Modeling of the Emulsion Terpolymerization of Styrene,α-Methylstyrene and Methyl Methacrylate

Modeling of the Emulsion Terpolymerization of

Styrene, a-Methylstyrene and Methyl Methacrylate

Sandrine Hoppe, Cornelius Schrauwen, Christian Fonteix, Fernand Pla*

Laboratoire des Sciences du Genie Chimique, UPR 6811 CNRS, ENSIC-INPL, 1, Rue Grandville, BP451,54001 NANCY Cedex, FranceFax: þ33 3 83 17 51 55; E-mail: [email protected]

Received: October 1, 2004; Revised: January 23, 2005; Accepted: February 22, 2005; DOI: 10.1002/mame.200400277

Keywords: emulsion terpolymerization; glass transition temperature; modeling; molecular weight distribution; number ofpolymer particles

Introduction

Emulsion polymerization is a complex heterogeneous pro-

cess commonly used to produce latex for applications in

important industrial domains such as coatings, paints, inks,

adhesives and rubbers. According to the recipe and the ope-

rating conditions used, the mechanisms governing the

polymerization dictate the characteristics of the resulting

products (macromolecules and latex). These characteris-

tics, for example molecular weight distribution, polymer

microstructure, glass transition temperature and particle

size distribution and morphology strongly govern the end

use properties of the products which are useful to a custo-

mer. It is therefore important to predict these properties

from the operating conditions used, which can be done using

the elaboration of an adapted simulator.

In this radical polymerization, the monomer is mainly

located in droplets dispersed in an aqueous phase which are

stabilized by a surfactant. The initiator is soluble in the

water phase which contains an excess of surfactant mainly

in its micellar form. In conventional emulsion polymeriza-

tions, during which monomers with very low water solubi-

lity are used, the polymerization starts in the aqueous phase

where the initiator decomposes and generates primary

radicals. These radicals propagate first in the aqueous phase

and then enter the surrounding micelles. This mechanism is

called micellar nucleation. With water soluble monomers,

additional water propagation can occur, leading to oligo-

meric radicals. When these radicals propagate beyond their

water solubility, they precipitate and are then stabilized by

the available emulsifier present in the medium. This mecha-

nism is called homogenous nucleation. Both mechanisms

Summary: This work deals with modeling the terpolymer-ization of styrene, a-methylstyrene and methyl methacrylatein the presence of an inhibitor. The model used is a ‘‘tendencymodel’’ based on the kinetics of the complex elementarychemical reactions both in the aqueous phase and in the par-ticles. It considers the reversible propagation of a-methyl-styrene and the main physical phenomena occurring duringthe process, i.e., (i) partitioning of monomers, surfactant andinhibitor between the aqueous phase, polymer particles, mo-nomer droplets and micelles; (ii) homogeneous and micellarnucleation; (iii) radical absorption and desorption; (iv) geland glass effects. The main kinetic parameters of the modelare estimated on the basis of batch experimental data in orderto be able to describe the complete picture of this complexprocess. The model can be used to predict (with good preci-sion) the global monomer conversion, number and weight-average molecular weight, the average diameter and numberof polymer particles and the glass transition temperature, andconsequently to study the effects of AMS on conversion andterpolymer and latex characteristics.

Comparison of experimental and simulated results of theweight-average molecular weight versus conversion for theemulsion terpolymerization of styrene, alpha methylstyreneand methyl methacrylate at 60 8C.

Macromol. Mater. Eng. 2005, 290, 384–403 DOI: 10.1002/mame.200400277 � 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

384 Full Paper

lead to polymer particles in which propagation, termination

and/or chain transfer reactions take place. The monomer

needed for propagation is provided by the monomer

droplets.

The optimization of emulsion polymerization processes

needs accurate models with low prediction errors. Among

the existing models, the most representative have recently

been reviewed by Gao and Pendilis.[1] They mainly con-

cerned homopolymerization and copolymerization and

were generally built to predict the polymerization kinetics

and the microstructure and properties of the resulting

products. The major problem is the extensive number of

parameters that must be considered. Therefore, depending

on the information sought, important approximations are

made to overcome the most complex and little understood

phenomena, such as particle nucleation, coagulation and/or

typical termination and complex chain transfer reactions.

Moreover, the growing industrial need for new polymer

based materials has recently led to some interesting studies

on terpolymerization processes. Among them, only a few

have been focused on modeling. The main systems studied

have been composed of hydrophilic and hydrophobic

monomers, such as acrylonitrile/styrene/a-cyannocinnamic

ester,[2] methyl methacrylate/ethyl acrylate/methacrylic

acid,[3] styrene/methyl methacrylate/methyl acrylate[4]

and vinyl acetate/methyl methacrylate/butyl acrylate.[5–7]

This paper deals with the elaboration of a tendency model

for the emulsion terpolymerization of styrene (STY),

a-methylstyrene (AMS) and methyl methacrylate (MMA).

As is well known, poly(a-methylstyrene) has a high Tg

(177 8C) but suffers from a low ceiling temperature

(61 8C),[8] which results in important depropagation during

its fabrication. Nevertheless, AMS can be involved in radi-

cal copolymerization and subsequently improves the heat

resistance and mechanical properties of the resulting

copolymers.[9] On the other hand, it changes the polymer-

ization kinetics considerably. Both radical homopolymer-

izations and copolymerizations using high levels of AMS

are slow. Moreover, in the case of the emulsion copolymer-

ization of AMS and MMA, Martinet et al.[10] clearly

showed that an increase in the initial concentration of AMS

led to a decrease in both kinetic rates and average molecular

weights. Similar results were also obtained by Castellanos

Ortega[11] for the system AMS/STY.

In the present work, the terpolymerization has been

studied using a batch reactor.

The objective for the model was to predict the global con-

version, the number and weight-average molecular weights,

the average diameter and the number of polymer particles

and the glass transition temperature of the terpolymer.

Subsequently, the effect of AMS on the global conversion

and on the terpolymer and latex characteristics was studied.

The model used takes into account the thermodynamic

properties of the surfactant, the inhibitor and the three

monomers, the water solubility of each being quite different

(MMA is partially soluble while STY and AMS are only

very slightly soluble in water). It also takes into account the

main chemical and physical phenomena occurring in both

the water phase and in the polymer particles and the

reversible propagation of AMS.

Mathematical Model

Basic Principles

The model developed in this work has the same general

structure as a previous one elaborated on in our labora-

tory.[12]

The establishment of the model required the use of seve-

ral assumptions to enhance the speed of convergence. Some

of these assumptions are made here without providing

justification, as they have been readily accepted and validat-

ed in the classical literature. Others will be accompanied

with any necessary explanation.

The major useful assumptions made can be summarized

as follows.

1. The monomer droplet diameter as well as that of the

growing particles are considered to be monodisperse.

2. Propagation, chain transfer to the monomer and ter-

mination reactions in the aqueous phase are neglected. This

assumption, which simplifies the model, is not perfectly

valid because, owing to its partial water solubility, MMA

could give rise to propagation in the aqueous phase and then

lead to the formation of a terpolymer rich in MMA units at

very low conversion. However, the resulting oligomeric

radicals would also contain STY and/or AMS units which,

in turn, would reduce the water solubility of these radicals

and favor homogenous nucleation. Moreover, due to the

high chain length of the macromolecules formed during

emulsion polymerization, it is acceptable to consider that

any possible deviations in the terpolymer composition

which may occur at low conversion are not significant

enough to affect the predictions expected in the present

tendency model.

3. Chain transfer reactions to the polymer are not

considered.

4. Coagulation between particles is neglected.

5. The reactor is assumed to be isothermal and perfectly

mixed.

Due to the complexity of this process, we chose not to

develop a single model, but instead several small ones. Each

of these evaluates a specific characteristic of the state pro-

cess versus the reaction time. We therefore make the hypo-

thesis that ‘‘all the statistic distributions are independent’’.

Theoretically, this assumption is not totally true, but it is

well known that it simplifies the problem without losing

precision for the model predictions.

In this work, the monomer emulsion is stabilized by use

of an excess of surfactant, S, which forms micelles. All

components are initially introduced into the reactor, except

Modeling of the Emulsion Terpolymerization of Styrene, a-Methylstyrene and Methyl Methacrylate 385

Macromol. Mater. Eng. 2005, 290, 384–403 www.mme-journal.de � 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

the initiator, and the polymerization begins with the shot of

initiator.

Kinetic Scheme

According to these assumptions, the model is based on the

following elementary chemical reactions.

a) In the Aqueous Phase

a1) Initiation

The initiator, I2, is thermally decomposed into radicals

denoted by I�aq with a dissociation rate coefficient kDA.

I2 ! 2I�aq ð1Þ

These radicals can react with monomer units, Miaq,

dissolved in the aqueous phase (mainly with MMA) to give

a radical R�iaq with a degree of polymerization equal to 1.

I�aq þMiaq ! R�iaq ð2Þ

According to Assumption 2, propagation, chain transfer

to the monomer and termination reactions in the aqueous

phase will be neglected. Therefore, the other mechanisms

involved in the aqueous phase will be as follows.

a2) Radical Inhibition

The monomers used in this work are stabilized by an

inhibitor. According to Odian’s assumptions,[13] we con-

sider that radicals, R�iaq, can react with the inhibitor, Zaq,

dissolved in the aqueous phase. This reaction leads to a

molecule P.

