/MeCl=60/40 v/v; Exper-

imental: [TiCl4] = 0.05mol/L (m), [TiCl4] = 0.025mol/L (s);

Temperature: �80�C; Simulation parameters: Set 4 (Table 1).


be noted here that the parameters estimated for k0, k�0,k1 and k2 in all four sets gave reasonable agreement with

experimental data. However, the model is very sensitive

to kp/k�1. Since kp and k�1 are highly correlated we

had good agreement between simulation and experimen-

tal data with either low or high kp data as long as kp/k�1was kept constant at 16.5L/mol. We do not intend to

argue that agreement between experimental and simu-

lated data alone can prove any proposed mechanisms.

Kinetic simulation can only disprove a proposed mecha-

nism if it shows serious discrepancy between experimen-

tal and simulated plots. In our particular case, if we

accept independently measured capping constants in

the range of 107s�1, good agreement can only be found

for high propagation rate constant (kp � 108L/mols).

2.5. Parameter estimability analysis using the ODE model

Attempts were also made to estimate the kinetic

parameters using the same batch reactor data [15] the

ODE version of the model in Tables 3–5 and the GREG

parameter estimation software [52]. Before using GREG

to estimate the parameters, it was confirmed that the Pre-

dici model and the ODE model yielded practically iden-

tical simulation results when using the same parameters;

representative examples for a Path A simulations are

shown in Figs. 1–3. In all other figures, only the PRE-

DICI results are shown. Pooled variances for measured

molecular weight and initiator and monomer concentra-

tions were determined from replicate experiments at two

different experimental settings (four replicate Path B runs

with [TMPCl]0 =0.004mol/L and [TiCl4]0 = 0.032mol/L,

and two Path A runs with [TMPCl]0 = 0.05mol/L and

[TiCl4]0 = 0.0125mol/L), and were supplied to the

GREG parameter estimation software [58].

Attempts to estimate all of the model parameters

simultaneously failed, even after the model was re-

parameterized in an attempt to improve the numerical

conditioning of the parameter estimation problem (i.e.,

expressing the model in terms of natural logarithms to

ensure that all parameters estimated would have the

same order-of-magnitude size). Simulations revealed

0 200 400 600 8000.0

0.5

1.0

1.5

2.0

[M],

mol

/L

time (s)

Predici ODE

Fig. 2. Comparison of experimental and simulated monomer

consumption plots for Path A: [TMPCl]0 = 0.05mol/L, [IB]0 =

2mol/L, [DtBP]0 = 0.007mol/L. Hx/MeCl=60/40 v/v; Experi-

mental: [TiCl4] = 0.05mol/L (m), [TiCl4] = 0.025mol/L (s);


0.0 0.2 0.4 0.6 0.80

20

40

60

80

100

Predici ODE

DP n,

DP w

Conversion

Fig. 3. Experimental and simulated molecular weight vs

conversion plots for Path A: Experimental: Mn (s), Mw (m),

[TMPCl]0 = 0.05mol/L, [TiCl4]0 = 0.025mol/L, [IB]0 = 2mol/L,

[DtBP]0 = 0.007mol/L. Hx/MeCl=60/40 v/v; Temperature:

�80�C; Simulation parameters: Set 4 (Table 1).


that nearly identical simulation results could be obtained

if certain parameters were adjusted in pairs. For exam-

ple, simulation results indicate that if the forward rate

constant k0 and the reverse rate constant k�0 are both

doubled, there are no significant changes in the model

predictions. The equilibrium position of the reaction is

maintained, and since both reactions are already very

fast, increasing their rates, together, has no discernable

effect on the molecular weights or reactant concentra-

tions. Similar conclusion applies to kp/k�1.

To further investigate the correlated effects of the

parameters on the model predictions and to determine

which subsets of the model parameters can be esti-

mated together, a detailed parameter estimability anal-

ysis [59] was performed, after including the correlations

in Eqs. (10) and (11) in the model. It was found that,

among the four parameters, k�0, k0, k1 and k2, it is

impossible to estimate any two parameters simultane-

ously. k�0 and k0 are correlated through the kinetic

mechanism (K0 = k0/k�0), and k1 and k2 are related

by the correlations defined in (10) and (11). Hence,

only one of these parameters can be included in the

set of parameters to estimate using the current set of

experimental data. If additional measurements were

available, e.g., measurements of the concentration of

the I * LA or Pn * LA intermediates, then more kinetic

parameters could be estimated. Without any additional

measurements to compare with model predictions,

however, multiple sets of values for these four param-

eters will give the same model predictions as the

parameter estimates in Set 4, which were used to sim-

ulate the experimental results.

2.6. Simulation of living IB polymerization using the

proposed model of Scheme 1

Figs. 1 and 2 show experimental [15] and simulated

initiator and monomer consumption plots for Path A.

