A Review of Modeling of Diffusion Controlled Polymerization Reactions

Review

A Review of Modeling of Diffusion ControlledPolymerization Reactions

Dimitris S. Achilias

A plethora of models have been developed quantifying the effect of diffusion-controlledphenomena on polymerization reactions. The most prominent approaches are reviewed here,including innovative ones that have emerged over the last decade. Free-radical and step-growth polymerizations are considered in away to show that similar models have beenused in both mechanisms. In free-radicalpolymerization the models proposed are sub-divided according to their theoretical back-ground into four categories: (i) based ona Fickian description of reactant diffusion;(ii) free-volume theory based; (iii) chain-length dependent; and, (iv) empirical. Thereversible addition-fragmentation techniqueis discussed, together with two industriallyimportant case-studies, solid state polycon-densation and epoxy-amine curing.

1 Introduction

In polymerization reactions, besides the conventional

chemical kinetics associated with the polymerization

mechanism, physical phenomena related to the diffusion

of various chemical reactive species also play an important

role. In fact, in free-radical polymerizations, as the reaction

goes from zero to complete conversion, the viscosity of the

reacting mixture increases by several orders of magnitude.

Classical low conversion free radical kinetics does not

apply. An autoacceleration in rate begins at 30–50%

conversion, caused by a drastic decrease in the rate of

D. S. AchiliasLaboratory of Organic Chemical Technology, Department ofChemistry, Aristotle University of Thessaloniki, GR-541 24Thessaloniki, GreeceFax: þ30 2310 997769; E-mail: [email protected]

Macromol. Theory Simul. 2007, 16, 319–347

� 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

chain termination due to severe diffusional limitations.

Other reactions also become diffusion-controlled as the

polymer phase approaches its glass transition tempera-

ture. Evidence of diffusion effects on reactivity and poly-

merization mechanism has been apparent since early

experimental studies.[1–3] Since then a large number of

papers have been published devoted to studying the effect

of diffusion-controlled phenomena on free-radical poly-

merization kinetics. Very interesting reviews have

been written some decades ago by North[4] (1968); Mita

and Horie[5] (1987); O’Driscoll[6] (1989); Litvinenko and

Kaminsky[7] (1994); and, Dube et al.[8] (1997). Therefore,

one could assume that the subject of this contribution

would possibly be over-studied. However, the advent of

new techniques to carry out radical polymerizations,

such as in supercritical carbon dioxide[9] or the revers-

ible addition-fragmentation transfer, RAFT,[10] as well

as the increasing interest in photo-polymerization of

DOI: 10.1002/mats.200700003 319

D. S. Achilias

Dimitris S. Achilias occupies the position ofAssistant Professor in Polymer Chemistry andTechnology at the Department of Chemistry,Aristotle University of Thessaloniki (AUTh), Thes-saloniki, Greece. He obtained his diploma inChemical Engineering from AUTh in 1985 andhis Ph.D. degree in 1991 from the same depart-ment. The title of his thesis was ‘‘Theoreticaland Experimental Investigation of Diffusion-Controlled Free-radical Polymerization Reac-tions’’ and the work was carried out under thesupervision of Prof. C. Kiparissides. From 1993 to1996 he worked as a post-doctoral researchassociate at the Chemical Process EngineeringResearch Institute in Thessaloniki and in 2000 asa government official in the Department ofEnvironmental Design/Ministry of the Interior,Public Administration and Decentralization/Greece. He was elected as a Lecturer at theDepartment of Chemistry, AUTh, in 2000 andas an Assistant Professor in 2004. Dr. Achiliascurrent research interests are the modelling offree-radical and step-growth polymerizationkinetics, chemical recycling of polymers andcrystallization kinetics of polymers. He has pub-lished 41 papers in international refereed jour-nals and presented over 40 papers ininternational and local peer reviewed confer-ences. He has been member of the editorialboard of ‘‘Macromolecules: an Indian Journal’’since 2006 and serves as a reviewer for severaljournals.

320

multifunctional monomers[11–13] or biomaterials[14] re-

quire the use of comprehensive kinetic models that should

include the effect of diffusional phenomena on the reac-

tion mechanism. Apart from free radical polymerizations,

which have been extensively studied, diffusion phenom-

ena play also an important role in step-growth reac-

tions,[15] although this has been less studied in literature.

The objective of this review is to summarize progress

made mainly during the last decade in modeling of

diffusion-controlled phenomena in not only free-radical

but also step-growth polymerization reactions. Similarities

and differences between the models used are highlighted.

Therefore, the paper is subdivided into two main sections

related to the two mechanisms considered. In the free

radical polymerization section, the effect of diffusion

phenomena on the termination, propagation and initia-

tion reactions is presented initially, followed by the

different modeling approaches proposed and extensively

used recently. In step-growth reactions, besides the

presentation of the models used for diffusion controlled

reactions, two industrially important case studies are

considered, namely solid-state polycondensation and

curing of epoxy-amine resins.



2 Free-Radical Polymerization

The chemistry and kinetics of free-radical polymerization

is described in detail in two excellent books published

recently.[16,17] A simple mechanism of free-radical poly-

merization can be described in terms of the following four

elementary reactions:

Initiation

I �!kd2I� (1)

I� þM �!kIR�

1 (2)

Propagation

R�n þM �!

kp

R�nþ1 (3)

Chain transfer to monomer

R�n þM �!

ktr;mR�

1 þ Dn (4)

Termination by combination/disproportionation

R�n þ R�

m�!ktc

Dnþm

�!ktdDn þ Dm

((5)

In the above kinetic scheme, the symbols I, I� and M

denote the initiator, radicals formed by the fragmentation

of the initiator and monomer molecules, respectively. The

symbols R�n and Dn are used to identify the respective

‘‘live’’ macroradicals and the ‘‘dead’’ polymer chains,

containing n monomer structural units, respectively.

To describe the progress of the reaction and molecular

weight or chain structural developments during polymer-

ization, population mass balance equations are derived for

all chemical species present in the reactor. These constitute

a set of simultaneous differential equations which are

usually solved numerically provided that the appropriate

rate constants of every elementary reaction are known.[18]

However, the term ‘rate constants’ is somewhat of a

misnomer,[8] as these so-called rate constants vary during

the course of any polymerization. This variation was

assumed in order to quantitatively describe the effect of

diffusion-controlled phenomena on the polymerization

kinetics. That means that ki’s appearing in Equation (1) to

(5), which are influenced by diffusional phenomena

are to be regarded as ‘‘apparent rate constants’’ or ‘‘rate

coefficients’’ rather than ‘‘rate constants’’.[11,19,20]

In free-radical polymerization, the presence of diffu-

sion limitations is so well documented that different

DOI: 10.1002/mats.200700003

A Review of Modeling of Diffusion Controlled Polymerization Reactions

manifestations have been given particular names. The

impact of diffusion on the termination step is labeled as

the Trommsdorff or gel-effect, while the effects on the

propagation and initiation reactions are known as the

glass and the cage-effect, respectively. Different theories

for the modeling of such effects have been proposed. These

are reviewed next.

2.1 Diffusion-Controlled Termination Reaction

Termination in free radical polymerization involves the

reaction of two macroradicals; therefore, it is now

generally considered as always being diffusion con-

trolled[19] from the very beginning of polymerization

and even in solvents with very low viscosity. Evidence in

this regard has been recently summarized[20] and includes

the following: (1) an inverse dependence of kt on the

medium viscosity as anticipated for a diffusion-controlled

process, (2) a decrease of kt values with pressure meaning a

positive activation volume (the inverse is expected for a

chemical reaction involving the combination of two

species) and (3) an analogy with the chemistry of small

radicals, where it was found that the Smoluchowski

equation accurately predicts the observed kt and, as

a result, if small molecule termination is diffusion-

controlled then macromolecular termination should be

also diffusion-controlled. The consequence is that the

termination rate coefficient values does not only depend

on temperature and pressure as all rate coefficients, but

also on many other parameters that can exert an effect on

the diffusive motion of the polymeric radicals being

terminated. These parameters include polymer weight

fraction, solvent viscosity, polymer-monomer-solvent

interactions, chain lengths of the macro-radicals involved

in the termination reaction, chain flexibility, dynamics of

entanglements and the MWD of the surrounding matrix

polymer, through which a radical chain end must diffuse

to encounter another radical chain end.[20,21] Not only are

these parameters quite numerous but they also are

interrelated and their exact effect on the macroradicals

mobility is not yet exactly known. Termination rate

coefficients are thus highly system specific and can easily

Scheme 1. Diffusion and reaction steps during termination of two m



vary by 6 to 7 decades during the course of polymeriza-

tion.[21]

It is for this reason that many different theoretical

approaches have found use in modeling the termination

rate constant. Excellent reviews appear in ref.[5,7,8,21] Each

of these models simulates a limited number of experi-

mental data, usually conversion and average molecular

weights, versus time and provide a description of reality,

which is rather impossible given the different physical

bases of each model. A complicating factor has been that

the experimental data used for modeling purposes may

have been taken under nonisothermal conditions,[22] a

detail not accounted for by such models. Some researchers

try to present detailed models based on the microscopic

phenomena occurring during termination, although not to

predict macroscopic properties over the entire conversion

range, while others are able to predict the full conversion

and average molecular weight time evolution although a

number of adjustable parameters are used. A single

all-encompassing model is still unfulfilled.[20]

2.1.1 Mechanism of Bimolecular Termination

Benson and North[23,24] first described a three staged

mechanistic picture of the diffusion-controlled nature

of the bimolecular termination reaction, depicted in

Scheme 1. This scheme suggests that for termination to

occur two polymer coils must first come into contact as a

result of center-of-mass or translational diffusion (step 1).

Once this contact has been made, a segmental reorienta-

tion of the two macro-radicals has to occur (step 2) to bring

both reactive chain ends in close proximity (i.e., within a

capture radius) to form a radical-radical encounter pair.

The third and final step comprises the actual termination

reaction itself, in which the two radical functionalities are

annihilated.

The above description of the sequence of events is a good

approximation when the concentration is dilute and

individual coils can be identified. As conversion increases,

the polymer chains and the macroradicals begin to form

entanglements and translational as well as segmental

diffusion are significantly retarded. The situation becomes

acroradicals.

www.mts-journal.de 321

D. S. Achilias

Figure 1. Indicative results of polymerization rate, Rp, conversion, X and �ln(1–X) versustime[26] from polymerization of MMA at 80 8C with AIBN 0.03 mol � L�1, presenting theclassification of reaction into four regimes.

322

more complex since, besides diffusing,

the radical size also continuously

increases with time via the propaga-

tion step. As a consequence, during the

lifetime of a radical, no single molecu-

lar weight can realistically represent

the radical, making it difficult to apply

the scaling concepts. Also, at high

degrees of conversion, besides center-

of-mass diffusion and segmental re-

orientation, the description of the

diffusion behavior of propagating poly-

mer coils can be further complicated by

other modes of diffusion, such as

reaction diffusion. The later is asso-

ciated with the diffusive motion of the

macroradical chain end as a result of

propagation, which lengthens the radi-

cal chain and eventually moves the

radical end in spatial position. How-

ever, all of these different modes of

diffusion are not equally important

over the entire range of conversion

Figure 2. Variation of termination rate coefficient, kt, with frac-tional monomer conversion for polymerization of methyl meth-acrylate at 0 (&) and 50 8C (&).[20,28]

and therefore the polymerization is divided into three[21]

or sometimes four conversion regimes.[25] These could be

classified from macroscopic measurements of the rate of

polymerization or monomer conversion versus time

(Figure 1). As has been reported,[25] in the first stage of

polymerization (low conversion), the conversion-time

curve, as well as polymerization rate Rp, versus time t,

follows ‘classical’ free-radical kinetics and all kinetic rate

coefficients remain constant. A plot of �ln(1–X) versus t is

almost linear. The crossover between regime I and II

denotes the onset of the gel-effect and it corresponds to

10–40% conversion. Regime II is characterized by a sharp

increase in the polymerization rate followed by an increase

in conversion. The maximum in the Rp versus t curve

marks the crossover between regimes II and III. In regime

III, the reaction rate falls significantly and the curvature of

the conversion versus time changes. Finally, at very high

conversions beyond 80–90%, the reaction rate tends

asymptotically to zero and the reaction almost stops

before the full consumption of the monomer (regime IV).

This is a situation that happens when the polymerization

temperature is below the glass transition temperature and

at this point the Tg of the monomer-polymer mixture

approaches the reaction temperature; thus, a glassy state

appears and it corresponds to the well known glass-effect.