R�iaq þ Zaq ! Pþ Z�aq ð3Þ

a3) Nucleation

Free radicals produced in the aqueous phase by thermal

decomposition of the initiator can nucleate micelles

swollen with monomer and create latex particles. Then, in

the particles, the captured radicals, R�iaq, become active

chains R�i .

R�iaq þ micelle! particleþ R�i ð4Þ

a4) Radical Absorption by the Particles

Free radicals present in the aqueous phase are also able to

enter into the latex particles and then initiate new active

chains.

R�iaq þ particle �! particleþ R�i ð5Þ

When the radical enters into the particle, the phenom-

enon is called capture. When it comes out of the particle, it

is called desorption.

b) In the Particles

b1) Propagation

The propagation rate constant is supposed to be indepen-

dent of the chain length and depends on the terminal mono-

mer unit of the propagating chain. Thus, a radical ending

with a monomer unit i can propagate with a monomer unit j

with a propagation rate constant kpij according to following

scheme.

R�i þMj ! R�j ð6Þ

b2) Depropagation

As previously mentioned, AMS is known to give depropa-

gation. But, even if its radical homopolymerization is

difficult due to its low reactivity and low ceiling tem-

perature, it can copolymerize with monomers like styrene

and methyl methacrylate.

b3) Termination Reactions

Termination reactions may occur by combination and

disproportionation as follows.

R�i þ R�j ! P ð7Þ

R�i þ R�j ! 2P ð8Þ

As for propagation, the corresponding termination rate

coefficients, kTCij and kTDij, are assumed to be independent

of the chain length.

b4) Radical Inhibition

Free radicals ended by a monomer unit i can react with a

molecule of inhibitor ZP with an inhibition rate coefficient

kZP.[13]

R�i þ Zp ! Pþ Z�p ð9Þ

b5) Chain Transfer to Monomer

Chain transfer to the monomer is the most prevalent reac-

tion of chain transfer in the absence of added chain transfer

agent. This reaction does not change the concentration of

the macroradicals but modifies the chain lengths and creates

dead macromolecules with a transfer rate constant kTRij.

R�i þMj ! Pþ R�j ð10Þ

b6) Radical Desorption

The only candidates for exit out of the particles are small

radicals resulting from chain transfer. This phenomena

involves three steps which are diffusion through the parti-

cle, penetration of the particle-water interface and diffusion

386 S. Hoppe, C. Schrauwen, C. Fonteix, F. Pla


https://www.researchgate.net/publication/281136697_In_Principles_Of_Polymerization?el=1_x_8&enrichId=rgreq-336a6ef6526cd348c138622745a4c970-XXX&enrichSource=Y292ZXJQYWdlOzIyOTU4MDY2MztBUzoxNDMxNjIzOTU4NjA5OTJAMTQxMTE0Mzk4MzkyOQ==


away from the surface of the particle. Desorption is the

complementary part of capture in absorption. It can be

simply schematized as follows.

R�i ! R�iaq ð11Þ

Reaction Rates

a) Initiator Decomposition

The initiator is consumed by thermal decomposition in the

aqueous phase. The corresponding reaction rate, RDA, is

given by

RDA ¼ kDAIAQ ð12Þ

where IAQ is the total number of moles of initiator in the

aqueous phase.

The reaction of the initiator free radicals with monomer i

is instantaneous, so its initiation rate is

2fRDAfAQi ð13Þ

where f is the free radical efficiency in aqueous phase and

fAQi the molar fraction of monomer i in the aqueous phase.

b) Inhibitor Consumption

The inhibitor, Z, is consumed through reactions with the

radicals present in the aqueous phase and in the latex

particles. Its reactivity is supposed to be independent of the

nature of the radicals. The corresponding rates of con-

sumption are as follows.

1. In the aqueous phase

RZAi ¼ ekCPiAAQi½Z�AQ ð14Þ

2. In the latex particles containing h radicals

RZPkZPi½Z�PNPhnhPi ð15Þ

where ekCPi and kZP are the inhibition coefficient rates in the

aqueous phase and in the latex particles, respectively, AAQi

is the total number of moles of free radicals (ended by a

monomer unit i) in the aqueous phase, [Z]P and [Z]AQ are the

inhibitor concentrations in the particles and in the aqueous

phase, respectively, NP is the number of particles expressed

by the total number of moles (total number of particles

divided by Avogadro’s number), h is the number of free

radicals in a particle, nh is the fraction of particles con-

taining h free radicals and Pi is the fraction of free radicals

ended by a monomer unit i.

X1h¼0

uh ¼ 1;X1h¼0

huh ¼ �nn andX3

i¼1

Pi ¼ 1 ð16Þ

�nn is the average number of free radicals per particle. This

gives the inhibitor total consumption rate.

RZP ¼ kZP½Z�PNP�nn ð17Þ

c) Micellar and Homogeneous Nucleation

The micellar nucleation rate, RMi, is

RMi ¼ dkCPiAAQiNMdMV�1AQ ð18Þ

where NM is the number of micelles, expressed as the total

number of moles, dM is the micellar diameter and VAQ is the

aqueous phase volume. d is the ratio between the nucleation

rate constant and the capture rate constant for radicals

ending with monomer i.

The homogeneous nucleation rate, RHi, is written as

RHi ¼ dkCPiAAQi

SW � CSCVAQ

nS

dMV�1AQ ð19Þ

where nS is the average number of surfactant molecules per

micelle, SW is the total number of moles of surfactant in

solution in the aqueous phase and CSC is the critical con-

centration of surfactant below which there is no homo-

geneous nucleation. We consider that homogeneous

nucleation requires radicals resulting from the initiator

decomposition and surfactant molecules. Thermodynamics

implies then that, simultaneously, monomer molecules ap-

pear in the resulting new particles. Thus the rate of homoge-

neous nucleation is a function of both radical concentration

and surfactant. If the surfactant concentration is too low,

homogeneous nucleation cannot develop.[14] In this study,

the value of CSC is close to 0.

The total nucleation rate with free radicals of monomer i,

RNi is

RNi ¼ RMi þ RHi ¼ dkCPiAAQi NM þSW � CSCVAQ

nS

� �dMV�1

AQ

ð20Þ

where d is obtained from: d ¼P3

i¼1 dif0i, with foi being the

initial molar fraction of monomer i (on all monomers) in the

reactor.

The total nucleation rate, RN, with all radicals is

RN ¼X3

i¼1

RNi ð21Þ

d) Radical Absorption

The classical absorption rate is given by

kidmi

AAQi

VAQ

� NP�nn

VP

Pioi

mdi

� �pd2

PNPNAV ð22Þ

where VP is the total particle volume, dP is the average

particle diameter and NAV is Avogadro’s number. dmi is

the ratio of the transfer resistance in the aqueous phase to

the overall transfer resistance of free radicals ended by

monomer unit i. oi is the fraction of R�i formed only by one

monomer unit i.



As the particle diameter is small, the corresponding

Sherwood number, Sh, is equal to 2

Sh ¼ kidP

Di

¼ 2 ð23Þ

where Di is the diffusion coefficient of R�i in the aqueous

phase.

So, the absorption rate is as follows,

kCPi

AAQi

VAQ

� NP�nn

VP

Pioi

mdi

� �NPdP with kCPi ¼ 2pNAV Didmi

ð24Þ

Then, the capture and desorption rates for monomer i are,

respectively,

RCPi ¼ kCPiAAQiNPdPV�1AQ and RDSi ¼ kCPi

NP�nn

VP

Pioi

mdi

NPdP

ð25Þ

The total capture and desorption rates for all monomers

are, respectively,

RCP ¼X3

i¼1

RCPi and RDS ¼X3

i¼1

RDSi ð26aÞ

RDS is an output rate for the particle phase. The input rate of

radicals for the aqueous phase is fRDS. f is the same factor as

that corresponding to initiation because the phenomenon

is the same.[9] It corresponds to the probability of the

existence of radicals in the aqueous phase.

In the case of particles with h radicals, we have

X3

i¼1

kCPiAAQinhNPdPV�1AQ ¼ RCPnh

andX3

i¼1

kCPi

NPhnh

VP

Pioi

mdi

NPdP ¼ RDS

hnh

�nnð26bÞ

e) Propagation

The propagation rate of monomer i is given by

kPij Mj

� �PNPhuhPi ð27Þ

where [Mi]P is the molar concentration of monomer i in the

particles. The reactivity ratio is given by rPij¼ kPii/kPij.