The agreement is very good. As discussed above, initia-

tor and monomer consumption is simultaneous under

the experimental conditions used. As a consequence,

the degree of polymerization increases only slightly with

conversion, as shown in Fig. 3. The Mn prediction is

quite good, while Mw is slightly overestimated. This

could be attributed to the very low molecular

weights––loss of fractions during precipitation is inevita-

ble. The MWD is slightly lower than 2 during the entire

reaction.

In Path B, the simulated monomer consumption

shows very good agreement with experimental data as

can be seen from Fig. 4. The predicted Mn also shows

a good fit to experimental data (Figs. 5 and 6). However,

the Mw value becomes progressively underpredicted

with increasing conversion. This points to the existence

of possible side reactions. Therefore linear semilogarith-

mic monomer consumption plots together with linear

Mn––conversion plots, which are currently considered

as proof for living conditions, are insufficient to demon-

strate living conditions. The additional criterion of

Mw�Mn = constant [5] should also be considered when

evaluating living polymerizations. This is evident from

the simulated plots obtained with an ideal model, but

the experimental data deviates from this prediction.

Introduction of various possible side reaction scenar-

ios such as slow initiation, termination, transfer and

slow protic initiation into our model was unsuccessful.

0 2000 4000 6000 80000.0

0.5

1.0

1.5

2.0

[M],

mol

/L

time (s)

Predici

Fig. 4. Comparison of experimental and simulated monomer

consumption plots for Path B: [TMPCl]0 = 0.004mol/L, [IB]0 =

2mol/L, [DtBP]0 = 0.007mol/L. Hx/MeCl=60/40 v/v; Experi-

mental: [TiCl4] = 0.064mol/L (m), [TiCl4] = 0.04mol/L (s);


0.0 0.2 0.4 0.6 0.80

100

200

300

400

500

600

Predici

DP n,

DP w

Conversion


conversion plots for Path B: Experimental: Mn (s), Mw (m),

[TMPCl]0 = 0.004mol/L, [TiCl4]0 = 0.04mol/L, [IB]0 = 2mol/L,

[DtBP]0 = 0.007mol/L, Hx/MeCl=60/40 v/v; Temperature:


0.0 0.2 0.4 0.6 0.80

100

200

300

400

500

600

Predici

DP n,

DP w

Conversion


conversion plots for Path B: Experimental: Mn (s), Mw (m),

[TMPCl]0 = 0.004mol/L, [TiCl4]0 = 0.064mol/L, [IB]0 = 2mol/

L, [DtBP]0 = 0.007mol/L. Hx/MeCl=60/40 v/v; Temperature:



When TMP-Cl and PIB 36-mer were used as initiators, it

was found that kp � 3 * ki [23], but simulations con-

ducted using a ki value of 1–2 * 108L/mols did not show

any noticeable difference when compared to simulations

with kp = ki. When slow irreversible termination was

introduced, a better fit was obtained for Mw in certain

experiments [54], but in others led to discrepancies in

the monomer consumption and Mn data. Introducing

chain transfer caused a simultaneous reduction in both

Mn and Mw values. More detailed studies are necessary

to find the cause of the Mw divergence.

2.7. Calculating the average run length �l0 from Path A

Puskas et al. [9] introduced a method to calculate the

average run length �l0, defined as the average number ofmonomer units incorporating in one active ionization

period, from MWD data using Eq. (12)

DPw

DPn¼ 1þ l0

ðDPnÞð12Þ

where �l0 ¼ ðkp=k�1Þ½M�0. Eq. (13) shows the form for

intermediate conversion.

�l ¼ 1þ l0 � 1

21þ ½M�

½M�0

� �ð13Þ

Subsequently Muller et al. [60] introduced another

method:

DPw

DPn¼ 1þ 1

b2

X� 1

� �ð14Þ

where X = conversion and b = k�1/kp[I]0.

Storey recently showed that average run length num-

bers could also be obtained from ‘‘rapid monomer con-

sumption’’ (RCM) data [28]. We now will show that �l0can also be obtained from Path A in IB polymerization.

At the end of the reaction in Path A, when the initiator

is consumed, polydispersity of 1.84 was obtained exper-

imentally and simulation yielded a polydispersity of 1.88

(Fig. 3). The conditions in Path A experiments, when

initiator and monomer consumption is simultaneous,


can be considered as the first ‘‘run’’ of a living polymer-

ization; initiation is complete only at the end of the reac-

tion. Using DPw/DPn = 1.88 and the DPn value of 40 at

full initiator conversion, �l0 ¼ 35 was obtained from Eq.