This rather peculiar behavior during polymerization

could in a great extent be explained by following the

change of kt with conversion. A variation of the termina-

tion rate coefficient, kt with monomer conversion for the

polymerization of MMA from experimental data has been

presented by Buback et al.[20,27,28] (Figure 2). Almost the



same variation of kt with X has been experimentally

observed by Zetterlund et al.[29,30] for the bulk polymer-

ization of styrene studied by Electron Spin Resonance

(ESR), Fourier-Transform Near-Infra-Red (FT-NIR) and Gel

Permeation Chromatography (GPC).

The data in Figure 2 are described as follows: at low

conversion a ‘plateau region’ is observed and kt remains

relatively constant [20] over approximately 15–20% con-

version. This conversion range has been assigned to

termination rate control by segmental diffusion.[23,24,31]

It is noteworthy that the initial conversion range in

which kt remains approximately constant is considerably

DOI: 10.1002/mats.200700003


dependent on the monomer type and to a less extent on

temperature. Monomers such as dodecyl acrylate,[32] or

dodecyl methacrylate[33] show an initial plateau region

extending to much higher values of X compared to MMA or

styrene.[29] This has been proposed to have its origin on the

effect of steric hindrance on the termination step.[33]

According to Buback et al.,[20] in this conversion regime

although the polymerizing medium viscosity is changing,

kt remains relatively constant due to the fact that

microscopic viscosity, which is the main influence on kt,

is different to macroscopic (bulk) viscosity and remains

constant. Monomers that show this type of behavior for

their termination rate coefficient were classified by Buback

et al.[27] as class ‘B’. On the other hand, this initial plateau

region is absent in the so-called class ‘A’ monomers, (of

which butyl acrylate is considered typical) and transla-

tional diffusion is the rate-controlling step from the very

beginning of the polymerization.[27] In the free-radical

crosslinking polymerization of dimethacrylate monomers

it was found that this initial plateau region ranges only to

approximately 0.5% conversion.[14]

As the conversion increases, beyond approximately 15%,

center-of-mass diffusion of macroradicals becomes the

rate-determining step and kt decreases as the viscosity of

the system increases. This brings about an increase in total

macroradical concentration and finally an increase in the

polymerization rate.

At the conversion interval from approximately 40–80%

(regime III), the observed decrease in kt is not so abrupt but

only gradual. At this stage, the center-of-mass motion of

radical chains becomes very slow and any movement

of the growing radical site is attributed to the addition of

monomer molecules at the chain end (Scheme 2). This

additional diffusion mechanism is so-called ‘reaction

diffusion’. The more flexible class ‘A’ monomers seem to

have a relatively high kp, whereas class ‘B’ monomers are

stiffer with much lower values of kp.[27] The higher the kp

value the more likely reaction-diffusion is to be rate

determining. This is the reason for the much wider

conversion range where termination is reaction-diffusion

controlled for class ‘A’ monomers compared to class ‘B’.

Since the activation energy for propagation far exceeds

Scheme 2. Schematic diagram illustrating the implicit movementof the macroradical and eventually termination by the reaction-diffusion mechanism.



that of center-of-mass diffusion, an increase in tempera-

ture causes the conversion range over which reaction-

diffusion is rate determining to become more extensive.[27]

It is interesting to note that in the photopolymerization

of multifunctional monomers leading to highly cross-

linked polymers, network formation causes the reaction-

diffusion termination mechanism to be much more

pronounced and to set on at lower conversions[11,34]

(sometimes as low as 2–3% conversion in the case of

Bis-GMA polymerization[35]). In these polymerizations,

since crosslinked networks are formed the motion of the

full radical chain is very much hindered and the only way

for radical movement diffusion in space is through the

addition of monomer molecules.

Finally, after approximately 80–90% conversion kt again

decreases markedly with conversion following the

decrease in the propagation rate coefficient (glass-effect).

Recently, O’Neil et al.[36,37] carried out a critical experi-

mental examination and provided a modeling insight into

the diffusion-limited cause of the gel-effect in FRP. They

proved that the theory that the onset of entanglement

between macromolecular chains causes the gel-effect is

incorrect. Trends concerning the effects of temperature,

polymer concentration and molecular weight on the

gel-effect onset conversion were not predicted by this

theory. In a subsequent paper,[38] the same authors

concluded that termination related to the gel-effect is

not controlled by chain-end segmental mobility but

instead is controlled by diffusion of the shortest radical

chain present in reasonable number. This gave credence

to the general idea of ‘‘short-long’’ termination pro-

cesses,[39,40] in which short radical chains control the

termination reaction.[41] This picture postulates that

during the gel-effect the termination of a long chain

becomes so hindered due to diffusional limitations that it

can only terminate when a ‘‘short’’ chain diffuses into its

vicinity. As the population of ‘‘short’’ chains is presumably

small, at this point the overall rate of termination

decreases strongly. According to this theory, the poly-

merization rate in the gel-effect region is independent of

the initiation rate, while the mean living chain length is

inversely proportional to the initiation rate. Faldi et al.[42]

have also suggested ‘‘short-long’’ termination as a means

of describing their experimental data on MMA polymer-

ization, which shows that the concentration dependence

of kt more closely resembles that of the self-diffusion of

short PMMA chains than that of the self-diffusion of longer

chains.

2.2 Diffusion-Controlled Propagation

Propagation involves the reaction of small monomer

molecules and only one large radical, thus propagation is


D. S. Achilias

Figure 3. Qualitative results on the variation of the propagationrate coefficient, kp, and initiator efficiency, f, with fractionalmonomer conversion.

324

much less hindered during the reaction and kp remains

relatively unaffected until very high conversions. At

conversions beyond 80–90% the reaction rate tends

asymptotically to zero and the reaction almost stops,

‘‘freezing’’ before the full consumption of the monomer.

This appears in polymerizations taking place at tempera-

tures below the glass transition temperature of the

polymer and is related to the well known glass-effect.

At this point even the mobility of small molecules, such as

monomers, is restricted and as a result the propagation

reaction is assumed to be controlled by diffusion

phenomena and decreases with conversion. At this stage

because kt is now determined by reaction-diffusion

mechanism and is proportional both to the propagation

rate coefficient and monomer concentration, it also

indicates a rapid drop. At the limiting conversion the

glass transition temperature of the monomer-polymer

mixture becomes equal to the reaction temperature.

Accurate measurement of the decrease in kp is difficult,

often being confounded by a decrease in initiator

efficiency. Nevertheless, with ESR techniques some

researchers have been able to measure this effect with a

minimum of modeling assumptions. No decrease with

conversion was observed in the low and intermediate

conversion levels for MMA, as observed by Shen et al.,[43]

Zhu et al.[44] and Carswell et al.[45] Near the glass point of

the mixture, however, the value of kp begins to drop

rapidly.[46] A qualitative picture of the variation of the

propagation rate coefficient with conversion according to

data presented in references[25,46] appears in Figure 3.

2.3 Diffusion-Controlled Initiation

For a number of years, the diffusion controlled propagation

reaction was assumed to be solely responsible for the

reduction in the reaction rate observed at high conver-

sions. However, later comparative experiments on MMA

bulk and emulsion polymerization revealed that the

initiation reaction could also be diffusion-controlled.[47]

Data on the change in propagation rate coefficient with

increasing conversion presented by Ballard et al.[47] during

the emulsion polymerization of MMA revealed that the

rate of emulsion polymerization is higher than that of bulk

polymerization at equal conversions. In addition, emulsion

polymerization can be carried out to full monomer

conversion, whereas bulk polymerization stops before full

monomer consumption. Since the main difference

between the two techniques lies in the mode of initiation

(initiation starts in the aqueous phase during emulsion

polymerization and in the organic phase during bulk) it

was concluded that the initiator efficiency should decrease

at high conversions for bulk polymerization. The initiator

decomposition rate constant, kd, is unlikely to be affected



by viscosity and hence can hardly depend on the viscosity

of the reaction medium. Thus, the initiator efficiency was

the parameter assumed to strongly depend on diffusion-

controlled phenomena. Further experimental results

showed that initiator efficiency can dramatically change

with monomer conversion.[43,44,48]

It is worthy to mention here that, according to Figure 3,

the initiator efficiency, f, starts dropping earlier than kp.[25]

Besides MMA, the same phenomenon has also been

experimentally observed for styrene.[30] This earlier effect

of diffusion phenomena on the primary radicals compared

to monomer molecules is considered responsible for the

short decrease in the total radical concentration measured

experimentally by ESR in the conversion range 60–80% for

MMA and St polymerization.[44,30] At the point where kp

becomes diffusion-controlled and decreases with conver-

sion, eventually kt also decreases and the total radical

concentration increases again.

Finally, it should be noted, that the term ‘‘cage effect’’

was originally used in the definition of the initiator

efficiency, f. This empirical parameter was introduced in

order to account for all side reactions that could take place

between primary radicals formed from the initiator

decomposition inside an ‘ideal’ cage before they escape

and react with monomer molecules. Thus, any modeling

approach for f should include the effect of both the

initiation reaction kinetics and the diffusion phenomena

taking place at high conversions

2.4 Diffusion-Controlled Models

In this section, different theories that have been proposed

for modeling diffusion controlled reactions in free radical

polymerization systems are presented in detail. From the

DOI: 10.1002/mats.200700003


wide literature on this subject only models that were

extensively used during the last decade are reviewed. For

earlier models one can resort to the reviews of de Kock

et al.,[21] Dube et al.,[8] Litvinenko and Kaminsky,[7] Mita

and Horie[5] and Tefera et al.[49]

Inclusion of a model to account for diffusion controlled

phenomena into a general modeling approach of a

free-radical polymerization reaction/reactor should meet

the following demands:

– Adequate fitting of the experimental data on conversion

(or polymerization rate) versus time and final product

properties (such as averages of the molecular weight

distribution, etc.);

– Allowance for the eventual presence of other com-

pounds in the reaction medium (such as solvent, chain

transfer agent, etc.);

– The possibility of running under reactor optimization

conditions, such as in nonisothermal conditions (opti-

mal temperature profile), or intermediate addition of

initiator(s);

– All elementary reactions should be considered as

diffusion controlled (i.e., the effect of diffusion-

controlled phenomena should be taken into considera-

tion for the termination, propagation, initiation and

chain transfer reactions);

– The number of adjustable parameters should be kept as

small as possible and they should have a physical

meaning; and,

– The use of critical break points denoting the onset of a

phenomenon should be avoided.

Even though the categorization of diffusion controlled

models in different groups is not definite due to their

hybrid modeling background, the following classification

has been set according to their theoretical background.

2.4.1 Models Based on a Fickian Description

of Reactant Diffusion

2.4.1.1 Modeling the Diffusion-Controlled Termination

and Propagation Reactions

The approach adopted in this model category resembles

that used in the classic monograph on the collision theory

of chemical reactions in liquids by North.[50] The following

description of the diffusion process was the basis for the

model originally derived by the group of Soong[51,52,53] and

extended by Achilias and Kiparissides[25,54] (CCS-AK). It is

subsequently presented taking also into considerations

the suggestions proposed by Litvinenko and Kaminsky[7]

and Zhu et al.[55] Although this model was developed more

than ten years ago, it has found a lot of applications during



the last decade in modeling of several polymerization

systems under different reaction conditions and polymer-

ization techniques.[9,10,14,25,56–70] It was also used in

advanced and nonlinear control studies of polymerization

reactions.[71–76]

In this modeling approach, diffusion effects are viewed

as an integral part of the termination, initiation and

propagation reactions from the beginning until the end of

polymerization. This eliminated the need for the use of

critical break-points denoting the onset of diffusion effects

and the associated segmentation of the model in different

parts.

Accordingly, in order to evaluate the effect of diffusion

limitations on the reaction rate between two chemical

species, A and B, one can consider the concentration of B

around a single molecule of A (Scheme 3). This molecule of

A is considered to be placed at the center of two concentric

spheres. The inner, with radius rAB, characterizes the

minimum separation within which all diffusive motions

have been completed and the two molecules can react.

At a large distance rb, the concentration of molecules B

approaches the unperturbed bulk concentration, CBb. If J is

the net average flux of B molecules towards A molecules,

the steady state solution of the Fick’s first law in spherical

coordinates leads to:[50,58]

4pr2DABdCB

dr

� �r

¼ J (6)

with boundary conditionsCB ¼ Cm

B ; r ¼ rAB

CB ¼ CbB; r ¼ rb

�

where DAB is the mutual diffusion coefficient of the

reactive molecules, usually taken as the sum of the two

self-diffusion coefficients (i.e., DAB¼DA þDB), CBm is the

concentration of B at the collision radius rAB and CBb the

bulk concentration (at infinite distance from A, rb).