The total rate of propagation is

RP ¼ NP�nnX3

i¼1

X3

j¼1

kPij½Mj�PPi ð28Þ

f) Depropagation

The depropagation rate of AMS (monomer 3) is

K kP33NPhuhP3

pk

1� p1

with k � 2 ð29Þ

where K is an equilibrium constant and pk the fraction of R�3ended by k units of monomer 3, i.e.:

X1k¼1

pk ¼ 1 andX1k¼2

pk

1� p1

¼ 1 ð30Þ

The total depropagation rate is

RDP ¼ KkP33NP�nnP3 ð31Þ

g) Termination

A coefficient,t, defined as the ratio between the rate of

termination by disproportionation and the rate of termina-

tion by combination is now introduced in order to char-

acterize the predominance between these two termination

mechanisms. These rates are, respectively,

tkTCijN2PV�1

P hðh� 1ÞuhPiPj and kTCijN2PV�1

P hðh� 1ÞuhPiPj

ð32Þ

According to the hypothesis assuming that the kinetic

coefficients do not depend on the chain length, it is accep-

table to consider that the termination rate coefficients

between a radical ended by a monomer unit i and a radical

ended by a monomer unit j are equal.

kTCij ¼ kTCji ð33Þ

These coefficients can be calculated using the homo-

polymerization termination coefficients of the monomers

concerned,

kTCij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikTCiikTCjj

pð34Þ

Thanks to these data, a global termination rate can be

defined as follows,

RT ¼ 2rT~nn with rT ¼ð1þ tÞN2

P

2VP

X3

i¼1

X3

j¼1

kTCijPiPj and

~nn ¼X1h¼2

hðh� 1Þuh ð35Þ

Then, the expression of the total termination by

combination rate is given by

RTC ¼RT

1þ tð36Þ

h) Transfer to Monomer

The transfer to monomer rate is

kTRij½Mj�PNPhuhPi ð37Þ

The transfer to monomer rate coefficients, kTRij, are

defined thanks to the corresponding homopolymerization

transfer rate coefficients

kTRij ¼ kTRji ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikTRiikTRjj

pð38Þ



Then, a global transfer rate can be defined as

RTR ¼ NP�nnX3

i¼1

X3

j¼1

kTRij½Mj�PPi ð39Þ

i) Influence of the Temperature

The Arrhenius’ law is used for the kinetic constants. A

similar expression is considered for the specific area aS.[15]

j) Glass and Gel Effects

The glass and gel effects equations used for propagation and

termination reactions were those proposed by Nomura.[16]

kPij ¼ k0Pij if wP � 0:7 and

kPij ¼ k0Pijexp

��aGl

ij ðwP � 0:7Þ�

if wP > 0:7 ð40Þ

kTCij ¼ k0TCij if wP � 0:32;

kTCij ¼ k0TCijexpð � bGe wP � 0:32ð ÞÞ if 0:32 < wP � 0:8

ð41Þ

kTCij ¼ k0TCijexpð�bGe 0:8� 0:32ð Þ� bGl wP � 0:8ð ÞÞ if wP > 0:8 ð42Þ

where bGe and bGe are the gel and glass coefficients of

termination reaction respectively and wP is the mass

fraction of polymer in the particles,

wP ¼

P3

i¼1

ðN0i � NTi � AAQiÞMi

P3

i¼1

ð½Mi�PVP þ N0i � NTi � AAQiÞMi

ð43Þ

The values of bGe and bGl were taken as identical for all the

termination reactions and result from a previous work.[17]

Thus, the propagation rate coefficient of styrene is related

to the weight fraction of styrene which is incorporated in the

polymer.

Surfactant Partition

The total number of moles of surfactant in the reactor is

given by

SO ¼ SP þ SM þ SD þ SW ð44Þ

where S0 is the initial number of moles of surfactant and SP,

SM, SD and SW are the number of moles of surfactant on the

particles, in the micelles, on the droplets and dissolved in

the aqueous phase, respectively.

The conditions are:

SW � CMC and if SW < CMC then NM ¼ 0 ð45Þ

SPaS ¼ NPNAVpd2P and 6VP ¼ NPNAVpd3

P ð46Þ

SM ¼ NMnS and 6VD ¼ SDaSdD ð47Þ

where CMC is the critical micelle concentration, aS is the

surface covered by one mole of surfactant and VD is the

droplet volume.

Volumes, Monomers and Inhibitor Partitions(Thermodynamic Equilibrium)

The volume of the reactor is given by

VR ¼ VR0 þX3

i¼1


1

rPi

� 1

ri

� �ð48Þ

where VR0 is the initial volume of the reactor, Mi the

molecular weight of monomer i, ri its density and rPi the

density of monomer i in the terpolymer. NTi is the total

number of moles of free monomer i in the reactor. N0i is the

initial mole number of monomer i. The volume of the

polymer is then

VPOL ¼X3

i¼1


rPi

ð49Þ

The volume of the aqueous phase is given by

VAQ ¼ VR � VD � VP ð50Þ

The thermodynamic balance of the three monomers and the

inhibitor between the aqueous phase, droplets and particles

is presented in Appendix 1.

Considering the thermodynamic equilibrium, we have

(see Appendix 1)

VP ¼s

s� 1VPOL

1

s¼ NTZMZ

rZðsVD þ VP þ KPZVAQÞþX3

i¼1

NTiMi

riðsVD þ VP þ KPiVAQÞ

8>><>>:

9>>=>>;

ð51Þ

where s, KPi and KPZ are the equilibrium constants.[16,18,19]

rZ is the density of the inhibitor and MZ is its molecular

weight. NTZ is the total number of moles of inhibitor.

According to Equation (50), Equation (51) is an equation

containing one unknown variable, VD. Relationship (52)

must be applied after Equation (51).

if VD � 0 then VD ¼ 0 and

1� VPOL

VP¼ NTZMZ

rZðVP þ KPZVAQÞþX3

i¼1

NTiMi

riðVP þ KPiVAQÞ

264

375

ð52Þ

According to Equation (46), Equation (48) is also an

equation containing one unknown variable, VP.

The concentrations of monomer i and inhibitor in the

particles are



½Mi�P ¼NTi

sVD þ VP þ KPiVAQ

and

½Z�P ¼NTZ

sVD þ VP þ KPZVAQ

ð53Þ

The concentration of inhibitor and the molar fraction of

monomer i in the aqueous phase are respectively

½Z�AQ ¼ KZP½Z�P and fAQi ¼KPi½Mi�PP3

j¼1

KPj½Mj�Pð54Þ

This new technique to determine the thermodynamic

equilibrium is interesting in order to reduce the numeric

instabilities during the computer solving. Indeed, in the

traditional technique[12] the volumes of each monomer and

of the inhibitor in each phase (monomer droplets, water and

particles) are unknown and are calculated from 12 equa-

tions [Equation (92), Equation (93) and Equation (94)].

Replacing a system of 12 algebraic equations by one of 2

equations with 2 unknown factors (VP and VD) in the com-

plete system of algebraic and differential equations is a

factor in the stability of the solution.

Balances in the Aqueous Phase

Balances in the aqueous phase are given, for the different

species, by the following equations.

Initiator:

dIAQ

dt¼ �RDA where IAQ ¼ I0 when t ¼ 0: ð55Þ

Free radicals of monomer i:dAAQi

dt¼ 2fRDAfAQi � RZAi � RCPi � RNi þ fRDSi

where AAqi ¼ 0 when t ¼ 0: ð56Þ

Inhibitor:

dNTZ

dt¼ �

X3

i¼1

RZAi � RZP where NTZ ¼ N0Z when t ¼ 0:

ð57Þ

Free monomer i:

dNTi

dt¼� 2fRDAfAQi þ RZAi

� NP�nn½Mi�PX3

j¼1

ðkPji þ kTRjiÞPj þ di3RDP ð58Þ

where di3 is the Kronecker symbol (di3¼ 0 if i< 3, d33¼ 1).

NTi¼N0i when t¼ 0.

Number of particles:

dNP

dt¼

X3

i¼1

RNi where Np ¼ 0 when t ¼ 0: ð59Þ

Alltheabovespeciesunitsaregiveninmoles intheaqueous

phase, including the number of particles. The use of the total

number of moles rather than the concentration allows us to

modify easily the model in the case of a fed-batch reactor.

Number of Radicals per Particle

In order to evaluate the reaction rates, one must also know

the average number of radicals per particle. This was

obtained by performing balances on the number of particles

containing, at any instant, h radicals. Taking into account

Equation (26b) this leads to recurrence formulas.

Then, for h¼ 0 and h¼ 1 (0 or 1 radical per particle), it

follows that

dðNPu0Þdt

¼ rTð2 � 1 � u2Þ � RCPu0 þ ðRZP þ RDSÞu1

�nnwhere

n0 ¼ 0 when t ¼ 0 ð60Þ

dðNPu1Þdt

¼ RN þ rTð3 � 2 � u3Þ � RCPu1 þ RCPu0

þ ðRZP þ RDSÞ2u2 � u1

�nnð61Þ

where n1¼ 1 when t¼ 0.

For h> 1:

dðNPuhÞdt

¼ rTDh þ RCPðuh�1 � uhÞ

þ ðRZP þ RDSÞðhþ 1Þuhþ1 � huh

�nnð62Þ

where nh¼ 0 when t¼ 0, and with Dh ¼ ðhþ 2Þðhþ 1Þuhþ2 � hðh� 1Þuh.

Theaveragenumberofradicalsperparticle is thengivenby

dðNP�nnÞdt

¼ RN þ RCP � RT � RZP � RDS ð63Þ

where �nn ¼ 1 when t¼ 0. Resolution of Equation (63)

allows, using Equation (16) to correct, if necessary, the

values of nh to be found. Indeed, the computing errors in nh

induce an important deviation on the value of ~nn which is

necessary for the calculation of the termination rates. The

corrections of nh thus make it possible to reduce the error in

these termination rates.

