ORIGINAL PAPER

An exhaustive study of chain-length-dependentand diffusion-controlled free radical and atom-transferradical polymerization of styrene

Mohammad Najafi & Hossein Roghani-Mamaqani &Mehdi Salami-Kalajahi & Vahid Haddadi-Asl

Received: 16 January 2010 /Accepted: 26 December 2010 /Published online: 1 February 2011# Springer Science+Business Media B.V. 2011

Abstract A finely tuned-up and high-performance MonteCarlo simulation, which takes diffusion-controlled and chain-length-dependent bimolecular termination reactions intoaccount, is developed to thoroughly simulate and comparefree radical (FRP) and atom transfer radical polymerization(ATRP) of styrene. It is found out that the termination rateconstant falls and eventually plateaus upon the increase of thechain length of radicals. In addition, average termination rateconstant greatly decreases during ATRP; nonetheless, itremains almost unchanged and smaller in FRP. Moreover,there is an accumulation of CuBr2 in the reactor as the ATRPproceeds, while the concentration of CuBr decreases andfinally plateaus. Polymer chains are entirely initiated at thebeginning of the ATRP, whereas initiation of chainscontinues throughout the free radical polymerization up tothe end of the reaction. Also, the dead polymers are muchlower in concentration (only 20%) in ATRP as compared toFRP (about 50%). In addition, a shift (toward highermolecular weight) in the location of the peaks of molecularweight distributions can easily be seen for the ATRP system,whilst the chain length distribution of free-radically gener-ated polymers remains the same throughout the free radicalpolymerization. The molecular weight distributions narrow

as the atom transfer radical polymerization progresses.Finally, the simulation results correspond closely to theexperimental data.

Keywords Chain length distribution . Chain-length-dependent and diffusion-controlled termination .MonteCarlo simulation . Styrene ATRP

Introduction

Because of its robustness and flexibility, the field of freeradical polymerization (FRP) has been a research topic ofinterest in the past twenty years. Yet, due to irreversibletermination reactions available therein, polymers withnarrow molecular weight distribution and well-definedtopology cannot easily be obtained by this process. Onthe other hand, living free radical polymerization (LFRP),also known as controlled/“living” radical polymerization,has attracted much attention over the recent years forproviding simple and robust routes to the synthesis of well-defined, low-polydispersity polymers and the fabrication ofnovel functional materials [1–4]. In this context, nitroxide-mediated polymerization [5–7], radical polymerization withreversible addition-fragmentation chain transfer (RAFT)[8–11], and atom transfer radical polymerization (ATRP)[12–16] have extensively been studied.

Mechanistically, LFRP is quite distinct from conven-tional free radical polymerization by the existence of anequilibrium reaction which swiftly switches growing chainsbetween active and dormant states so as to lessen theinstantaneous concentration of free radicals and therebysuppressing the bimolecular irreversible termination reac-tions [17]. Atom transfer radical polymerization, whichuses a transition metal complex as its stimulus in the

M. Najafi :M. Salami-KalajahiPolymer Science and Technology Division,Research Institute of Petroleum Industry (RIPI),1485733111 Tehran, Iran

H. Roghani-Mamaqani :M. Salami-Kalajahi :V. Haddadi-Asl (*)Department of Polymer Engineering and Color Technology,Amirkabir University of Technology,Tehran, Irane-mail: haddadi@aut.ac.ir

J Polym Res (2011) 18:1539–1555DOI 10.1007/s10965-010-9559-1

equilibrium process, has been widely used to develop newpolymers. Unlike other techniques, it takes the advantage ofhalogen exchange to swap a growing radical with adormant species (Scheme 1) [18, 19].

In addition to experimental studies performed on thedevelopment of the kinetics of living radical polymerization(and in particular ATRP), several attempts have been made toanalytically model its behavior. For example, molecularweight averages have been calculated using moment equationsin different living free radical polymerizations [20–29].Molecular weight distributions have also been obtained byemploying PREDICI package, which is known as a powerfultool for solving series of differential equations [30–35].However, unfortunately, these tools are rather abundant withsome limitations (such as requiring many assumptions) whichmostly cause them to lead to average and approximate values.