Notice that in deriving Equation (6) it is implicitly

assumed that the diffusion coefficient DAB is constant (i.e.,

independent of the radius, r). According to Buback et al.,[20]

in general D cannot be assumed to be independent of r and

using Equation (6) means that D represents an ‘effective’

diffusion coefficient. Alternatively, if Equation (6) holds

and is used with D as the mutual translational diffusion

coefficient of two macroradicals (as it was considered

when this model was developed) then the radius rAB

should denote an ‘effective’ reaction radius.

At steady state, the diffusive mass flux, J, is assumed

equal to the rate of depletion of molecules B due to

chemical reaction, thus:

J ¼ k0CmB

1

NA(7)


D. S. Achilias

Scheme 3. Schematic diagram illustrating the diffusion controlledphenomena on the propagation (a), termination (b) and initiation(c), reactions.

326

where NA is Avogadro’s number and k0 an intrinsic

reaction rate constant.

Combining the solution of Equation (6) with Equa-

tion (7) and assuming that rb� rAB, is the following is

obtained:

Macrom

� 2007

CmB ¼ Cb

B

1 þ k0=4pNArABDAB(8)

If the symbol keff is used to denote the effective

(observed) rate coefficient of the reaction (i.e., that

corresponding to the reaction rate expressed at the bulk

ol. Theory Simul. 2007, 16, 319–347

WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

concentration of B) then the overall reaction rate is

expressed as keff CBb. This rate equals k0 CB

m, so that:

keffCbB ¼ k0C

mB (9)

Combining Equation (8) and (9), the effective rate

coefficient is expressed as:

1

keff¼ 1

k0þ 1

4pNArABDAB(10)

The last term in Equation (10), 4pNArABDAB, is the

well-known simple formula of Smoluchowski for the

diffusion controlled reaction between two solid spherical

particles. Then, Equation (10) expresses the parallel

summation of two terms: one kinetically controlled (k0)

and the other diffusion controlled (4pNArABDAB). As will be

shown later in the paper, Equation (10) constitutes the

‘cornerstone’ of any model on the diffusion control of rate

coefficients.

According to this theory (Scheme 3a), if the diffusion

molecule B is a monomer, then keff represents the effec-

tive propagation rate coefficient, which is therefore

expressed as:

1

kp;eff¼ 1

kp;0þ 1

4pNArpDMP(11)

with rp denoting the radius of interaction for propagation.

The mutual diffusion coefficient, DAB, between a

macroradical chain end and a monomer molecule, DMP

in this case, is assumed equal to the sum of the two self

diffusion coefficients, that is:

DMP ¼ DAB ¼ Dm þ Dp (12)

However, the diffusion coefficient of the macroradical

chain end, DP is much slower compared to the self diffusion

coefficient of the monomer molecule, Dm,[58] permitting

thus the assumption, DAB ¼DMPffiDm. Indeed, for example,

for MMA[77] the value measured for Dm initially is in the

order of magnitude of 10�5–10�6 cm2 � s�1 (at 50 8C) and at

90% conversion is lowered to 10�9 cm2 � s�1 compared to Dp

which is in the order of magnitude of 10�13 cm2 � s�1.

Continuing, if the diffusion molecule B is a macroradical

(Scheme 3b) then keff represents the effective termination

rate coefficient, which is therefore expressed as:

1

kdift;eff

¼ 1

kt;0þ 1

4pNArtDp;xy(13)

with rt denoting the radius of interaction for termination

and Dp,xy the mutual diffusion coefficient between two

DOI: 10.1002/mats.200700003


polymerizing chains with degrees of polymerization x and

y, respectively. Again, the mutual diffusion coefficient is

given by the sum of the self diffusion coefficients of the

two species:

Macrom

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Dp;xy ¼ Dp;x þ Dp;y (14)

In Equation (11) and (13), kt0 and kp0 represent the

intrinsic rate constants for termination and propagation,

respectively. Recently, a great effort was undertaken by an

IUPAC working party in proposing benchmark values for

these rate constants for a number of polymerizing

systems.[20,78–83] These are also summarized by Moad

and Solomon.[16]

Evaluation of the physical parameters, DAB and rAB

involved in Equation (11) and (13) is discussed in later

sections.

2.4.1.2 Modeling Reaction-Diffusion

In order to take into consideration the effect of reaction-

diffusion according to the mechanism reported previously,

two approaches have been used in literature. In the first

approximation a second term is added in the termination

rate coefficient so that the overall effective termination

rate coefficient, kt,eff is expressed as:[25,59]

kt;eff ¼ kdift;eff þ krd

t (15)

where ktrd was assumed proportional to the frequency of

monomer addition to the radical chain end and was

defined as:

krdt ¼ Akp;eff ½M� (16)

where A is a proportionality constant.

Buback et al.[27] examined different equations for

evaluating this parameter. Russell et al.[84] defined an

upper and a lower bound for this parameter, as:

Amin ¼ 4

3pa2s (17)

and,

Amax ¼ 8

3pa3j1=2

c (18)

where, a is the root-mean-square end-to-end distance per

square root of the number of monomer units, s is the

Lennard-Jones diameter and jc the entanglement spacing

of the pure polymer.

The second way of including the reaction-diffusion

mechanism in the termination reaction, is in the macro-

ol. Theory Simul. 2007, 16, 319–347


radical diffusion coefficient, Dp,x. Accordingly, Dp,x is

assumed to be the sum of the center-of-mass diffusion

coefficient, Dpcom,x and the diffusion coefficient due to

reaction-diffusion, Dp,rd:[10,14]

Dp;x ¼ Dpcom;x þ Dp;rd (19)

For the calculation of the polymer diffusion coefficient

due to reaction-diffusion, it has been proposed that it must

be proportional to the frequency of monomer addition to

the live radical chain.[84] The following equation was

proposed by Russell et al.[84] and slightly modified by

Buback et al.:[85]

Dp;rd ¼ b kp;eff ½M�a2

6(20)

where the term b has been added to take into considera-

tion both the totally-flexible and the rigid-chain limits. Its

value was proposed to be around 0.6.[85] The estimation of

Dpcom,x is provided in a subsequent section.

2.4.1.3 Modeling the Diffusion Controlled

Initiation Reaction

Using the same basic idea described above, the effect of

diffusion-controlled phenomena on the initiation reaction

was considered by letting the initiator efficiency, f change

during the reaction and depend on the diffusion coefficient

of the primary initiator radicals DI (Scheme 3c):[25]

1

f¼ 1

f0þ r3

2

3r1

ki0½M�f0

1

DI(21)

where r1 and r2 are the radii of the two concentric spheres

in which diffusion is assumed to occur, ki0 is the intrinsic

chain initiation rate constant and f0 the initial initiator

efficiency.

In Equation (21) the effect of possible loss of primary

radicals through the recombination reactions is considered

in f0, while as DI decreases at high conversions the overall

f also decreases.

Shen et al.,[43] based on experimental measurements

using EPR, determined a value of f as a function of

conversion during polymerization of MMA. The following

expression was proposed accounting for diffusion and

combination of the primary radicals in the ‘‘cage’’:

1

f¼ 1 þ r0k0

DI(22)

Taking the initiator efficiency and diffusion coefficient

at zero conversion, as f0 and DI0, respectively, the following


D. S. Achilias

328

expression for f was derived; it was also used by Muller

et al.:[58]

Macrom

� 2007

1

f¼ 1 � DI0

DI1 � 1

f0

� �(23)

An equation similar to Equation (22) was used recently

in modeling dimethacrylate free-radical polymeriza-

tion:[14]

1

f¼ 1 þ C

DI(24)

where C was set as an adjustable constant.

All of Equations (21)–(24) show a proportional depen-

dence of f on the diffusion coefficient of the primary

initiator radicals, DI. The evaluation of DI is discussed in the

following section.

2.4.1.4 Calculation of Diffusion Coefficients

The estimation of the diffusion coefficients of small

molecules in polymer systems has been a matter of

interest for several decades and by many investigators. A

recent review can be found in Masaro and Zhu.[86] Several

models have been proposed based on obstruction effects or

hydrodynamic theories. However, the model that found a

great applicability in polymerization reactions was that

based on the free volume theory. It is beyond the scope of

this paper to present details on different models, although

the interesting reader can refer to Masaro and Zhu for

details.[86] The first model based on the free volume theory

was proposed by Fujita.[87] According to this model the

self-diffusion coefficient of a small molecule diffusing in a

binary monomer-polymer system was defined as:

D ¼ ART exp � B

Vf

� �(25)

where A is a proportionality factor, R the gas constant, T

temperature, B a constant depending only on particle size

and not on polymer concentration or temperature and Vf

the free volume of the mixture calculated from the

contributions of both the diffusing molecule and the

polymer:

Vf ¼X

’iVf;i (26)

where Vf,i is the free volume contribution of every

component in the mixture (i.e., i¼ 1 for the diffusing

molecule, i¼ 2 for polymer, etc.) and wi is the volume

fraction of every component i.

ol. Theory Simul. 2007, 16, 319–347


A major contribution to the development of the free

volume theory was made later by Vrentas and Duda

et al.[88–91] With numerous improvements, the free volume

theory of Vrentas and Duda takes into account several

physical parameters such as temperature, activation

energy, polymer concentration, solvent size and molecular

weight of the diffusant. In the case of a ternary system

consisting of the monomer (i¼ 1), the primary initiator

radicals (i¼ 2) and the polymer (i¼ 3), the diffusion

coefficient of the monomer can be expressed by the

following equation:[25]

Dm ¼ Dm0 exp � Em

RT

� �exp � gV�

1MJ1

VF

X3

i¼1

vi

MJi

!(27)

where Dm0 is a pre-exponential factor, Em is the activation

energy, g represents the overlap factor, V1� is the specific

volume needed for one jumping unit of component i

having molecular weight MJi, vi is the weight fraction of

component i and VF is the free volume of the mixture,

expressed as:

VF ¼X3

i¼1

viV�i Vf;i (28)

where Vfi denotes the fractional free volume of the

component i, that can be calculated using the glass

transition temperatures, Tg,i and the difference in thermal

expansion coefficient between the liquid and the glassy

state of component i, ai, from:

Vf;i ¼ Vg;i þ aiðT � Tg;iÞ (29)

where Vg,i is the free volume of component i at the glass

transition temperature.

The estimation of the diffusion coefficient of any other

small molecule, such as the primary radicals formed from

the initiator decomposition is straightforward:

DI ¼ DI0 exp � EI

RT

� �exp � gV�

2MJ2

VF

X3

i¼1

vi

MJi

!(30)

The calculation of the polymer diffusion coefficient Dp,

is not so straightforward since the molecular weight or

chain length of the diffusing macro radical should be

considered. As will be discussed later, a better approxima-

tion is to use the individual chain length of every diffusing

macroradical. However, since it very much complicates

the calculation required in order to estimate the change

of macroscopic polymer properties (average molecular

DOI: 10.1002/mats.200700003


weight number of terminal double bonds, etc.) and

conversion versus time, the assumption made by AK[25]

was that the diffusion coefficient of the polymer to be

inversely dependent on an average cumulative molecular

weight of the polymer. Thus, the contribution of the center

of mass diffusion of polymer according to the extended

free volume theory is given by:

Macrom

� 2007

Dpcom ¼ Dp0

Mnx

exp � gV�3MJ3

VF

X3

i¼1

vi

MJi

!(31)

where, the weight average molecular weight, Mw was

used as an average molecular weight in place of Mx and the

superscript n was set equal to 2, according to the reptation

theory.[54]

The free-volume theory has the main advantage of

providing equations for various diffusion coefficients

which can be applied to the whole range of monomer

conversion based on quantities that can be independently

measured from the physical properties of the system.[54,92]

However, its main drawback is that it requires a rather

large number of parameters which may not be available

for some systems. It has been experimentally evaluated

not only by Vrentas and Duda[88,89] and Vrentas and

Vrentas[90] but also by Faldi et al.[77] in polymerization

systems. O’Neil et al.[37] found that the free-volume theory

may be an appropriate basis for modeling the gel-effect.

However, it cannot by itself be used to predict accurately

conversion-time results for a broad range of conditions and

is not a molecular level theory since it does not account for

radical chain length effects in the rate of termination.[37]

2.4.1.5 Calculation of the Reaction Radius

The estimation of the ‘effective’ radius in which the

termination of macroradicals could be possible is not so

straightforward. AK[25] in their original model used the

following equation taken from Soh and Sundberg:[93]

rt ¼1

tln

1000t3

NA½R��p3=2

� �� 1=2

(32)

with,

t ¼ 3=2jca2

� �1=2(33)

where jc is the entanglement spacing and a is the

root-mean-square end-to-end distance per square root of

the number of monomer units.