Molecular Weight Distribution

a) Radical End Distribution in the Particles

The fraction of radicals ending in monomer i is Pi. Then

dðNP�nnPiÞdt

¼ RCPi þ RNi � RZPPi

þ NP�nn½Mi�PP3

j¼1

ðkPji þ kTRjiÞPj

� NP�nnPi

P3

j¼1

ðkPij þ kTRijÞ½Mj�P

�ð1þ tÞN2P~nn

VP

Pi

X3

j¼1

kTCijPj � RDSi

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

ð64Þ

where Pi¼ f0i when t¼ 0.



For monomer 3, there is a distribution of unit number at

the end of the radical given by

dðNP�nnP3p1Þdt

¼ RCP3 þ RN3 � RZPP3p1

þ NP�nn½M3�PP3

j¼1

ðkPj3 þ kTRj3ÞPj

� NP�nnP3p1

P3

j¼1

ðkP3j þ kTR3jÞ½Mj�P

� ð1þ tÞN2P~nn

VP

P3p1

X3

j¼1

kTC3jPj

� kP33NP�nnP3½M3�P þ RDPp2

1�p1� RDS3

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

ð65Þ

where p1¼ 1 when t¼ 0.

pk, the fraction of radicals ended by an AMS unit with k

AMS consecutive units at their extremity, is given by

dðNP�nnP3pkÞdt

¼ �RZPP3pk þ kP33NP�nn M3½ �PP3pk�1

þ RDP

pkþ1 � pk

1� p1

� NP�nnP3pk

P3

j¼1

ðkP3j þ kTR3jÞ Mj

� P

� ð1þ tÞN2P~nn

VP

P3pk

X3

j¼1

kTC3jPj

8>>>>>>>>>>><>>>>>>>>>>>:

9>>>>>>>>>>>=>>>>>>>>>>>;

ð66Þ

where pk¼ 0 when t¼ 0.

For the radical desorption we must evaluate oi

dðNP�nnPioiÞdt

¼ RCPi þ RNi � RZPPioi

þ NP�nn½Mi�PP3

j¼1

kTRjiPj þ RDPP3o�3di3

� NP�nnPioi

P3

j¼1

ðkPij þ kTRijÞ½Mj�P

� ð1þ tÞN2P~nn

VP

Pioi

X3

j¼1

kTCijPj � RDSi

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

ð67Þ

where oi¼ 1 when t¼ 0, and where o�3 corresponds to a

radical with only 2 moles of monomer 3. In order to avoid its

calculation we can approximate o�3 by o�3 ¼ o3

p2

p1

. This

balance is similar to that used by Nomura[20] to define the

desorption rate. It is however more complete because it does

not neglect any concerned reaction.

b) Monomer Unit Distribution in the RadicalsPresent in the Particles

The physical and mechanical properties of polymers

depend strongly on their molecular weight distributions

(MWD). Therefore, one of the aims of the model was to

determine this characteristic. This was achieved by using

the method of moments developed by Villermaux and

Blavier[21] for the case of bulk radical homopolymerization.

To apply this method to the case of emulsion polymer-

ization, a description of the evolution of the instantaneous

distributions of the degree of polymerization of both macro-

radicals and macromolecules is required. For this, we must

first calculate the average degree of polymerization of the

radicals present in the particles. This is possible because

the molecular weights of these three monomers are very

close (MMA¼ 100 g �mol�1, STY¼ 104 g �mol�1,

AMS¼ 118 g �mol�1) and, as demonstrated in this case

(see Appendix 2), the calculated number-average molecular

weight is exact while the calculated weight-average molec-

ular weight is a good approximation.

Alone, the free radicals with more than one monomer

unit (with no discrimination between MMA, STYand AMS)

can eventually depropagate (if they are terminated with

more than one AMS unit). The reaction rate RDP takes into

account the proportion of free radicals which can depro-

pagate. If the probabilities are independent, the depropaga-

tion rate of radicals with l monomer elements (l> 1) is

proportional to �wwl, that is to say to CD�wwl. The total

depropagation rate is

RDP ¼X1l¼2

CD�wwl ¼ CDð1� �ww1Þ ð68Þ

So, the depropagation rate of radicals with l monomer

units (l> 1) is

CD�wwl ¼ RDP

�wwl

1� �ww1

ð69Þ

The balance of radicals with 1 and l monomer units

(l> 1) is

dðNPnw1Þdt

¼ RN þ RCP þ RTR � ðRP þ RTR þ RT þ RZPÞ�ww1

þ RDP

�ww2

1� �ww1

� RDS ð70Þ

where �ww1 ¼ 1 when t¼ 0

dðNPnwlÞdt

¼ RP�wwl�1 � ðRP þ RTR þ RT þ RZPÞ�wwl

þ RDP

�wwlþ1 � �wwl

1� �ww1

ð71Þ

where �wwl is the fraction of free radicals with l monomer

units, (�wwl ¼ 0 when t¼ 0).

The first moment is

l1 ¼X1l¼1

l�wwl withX1l¼1

�wwl ¼ 1 ð72Þ

The sum of Equation (70) and Equation (71) for all values

of l gives Equation (63). The sum of Equation (70) and

Equation (71) multiplied by l, for all values of l, gives

Equation (73).

dðNp�nnl1Þdt

¼ RN þ RCP þ RP � RDP � RDS

þ RTRð1� l1Þ � ðRZP þ RTÞl1 ð73Þ

with l1¼ 1 when t¼ 0.



c) Monomer Unit Distribution in the Polymer Chains

If NC is the mole number of polymer chains (free radicals

and dead macromolecules), wl is the fraction of polymer

chains with l monomer units.

dðNCw1Þdt

¼ RN þ RCP þ RTR � ðRP þ RTCÞ�ww1

þ RDP

�ww2

1� �ww1

� RDS ð74Þ

where w1¼ 1 when t¼ 0.

dðNCwlÞdt

¼ RP�wwl�1 þ1

2RTC

Xl�1

j¼1

�wwj�wwl�j

� ðRP þ RTCÞ�wwl þ RDP

�wwlþ1 � �wwl

1� �ww1

ð75Þ

where wl¼ 0 when t¼ 0 and with the corresponding kth

moment

X1l¼1

wl ¼ 1;X1l¼1

lwl ¼ L1 andX1l¼1

l2wl ¼ L2 ð76Þ

and the following properties

X1l¼2

ððl� 1Þ2 � l2Þ�wwl ¼ �ww1 � 2l1 þ 1

¼ �ð1� �ww1Þ � 2ðl1 � 1Þ ð77Þ

X1l¼1

l2Xl�1

j¼1

�wwj�wwl�j ¼X1l¼1

X1j¼1

ðlþ jÞ2�wwl�wwj ¼ 2l2 þ 2l21

ð78Þ

Hence, Equation (79) is the sum of Equation (74) and

(75) for all values of l. The sum of Equation (74) and (75)

multiplied by l, for all values of l, gives Equation (80). The

sum of Equation (74) and (75) multiplied by l2, for all

values of l, gives Equation (81).

dNC

dt¼ RN þ RCP þ RTR � RDS �

RTC

2ð79Þ

where NC¼ 0 when t¼ 0.

dðNCL1Þdt

¼ RN þ RCP þ RTR þ RP � RDP � RDS ð80Þ

where L1¼ 1 when t¼ 0.

dðNCL2Þdt

¼ RN þ RCP þ RTR þ RPð2l1 þ 1Þ

� RDP 2l1 � 1

1� �ww1

þ 1

� �þ RTCl

21 � RDS ð81Þ

where L2¼ 1 when t¼ 0.

In order to avoid the calculation of �ww1 which is very

small, it is possible to simplify this last relationship. For

this, we approximate �ww1 by 1/l1:

dðNCL2Þdt

¼ RN þ RCP þ RTR þ ðRP � RDPÞð2l1 þ 1Þ

þ RTCl21 � RDS ð82Þ

d) Average Molecular Weights of the Polymer Chains

Considering that the molecular weights of all the monomers

are similar, it is possible to define the average molecular

weight of the monomer unit as

M ¼

P3

i¼1


P3

i¼1

ðN0i � NTi � AAQiÞð83Þ

Then, the number-average molecular weight of the

polymer chain is

Mn ¼ML1 ð84Þ

and the weight-average molecular weight of the polymer

chain is (see Appendix 2)

Mw ¼ML2

L1

ð85Þ

Glass Transition Temperature

The control of the product quality during emulsion poly-

merization requires the monitoring of a large set of parame-

ters related to end-use properties. Some of these properties

are often described through distributed characteristics such

as the molecular weight distribution (MWD) and/or the

copolymer composition distribution (CCD). This is, for

example, the case for the glass transition temperature, Tg,

which depends strongly on the microstructure of the macro-

molecules, particularly for copolymers. To complete the

model, the glass transition temperature is evaluated

according to Fox’s equation.[22,23]

Tg þa

Mn

��1

¼X3

i¼1

Wi

Tgi

ð86Þ

with

Wi ¼ðN0i � NTi � AAQiÞMiP3

i¼1


ð87Þ

Conversion and Copolymer Composition

The global mass conversion is

X ¼

P3

i¼1

ðN0i � NTiÞMi

P3

i¼1

N0iMi

ð88Þ



The residual mass fraction of monomer i is

Fi ¼NTiMiP3

i¼1

NTiMi

ð89Þ

when i< 3 (MMA and STY), and

F3 ¼ 1� F1 � F2 ð90Þ

Experimental Part

Starting Compounds

The chemicals required to carry out the emulsion polymeriza-tion consisted of the following monomers. Styrene (STY) anda-methylstyrene (AMS) previously stabilized with 5 ppm and2.5 ppm of 4-terbutylcatechol (inhibitor), respectively, wereobtained from Sigma-Aldrich. Methyl methacrylate (MMA)was provided by ATOFINA, with a purity higher than 99.5%and without any inhibitor. The initiator used was potassiumpersulfate (KPS) which was obtained from Sigma-Aldrich andwas soluble in water. The surfactant used was sodium dodecylsulfate (SDS) from Sigma-Aldrich. The inhibitor used toquench the polymerization reaction in the withdrawn sampleswas hydroquinone and deionized water was used.