(13). Using �l0 ¼ ðkp=k�1Þ½M�0 [8], kp/k�1 = 17.5L/mol iscalculated. This is in excellent agreement with that pub-

lished by Kaszas and Puskas [17] (kp/k�1 = 16.5L/mol)

and Schlaad et al. [23] (kp/k�1 = 16.4L/mol). Thus �l0can be obtained experimentally under Path A condi-

tions, where initiation will compete with propagation.

Under these conditionsMn andMw will remain approx-

imately constant, with the polydispersity for the most

probable distribution of MWD = 2. The MWD will start

to narrow only after initiation is complete.

3. Conclusions

In conclusion, we can state that the reaction mecha-

nism proposed in Scheme 1 reconciles seemingly contra-

dictory experimental findings (e.g., shifting TiCl4 order

and discrepancy in rate constant values obtained with

diffusion clock methods and kinetic measurements)

and can be used to simulate experimental results with

rate constants that satisfy the correlations imposed by

the mechanism and/or independently measured rate con-

stants. This by no means can be considered as proof of

the proposed mechanism. The existence and chemical

nature of the proposed intermediates must be proven

independently. However, in the absence of another

model that satisfies all the contradictory requirements

this model does not seem to be unreasonable. In addi-

tion, large-scale production data (confidential) demon-

strates that the rate constant of propagation must be

very high.

Acknowledgment

Financial support of Bayer Inc., Canada and

NSERC Canada is gratefully acknowledged.

Appendix A. Path A

Initiator consumption rate. From material balances

on the initiator and the intermediates, we have:

d½I�dt

¼ �k0½I�½LA� þ k�0½I � LA� ðA:1Þ

d½I � LA�dt

¼ k0½I�½LA� � k�0½I � LA� � k1½I � LA�

þ k�1½IþLA�� ðA:2Þ

To obtain the net rate of consumption of both of these

initiator-derived species together to form ionized species

that can polymerize, Eqs. (A.1) and (A.2) may be added

together to get:

d½I�dt

þ d½I � LA�dt

¼ dð½I� þ ½I � LA�Þdt

¼ �k1½I � LA� þ k�1½IþLA�� ðA:3Þ

[I] + [I * LA] = [I]u, where [I]u is the concentration of

unreacted initiator, thus Eq. (A.4) defines the rate of

initiator consumption and polymer formation:

� d½I�udt

¼ d½Pn�dt

ðA:4Þ

If the reactions involving k0 and k�0 are very fast,

then we will have an initial rapid equilibration, after

which:

½I � LA� ¼ k0k�0

½I�½LA� ðA:5Þ

so that Eq. (A.3) becomes:

d½I�udt

¼ �k1K0½I�½LA� þ k�1½IþLA�� ðA:6Þ

Now, if [I+LA�] is small due to the fast propagation

reaction of the ionized species, and [LA] � [LA]0, we

have:

d½I�udt

¼ �k1K0½I�½LA�0 ðA:7Þ

Expanding (A.7) with 1/[I]u on both sides and substitut-

ing (A.5) gives:

1

ð½I�uÞ� d½I�udt

¼ �k1K0½I�½LA�0ð½I� þ ½I � LA�Þ ¼

�k1K0½LA�0ð1þ K0½LA�0Þ

ðA:8Þ

Assuming K0[I]0 � 1 (A.8) can be simplified to give

(A.9):

1

ð½I�uÞ� d½I�udt

¼ �k1K0½I�½LA�0 ðA:9Þ

Integrating (A.9) between the limits of [I]u,0 = [I]0 and

[I]u,t = [I]t + [I * LA]tZ ½I�u;t

½I�u;0

d½I�u½I�u

¼ �k1K0½LA�0Z t

t¼0dt ðA:10Þ

we get the simplified rate Eq. (A.10)

ln½I�0½I�u;t

¼ k1K0½LA�0t ðA:11Þ

This is equivalent to Eq. (6) in the text.

Propagation rate. For Path A, the rate of propaga-

tion is described by Eq. (A.12):

Rp ¼ � d½M�dt

¼ kp½M�ð½IþLA�� þ ½Pþn LA

��Þ ðA:12Þ

The consumption rate of the active propagating species

is given in Eq. (A.13):


dð½IþLA�� þ ½Pþn LA

��Þdt

¼ k1½I � LA� � k�1½IþLA��

þ k1½Pn � LA� � k�1½Pþn LA

��ðA:13Þ

The combined change in initiator and polymer concen-

trations with time can be written as follows:

dð½I� þ ½Pn�Þdt

¼ �k0½I�½LA� þ k�0½I � LA�

� k0½Pn�½LA� þ k�0½Pn � LA� ðA:14Þ

Since the reactivities of the initiator and the growing

chains are about equal, the sums of the concentrations

of the active cationic species and also the dormant spe-

cies are constant and the quasi-steady state assumption

(QSSA) can be applied:

dð½IþLA�� þ ½Pþn LA

��Þdt

ffi 0 ðA:15Þ


ffi 0 ðA:16Þ

With this and assuming fast equilibria, the concentration

of active cationic species is derived as follows:

½IþLA�� þ ½Pþn LA

�� ¼ K1ð½I � LA� þ ½Pn � LA�Þ ðA:17Þ

½I � LA� þ ½Pn � LA� ¼ K0ð½I� þ ½Pn�Þ½LA� ðA:18Þ

Substituting (A.18) into (A.17), we have

½IþLA�� þ ½Pþn LA

�� ¼ K1K0ð½I� þ ½Pn�Þ½LA� ðA:19Þ

where K1 = k1/k�1 and K0 = k0/k�0.

Substituting (A.18) into (A.12) yields

Rp ¼ � d½M�dt

¼ kpK1K0½M�ð½I� þ ½Pn�Þ½LA� ðA:20Þ

Since the concentration of active cationic species

([I+LA�] and ½Pþn LA

��) is very small, simplified mass

balances for the initiator and Lewis acid can be written

as follows:

½I�0 ¼ ½I� þ ½Pn� þ ½I � LA� þ ½Pn � LA� ðA:21Þ

½LA�0 ¼ ½LA� þ ½I � LA� þ ½Pn � LA� ðA:22Þ

Substituting (A.17) into (A.20) and (A.21) yields:

½I� þ ½Pn� ¼½I�0

1þ K0½LA�ðA:23Þ

and

½LA� ¼ ½LA�01þ K0

½I�01þ K0½LA�

ðA:24Þ

If the concentration of free Lewis acid is small, i.e.,

K0[LA]� 1, (A.23) and (A.24) can be simplified:

½I� þ ½Pn� � ½I�0 ðA:25Þ

½LA� � ½LA�01þ K0½I�0

ðA:26Þ

With K0[I]0 � 1, [LA] � [LA]0.

Substituting (A.25) and (A.26)into Eq. (A.20) gives

the general form of the differential polymerization rate

equation:

Rp ¼ � d½M�dt

¼ kpK1K0½M� ½I�0½LA�01þ K0½LA�0

ðA:27Þ

Assuming K0[LA]0 � 1 again, we can simplify and inte-

grate the rate equation to get (A.28):

ln½M�0½M� ¼ kpK0K1½I�0½LA�0t ðA:28Þ

(A.28) is equivalent to Eq. (7) in the text.

Appendix B. Path B

Propagation rate. The rate of polymerization for Path

B can be derived similarly to Path A:

Rp ¼ � d½M�dt

¼ kp½M�ð½IþLA�2 � þ ½Pþ

n LA�2 �Þ ðB:1Þ

dð½IþLA�2 � þ ½Pþ

n LA�2 �Þ

dt¼ k2½I � LA�½LA�

� k�2½IþLA�2 �

þ k2½Pn � LA�½LA�� k�2½Pþ

n LA�2 � ðB:2Þ


¼ �k0½I�½LA� þ k�0½I � LA�

� k0½Pn�½LA� þ k�0½Pn � LA� ðB:3Þ

dð½IþLA�2 � þ ½Pþ

n LA�2 �Þ

dtffi 0 ðB:4Þ


ffi 0 ðB:5Þ

Since the reactivities of the initiator and the polymer are

close to equivalent, the rate of consumption of the initi-

ating species is equal to the rate of the formation of the

growing species.

½IþLA�2 � þ ½Pþ

n LA�2 � ¼ K2ð½I � LA� þ ½Pn � LA�Þ½LA�

ðB:6Þ

½I � LA� þ ½Pn � LA� ¼ K0ð½I� þ ½Pn�Þ½LA� ðB7 ¼ A7Þ

½IþLA�2 � þ ½Pþ

n LA�2 � ¼ K2K0ð½I� þ ½Pn�Þ½LA�2 ðB:8Þ


where K2 = k2/k�2.

Substituting (B.8) into (B.1) yields:

Rp ¼ � d½M�dt

¼ kpK1K0½M�ð½I� þ ½Pn�Þ½LA�2 ðB:9Þ

The concentrations of the ionic species are very small,

thus the simplified mass balances established in Eqs.

(A.20) and (A.21) and Eqs. (A.22)–(A.25) will apply.

With these we get the form given below:

Rp ¼ � d½M�dt

¼ kpK1K0½M� ½I�0½LA�2

0

ð1þ K0½I�0Þ2

ðB:10Þ

Assuming K0[LA]0 � 1 again and integrating, we get

(B.11):

ln½M�0½M� ¼ kpK0K1½I�0½LA�

2

0t ðB:11Þ

Eq. (B.11) is equivalent to Eq. (9) in the text.

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