According to the so-called flexible-chain limit used by

Maxwell and Russell,[94] a polymer chain is characterized

ol. Theory Simul. 2007, 16, 319–347


by nodes of entanglement every jc monomer units (it is

these nodes that restrict the center of mass motion of the

chain as a whole). Then, the radius of interaction for

termination will be given by the distance of the chain end

from the node of entanglement closest to it, that is:

r2t ¼ jca

2 (34)

Furthermore, jc can be either considered to be constant

or conversion dependent and decrease with an increase of

the polymer volume fraction, wp, according to:

1

jc¼ 1

jc0þ

2’p

Xc0(35)

where Xc0 is the critical degree of polymerization for the

entanglement of polymer chains and jc0 the entanglement

spacing at zero polymer conversion.

According to Buback et al.,[27] jc should scale with the jc0,

which is the entanglement spacing for pure polymer,

according to:

jc ¼jc0

’xp

(36)

where x lies in the range 0< x< 2. In a recent publication

by Buback et al.[85] the x value was set equal to 2.

A sensible estimate of rp is the size of a monomer

molecule (the free radical chain end is also a monomer

unit), which according to Maxwell and Russell[94] can be

approximated by the Lennard-Jones diameter of the

monomer, sm, (i.e., rp ¼ sm).

2.4.1.6 Extensions of the CCS-AK Modeling Approach

Fleury et al.[95] and Nising and Meyer[96] modified the CCS

model in order to simplify it and to adapt it for the

polymerization of MMA above its Tg. The following

equation was used for the effect of conversion, X, on the

termination rate coefficient, kt:

1

kt¼ 1

kt;0þ g

expða� bXÞ (37)

where a and b are adjustable parameters estimated from a

number of experiments and related to the reaction

temperature, initial initiator concentration, solvent con-

tent and Tg. kp was assumed to be unaffected during the

reaction since T> Tg. Furthermore, Fenouillot et al.[97,98]

modified Equation (37) in order to facilitate the estimation


D. S. Achilias

330

of parameters a and b:

Macrom

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1

kt¼ 1

kt;0þ 1

kt;0 exp½bðXc � XÞ� (38)

where Xc is an adjustable parameter estimated as a

function of temperature and initial chain transfer agent

concentration and b a parameter depending only on

temperature. A very good fit to experimental data was

achieved.

Moreover, the group of Gupta in a series of papers

extended the CCS-AK model, in a semi-theoretical

way,[99–103] to account for reactors operating under

non-isothermal conditions or in cases with intermediate

addition of compounds (semi-batch operation). These

authors started from the basic equation of the CCS model–

Equation (11), (13) and (21) – and finally came to the

following equations describing the effect of diffusional

phenomena on the termination and propagation rate

coefficients as well as on initiator efficiency.

1

kt¼ 1

kt;0þ utðTÞm2

nðl0=VlÞ1

exp½�ðx� xrefÞ�(39)

1

kp¼ 1

kp;0þ upðTÞðl0=VlÞ

1

exp½�j13ðx� xrefÞ�(40)

1

f¼ 1

f01 þ uiðTÞðM=VlÞ

1

exp½�jI3ðx� xrefÞ�

� �(41)

where the subscript 0 denotes intrinsic rate constants, mn

is the polymer number average chain length and l0 the

zero moment of the polymer CLD, meaning polymer

concentration;

x ¼ gV�3MJ3

X3

i¼1

viV�i

V�i MJi

1

VF

� �(42)

xref ¼g

Vfp(43)

ji3 ¼ V�i ðMWiÞV�pMJp

(44)

where i¼ 1, 2, 3 refer to monomer, solvent and polymer,

respectively, and ut, up and ui are adjustable parameters

depending on temperature; the free volume of the mixture

Vf is given by Equation (28). It should be kept in mind that,

when dealing with modeling of semi-batch reactors and/

or nonisothermal conditions, the parameters in Equa-

tion (39) to (44) denote the ‘current’ or ‘local’ values. Details

can be found in the original manuscripts.[99–103]

ol. Theory Simul. 2007, 16, 319–347


Experimental results of monomer conversion and

average molecular weights were in reasonable agreement

with model predictions.

2.4.2 Semi-Empirical Models Based on the

Free-Volume Theory

A semi-empirical model based on the free-volume theory

was originally developed by the group of Hamielec and

later extended and used by the groups of Hamielec,

Penlidis, Kiparissides and Vivaldo-Lima.[104–116] According

to this model, two distinct regimes are considered for the

effect of diffusion-controlled phenomena on the termina-

tion rate coefficient. A parameter K is calculated and

compared to a critical value Kcr used to define the onset of

diffusion controlled phenomena on the termination

reaction:[111]

K ¼ Mmw exp

A

Vf

� �; Kcr ¼ Acr exp

Ecr

RT

� �(45)

where, Mw is the polymer cumulative weight-average

molecular weight, Vf is the total free volume, and m, A, Acr

and Ecr are parameters dependent upon monomer type.

If K is less than Kcr (at the initial stages of polymeriza-

tion) the termination rate coefficient is dominated by

segmental diffusion. With increasing polymer concentra-

tion, the reaction medium becomes a poorer solvent for the

polymer. The coil size of propagating radical chains

decreases and, effectively, there is a higher concentration

gradient across the coil. The segmental diffusion of the

radical site out of the coil increases and in turn, produces

an increase in the termination rate coefficient shown by

the following equation:

kt;seg ¼ kt0ð1 þ dcCpMWÞ (46)

where kt0 is the chemically controlled rate constant, dc is a

segmental diffusion parameter, Cp is the polymer con-

centration and MW the molecular weight of the monomer.

When K becomes equal to Kcr defines the onset of

autoacceleration region known as the ‘‘gel’’ effect. At this

point in the reaction, Vf and Mw are taken to be at their

critical values, Vf,cr and Mw,cr and the termination rate

coefficient decreases according to:

kT ¼ kt0Mw;cr

Mw

� �n

exp �A1

Vf� 1

Vf;cr

� �� (47)

where n and A are monomer dependent adjustable

parameters.

(a) The overall termination rate coefficient, kt was

estimated from a serial summation of three individual

DOI: 10.1002/mats.200700003


termination rate coefficients, including segmental diffu-

sion, kt,seg, translational diffusion, kT and ‘‘reaction

diffusion’’ control, kt,rd, that is:[8,108]

Macrom

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kt ¼ kt;seg þ kT þ kt;rd (48)

where kt,rd was estimated according to Equation (16)–(18)

with the two bounds handled with the following

expression:

kt;rd ¼ kt;rd;minX þ kt;rd;maxð1 � XÞ (49)

where X is the conversion.

According to Vivaldo-Lima et al.,[115] the serial summa-

tion provides more accurate and reliable prediction

compared to the parallel summation model used by AK.[25]

The propagation rate coefficient is similarly affected by

a diffusion-controlled process and will start to decrease

when the free volume is below a critical value of the

monomer free volume, Vf,crM:

kp ¼ kp0 exp �B1

Vf� 1

Vf;crM

� �� (50)

where kp0 is the chemically controlled propagation rate

constant and B is a monomer specific parameter.

Towards the end of the reaction the initiator efficiency

was also assumed to be diffusion controlled. As was

mentioned earlier, at high conversions the efficiency

begins to drop as initiator radicals are increasingly

hindered from moving out of their cage due to the

growing presence of larger molecules. The efficiency was

expected to decrease dramatically when a critical free

volume for the particular initiator is reached, Vf,crEff:

f ¼ f0 exp �C1

Vf� 1

Vf;crEff

� �� (51)

where f0 is the initial initiator efficiency and C is a

constant.

The fractional free volume of the mixture, Vf, is

expressed as the sum of the individual fractional free

volumes, Vf,i, of the monomer, polymer, solvent or other

component present in the reaction mixture weighted on

their volume fraction, wi, according to:

Vf ¼Xi

Vf;i’i; i ¼ m; p; s; etc: (52)

where the subscripts m, p, s refer to monomer, polymer

and solvent, respectively. The free volume of every

component and its volume fraction are expressed as:

Vf;i ¼ 0:025 þ aiðT � Tg;iÞ; ’i ¼ Vi=V (53)

ol. Theory Simul. 2007, 16, 319–347


where, Vi is the volume of each component and V the total

reaction volume.

Equation (53) comes from Equation (29) by setting

Vg,i¼ 0.025.

2.4.3 Models Based on Chain-Length Dependent Rate

Coefficients

2.4.3.1 Chain Length Dependent Termination

According to this modeling approach, during the termina-

tion reaction in a polymerizing system, since macro-

radicals of a variety of sizes are present, one single rate

coefficient, kt is not adequate to describe all the termina-

tion interactions occurring at any instant. Thus, termina-

tion is better described in terms of kti,j the rate coefficient

for termination between radicals of degree of polymeriza-

tion i and j respectively.[19,20] Hence, the termination rate

coefficient should depend on the sizes of the two

terminating chains. If this is the case then all polymerizing

radicals can not be grouped in one value [R], but in

modeling FRP kinetics one must consider each [Ri],

independently. However, the basic equation denoting

the time variation of radicals still holds if kt is replaced by

hkti, the overall or average termination rate coeffi-

cient,[117,118] that is:

d½R�dt

¼ 2fkd½I� � 2 kth i½R�2 (54)

The chain-length averaged termination rate coefficient

is defined according to:

kth i ¼X1i¼1

X1j¼1

ki;jt

½Ri�½Rj�½R�2

; ½R� ¼Xall i

½Ri� (55)

Since the value of hkti is determined from the values of

[Ri], initiation, propagation and transfer all play a role in

determining the value of hkti. It follows from Equation (55)

that knowledge of the individual values of all kti,j is in

principle required to estimate the values of hkti. In order to

estimate how the termination rate coefficient varies with i

and j, many models have been proposed. In this review,

three of the most common models for kti,j are presented

next.[16,19]

(a) The geometric mean model (GMM) for termination,

which is:[19,117]

ki;jt ¼ k1;1

t ðffiffiffiij

pÞ�e ¼ k1;1

t ðijÞ�e=2 (56)


D. S. Achilias

332

kt1,1 is the rate coefficient for termination between two

radicals of chain length i¼ 1, while the exponent e

quantifies the strength of the chain length dependence

of termination (e¼ 0 gives ki;jt ¼ k1;1

t ; i.e., chain-

length-independent termination).

When the two terminating radicals have the same

length i, then Equation (56) is reduced to Equation (57):

Macrom

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ki;it ¼ k1;1t i�e (57)

Various workers have independently carried out math-

ematical explorations of the FRP equations using Equa-

tion (56).[119–124] An interesting result which has been

obtained correlates hkti with the number average degree of

polymerization, DPn with Equation (58):[123]

kth i � 1=ðDPnÞe (58)

It has been proposed that this result holds also for other

termination models (i.e., not just the geometric mean). This

equation says that a log-log plot of hkti versus DPn will

have a slope equal to e, an approach that has been used for

the experimental determination of e.[123]

The GMM has no physical basis but has been suggested

to best approximate the functional form of the segmental

diffusion process.[16]

(b) The harmonic mean model (HMM), that has been

shown to be the functional form expected for kti,j if

chain-end encounter upon coil overlap is the rate

determining step for termination.[19]

ki;jt ¼ k1;1

t

2ij

iþ j

� ��e

(59)

(c) The ‘diffusion mean’ model (DMM) is of the

functional form expected if translational diffusion is rate

determining;[16] it is known to provide a reasonable

description of the termination kinetics of small radicals:[16]

ki;jt ¼ 0:5k1;1

t ði�e þ j�eÞ (60)

The DMM follows from the long-time limit of the

Smoluchowski equation for a diffusion controlled rate

coefficient, which as applied to termination is:[19,40]

ki;jt ¼ 2pspspinðDi þ D jÞNA (61)

where s is the radical-radical separation at which the

termination reaction becomes inevitable (considered

chain-length independent), pspin denotes the probability

of reaction upon encounter (probability that the radicals

are in a singlet state while their separation is s), Di is the

ol. Theory Simul. 2007, 16, 319–347


diffusion coefficient relevant to termination of a species Ri.

Notice that Russell[19] assumes that Di is different from Di

which denotes a dead polymer species with chain length i.

If it is assumed that Di i�e then Equation (61) follows

from Equation (60). However, these equations should differ

if a different dependence of Di on the chain length i is used.