Equipment for Polymerization

The reactor used was a 1 l jacketed glass batch reactor equippedwith a stirrer, a reflux condenser, a cryostat, a sampling deviceand an inlet system for nitrogen. The stirrer used was composedof a pitch blade turbine.

Samples were withdrawn from the reactor at appropriatetime intervals and polymerization was short stopped withhydroquinone.

Characterization of Latex and Macromolecules

In order to follow the polymerization, analytical methods havebeen developed to obtain experimental data. During the poly-merization, samples were taken and put into glass vesselscontaining a small amount of hydroquinone to stop the reac-tion. The resulting products were then characterized bychemical and physical analyses.

Global Conversion

The global monomer conversion was determined gravimetri-cally using a Mettler Toledo HG 53 halogen moisture analyzer.About 1 g of latex was placed on an aluminium plate that wasintroduced into the halogen moisture analyzer and heated to175 8C to evaporate the water and residual monomers com-pletely. The mass of the final dried sample was automaticallymeasured. After correction of the remaining amounts ofinitiator and surfactant, the global conversion was determined.

Residual Monomer Titration

Gravimetry gives access to the total amount of synthesizedterpolymer and allows only the evaluation of the global

conversion of the monomers versus the polymerization time.For better control of the consumption of each monomer duringpolymerization, a more precise titration of the monomers isrequired. To obtain this information, gas chromatography wasperformed using a DELSI NERMAG DN 200 chromatographequipped with a capillary column HP FFAP (length 10 m,diameter 0.53 mm) and a stainless pre-column filled with glassfiber. Analysis was carried out under the following operatingconditions: injection temperature 175 8C; column temperature80 8C; detector temperature 175 8C; gas vector helium, flowrate 4 ml �min�1.

Average Particle Size

The average particle size was determined using a Malvern 4700quasielastic light scattering apparatus. After dilution of thesamples with deionized water, the average particle diameterwas measured. This value, together with that of the overallconversion, was then used to estimate the number of polymerparticles per liter of latex.

Number- and Weight-Average Molecular Weight

The number- and weight-average molecular weights weredetermined by size exclusion chromatography (SEC) using amulti-angle laser light scattering (MALLS) apparatus (DawnDSP-F) and a differential refractometer (Waters 410, Milli-pore) as detectors. Elutions were performed at 35 8C withtetrahydrofuran (THF) containing 2,6-di-tert-butyl-4-methyl-phenol as a stabilizer. The flow rate was 1 ml �min�1. Theconcentrations of the polymer solutions and the correspondinginjected volumes were 1 g � l�1 and 25 ml respectively. Prior tothe chromatography, THF and the polymer solutions werepassed through a nylon filter with a porosity of 0.45 mm.The SEC assembly consisted of a degasser, a Waters 510,Millipore pump, a U6K, a Millipore injector, a precolumn,two chromatographic columns assembled in series and filledwith linear ultrastyragel and an electric oven to control thetemperature of the columns.

Data from the two detectors were acquired and computed byuse of Astra software from Wyatt Technology which allowedthe determination of the molecular weight distribution andthe number- and weight-average molecular weights of thesamples.

Glass Transition Temperatures

Glass transition temperatures were determined using a diffe-rential scanning calorimeter (DSC Pyris 1 Perkin Elmer).Temperature scanning was carried out between �50 8C and60 8C with a scan rate of 10 8C �min�1.

Results and Discussion

Parametric Identification

The aim of the model was to predict satisfactorily and

simultaneously the global and partial monomer conversion,

the number- and weight-average molecular weights and the



average particle diameter and concentration versus the

process operating conditions (i.e., temperature, surfactant,

initiator and monomer concentrations). The model para-

meters were determined by minimization of a maximum

likelihood criteria, J, with the experimental data.[24]

J ¼X5

a¼1

NaLnXNa

b¼1

ðxaðtabÞ � xxaðtab; yÞÞ2 ð91Þ

where Na is the number of measurements of the variables xa,

tab is thebth time of measurement of the variable xa, and xxa is

the value of xa predicted by the model using the values y of

the unknown parameters. In this relation, the five variables

xa are the global mass conversion X, the number of particles

NP, the weight-average molecular weight Mw, and two of

the three residual mass fractions of monomers Fi (the sum

being equal to 1, we have chosen i¼ 1 and 2). The seventeen

unknown parameters of the vector y are simultaneously

obtained by minimization of J using a genetic algorithm,[25]

using all the measurements. This class of algorithm is adapt-

ed to the determination of more than 17 parameters.[12] The

optimized values are presented in Table 1. Four temper-

atures (60 8C, 65 8C, 70 8C and 80 8C) and several initial

monomer compositions were used. The number of measure-

ments per variable Na is given in Table 2. Several parame-

ters, given in Table 3, were taken from the literature. It

should be noticed that the kinetic constants in the aqueous

phase are expressed as functions of the diffusion coefficient,

Di. However, as is well known in such cases, the predictions

of the model are not sensitive to the values of Di, but only to

the ratios of these constants.

The values of the propagation rate constant obtained in

this work (kP22¼ 262 m3 � kmol�1 � s�1) is close to values

available in the literature: 176 m3 � kmol�1 � s�1 at 60 8C(Arzamendi et al.[19]); 240 l �mol�1 � s�1 at 50 8C (Gil-

bert[15] for styrene).

The transfer rate constant to AMS is similar to that given

by Martinet[10] where kTR33¼ 9.97 10�2 m3 � kmol�1 �s�1 (0.11 m3 � kmol�1 � s�1 in this work) at 60 8C. In the

same way, the activation energy of the transfer reaction to

AMS is of the same order of magnitude as that obtained by

Martinet[10] who indicated that, for AMS, the activation

energy is three times higher than that of other classical

monomers.

Concerning the nucleation efficiency of each monomer,

the results show that the nucleation and, a fortiori, the num-

ber of particles depend much more on the initial concentra-

tion of MMA and AMS than on the initial concentration of

STY. In studies on the copolymerization of MMA and STY,

Nomura et al.[20] found that the nucleation efficiency of

MMA is four times higher than that of STY. Table 1 clearly

shows that our results are in agreement with those of

Nomura.

On the other hand, the desorption coefficient of AMS is

much higher than that of STY and MMA. These results

confirm the tendency of AMS to desorb and produce a low

number of radicals per particle.[10]

The average initiator efficiency, equal to 0.3, is much

lower than that obtained only with MMA by Gilbert,[15]

who also gave an efficiency of 0.03 for the system KPS/

STY, showing that the presence of styrenic monomers gives

Table 1. Results of the parametric identification.

Parameter Meaning Value

kP22 (60 8C) Propagation constant of STY 262 m3 � kmol�1 � s�1

kP33 (60 8C) Propagation constant of AMS 66 m3 � kmol�1 � s�1

kTR22 (60 8C) Transfer constant of STY 0.0261 m3 � kmol�1 � s�1

kTR33 (60 8C) Transfer constant of AMS 0.0997 m3 � kmol�1 � s�1

ETR33 Transfer activation energy of AMS 177 000 kJ � kmol�1

d1 Nucleation efficiency of MMA 0.0935d2 Nucleation efficiency of STY 0.0184d3 Nucleation efficiency of AMS 0.0623f Average initiator efficiency 0.3e Inhibition efficiency 3.02KZP Inhibition constant in particles 63.46 m3 � kmol�1 � s�1

KPZ Inhibitor partition constant (water/particles) 0.0184all aGl

ij except aGl11 Glass coefficient for propagation 5.73

t Termination ratio 0.38a Flory constant for Tg 18.1 106 kg 8C � kmol�1

dm2 Desorption coefficient for STY 0.12dm3 Desorption coefficient for AMS 8.68

Table 2. Number of measurements, Na.

Temperature Total massconversion

Mass conversionof monomer 1

and 2

Weight-averagemolecular

weight

60 8C 53 30þ 30 3565 8C 19 12þ 12 970 8C 34 23þ 23 2280 8C 15 10þ 10 8Na (total) 121 75þ 75 74



rise to a significant decrease in the initiator efficiency.