On the other hand, Monte Carlo method, owing to thestatistical nature of chain growth and chain terminationreactions, is an excellent technique widely utilized tosimulate conventional free radical (co)polymerization sys-tems [36–38] as well as living free radical reactions [39–43]. Moreover, Monte Carlo method enjoys some advan-tages over analytical modeling: it is not restricted to solvingstiff differential equations as opposed to moment equations;nor it is limited to strict assumptions which mainly makemost of analytical methods result in approximate outcomes;besides, it can be simply implemented to different poly-merization processes; Monte Carlo method is also capableof calculating chain length distributions, chemical compo-sition distribution, and sequence length distribution ofcomonomers throughout the (co)polymerization since itcan store the whole information of each individual polymerchain virtually generated during the reaction. Finally, it canvisualize the microstructure of the macromolecules. Never-theless, unfortunately, simulation programs based on MonteCarlo method, if not implemented properly, usually take along time and a huge amount of memory to complete.

A review of related literature indicates that theMonte Carlosimulations of ATRP—and particularly styrene ATRP—presented so far [43–45] have largely taken account ofequilibrium, propagation, termination, and transfer reactions,and thermal initiation of styrene and chain end degradationof dormant chains and radicals (as shown in Scheme 2) havenot been taken into consideration. Chain end degradation of

dormant chains and radicals, by reason of dealing with keyelements of the reaction (radicals, dormant chains, andcatalyst in higher oxidation state) can greatly influence theliving nature of the ATRP, and thereby changing themicrostructure of the polymer. Moreover, most simulationspresented have ignored the effect of chain length ontermination rate constant; this can greatly affect thecontribution of termination reaction and accordingly thelivingness of the reaction. Additionally, to the best of ourknowledge, the simultaneous implementation of diffusion-controlled and chain-length-dependent terminations to thesimulation of styrene ATRP has not been reported yet.

To this end, by employing a thorough kinetic schemeincluding chain end degradation and thermal initiationreactions, the ATRP of styrene is simulated by using a MonteCarlo method in this contribution [46]. Additionally, a hybridmodel is introduced to the simulation to simultaneouslyconsider diffusion-controlled and chain-length-dependentterminations over the course of the polymerization. Lastly,the simulation results are compared with experimental dataso as to verify the accuracy of the simulation.

Simulation description

The kinetic scheme used in the simulation of ATRP comprisesactivation, deactivation, thermal initiation, propagation, ter-mination by combination, transfer to monomer, chain enddegradation type A (dormant chain degradation), and chainend degradation type B (radical degradation) reactions [46].The elementary reaction steps of styrene ATRP used in thesimulation are listed in Eqs. 1 to 9. The FRP mechanism isalso given in Eqs. 10 to 14. The kinetic parameters andinitial concentrations inputted to the simulation are alsogiven in Table 1.

RX þMnt Y=L ���!kact R� þ X �Mnþ1

t Y=L ð1Þ

3M �����!ktherm R� þ R0 � ð2Þ

R� þ X �Mnþ1t Y=L ������!kdeact RX þMn

t Y=L ð3Þ

PnX þMnt Y=L ���!kact P �

n þ X �Mnþ1t Y=L ð4Þ

P �

n þM ���!kp P �

nþ1 ð5Þ

P �

n þM ���!ktrM Pnþ1 þ R� ð6ÞScheme 1 Halogen exchange in ATRP reaction

1540 M. Najafi et al.

P �

m þ P �

n ���!ktc Pmþn ð7Þ

PnX þ X �Mnþ1t Y=L �����!kceda Pn þ X �Mnþ1

t Y=Lþ HX ð8Þ

P �

n þ X �Mnþ1t Y=L �����!kcedb Pn þMn

t Y=Lþ HX ð9Þ

I ���!kd 2R� ð10Þ

3M ������!ktherm R� þ R0 � ð11Þ

P �

n þM ����!kp P �

nþ1 ð12Þ

R= H or Pn

+ MtnY/L + X-Mtn+1Y/L

RRX

kact

kde act

+

Pn Pn+1

kp

M

3ktherm +

M2

Equilibrium Reaction Thermal Initiation

Propagation

Pn

ktc

Termination by Combination

PnPm

+

Pn

ktrM

M

Transfer to Monomer

Pm

+

+ Pn+1

+ X-Mtn+1Y/L + X-Mtn+1Y/L

RR

X

kceda

Chain End Degradation

+ HX + X-Mtn+1Y/L + MtnY/L

RR

kcedb+ HX

R= H or Pn R= H or Pn

. ..

.

. .

. .

.

. .