For example, Russell et al.[40,125–128] in previous publica-

tions assumed that the diffusion coefficient Di constitutes

of two terms related to two mechanisms of motion: center

of mass diffusion as a whole with diffusion coefficient,

Dicom and diffusion by propagational growth of the chain

end (reaction diffusion), with diffusion coefficient Dird; that

is:

Di ¼ Dcomi þ Drd

i (62)

Dird was specified according to Equation (63) which

comes from Equation (20) by setting b¼ 1.

Drdi ¼ 1=6kp½M�a2 (63)

The following semi-empirical equation for the chain

length dependence of Dicom was proposed:[127]

Dcomi ðwpÞ ¼

DmonðwpÞi�a ; i Xc

DmonðwpÞX�ða�bÞc i�b ; i > Xc

�(64)

where, wp is the polymer weight fraction (representing

conversion of monomer to polymer), Dmon is the diffusion

coefficient of monomeric free radicals, Xc is a critical chain

length distinguishing ‘‘short’’ from ‘‘long’’ chains and a

and b are exponents.

Use of the Smoluchowski model to describe termination

in FRP has been criticized because the translational

diffusion process does not describe the motion of long

polymer molecules in solution and is not an appropriate

model for describing the approach of the chain ends in the

rate-determining step. The GMM while empirical in origin

was thought to provide a more appropriate description.

Recently, Ship et al.[129] compared the Smoluchowski DMM

and the GMM using PLP-MWD experiments in combina-

tion to kinetic simulations. They found that simulated

MWDs obtained using the DMM provide a better fit to the

experimental MWDs than those obtained using the GMM

and therefore it was suggested that the Smoluchowski

model has greater applicability. However, the data

presented in this study did not indicate any need to

include a spin probability factor in Equation (61) as was

assumed by Russell et al.[39,40,125–127]

While many data are suggestive of chain length

dependence, the data are not usually suitable for or have

not been tested with respect to model discrimination.[16]

For the situation where the chain length of one or both of

DOI: 10.1002/mats.200700003


the species is ‘small’ and conversion of monomer to

polymer is low the termination kinetics should be

dominated by the rate of diffusion of the shorter chain.

While the chain remains short the time required for the

chain reorganization to bring the reacting centers together

will be insignificant and center of mass diffusion can be

the rate-determining step. As the chain becomes longer,

segmental diffusion will become more important. Thus,

it is expected that kti,j should lie between an upper limit

predicted by the Smoluchowski DMM and a lower limit

predicted by the GMM with the value being closer to the

geometric mean value for higher chain lengths as shown

in Figure 4.[16]

In general, for e< 1 it has been shown that the DMM

value is between the other two[19,117] kti,j (GMM) kt

i,j

(DMM) kti,j (HMM).

The above rate Equation (56), (59) and (60) depend on

one parameter e and they all reduce to the same

expression, Equation (57), when the two terminating

radicals have the same length, i.

Recent work has allowed values of kt1,1 and e for bulk

polymerization in dilute solution to be estimated. This

work suggests values of kt1,1 ffi 1� 108 mol � L�1 � s�1 and

effi 0.15–0.25 for both MMA and St,[117,130] which is close to

the value that theory predicts for the case of the rate-

determining step for termination being that of chain-end

encounter of proximate macroradicals, which for good

solvents is e¼ 0.16.[131] Using the time-resolved single-

pulse pulsed-laser-polymerization technique (SP-PLP),

Buback et al.[132] were able to measure kt values up to

Figure 4. Chain length dependence of ki;jt predicted by the

Smoluchowski model (DMM) (Equation (60)) with e¼0.5 andk1;1

t ¼ 109 or the geometric mean model (Equation (56)) withe¼0.2 and k1;1

t ¼ 108. The values were taken from Moad andSolomon.[16]



high degrees of monomer conversion. They verified the

value of e as being close to 0.16 for methacrylates but only

at low degrees of monomer conversion. At conversion

above 20%, the exponent e was found to increase

significantly with increasing conversion and values close

to unity were given for MMA at 50–60% conversion.

While the value of e is as anticipated the value of kt1,1

deduced from the Smoluckowski equation was almost an

order of magnitude smaller (i.e., kt1,1ffi 109 L �mol�1 � s�1)

than that experimentally measured. Therefore, Smith

et al.[117] suggested a slightly modified model consisting

of different power-law behavior above and below a critical

chain length ic, which in the absence of specific informa-

tion was assumed equal to 100. The equation for dilute

solution was:

ki;it ¼ k1;1t i�es ; i ic

k1;1t i

�ðes�elÞc i�el ; i > ic

(and

ki;jt ¼

ffiffiffiffiffiffiffiffiffiffiffiffiki;it k

j;jt

q (65)

where es is the scaling exponent for short radicals (�0.5)

and el the scaling exponent for long radicals (�0.16).

Interestingly, k1;1t i

�ðes�elÞc ¼ 109 � (100)�(0.5–0.16)¼ 2.1� 108 L �

mol�1 � s�1, very close to the experimentally measured

value. The basis of this so-called ‘‘composite’’ model was

that the rate determining step for termination should be

different for short and long chains. Buback et al.[133]

experimentally verified these values for MMA. The results

of this model, in that the chain-length dependence is more

pronounced for shorter than for longer radicals (i.e., es > el)

was recently also verified for other systems, such as

dodecyl methacrylate, cyclohexyl methacrylate and benzyl

methacrylate.[134] Finally, it is obvious that Equation (65) is

similar to Equation (64).

Concerning the variation of hkti with the concentration

of initiator, [I], monomer [M] and kt1,1 the following result

had come out:[19,118]

kth i � ð2fkd½I�Þaðkp½M�Þ�2aðk1;1t Þ1þa (66)

and consequently the rate of polymerization becomes:

Rpol � ð2fkd½I�Þ0:5ð1�aÞðkp½M�Þð1þaÞðk1;1t Þ�0:5ð1þaÞ (67)

where a e/(2–e), the equality holds as long as e is not too

large and there is negligible transfer.

In Equation (67) if a is set equal to zero (meaning e¼ 0,

and not a chain length dependent termination rate

coefficient) then the so-called classical rate law for FRP

can be obtained, meaning a dependence of Rpol on [I] and

[M] to the 0.5 and 1 power, respectively. The same result is

obtained when transfer is dominant.[118] If e> 0 then


D. S. Achilias

334

polymerization rate has an order of< 0.5 with respect to

[I] and of >1 with respect to [M].

Inclusion of chain-length-dependent rate coefficients in

any free radical polymerization kinetic model requires the

solution of an infinite set of differential equations

describing the evolution of the concentration of every

radical and ‘‘dead’’ polymer with chain length i. Otherwise,

the full MWD is characterized by solving only for the first

few moments. Therefore, a significantly higher computa-

tional effort is required and consequently it should be used

only if needed. Bamford[121,135,136] has proposed a general

treatment for solving polymerization kinetics with chain

length dependent termination rate coefficient. Calculation

of the molecular weight averages and eventually of the full

MWD in polymerization with chain length dependent

termination was presented by Rivero.[62] In contrast,

Hutchinson,[46] did not include a chain-length dependent

termination in his model of MMA-EGDMA crosslinking

polymerization because he found that when this model

was compared to experimental results, the representation

did not completely capture the essential nature of the

process.

2.4.3.2 Chain Length Dependent Propagation

In contrast with termination, the propagation rate

coefficient is usually assumed to be chain-length-

independent. Although some experimental studies already

indicated that the first few addition steps are much faster

than in propagation involving long chains, because in FRP

long chains are produced the assumption of chain-

length-independent kp was assumed to be valid.[137,138]

In fact it has been shown that on the basis of transition

state theory it should be expected that kp1 ffi 10kp, where

kp1 denotes the propagation rate constant of a monomeric

radical, with kpi converging to the long-chain value, kp, by

about iffi 10.[139] Experimental measurements of kp1 and

kp2 are in line with these predictions, as summarized by

Moad and Solomon.[16] Experiments involving rather short

chains (MMA polymerization in the presence of large

amounts of dodecyl mercaptan) revealed that kp seems to

show a weak chain-length-dependence out to chain

lengths in the hundreds.[118,140] This effect was interpreted

to be caused by a progressive displacement of monomer

from the proximity of the radical chain end by the rest of

the chain, thus decreasing the local monomer concentra-

tion which results in a formal decrease of kp if the average

monomer concentration is inserted into its calculation.[141]

More recent experiments by the group of Olaj confirmed

the long range chain-length dependence of kp and

suggested that it should be taken into consideration when

the chain-length-dependence of the termination rate

coefficient is studied[142] and/or in advance modeling of



free radical polymerization reactions.[141,143] According to

Olaj et al.,[141,143] the following Langmuir-type expression

is appropriate for the dependence of kp on the chain

length i:

kip ¼ k0p � A

Bþ ii (68)

as well as the exponential form:

kip ¼ ðk0p � k1p Þ exp½�ki� þ k1p (69)

where A, B, kp0, kp

1 and k are constants. This is better than

describing the chain length dependence of kp by the rather

unphysical power law dependence kpi¼ kp

1i�a.

In contrast, careful experiments on the styrene and

methyl methacrylate polymerization, using the PLP-MWD

technique, revealed an inconsistency between the experi-

mental data of kp and the form of chain length dependence

shown in Equation (68) and (69).[129] Although these

authors[129] had previously proposed an equation for

the chain length dependence of kp[144] they assumed kp to

be independent of chain length since the experimental

scatter in measurements was too large for a definite

answer.

Finally, very recently Smith et al.[137,138] proposed a

chain-length-averaged propagation rate coefficient, hkpidefined by Equation (70), in which kp

i is the rate coefficient

of an i-meric radical [Ri] adding to a monomer mole-

cule:[137,138]

kp

¼X1i¼1

kip½Ri�½R� ; ½R� ¼

Xall i

½Ri� (70)

Based on available experimental and theoretical data

the following functional form for kpi was proposed:

kip ¼ kp 1 þ C1 exp � ln 2i1=2

ði� 1Þh in o

with

C1 ¼k1

p � kp

� �kp

(71)

where kp denotes the long chain propagation rate

coefficient and C1 is the factor by which kp1 exceeds kp;

i1/2 is a measure which dictates the chain-length

dependence of kpi and is analogous to the ‘half life’ of first

order kinetics. For MMA polymerization at 60 8C it was

found that C1 ¼ 15.8 and i1/2 ¼ 1.12. According to Equa-

tion (71), kpi starts at an initially high-value (i.e., the rate

coefficient for the addition of a monomeric radical to

monomer) that then exponentially decays over a relatively

small range of chain length to its long chain value.

DOI: 10.1002/mats.200700003


2.4.3.3 Chain Length Dependent Transfer Reactions

Since the chain transfer to monomer reaction competes

with propagation involving the same reactants, in general

it is expected to have similar chain length dependence to

that for propagation.[19] Russell[40] assumed the following

expression.

Macrom

� 2007

ktr;0

ktr¼ kp;0

kp¼ 1 þ kp;0

kp;diff(72)

However, this requires additional experimental exam-

ination. The interesting point is that from Equation (72)

one should expect that the ratio ktr,m/kp ¼ ktr,m0/kp,0 to be

close to chain length independent, even for chain lengths

such that both individual rate coefficients are chain length

dependent.

2.4.4 Models Based on a Combination of the Free

Volume Theory and Chain Length Dependent Rate

Coefficients

2.4.4.1 The Model of Buback et al.

In order to simulate experimental data on the effect of

polymer conversion on the termination rate coefficient,

Buback et al. proposed the following equation for the

estimation of kt over the full monomer conver-

sion:[27,28,31,145]

1

kth i ¼1

kt;TDþ 1

kt;SDþ kt;RD (73)

kt ¼�
In Equation (73) the effect of segmental, translational
and reaction diffusion are considered using the rate

coefficients kt,SD, kt,TD and kt,RD, respectively. The transla-

tional diffusion termination rate coefficient, kt,TD was

expressed in terms of the viscosity of the polymerizing

medium, h and the viscosity of pure monomer (i.e., at

X¼ 0), h0, from:

kt;TD ¼ k0t;TD

h0

h(74)

where kt,TD0 refers to the (hypothetical) rate coefficient of

translational diffusion controlled termination at zero

conversion,

kt,SD denotes the termination rate coefficient under

segmental diffusion, which was considered constant.