Nevertheless, the optimal value given here is an average

obtained for the different experimental compositions used

in this study.

Moreover, the glass effect coefficient of STYand AMS is

lower than that corresponding to the homopolymerization

of MMA.[10] This is in agreement with the well known much

stronger glass effect of MMA.[15]

Conversion and Residual MassFractions of Monomers

Experiments were carried out with an initial KPS concen-

tration of 1.4 g � l�1 and an initial SDS concentration of

8.2 g � l�1. As previously mentioned, different initial

monomer compositions and temperatures were used. The

AMS fraction was limited to 30% in order to obtain a high

conversion. The simulated data resulting from the model are

compared to the corresponding experimental results ob-

tained for conversion, average molecular weights and glass

transition temperatures.

Figure 1 shows the time evolution of the global conver-

sion for experiments carried out at 60 8C with different

initial monomer concentrations. As expected, when the

AMS concentration is increased, the conversion rate is

lower. This is in agreement with previous results.[10] On the

other hand, for a given concentration of AMS, the total

polymerization rate increases when the concentration of

MMA is increased. This is due to the high reactivity of

MMA compared to STY.

Figure 2 and 3 show the evolution of the global conver-

sion for experiments carried out at 70 and 80 8C, respecti-

vely, with different initial monomer concentrations. As with

the experiments carried out at 60 8C, the polymerization

rate decreases when the AMS concentration increases. The

comparison of these three figures clearly shows that, for

the same monomer composition, the polymerization rate

increases with temperature and this is despite the fact that

the reaction temperature is higher than the AMS ceiling

temperature.

Moreover, these three figures show the good agreement

between the simulated and experimental results.

The residual monomer mass fractions were obtained by

gas chromatography measurements. Figure 4 shows the

time evolution of the composition of the residual mixture at

60 8C for different initial percentages of MMA/STY/AMS.

The kinetic scheme and the reactivity ratios used appear to

Table 3. Parameters from the literature.

Parameter Value Reference

kDA (50 8C) 7.63 10�7 s�1 [17]EDA 135 000 kJ � kmol�1 [15]D1 1.7 10�9 m2 � s�1 [26]D2¼D3 4.1 10�9 m2 � s�1 [19]dm1 (50 8C) 4 10�4 [17]kP11 (50 8C) 492 m3 � kmol�1 � s�1 [17]EP11 22 200 kJ � kmol�1 [15]EP22 32 500 kJ � kmol�1 [15]EP33 54 300 kJ � kmol�1 [10]r12 0.46 [27]r21 0.52 [27]r23 1.3 [28]r32 0.63 [29]r13 0.51 [10]r31 0.53 [10]a11

Gl 17.13 [10](1þ t)kTC11 (50 8C) 2.9 104 m3 � kmol � s�1 [17](1þ t)kTC22 (50 8C) 7 1010 m3 � kmol�1 � s�1 [30](þ t)kTC33 (50 8C) 4 109 m3 � kmol�1 � s�1 [10]ET11 3 600 kJ � kmol�1 [31]ET22 9 000 kJ � kmol�1 [30]ET33 8 360 kJ � kmol�1 [10]bGe 11.46 [10]bGl 3.78 [10]kTR11 (50 8C) 0.0148 m3 � kmol�1 � s�1 [17]ETR22 45 900 kJ � kmol�1 [15]ETR33 55 900 kJ � kmol�1 [15]K (60 8C); K (70 8C) 7.1 kmol �m�3; 9.1 kmol �m�3 [8]dM 5 nm [15]dP 10 000 nm [15]nS 62 [32a]aS 0.57 nm2/molecule [32b]md1¼md2¼md3 39 [31]s 1.3 [19]KP1 1/40 [19]KP2 1/2512 [19]KP3 1/251 [10]M1 100 kg � kmol�1 [33]M2 104 kg � kmol�1 [33]M3 118 kg � kmol�1 [33]MZ 198 kg � kmol�1 [33]r1 965.9–1.213 T (8C) kg �m�3 [34a]r2 906 kg �m�3 [33]r3 928.3–0.9 T (8C) kg �m�3 [34b]rZ 1 330 kg �m�3 [33]rP1 1 190 kg �m�3 [33]rP2 1 070 kg �m�3 [33]rP3 1 150 kg �m�3 [33]Tg1 105 8C [33]Tg2 100 8C [33]Tg3 177 8C [26]

Figure 1. Experimental and simulated results of the conversionversus reaction time for polymerizations carried out at 60 8C,withdifferent initial monomer compositions.



be coherent. For a high initial percentage of AMS (28%),

the time evolution of the residual composition depends on

the ratio of MMA/STY. Thus, the curves obtained for two

different initial percentages of MMA/STY (52/20: full

square points; 23.2/48.8: hexagonal points) are quite diffe-

rent. In the second case, the residual mass fraction of AMS

increases while it is almost constant for the first composi-

tion. This is due to the difference in the reactivity ratios of

the two systems (STY/AMS and MMA/AMS). As the

former is lower, the corresponding residual mass fraction of

AMS is higher.

Similar results were obtained at 70 and 80 8C (Figure 5

and 6).

Weight-Average Molecular Weights

Figure 7, 8 and 9 present the evolution of the weight-

average molecular weight versus the global conversion for

experiments carried out at 60 8C, 70 8C and 80 8C, respec-

tively. The simulated data are slightly overestimated by the

model but the shapes of all the curves are similar and show

(i) that the average molecular weights are almost constant

during the polymerization and (ii) that they are higher when

the initial fraction of AMS is decreased. This is due to the

low reactivity of AMS and to transfer reactions that are

more important with this monomer.

Moreover, for the same initial monomer composition, the

average molecular weight decreases when the temperature

increases.

Average Diameter and Number of Polymer Particles

Table 4 compares the experimental and simulated final

average diameter and number of polymer particles per liter

obtained for experiments carried at 60, 70 and 80 8C.



Figure 4. Experimental and simulated residual monomer massfractions for polymerizations carried out at 60 8C with differentinitial percentages of MMA/STY/AMS.




Again, a good agreement is observed between the experi-

mental and simulated values. In all cases, the final average

diameters do not exceed 50 nm. Similar results were

observed by Martinet[10] and Castellanos Ortega.[11]

Moreover, an increase in the initial fraction of AMS leads

to an increase in the final number of particles. This is

probably due to radical desorption which could inhibit the

growth of nucleated particles. Then, less surfactant is

consumed for particle stabilization and is therefore avail-

able in the form of micelles to create new particles.

On the other hand, for the same initial monomer compo-

sition it appears that the number of particles increases with

reaction temperature. This is due to the initiation and

desorption rates which increase with temperature.

Glass Transition Temperature

Table 5 again shows acceptable agreement between the

experimental and simulated values of the glass transition

temperature (Tg) for experiments carried out at 60, 70 and

80 8C. As expected, an increase in the initial fraction of

AMS leads to an increase in Tg.

Conclusion

In this work, a dynamic reactor model has been developed

for the batch emulsion terpolymerization of styrene, a-

methylstyrene and methyl methacrylate in the presence of

an inhibitor. This model takes into account the depropaga-

tion reactions of a-methylstyrene occurring during the

process.

Based on the kinetics of the complex elementary chem-

ical reactions occurring both in the aqueous phase and in the

particles, this model has been elaborated on using the latest

theories in development for emulsion polymerization, i.e.,

particle nucleation, radical absorption and desorption, the

partitioning of monomers, surfactant and inhibitor between

the monomer droplets, the aqueous phase and the polymer

particles. It also considers the gel and glass effects occur-

ring during the polymerization.

The outputs of the model are the time evolution of the

conversion, average molecular weight, average diameter

and number of polymer particles and the glass transition

temperature.


Figure 7. Experimental and simulated results of the weight-average molecular weight versus conversion for polymerizationscarried out at 60 8C with different initial monomer composi-tions.

Figure 8. Experimental and simulated results of the weight-average molecular weight versus conversion for polymerizationscarried out at 70 8C, with different initial monomer composi-tions.

Figure 9. Experimental and simulated results of the weight-average molecular weight versus conversion for polymerizationscarried out at 80 8C, with different initial monomer composi-tions.



The model parameters were estimated by minimization

of a maximum likelihood criterion with well controlled and

designed experiments. Good agreement between simulated

and experimental data was achieved.

Moreover the method used in this work for the calcula-

tion of the thermodynamic equilibrium helped the simulator

to avoid numerical instabilities.

The model also allowed the effect of AMS on this terpoly-

merization to be studied, which can be summarized as

follows. The activation energy of the transfer reaction to

AMS is higher than that of the other monomers. The

nucleation and the number of particles strongly depends on

the amount of MMA and AMS in the medium. The value of

the AMS desorption coefficient confirms the tendency of

AMS to desorb and to generate low numbers of radicals by

particles. An increase in the initial concentration of AMS

leads to: (i) a decrease of the terpolymerization rate; (ii) an

increase in the final number of particles, probably due to

radical desorption which could inhibit the growth of nucle-

ated particles; (iii) an increase in Tg. The average molecular

weight remains almost constant during the terpolymerization

and are higher when the initial fraction of AMS is decreased.