Scheme 2 Mechanism of styrene ATRP

Coefficient/Parameter Expression/Value Reference

kact (L.mol−1.s−1) 0.45 [47, 48]

ktherm (L2.mol−2.s−1) 2.19×105 exp(−13800 K/T) [49]

kdeact (L.mol−1.s−1) 1.1×107 [48]

kp (L.mol−1.s−1) 4.266×107 exp(−3910 K/T) [49]

k0tc (L.mol−1.s−1) 3.82×109 exp(−958 K/T) [49]

ktrM (L.mol−1.s−1) 2.31×106 exp(−6377 K/T) [49]

kceda (L.mol−1.s−1) 1.0×10−4 [46]

kcedb (L.mol−1.s−1) 1.63×103 [46]

kd (s−1) 1.04×1015 exp(−33500 /RT)a [50]

[M]0 (mol.L−1); ATRP 1 this work

[RX]0 (mol.L−1); ATRP 0.01 this work

1M 1n1t 1Y=L

� �0(mol.L−1); ATRP 0.01 this work

[M]0 (mol.L−1); FRP 5 this work

[I]0 (mol.L−1); FRP 0.05 this work

Table 1 Kinetic rate constantsand initial concentrations usedin simulation of styrene ATRP at110 °C

a R=1.987 (cal.mol−1 .K−1 )

An exhaustive study of chain-length-dependent and diffusion-controlled 1541

P �

n þM �����!ktrM Pnþ1 þ R � ð13Þ

P �

m þ P �

n �����!ktc Pmþn ð14ÞAs viscosity of the system noticeably increased during

the polymerization, a gel effect correlation was imple-mented to the simulation; the diffusion-controlled rateconstant of termination, kgeltc , varies with monomer conver-sion in the reaction system and the rate coefficient at zeroconversion (k0tc) according to Eq. 15 [51].

kgeltcL

mol:s

� �¼ k0tc � exp �2 Axþ Bx2 þ Cx3

� �� � ð15Þ

where x is monomer conversion; A, B, and C are correlatedto the reaction temperature, T [°K], as following:

A ¼ 2:57� 5:05� 10�3T

B ¼ 9:56� 1:76� 10�2T

C ¼ �3:03þ 7:85� 10�3T

It is well-known that the rate of bimolecular reactionsbetween radicals drastically drops when the length ofinvolving radicals increases. Therefore, a model is neededto calculate the termination rate constant of two radicalswith lengths of i and j (where both i and j are larger thanone), i.e. ki;jtc ; such a model is usually developed based onthe termination rate constant of two homo-length radicals(ki;itc ) [52–56]. The termination of short radicals of the samelength happens according to Eq. 16:

ki;itcL

mol:s

� �¼ k0tc � i�eS ; i � ic ð16Þ

This k0tc is the real rate coefficient for terminationbetween two monomeric free radicals in a system whichis not diffusion-controlled. i and ic are respectively chainlength and critical chain length indicating the short lengthboundary; es is also reported to be equal to 0.5 [56].

However, for “long” radicals the following equation isproposed:

ki;itcL

mol:s

� �¼ k0tc � icð Þ�eSþeL � i�eL ; i > ic ð17Þ

where, the k0tc � ðicÞ�eSþeL term herein gives continuity withEq. 16 at the “crossover” chain length (ic); eL and ic are equalto 0.16 and 100 respectively [56]. To summarize, the

following composite model is used for a system which ischain-length dependent:

ki;itcL

mol:s

� �¼ k0tc � i�0:5; i � ic

ki;itcL

mol:s

� �¼ k0tc � icð Þ�0:34 � i�0:16; i > ic

ð18Þ

Values of the cross-termination rate coefficients (ki;jtc ),where i ≠ j, are calculated based on diffusion, harmonic,and geometric means of homo-termination rate constants assummarized below:

ki;jtc hmð Þ L

mol:s

� �¼ k0tc �

2ij

iþ j

� ��0:5

; i � ic

ki;jtc hmð Þ L

mol:s

� �¼ k0tc � icð Þ�0:34 � 2ij

iþ j

� ��0:16

; i > ic

ð19Þ

ki;jtc dmð Þ L

mol:s

� �¼ 1

2� ki;itc þ kj;jtc� � ð20Þ

ki;jtc gmð Þ L

mol:s

� �¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiki;itc � kj;jtc

qð21Þ

The overall rate coefficient for termination at any instantof the polymerization is correlated to ki;jtc values (eitherdiffusion, geometric or harmonic mean) by the followingform:

ktcL

mol:s

� �¼X1i¼1

X1j¼1

ki;jtc½Ri� � ½Rj�½R� � ½R� ð22Þ

where, [Ri] and [Rj] stand for the concentration of chainswith the length of i and j respectively. [R] is also the totalconcentration of all radical species. As this statementimplies, Eq. 22 is not restricted to steady states.