Finally, kt,RD refers to the termination rate coefficient

under reaction diffusion control, which is given as a

function of conversion, X, the propagation rate coefficient,

ol. Theory Simul. 2007, 16, 319–347


kp and a reaction-diffusion constant, CRD, from Equa-

tion (75), which is similar to Equation (16):

kt;RD ¼ kpCRDð1 � XÞ (75)

Several different methods for calculating CRD had been

proposed[27] and their values for a number of monomers

were tabulated. However, in general, they failed in

predicting the experimentally observed value.

Buback et al.[146] also proposed an equation for the

variation of the initiator efficiency with conversion. Based

on the idea that initiation follows as a result of primary

radical fragments diffusing away from each other they

proposed the following equation:

1

f¼ 1 þ Dterm

DI(76)

where Dterm is considered constant and DI is expressed

by the generalized free-volume theory of Vrentas and

Duda.

2.4.4.2 Extensions to Buback’s Model

Based on the original Buback model and the three-stage

process of radical termination, Tefera et al.[49,147,148]

published the following free-volume-based model. The

apparent termination rate coefficient, kt was expressed

as a function of the intrinsic rate constant, kt,R, the

translational diffusion coefficient, kTD, and the reaction

diffusion coefficient, kRD, in an equation similar to

Equation (73), which after some manipulation was

converted to the following final equation:

1

kt;0 � kRD;0þ 1

kTD;0Mn

w exp g1XVf

� ��Mn

w;0

h i��1

þkRD

(77)

where the subscript 0 refers to initial conditions,

XVf ¼ 1=Vf � 1=Vf;0, g1, kTD,0 and n are used as adjustable

parameters and kRD was estimated from Equation (16)

with the value of A set equal to 1.

A similar expression was proposed for the effect of

diffusion controlled phenomena on kp

kp

kp0¼ 1 þ kp0

kp;D0expðg6XVf Þ � 1 �� 1

(78)

again kp,D0 and g6 were used as adjustable parameters.

Finally, for modeling the change of radical efficiency

during polymerization the following empirical equation


D. S. Achilias

336

was proposed:

Macrom

� 2007

f ¼ 2f0

1 þ expðg2XVf Þ(79)

where g2 is another adjustable parameter.

The model described well available experimental

data on the MMA and styrene polymerization under

different initial initiator concentrations and temperatures

with or without the presence of a chain transfer

agent.[147,148]

Recently, Nising and Meyer[96] compared the Fleury

et al.[95] with the Tefera et al.[49,147] models in the high

temperature polymerization of MMA. They concluded that

the first model better describes the experimental data and

needs fewer adjustable parameters compared to the

second.

Furthermore, Panke[149] also used a model similar to

Buback’s Equation (73) but in the expression of the rate

coefficient for translational diffusion, kt,TD he proposed

the following equation depending not only on the free

volume of the mixture but also on the molar mass of the

macroradical and ‘‘dead’’ polymer:

kt;TD ¼ k0TD

exp �V�t

1Vf

� 1V0

f

� ��

MwMw

� �n (80)

ktp ¼

where kTD0, Vt

� and n were adjustable parameters, Mw was

used to denote the instantaneous molar mass representing

the molar mass of the macroradicals and Mw the

cumulative molar mass (meaning that of the ‘‘dead’’

polymer).

This model was used to simulate experimental data of

bulk and solution polymerization of MMA over an

extended temperature and initial initiator concentration

range.[149]

2.4.4.3 Inclusion of Chain Length Dependence[85,150]

In order to form a comprehensive equation for the

diffusion-controlled termination rate coefficient between

two macroradicals of chain lengths n and m, kt(n,m),

Buback and Kaminsky[85,150] proposed the following

Equation (81). Accordingly, kt was assumed to be both

chain-length dependent and conversion dependent using

an additive model, which they considered to be physically

more realistic:

ktðn;mÞ ¼ 0:5ktð1; 1Þ vðnÞ þ vðmÞð Þ þ ktp (81)

which can be obtained from the well-known Smolu-

chowski equation.

ol. Theory Simul. 2007, 16, 319–347


The final equation provided for v(n) allowed the

inclusion of a chain-length dependence of kt also in the

non-entangled region. The following equation was

derived, which is similar to Equation (64) or (65):

vðnÞ ¼ n�b ; n ne

n�ðb�gÞe n�g ; n > ne

�(82)

where b and g are constants and ne denoting a critical

chain length where entanglements appear.

In order to account for the conversion dependence of the

diffusion coefficient of small molecules, in the initial stage

of polymerization, the expression from free volume theory

was applied:

ktð1; 1Þ ¼ kt0 exp �B1

Vf� 1

Vf;0

� �� (83)

where kt0 is the value of the termination rate coefficient at

zero monomer conversion, B an adjustable parameter, Vf

the fractional free volume of the reaction mixture given by

Equation (52) and Vf,0 the free volume at zero conversion

(equal to the monomer free volume in bulk polymeriza-

tion).

Equation (83) is similar to Equation (50) used in the

free-volume-based semi-empirical approach. Furthermore,

the reaction-diffusion controlled termination rate coeffi-

cient, ktp, was estimated from an equation similar to

Equation (75) as:

kpC0RDð1 � XÞ with C0

RD ¼ 8p

3z½M�0a3n1=2

e NA (84)

where [M]0 is the initial monomer concentration and z

a parameter that accounts for flexibility limitations of

the free-radical chain and a the size of the monomeric

unit.

If the termination rate coefficient is assumed chain-

length dependent then employing the continuous model-

ing approach the following balance equation for the time

evolution of the concentration of free radicals of chain

length n holds:

@Rðn; tÞ@t

þ kpM@Rðn; tÞ

@n

¼ RIn þ ktr;mMR� �

dðnÞ � Rðn; tÞ

� ktr;mM þZ10

ktðn;mÞRðm; tÞdm

0@

1A (85)

where, ktr,m is the rate coefficient of chain transfer to

monomer, d(n) is the Dirac delta function, RIN the initiation

rate (usually, RIN ¼ 2fkdI) and R ¼R1

0 Rðn; tÞdn is the

overall free radical concentration.

DOI: 10.1002/mats.200700003


Therefore, using this modeling approach, in order to

simulate a polymerization reaction during the full

conversion range, the integro-partial differential Equa-

tion (85) should be solved, which requires complicated

mathematical procedures.

2.4.5 Fully Empirical Models

Gupta et al., in order to carry out optimization studies,

initially used their model given in Equation (39)–

(44).[151–152] However, the predictions of this model

were very sensitive to the values of the parameters

involved and it did not work satisfactorily for on-line

optimizing control applications.[153] Therefore, in a series

of recent papers[154–156] these authors used the following

empirical models to describe the effect of conversion on kp

and kt:

Macrom

� 2007

kt ¼ kt;0 expðA1 þ A2X þ A3X2 þ A4X

3Þ (86)

kp ¼ kp;0 expðB1 þ B2X þ B3X2 þ B4X

3Þ (87)

where Ai and Bi are functions of temperature.

These correlations were previously proposed by

Curteanu et al.[157–160] with all Ai (i¼ 1, . . ., 4) and Bi(i¼ 1, . . ., 4) set as empirical constants determined from

fitting to experimental data. Different sets of values were

estimated at different polymerization temperatures and

initial initiator concentrations.[160] The model was also

successfully applied in the free-radical polymerization of

methyl methacrylate under semi-batch and non-

isothermal reactor conditions.[160] This model, although

purely empirical with a large number of adjustable

parameters, seems to be preferred when one needs a

robust model to carry out optimization, or control

simulations.

Recently Qin et al.,[161,162] proposed a so-called three-

stage polymerization model (TSPM) on the basis of

treatment of available from literature experimental

kinetic results. The whole course of the bulk FRP was

divided into three different stages (i.e., low conversion

stage, gel-effect and glass-effect stage) and classical free

radical polymerization equations meaning linear plots of

�ln(1–X) vs. exp(�kdt/2) were applied in each stage.

Although this modeling approach fitted rather well the

experimental data on the polymerization of MMA and

styrene under a variety of initial initiator concentrations

and temperatures, it still could be considered as a

good-data-fitting procedure rather than a predictive

method.

ol. Theory Simul. 2007, 16, 319–347


2.5 A Case Study – Reversible Addition-FragmentationTransfer Polymerization

Controlled/living radical polymerization has attracted

much attention in recent years due to its potential for

developing new polymer products.[17] Living polymeriza-

tion offers good potential for the control of polymer

molecular weight and functionalization. There are three

major types of living radical polymerization developed so

far, namely, nitroxide-mediated polymerization (NMP)

(also termed stable free-radical polymerization), atom-

transfer radical polymerization (ATRP) and reversible

addition-fragmentation polymerization (RAFT). In this

section, RAFT polymerization is discussed as a case study

of the application of diffusion-controlled models (pre-

sented above) in polymerization kinetics. RAFT polymer-

ization has emerged during the last ten years and drawn

particular attention due to the large versatility of the

process and the peculiar chemistry involved. It is for these

reasons that several models have been published recently

to simulate the kinetics of the reaction.[16,17] Since radical-

radical termination is suppressed in processes such as

ATRP, or NMP, most of these models did not take diffusion-

controlled reactions into their schemes. However, in any

processes that involve propagating radicals, there will be a

finite rate of termination commensurate with the

concentration of propagating radicals and the reaction

conditions.[16] Therefore, it was very recently recognized

that diffusion phenomena on the rate coefficients may

play an important role during polymerization. Indepen-

dently, the three main approaches presented above were

investigated and are briefly commented on next.

Wang and Zhu[163] were among the first to incorporate

diffusion-controlled rate coefficients in order to investi-

gate the effect of diffusion-controlled radical deactivation

and termination on the ‘livingness’ of the RAFT process.

They used the general parallel summation equation

presented in Equation (10), with the following free-volume

based semi-empirical expression to account for diffusion

limitations:

kD ¼ k0DðrN;totÞ2 expð�1=VfÞ (88)

kD0 was considered as an adjustable parameter and rN,tot

denoted the number average chain length of the total

chains.

The effect of diffusion controlled reactions on the

polymerization rate and molecular weight distribution

was thoroughly demonstrated.[163] The model of Wang

and Zhu was an extension of the diffusion-controlled

model presented by Delgadillo-Velazquez et al.[164] on

ATRP reactions. The latter authors used the semi-empirical

free-volume approach to study the effect of diffusion-

controlled reactions on ATRP. Equation (47), (48), (50), (52)


D. S. Achilias

338

and (53) were used with the only exception that the term

(Mw,cr/Mw)n in Equation (47) was replaced by (rn/rw)x/2,

where rn and rw are the number and weight average chain

lengths of the polymer radicals. Chain-length dependence

of reactions among large macromolecules was considered

by using a number and a weight average kinetic rate

coefficient for such reactions. Exactly the same modeling

approach was later used by Al-Harthi et al.[165] and some

case studies were investigated. Their model showed that

diffusion-limited termination reactions produce polymers

with smaller polydispersities, while diffusion-limited

propagation reactions have the opposite effect. According

to their model simulations they concluded that diffusion-

controlled reactions could be ignored in the case studies

examined (i.e., solution polymerization of MMA and St and

bulk polymerization of butyl acrylate).

Furthermore, Peklak et al.[10] used the model presented

by Equation (10), (14), (19), (20) and (27)–(29) and with a

chain-length dependent center-of-mass diffusion coeffi-

cient, Dpcom,x, according to:[10]

Macrom

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Dpcom;x ¼ Dmxf ðfpÞ (89)

where wp is the polymer volume fraction.

Model predictions compared well to experimental

results of MMA polymerization. In addition, the retarda-

tion observed in living systems at large conversions was

well predicted.

Later, Peklak and Butte[166] examined the effect of

diffusion limitations on RAFT polymerization using two

approaches. They started from Equation (10) and con-

sidered either a chain-length-dependent diffusion coeffi-

cient of the macromolecules or a single, Dp, depending on

the number average degree of polymerization. In the first

case, the polymer diffusion coefficient scaled with the

radical chain-length i in a form similar to that presented in

Equation (57), as D i�e, with e an exponent ranging

between 0.5–2.5. In the second approach it was assumed

that all chains regardless of their individual length may

react with one rate coefficient scaling with the number

average degree of polymerization in a way similar to

Equation (58). This discrimination was considered based

on the higher computational effort that is required when a

chain-length dependent rate coefficient is used. The

behavior of these two models was investigated with

reference to three case studies, with special emphasis on

the conversion curve as well as average chain lengths and

polydispersity of the MWD. Even though it was expected

that for living polymerizations the results of the two

models would be almost equal due to the typically small

polydispersity of these systems, it was found that

differences between the two models could actually be

significant. It was also concluded that above 40%

conversion different conversion curves could be obtained

ol. Theory Simul. 2007, 16, 319–347


and at even higher conversions also different polydisper-

sities.[166]

2.6 Concluding Remarks – Recommendations

The following remarks conclude this section, some

recommendations are also provided here. From the

aforementioned model presentation, it seems that almost

all elementary reactions should be considered as diffusion-

controlled. The models presented have different physical

assumptions and different expressions for kt, kp and f, but

nevertheless they have been used to fit rather well the

experimental time-conversion and average molecular

weight-conversion data. This is probably due to the

existence of some adjustable parameters in these models

together with the compensating effect of various compli-

cating factors. As our knowledge of the phenomenon tends

to become more complete, the number of adjustable

parameters used is diminished. On the other hand, models

based on a detailed knowledge of microscopic phenomena

taking place during polymerization are usually difficult to

apply, since they require a rather large computational

effort.