This is due to the low reactivity of AMS and to transfer

reactions that are more important with this monomer.

List of Symbols

Reactions

RDA (kmol � s�1) Thermal decomposition of initiator in

the aqueous phase

RZAi (kmol � s�1) Inhibitor consumption by reaction with

free radicals of monomer i

RZP (kmol � s�1) Inhibitor consumption in particles

RMi (kmol � s�1) Micelles nucleation rate due to free

radicals ended by monomer i

RHi (kmol � s�1) Homogeneous nucleation rate due to

free radicals ended by monomer i

Table 4. Experimental and simulated particle diameters, dp, and number of polymer particles per liter of final latex resulting frompolymerizations carried out at 60 8C, 70 8C and 80 8C, with different initial monomer compositions.

Initial monomercomposition MMA/STY/AMS

Experimental dp Simulated dp Experimental NPNAV

VRSimulated NPNAV

VR

wt.-% nm nm part. � l�1 part. � l�1

60 8C56/36/8 52.6 52.9 1.84 1018 1.71 1018

60/30/10 52 52.2 1.85 1018 1.84 1018

50/40/10 47.8 51.7 2.57 1018 1.99 1018

44.3/42.1/13.6 53.2 49.0 2.02 1018 2.37 1018

23.2/48.8/28 41.8 41.1 2.52 1018 3.75 1018

52/20/28 42.8 41.9 3.48 1018 3.76 1018

70 8C56/36/8 40.9 43.6 3.2 1018 2.65 1018

54.4/35.3/10.3 43.0 42.5 2.9 1018 3.03 1018

44.3/42.1/13.6 40.5 42.1 3.4 1018 3.06 1018

50/20/30 32.4 34.3 8.7 1018 7.36 1018

80 8C56/36/8 39.6 39.5 4.15 1018 4.19 1018

44.3/42.1/13.6 37.0 35.5 5.2 1018 5.91 1018

Table 5. Experimental and simulated glass transition temperatures, Tg (8C), of terpolymers resulting from polymerizations carried out at60, 70 and 80 8C, with different initial monomer compositions.

Initial monomer compositionMMA/STY/AMS

60 8C 70 8C 80 8C

wt.-% Tg Experim. Tg Simul. Tg Experim. Tg Simul. Tg Experim. Tg Simul.

8C 8C 8C 8C 8C 8C

56/36/8 103.1 106.2 106.2 104.7 105.0 101.460/30/10 108.9 107.554.4/35.3/10.3 106.2 105.544.3/42.1/13.6 106.4 108.6 107.0 105.8 102.5 99.552/20/28 112.5 116.250/20/30 111.4 107.823.2/48.8/28 111.5 111.9



RNi (kmol � s�1) Total nucleation rate due to free radicals

ended by monomer i

RN (kmol � s�1) Total nucleation rate

RCPi (kmol � s�1) Capture rate of free radicals formed by 1

monomer unit i

RCP (kmol � s�1) Overall capture rate of free radicals by

particles

RDSi (kmol � s�1) Desorption rate of free radicals formed

by 1 monomer unit i

RDS (kmol � s�1) Desorption rate of free radicals from

particles

RP (kmol � s�1) Total propagation rate

RDP (kmol � s�1) Total depropagation rate

RT (kmol � s�1) Total termination rate

RTR (kmol � s�1) Total transfer to monomer rate

Kinetic Constants

kDA (s�1) Thermal decomposition of initia-

tor constant in the aqueous phase

f Free radical efficiency in aqueous

phase

kCPi (m2 � kmol�1 � s�1) Capture rate coefficient of free

radicals from monomer i by

particles

e (m) Ratio of inhibition in aqueous

phase and capture rate coeffi-

cients

kZP (m3 � kmol�1 � s�1) Inhibition rate coefficient in par-

ticles

d Overall ratio of nucleation and

capture rate coefficients

di Ratio of nucleation and capture

coefficients due to monomer unit i

dmi Ratio of transfer resistance in

aqueous phase on overall transfer

resistance of free radicals ended

by monomer unit i

Di (m2 � s�1) Diffusivity of free radicals form-

ed by 1 unit i in aqueous phase

ki (m � s �1) Transfer coefficient of free radi-

cals formed by 1 monomer unit i

kPij (m3 � kmol�1 � s�1) Propagation rate coefficient of

monomer j with free radical

ended by i

rij Reactivity ratio

aijGl Glass coefficient of propagation

reaction of monomer j with

radical i

kTCij (m3 � kmol�1 � s�1) Termination by combination rate

coefficient (radicals ended by i

and j)

t Ratio of the termination rates by

disproportionation and combina-

tion

bGe Gel coefficient of termination

reaction

bGl Glass coefficient of termination

reaction

rT (kmol � s�1) Total termination rate specific

coefficient

kTRij (m3 � kmol�1 � s�1) Transfer to monomer rate coeffi-

cient (radicals ended by i-mono-

mer j)

K Ratio of depropagation-propaga-

tion rate coefficient

y Vector of the identifiedparameters

Product Quantities

IAQ (kmol) Total number of moles of initiator in

aqueous phase

I0 (kmol) Initial total number of moles of

initiator in aqueous phase

fAQi Molar fraction of monomer i in the

aqueous phase

f0i Initial molar fraction of monomer i in

the reactor

AAQi (kmol) Total moles number of free radicals

from monomer i in aqueous phase

[Z]AQ (kmol �m�3) Inhibitor concentration in aqueous

phase

[Z]P (kmol �m�3) Inhibitor concentration in particle

phase

NP (kmol) Total number of moles of particles

(number of particles/NAV)

Pi Molar fraction of free radicals ended

by monomer unit i in particles

NM (kmol) Total number of moles of micelles

(number of micelles/NAV)

SW (kmol) Total number of moles of surfactant in

aqueous phase

CMC (kmol �m�3) Critical micellar concentration of

surfactant

CSC (kmol �m�3) Critical surfactant concentration for

homogeneous nucleation

oi Fraction of free radicals ended by i

formed by 1 unit i in particles

o30 Fraction of free radicals formed by

only 2 AMS in particles

[Mj]P (kmol �m�3) Monomer j concentration in particle

phase

pk Fraction of free radicals ended by

monomer 3, ended by k units of 3

SP (kmol) Total number of moles of surfactant

on the particles

SM (kmol) Total number of moles of surfactant in

the micelles

SD (kmol) Total number of moles of surfactant

on the droplets



S0 (kmol) Total number of moles of surfactant in

the reactor

h Number of free radicals in a particular

particle

nh Fraction of particles with h free

radicals

�nn Average number of free radicals in a

particle

~nn Average number of pairs of free

radicals in a particle

NTi (kmol) Total number of moles of free mono-

mer i in the reactor

N0i (kmol) Initial total number of moles of free

monomer i in the reactor

NTZ (kmol) Total number of moles of inhibitor in

the reactor

N0Z (kmol) Initial total number of moles of

inhibitor in the reactor

�wwl Fraction of free radicals formed by lmonomer units

wl Fraction of polymer chains of lmonomer units

Wi Global mass fraction of monomer i in

the polymer

wP Mass fraction of polymer in the

particles

Fi Residual mass fraction of monomer i

X Total mass conversion

xa; xxa Measurement and model prediction

of the ath variable

Product Properties

dP (m) Average particle diameter

l1 Average number of monomer units per

free radical

L1 Average number of monomer units per

polymer chain

L2 Second momentum of number of

monomer units per polymer chain

Tg (8C) Glass transition temperature of the

copolymer

Tgi (8C) Glass transition temperature of the

homopolymer i

M(kg � kmol�1) Average molecular weight of one

monomer unit

Mn (kg � kmol�1) Number-average molecular weight of

the polymer chains

Mw (kg � kmol�1) Weight-average molecular weight of

the polymer chains

Others

NAV (kmol�1) Avogadro number (number of molecules

or particles in 1 000 moles)

p 3.1415927. . .dij Kronecker number (¼1 if i¼ j and 0 if not)

Na Number of measurements of the ath

variable

tab bth time of measurement of the ath variable

J Maximum likelihood criteria for para-

metric identification

Dimensional Constants

dM (m) Micelle diameter

dD (m) Droplet diameter

nS Number of surfactant molecules per

micelle

aS (m2 � kmol�1) Surface engaged by a surfactant

Volumes

VAQ (m3) Volume of aqueous phase

VP (m3) Volume of particle phase

VD (m3) Volume of droplet phase

VR (m3) Engaged volume of the reactor

VR0 (m3) Initial engaged volume of the reactor

VPOL (m3) Total volume of polymer

Thermodynamic Constants

mdi Equilibrium constant of free radicals ended by i

between aqueous and particle phases

s Ratio of the volumetric fraction of the monomer i,

or the inhibitor in the droplets by the volumetric

fraction of the monomer i, or the inhibitor in the

particles

KPi Ratio of the volumetric fraction of the monomer i in

the aqueous phase by the volumetric fraction of the

monomer i in the particles

KPZ Ratio of the volumetric fraction of the inhibitor in the

aqueous phase by the volumetric fraction of the

inhibitor in the particles

Physical Constants

Mi (kg � kmol�1) Molecular weight of monomer i

ri (kg �m�3) Density of monomer i

rPi (kg �m�3) Density of polymer i

rZ (kg �m�3) Density of the inhibitor

MZ (kg � kmol�1) Molecular weight of the inhibitor

Appendix 1

The thermodynamic equilibrium of the 3 monomers and the

inhibitor (Z) between the aqueous phase (AQ), the droplets

(D) and the particles (P) is given by the following



relationships:

VAQi

VAQ

¼ KPi

VPi

VP

ð92Þ

VDi

VD

¼ sVPi

VP

;when the droplets are present; 8i 2 f1; 2; 3; Zg

ð93Þ

VAQi, VPi and VDi are the volumes of monomer i in the three

phases, respectively.