Modeling of high and very-high conversion polymeriza-

tion has significantly improved during last decade, by

including the effect of reaction-diffusion on the termina-

tion rate constant, together with the effect of diffusion on

the propagation rate constant and mainly on initiator

efficiency. It has been proved that the effect of diffusion-

controlled phenomena on f compared to kp start earlier and

are more important.[25,147]

In general, purely empirical models can fit very well

experimental data on specific reaction/reactor conditions

on which the parameters needed have been estimated.

However, they fail in predicting data under different

experimental conditions (meaning different reaction

temperatures, initial initiator concentration, type of

initiator, type of reactor operation, etc.) Furthermore, they

do not give any insight into the physical picture of the

system.

Semi-empirical models are approaches to fill the gap

between the detailed and purely empirical models and

they try to reach at least equal qualitative and quantitative

agreement between the experimental results and the

model simulation, while keeping the number of adjustable

parameters as small as possible.[49]

A reliable model for the variation of kt and possibly kp

during polymerization should include both a chain-length

and a conversion dependence. It seems that no single

power-law could describe the chain-length-dependence

of the termination reaction. Instead different exponents

were observed depending on chain-length.[21] For the

conversion dependence some free-volume based model is

DOI: 10.1002/mats.200700003


possibly adequate, since it includes also the effect of other

physical important parameters, such as the glass transi-

tion temperature, jumping unit size, etc.[37] From the most

recent and complete approaches, those of Muller et al.[58]

and Buback et al.[150] should be considered for guidance.

Both these modeling approaches look like the radius of the

same wheel starting from different parts in the perimeter

but ending at the same point in the central.

Finally, apart from the effect of the chain length of the

diffusion macroradical, the chain length of the surround-

ing macromolecules should be also considered in models,

since their number is much greater compared to macro-

radicals and they could possibly restrict their diffusion.

Although, it should be kept in mind that inclusion of

chain-length-dependent rate coefficients in any free

radical polymerization kinetic model requires the solution

of an infinite set of differential equations describing the

evolution of the concentration of every radical and ‘‘dead’’

polymer with chain length i and thus the computational

effort is greatly increased.

3 Step-Growth Polymerization

3.1 Diffusion-Controlled Step Growth Polymerization

The kinetics and reaction engineering of step-growth

polymerization can be found in several polymer textbooks,

such as the one published by Gupta and Kumar.[167] In

most step-growth polymerization models, one common

assumption is that all polymer chains of the same

functional group are equally reactive (‘‘equal reactivity

assumption’’), regardless of their molecular weight or any

diffusion limitations. However, under certain conditions

(e.g., on rigid-rod-like molecules) diffusion effects may play

an important role during polymerization. Although

important, diffusion-effects on step-growth are somewhat

less studied compared to free radical polymerization.

Models developed to account for diffusion effects on step

growth polymerization are reviewed next.

Gupta et al.[168] studied a nonlinear homopolymeriza-

tion and assumed that chains can only react if they are less

than a specified distance apart. Their approach was used to

describe changes in the weight average molecular weight,

but no information on polydispersity was reported. The

work of Oshani and Moreau[169] assumes that the

influence of chain length i on the effective rate coefficient

ki is dramatic in order to obtain an analytic solution for

linear homo-polymerization. The effect of segmental

diffusion on irreversible, step-growth polymerizations of

ARB type monomers was studied by Kumar et al.[167,170]

Recently, Guzman et al.[15] presented a detail study of

diffusion effects in step-growth polymerization via a

chain-by-chain simulation, meaning that they kept track



of the concentration of every chain length so that

additional approximations were not necessary to integrate

the population balance equations. Both homopolymeriza-

tion and A2 þB2 step-growth polymerizations were

considered. The following model was proposed, which is

actually a successful combination of two of the models

derived above. As an example the synthesis of nylon via

self-condensation of an amino acid was considered

according to the following reaction:

i H2N�R�COOH ! H�½NH�R�CO�i�OH þ ði� 1ÞH2O

or,

ðABÞi þ ðABÞj �!kij ðABÞiþj þ H2O (90)

where A and B represents the reactive functional groups

(H2N– and –COOH) and kij the effective polymerization

rate coefficient which was assumed to depend on the chain

lengths i and j of the two chains involved in the reaction.

The effective rate coefficient, kij, was assumed equal to

the parallel summation of the intrinsic reaction rate

constant kint and the diffusional rate coefficient kdij,

according to:

1

kij¼ 1

kintþ 1

kdij(91)

Equation (91) has the same form as Equation (10) above.

In order to estimate the diffusion rate coefficient kdij, a

functional form similar to that presented in the second

right-hand side of Equation (10) was used, with rAB ¼ riþ rj

and DAB¼DiþDj for the two chains with lengths i and j,

respectively. The trapping radius ri was set equal to the

radius of gyration of a polymer chain with i repeating

units,[171] therefore it was assumed to be chain length

dependent. For an ideal coil:

ri �ffiffii

p(92)

For the chain length dependence of the diffusion

coefficient Di an equation similar to Equation (64) was

used with a¼ 1 and b¼ 2 for both nonentangled and

entangled chains, respectively. [171] By lumping together

all the proportionality constants into one parameter w, the

following final equation was derived as a function of the

Hatta number (Ha):

1

kdij¼ 1

kintHaði; jÞ ¼ 1

kint

fffiffii

pþ

ffiffij

p� �i�l þ j�lð Þ

(93)

where w is a lumped constant and l¼ 1 below entangle-

ment and 2 otherwise.


D. S. Achilias

340

In order to simplify Equation (93), the following

expression was proposed resulting in almost the same

dependence of final properties on conversion:

Macrom

� 2007

1

kdij¼ 1

kint’

minði; jÞNent

� �l�1=2

(94)

where min(i,j) is the minimum of i and j, and w a

proportionality constant.

Again, the problem persists that using chain length

dependent kinetic rate coefficients means that one has to

solve the full population balance equations for all

macromolecular chains with no averaging. Thus, theoret-

ically an infinite set of ordinary differential equations

must be solved. In practice, accurate results were obtained

by solving a very large but finite set of equations. An

estimate for the number of equations required could be

obtained from the analytical solution of the equal reac-

tivity case.[15] For most simulations it was found satis-

factorily to use 10 times the number average degree of

polymerization meaning that for a typical value of 1 000

one has to solve simultaneously approximately 10 000

ODEs!

Exactly the same model as presented in Equation

(91)–(93) was later used by Yan et al.[172,173] in studying

diffusion effects on chain extension reactions, using

carboxyl-terminated polyamide 12 with bisoxazolines.

3.2 Modeling Diffusion Controlled Solid StatePolycondensation (SSP)

Because of the industrial importance of SSP, mathematical

modeling and process simulation have been employed to

gain a better understanding of the relevant mechanisms

and to predict the influence of different parameters on the

SSP rate. The advantages of SSP, compared to melt phase

polymerization, are low operating temperatures, which

restrain side reactions and thermal degradation of the

product, inexpensive equipment and uncomplicated and

environmentally sound procedures. Lower polymerization

temperatures during SSP are required also when reversible

reactions are taking place (such as in nylon or PET

production) to drive the reaction in the forward direction,

promoting the production of higher average molecular

weight products. Thus, in a typical, industrial PET reactor a

number average chain length of about 100 is achieved,

while SSP takes it to over 300.

The models that have been developed for SSP include a

variety of polymers, mainly belonging to the families of

polyesters (i.e., PET, PBT, etc.), or polyamides (PA-6 and

PA-6,6).[174] These are reviewed next.

ol. Theory Simul. 2007, 16, 319–347


3.2.1 Solid State Polycondensation of Polyamides

The CCS model[51] was used as a starting point for the

description of diffusion limitations in SSP of poly-

amides.[175–179] Kaushik and Gupta[175] presented a

theoretical model for the effect of segmental diffusion

on the SSP of Nylon 6. The following kinetic scheme was

supposed:

Ring opening

C1 þW Ðk1

k1=K1

P1 (95)

Polycondensation

Pn þ Pm Ðk2

k2=K2

Pnþm þW (96)

Polyaddition

Pn þ C1 Ðk3

k3=K3

Pnþ1 (97)

Only the forward step of the polycondensation reaction,

k2, was assumed to be susceptible to diffusional limita-

tions. Using the same physical picture as that of CCS the

authors[175] finally came to the following equation relating

the polycondensation rate coefficient to the diffusivity of

the polymer molecules, D and the effective diffusion

radius, rm:

k2

k2;0¼ 1 � ðr2

m=3DÞk02½W�f1 � ðl1=l0Þg1 þ ðr2

m=3DÞk2;0l0(98)

where k2,0 is the intrinsic rate constant and k02 ¼k2/K2 ¼ k02,0, [W] is the concentration of water and l0

and l1 denote the zero and first moment of the polymer

MWD, respectively.

The termr2

m3D was calculated based on an empirical

parameter, u according to:

r2m

3D¼ u

D0

D(99)

with,

logD0

D¼ AðTÞ

Vfþ BðTÞ (100)

where u, A(T) and B(T) were considered to be adjustable

parameters and Vf is the free volume fraction which for

DOI: 10.1002/mats.200700003


step growth polymerization reactions is given by:

Macrom

� 2007

Vf ¼ 0:025 þ ðal � agÞðT � TgÞ (101)

Exactly the same model was later used by Xie in a

detailed simulation of Nylon 6 SSP.[179]

Subsequently, Kulkarni and Gupta[176] improved their

model by including the effect of diffusion-controlled

limitations on the reverse polycondensation reaction rate

constant, k02, as well as the forward step of the

polyaddition reaction, k3. Accordingly:

1

ki¼ 1

ki;0þ

r2m;i

3Di

!½P�b (102)

with ki set equal to k2, k02 and k3 for the forwards and

reverse polycondensation reaction and forward polyaddi-

tion reaction, respectively. The corresponding diffusion

coefficients Di are that of the polymer, Dp, water, Dw and

monomer, Dm and were calculated based on the extended

free-volume theory of Vrentas and Duda.[88,89] [P]b is the

local bulk polymer concentration.

The final equations derived were:

1

k2¼ 1

k2;0þ u1ðTÞm2

nl01

exp½�xþ xref �(103)

1

k02¼ 1

k02;0þ u2ðTÞl0

1

exp½j23ð�xþ xrefÞ�(104)

1

k3¼ 1

k3;0þ u3ðTÞl0

1

exp½j13ð�xþ xrefÞ�(105)

where the subscript 0 denotes intrinsic rate constants, mn

is the polymer number average chain length and l0 the

zero moment of the polymer CLD, meaning polymer

Scheme 4. Transesterification/polycondensation a) and esterification b).

concentration; u1, u2 and u3 were adjustable

parameters and x and xref are given by

Equation (42) and (43), respectively.

Notice that Equation (103) and (104) are

similar to Equation (39) and (40), used to

model the effect of diffusion controlled

phenomena on the termination and pro-

pagation rate coefficients in free radical

polymerization. Almost the same diffusion

model was recently used by Li et al.[177,178]

in the simulation of the SSP of Nylon-66

with the only exception that the polymer

concentration, l0 was replaced by that

in the amorphous phase, that is, l0/(1–xc)

(xc is the degree of crystallinity) since

polymer end-groups, monomers, conden-

sates and catalysts exist exclusively in the

ol. Theory Simul. 2007, 16, 319–347


amorphous phase. Good agreement of simulations to

experimental data was achieved. Crystalline and amor-

phous phases were also treated separately in the model

used by Mallon and Ray.[180]

3.2.2 Solid State Polycondensation of Polyesters

The most extensively studied polyester, discussed also in

this section, is poly(ethylene terephthalate) (PET). The solid

state polycondensation of PET was recently presented in

an excellent and detailed review by the group of

McKenna.[181] The main reactions involved in the SSP of

PET include the ones shown in Scheme 4.

During SSP due to the low reaction temperature, chain

mobility is relatively low compared to corresponding melt

polymerization process. However, polymer chains have

translational degrees of freedom, which are necessary for

reactive end-groups to diffuse in the amorphous phase to

approach and react. Therefore, the reaction rate coefficient

between two reactive polymer chains was assumed

proportional to the chain mobility according to:[182]

ki ¼ Ai

X2n;ref

X2n

exp�Ep

RT

� �exp

�EiRT

� �; i ¼ 1; 2 (106)

where the first exponential term denotes the effect of

temperature on the intrinsic kinetic rate constant while

the second, the effect of temperature on the chain-end

translational mobility. Ai and Ei are the well-known

pre-exponential factor and activation energy according to

the Arrhenius expression, whereas Ep is the activation

energy of translational motion. Xn,ref is an arbitrarily

chosen reference number average degree of polymeriza-

tion and an inverse squared dependence on the number

average degree of polymerization was assumed. The latter


D. S. Achilias

342

was taken from the reptation theory assuming that the

translational mobility of a polymer chain decreases with

increasing their length, owing to chain entanglements.[182]

The same approach was later used by Wang and Deng,[183]

and Kim and Jabarin,[184] who proposed a comprehensive

model for the SSP of PET to account for the influence of

reactive chain mobility on reaction rate.

Devotta and Mashelkar[185] also considered that the rate

coefficient for transesterification was proportional to the

mobility of the macromolecules. They assumed that since

the dimensions of the reactive end groups are approxi-

mately the same as those of a molecule of ethylene glycol,

then the diffusion of these end groups in the amorphous

phase of the PET is similar to the process of diffusion of EG.

Thus, they assumed that the mobility of the end-groups

and consequently the value of the rate coefficient

depended on the free volume of the polymer that decreases

with reaction time. According to Gantillon et al.,[181] if it is

considered that PET is a semi-crystalline material, then it is

not realistic to consider that the entire macromolecule

could diffuse through the matrix on a time scale

proportionate with that of the polycondensation. Also

not only the local viscosity on the amorphous phase is very

high, but a large fraction of macromolecules remain

trapped in crystalline occlusions in the polymer matrix.

Thus they concluded that the Devotta and Mashelkar

representation sounded the most realistic from a physical

point of view, since only the chain ends will be able to

move and it will be this movement that determines the

rate of reaction.

The mobility of the macromolecular chains during SSP

depend not only on temperature and molecular weight of

the polymer but also on the morphology of the crystalline

phase of the material since reactive chain ends are

concentrated in the amorphous zones of the polymer

matrix.[181] In polymerizing systems with a high degree of

crystallinity, reactive end-groups could be trapped inside

the crystalline occlusions and thus either do not react at

all or react very slowly. In contrast, if the amorphous parts

of the matrix are distributed in such a way that the

reactive end groups are well dispersed, and if the distance

they need to diffuse in order to react with neighboring

groups is short, then the rate of reaction will be relatively

high.

In order to describe the effect of the end group diffusion

limitation on both esterification and polycondensation

reaction, Wu et al.[186] used a model similar to that

proposed by Chiu et al.[51] for free radical polymerization

reactions. Both reactions rates were calculated as a

function of a diffusion parameter, Q, defined in a similar

way to Equation (99), as:

Macrom

� 2007

Q ¼ r2m

3D¼ u

D0

D(99b)

ol. Theory Simul. 2007, 16, 319–347


where rm and D have the same meaning as in the

original CCS model and u is an empirical parameter

obtained from fitting to experimental data.

A different expression was used to estimate the

diffusion coefficient D:

logD0

D¼ �17:4ðT � TgÞ

51:6 þ T � Tg(107)

where Tg is the glass transition temperature of the polymer.

3.2.3 Simple Models

In order to account for the effect of diffusional limitations

or chain mobility restrictions on the reactive ends in SSP

kinetics, Ma and Agarwal[187,188] used the concept of

temporarily inactive hydroxyl and carboxyl end-groups.

This leads to an apparent inability of a fraction of hydroxyl

and carboxyl groups to participate in the reactions due to

the limited extent of mobility of some chain-ends as a

result of their being restricted by relatively short chain

segments linking them to crystalline parts, or as a result of

their having been incorporated in crystalline parts as

defects. Then, the classical kinetic rate expressions can be

written for the reactions in Scheme 4 but the actual

concentration of OH and COOH should be expressed as:

[OH]t ¼ [OH]� [OH]i and [COOH]t ¼ [COOH]� [COOH]i

where [OH], [COOH] and [OH]i, [COOH]i denote the

concentration of the total and temporarily inactivated

OH and COOH end-groups, respectively. This modeling

concept originally proposed by Duh[189] was later on

successfully used by several authors.[190,191]

3.3 Modeling Diffusion-Controlled Cure Kinetics ofEpoxy-Amine Resins

Another well-studied system involving diffusion con-

trolled reactions is the curing of epoxy-amine resins.

Curing of thermoset polymers involves the transformation

of a low molecular weight liquid mixture to a crosslinked

network. Thus, the reaction is initially chemically con-

trolled, while later, the increasing size and complexity of

the polymer network restrict diffusion and curing becomes

diffusion controlled. In addition, the curing process causes

a reduction in the free volume of the mixture and an

increase in the glass transition temperature. When the

material is cured at a temperature much greater than

the Tg of the fully cured polymer, diffusion may not affect

the overall rate of reaction. However, since in many

applications it is a prerequisite that curing must take place

at ambient temperature (usually below the glass transi-

tion temperature of the polymer being formed) the

reaction does not reach complete conversion and diffusion

DOI: 10.1002/mats.200700003


phenomena influence the overall reaction rate and

determine the final degree of cure. As the conversion

increases and the Tg of the mixture approaches the

reaction temperature, the diffusivity of the reactive

functional groups becomes seriously restricted. Thus, the

curing reaction decreases considerably even when there

may be a significant level of amino and epoxide groups

available for reaction. Finally, as it was reported for the

glass effect in free radical polymerization, once the

material has vitrified the reaction effectively ceases.[192]

A simple set of equations describing epoxy-amine poly-

merization involves three main reactions and may be

expressed as shown in Scheme 5.[193]

Chern and Poehlein[194] used a semi-empirical equation

based on the reduction of the free volume of the mixture.

They started from an equation similar to Equation (50) and

assuming a linear relationship between conversion, X, and

1/Vf they derived the following expression for the rate

coefficients used in Scheme 5:

Schrea

Macrom

� 2007

k ¼ kchem expð�CðX � XcÞÞ (108)

where kchem is a chemically controlled Arrhenius-type

reaction rate constant, C is a constant that depends on the

structure, system and curing temperature and Xc is a

critical conversion, which may depend on temperature.

Cole et al.[195] noticed that Equation (108) corresponds to

a rather abrupt onset of diffusion control at X¼Xc and

eme 5. Primary amine addition a), secondary amine addition b)ction c).

ol. Theory Simul. 2007, 16, 319–347


they defined the overall effective rate coefficient of the

reaction from a parallel summation of the chemical and

diffusional controlled rate coefficient, according to an

equation similar to Equation (10):

and et

1

keff¼ 1

kchemþ 1

kdiff(109)

where in place of kdiff the kinetic rate coefficient defined in

Equation (108) was used.

Thus, the final Cole-Poehlein diffusion model gives:

keff ¼ kchem1

1 þ expðCðX � XcÞÞ(110)

For values of X significantly lower than Xc the term

kchem/kdiff is approximately equal to zero and keff/

kchem ffi 1. This means that diffusion-controlled is negli-

gible and the overall reaction kinetic rate coefficients are

reaction-limited. When X approaches Xc, keff/kchem begins

to decrease, reaching 0.5 when X¼Xc. Beyond this point, it

continues to decrease eventually approaching zero, so that

the reaction becomes very slow and effectively stops.

Therefore, Xc seems to correspond to a critical conversion

above which vitrification is efficient, that is, the vitrifica-

tion conversion. Several authors have recently used

Equation (110) in modeling of the epoxy-amine cur-

ing.[193,196–198] Concerning the variation of C with tem-

herification

perature, no discernible trend was

observed in the literature.[193,195,197] In

contrast, the critical conversion, Xc, was

found always to increase with tempera-

ture.[193,195–198] This was attributed to

the increased mobility of molecules

caused by the increased temperature

and thus the shift of the onset of

diffusion control phenomena to higher

values. Furthermore, it was found that

under certain condition (i.e., a stoichio-

metric ratio of epoxy to amine and low

reaction temperatures) the value of Xc

resembles the gelation point (conver-

sion at which a 3-dimensional network

is formed).[193]

Later on, Fournier et al.[199] modified

Cole’s equation as follows:

keff
¼ kchem2
1 þ exp ðX � XfÞ=b½ � � 1

� �(111)

where Xf denotes the final degree of

curing and b is a constant.


D. S. Achilias

344

A comparison of the results obtained using Equation

(110) with those from Equation (111) and arbitrarily

chosen values for the parameters C¼ 22, Xc ¼ 0.64 and

Xf ¼ 0.82, b¼ 0.09 was carried out by Karayannidou

et al.[193] It was observed that both equations corresponded

to a gradual decrease with respect to the degree of curing.

While the Cole-Poehlein model presented a gradual change

during very high degree of X, the Fournier et al. model

showed a monotonically decrease with values of k/kchem

less than zero when X>Xf.

Another more fundamental chemico-diffusion model

used extensively in epoxy-amine curing is that proposed

by the group of Cook.[192] They started from the parallel

summation Equation (109) with kchem dependent only on

temperature and kdiff obtained from the Smoluchowski

equation. Thus, the basic equation was similar to Equa-

tion (10) derived in free-radical polymerization. Moreover,

in order to estimate kdiff the following assumption was

made: for the step-growth polymerization of an amine-

cured epoxy the overall diffusion of reactive groups toward

each other should be governed by the diffusion of chain

segments. Thus the overall diffusion coefficient, D, is

expected to be inversely proportional to the relaxation

time of polymer segments, which enables a model based

on the free-volume concept and a description similar to the

Williams-Landel-Ferry (WLF)-type equation:[192]

Macrom

� 2007

lnD

DTg

!¼ C1ðT � TgÞ

C2 þ T � Tg(112)

where, Tg is the glass transition temperature, DTg is the

diffusion coefficient at Tg and C1 and C2 are constants.

Equation (112) is similar to Equation (107) reported in the

previous section.

Accordingly, the final equation for the diffusion-limited

kinetic rate coefficient, kdiff was given by:[192]

kdiff ¼ kdiffðTgÞ expC1ðT � TgÞC2 þ T � Tg

� �(113)

where, kdiff(Tg) is the diffusion rate coefficient at Tg.

According to Wise et al.,[192] since the reactive species

during curing, epoxide and amine groups are in close

proximity to one another throughout the reaction, kdiff

should refer to the segmental diffusion required to bring

species within a collision radius. Thus, kdiff was assumed

independent of the size of the molecule to which the

reactive group was attached. This model[192] while more

theoretically correct, it requires the determination of the

change of the glass transition temperature as a function of

curing conversion, which is not always known. However, it

was extensively used recently in modeling the effect of

diffusion phenomena in epoxy-amine curing.[200–203]

ol. Theory Simul. 2007, 16, 319–347


4. Conclusion

In this review paper, different models proposed and

extensively used recently in quantifying the effect of

diffusion-controlled phenomena in polymerization were

presented. Modeling of diffusion in reactive systems faces

both theoretical and numerical challenges and different

authors have used different approximations to overcome

these obstacles. Both, the extensively studied free-radical

mechanism and the less investigated step-growth were

considered. Similarities and differences between the

modeling approaches used were highlighted. Solid-state

polycondensation and epoxy-amine curing were exam-

ined as special cases-studies in step-growth and RAFT in

free-radical polymerization. It seems that models using

chain length dependent kinetic rate coefficients describe

better the phenomena from a physical point of view.

However, the problem persists that using this modeling

approach one has to solve the full population balance

equations for all macromolecular chains with no averag-

ing. Thus, theoretically, an infinite set of ordinary

differential equations would be required to be solved

increasing considerably the computational effort. On the

other hand, for optimization and control studies very

simple empirical models are preferred due to their

robustness.

Received: January 4, 2007; Revised: March 12, 2007; Accepted:March 15, 2007; DOI: 10.1002/mats.200700003

Keywords: Diffusion-controlled reactions; modeling; free-radical;step-growth

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A Review of Modeling of Diffusion Controlled Polymerization Reactions

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