If Vi is the total volume of monomer i in the reactor, we

have:

Vi ¼ VAQi þ VPi þ VDi ¼ NTi

Mi

ri

ð94Þ

Therefore:

VPi

VP

¼ Vi

VP

Vi

VPi

¼ Vi

VP

VAQi

VPi

þ 1þ VDi

VPi

� �

¼ Vi

KPiVAQ þ VP þ sVD

¼ NTiMi

riðKPiVAQ þ VP þ sVDÞð95Þ

which gives:

VPZ þP3

i¼1

VPi

VP

¼ NTZMZ

rZðKPZVAQ þ VP þ sVDÞ

þX3

i¼1

NTiMi

riðKPiVAQ þ VP þ sVDÞ

¼ VP � VPOL

VP

ð96Þ

In the absence of droplets, this relation is true with VD¼ 0.

Moreover, in the presence of droplets, we have:

VDZ þP3

i¼1

VDi

sVD

¼VPZ þ

P3

i¼1

VPi

VP

¼ 1

s¼ VP � VPOL

VP

ð97Þ

Then, the monomer i concentration in the particles is:

½Mi�P ¼VPiri

VPMi

¼ NTi

KPiVAQ þ VP þ sVD

ð98Þ

and in the aqueous phase is:

fAQi ¼

VAQiri

VAQMiX3

j¼1

VAQjrj

VAQMj

¼ KPi½Mi�PP3

j¼1

KPj½Mj�Pð99Þ

The corresponding inhibitor concentration in the parti-

cles is:

½Z�P ¼VPZrZ

VPMZ

¼ NTZ

KPZVAQ þ VP þ sVD

ð100Þ

and in the aqueous phase is:

½Z�AQ ¼VAQZrZ

VAQMZ

¼ KPZ

VPZrZ

VPMZ

¼ KPZ ½Z�P ð101Þ

Appendix 2

Lemma 1

Let a, b, g be respectively the number of MMA (index 1),

STY (index 2) and AMS (index 3) monomer units of the

same polymer chain.Dabg is the fraction of polymer chains

with a MMA units, b STY units, and g AMS units.

The exact calculation of Mn is possible using the

following relations:

M ¼ �aaM1 þ �bbM2 þ �ggM3

�aaþ �bbþ �ggð102Þ

Mn ¼ML1 ð103Þ

and the definition of the average number of monomer units

per polymer chain, L1 is

L1 ¼ �aaþ �bbþ �gg ¼X1a¼0

X1b¼0

X1g¼0

ðaþ bþ gÞDabg ð104Þ

where:

�aa ¼X1a¼0

X1b¼0

X1g¼0

aDabg ð105Þ

�bb ¼X1a¼0

X1b¼0

X1g¼0

bDabg ð106Þ

�gg ¼X1a¼0

X1b¼0

X1g¼0

gDabg ð107Þ

with:

D000 ¼ 0 andX1a¼0

X1b¼0

X1g¼0

Dabg ¼ 1 ð108Þ

Proof

The definition of the number-average molecular weight of

the polymer chains is:

Mn ¼X1a¼0

X1b¼0

X1g¼0

ðaM1 þ bM2 þ gM3ÞDabg ð109Þ



This equation gives

Mn ¼ �aaM1 þ �bbM2 þ �ggM3 ð110Þ

which gives the definition of �MM in lemma 1 by use of the

relationship between Mn and L1.

In practice we have

�aa ¼ N01 � NT1 � AAQ1 ð111Þ

�bb ¼ N02 � NT2 � AAQ2 ð112Þ

�gg ¼ N03 � NT3 � AAQ3 ð113Þ

when a, b and g are expressed in mole numbers.

Lemma 2

A good approximation of the weight-average molecular

weight is:

Mw ¼ML1

L2

ð114Þ

where

L2 ¼X1a¼0

X1b¼0

X1g¼0

ðaþ bþ gÞ2Dabg ð115Þ

with the relative error

1

L2

X1a¼0

X1b¼0

X1g¼0

�ða� �aaÞ

�M1

M� 1

�þ ðb� �bbÞ

�M2

M� 1

�

þ ðg� �ggÞ�

M3

M� 1

��ða� �aaÞ

�M1

Mþ 1

�

þ ðb� �bbÞ�

M2

Mþ 1

�þ ðg� �ggÞ

�M3

Mþ 1

��Dabg ð116Þ

Proof

The definition of Mw is given by:

MwMn ¼X1a¼0

X1b¼0

X1g¼0

ðaM1 þ bM2 þ gM3Þ2Dabg ð117Þ

This equation gives, with the definition of L2:

MwMn �M2L2 ¼

X1a¼0

X1b¼0

X1g¼0

ðaðM1 �MÞ

þ bðM2 �MÞ þ gðM3 �MÞÞðaðM1 þMÞ þ bðM2 þMÞ þ gðM3 þMÞÞDabg ð118Þ

However,

P1a¼0

P1b¼0

P1g¼0

ðaðM1 �MÞ þ bðM2 �MÞ þ gðM3 �MÞÞ

ð�aaðM1 þMÞ þ �bbðM2 þMÞ þ �ggðM3 þMÞÞDabg ¼ 0

ð119Þ

because �aaðM1 þMÞ þ �bbðM2 þMÞ þ �ggðM3 þMÞ is inde-

pendent of a, b and g.

So

MwMn �M2L2 ¼

X1a¼0

X1b¼0

X1g¼0

ðaðM1 �MÞ þ bðM2 �MÞ

þ gðM3 �MÞÞð a� �aað ÞðM1 þMÞþ ðb� �bbÞðM2 þMÞ þ g� �ggð ÞðM3 þMÞDabgÞ

ð120Þ

In the same way �aaðM1 �MÞ þ �bbðM2 �MÞ þ �ggðM3 �MÞ is

independent of a, b and g. So

X1a¼0

X1b¼0

X1g¼0

ð�aaðM1 �MÞ þ �bbðM2 �MÞ þ �ggðM3 �MÞÞ

ðða� �aaÞðM1 þMÞ þ ðb� �bbÞðM2 þMÞþ ðg� �ggÞðM3 þMÞÞDabg ¼ 0 ð121Þ

and

MwMn �M2L2 ¼

X1a¼0

X1b¼0

X1g¼0

ðða� �aaÞðM1 �MÞ

þ ðb� �bbÞðM2 �MÞ þ ðg� �ggÞðM3 �MÞÞð a� �aað ÞðM1 þMÞ þ ðb� �bbÞðM2 þMÞþ ðg� �ggÞðM3 þMÞÞDabg

ð122ÞFinally,

Mw �M L2

L1

M L2

L1

¼ 1

L2

X1a¼0

X1b¼0

X1g¼0

��a� �aa

��M1

M� 1

�

þ ðb� �bbÞ�

M2

M� 1

�þ ðg� �ggÞ

�M3

M� 1

��ða� �aaÞ

�M1

Mþ 1

�þ ðb� �bbÞ

�M2

Mþ 1

�

þ ðg� �ggÞ�

M3

Mþ 1

��Dabg ð123Þ

Relative Error Estimation on theWeight-Average Molecular Weight Calculation

The values of

ðða� �aaÞðM1 �MÞ þ ðb� �bbÞðM2 �MÞþ ðg� �ggÞðM3 �MÞÞDabg



and

ðða� �aaÞðM1 þMÞ þ ðb� �bbÞðM2 þMÞþ ðg� �ggÞðM3 þMÞÞDabg

are small, and

X1a¼0

X1b¼0

X1g¼0

ðaðM1 �MÞ þ bðM2 �MÞþ gðM3 �MÞÞDabg¼0

ð124Þ

X1a¼0

X1b¼0

X1g¼0

ðða� �aaÞðM1 þMÞ þ ðb� �bbÞðM2 þMÞ

þ ðg� �ggÞðM3 þMÞÞDabg¼0 ð125Þ

so the estimation of the weight-average molecular weight

by

Mw ffiML2

L1

ð126Þ

is good.

In order to estimate the relative error we can replaceMi

Mþ 1 by 2 and Mi

M� 1 by C (the value of C is about 0.07).

We obtain:

Mw �M L2

L1

M L2

L1

�� ffi 2C 1� L2

1

L2

� �ð127Þ

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Modeling of the Emulsion Terpolymerization of Styrene,α-Methylstyrene and Methyl Methacrylate

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Transcript of Modeling of the Emulsion Terpolymerization of Styrene,α-Methylstyrene and Methyl Methacrylate