For simultaneously calculating the effect of systemviscosity and chain-length dependency of termination ratecoefficient, a hybrid model is developed by integrating geleffect (Eq. 15) into the equations used for calculating chain-length dependency of termination rate constant (Eqs. 16 to22). The final form of this relation is given by Eq. 23.

kgeltcL

mol:s

� � ¼ P1i¼1

P1j¼1

ki;jtc½Ri��½Rj�½R��½R�

!�

exp �2 2:57� 5:05� 10�3T� �

xþ 9:56� 1:76� 10�2T� �

x2 þ �3:03þ 7:85� 10�3T� �

x3 � � ð23Þ

1542 M. Najafi et al.

The Monte Carlo methodology employed herein isadopted from the well-known Gillespie’s algorithm [57].A simulation volume is considered to calculate the numberof different reactants based on their concentrations (seeEq. 24) and to transform the experimental reaction rateconstants into stochastic rate constants (Eqs. 25 to 27) asrequired in the Gillespie’s algorithm.

XM ¼ ½M �NAvVs ð24Þwhere [M] is the monomer molar concentration; NAv and Vs

also stand for Avogadro’s number and the size of thesimulation volume respectively. A similar approach is usedto calculate the number of other reactants and/or productsgenerated in the simulation. According to Gillespie’salgorithm [57], the stochastic rate coefficients are trans-formed from experimental rate constants as reads:

kMC ¼ kexp; for first order reactions ð25Þ

kMC ¼ kexp

NAv � Vs; for bimolecular between different species

ð26Þ

kMC ¼ 2kexp

NAv � Vs; for bimolecular reactions between

similar speciesð27Þ

It should be noted that since the stochastic ratecoefficients (kMC) have units of reciprocal time, experimen-tal rate coefficients (kexp) for bimolecular reactions must bedivided by the product NAv × Vs.

The probability of any reaction (pk) happening at aparticular time is calculated by Eq. 28 and a condition inthe standard form of Eq. 29 is used to decide which reactiontype would occur:

pk ¼ akPNk¼1

ak

ð28Þ

Xn�1

k¼1

pk< r1 �

Xnk¼1

pk

ð29Þ

where ak is the reaction rate of kth reaction; N is the totalnumber of reaction types available in the system; n is thenumber of reaction type selected to proceed and r1 is arandom number uniformly distributed in the interval (0,1].

To determine the time interval (τ) between two succes-sive reactions, the following relation is used:

t ¼ 1PNk¼1

ak

ln1

r2

� �ð30Þ

where r2 is another random number generated by the samesubroutine.

A subroutine on the base of improved Mersenne Twisteralgorithm is used in the simulation to produce randomnumberswith long period lengths; this condition should be met to avoidthe repetition of random numbers in polymerization reactions[58]. It is written in C++ language and the random numbersgenerated therefrom satisfy the tests of uniformity and serialcorrelation with high resolution. It should also be noted thatthe cycle length of this random number generator is longenough, namely 2216091-1, to fulfill the needs of thissimulation. Two suitable and optimized programs accordingto the flow charts presented in Scheme 3 and Scheme 4 arewritten in C++ language and compiled on a 64-bit openSuSELinux operating system to manage the huge amount ofmemory requested by the program. The average run time ofthe FRP and ATRP programs for a simulation volume of 1×10-14 on a computer equipped with two 2.0 GHz CPU’s and4 GB memory is approximately 1 and 3.5 hours respectively.

Results and discussion

It is well known that the prospect of the development of“living” free radical polymerizations is that polymers withwell-defined microstructure and properties can be synthe-sized. Amongst these properties, controlling molecularweight and polydispersity index of the products are ofmore interest to researchers. On the other hand, it isobvious that in radical polymerizations the molecularweight of chains increases throughout the reaction andconsequently raises viscosity of the system. In addition, asmacromolecules become longer, their hindrance effect andlower mobility reduce the reaction rates of bimolecularreactions between growing polymer chains, namely radicaltermination. In detail, the reaction rates of bimoleculartermination between long chains are almost an order ofmagnitude smaller than those of monomeric species. Geleffect even more abates the bimolecular termination too.Therefore, to consider these phenomena, we apply acomposite model to our Monte Carlo simulation methodto simultaneously take account of both gel effect and chain-length dependency of termination reactions. In the latter, acritical chain length, around which the dependency oftermination rate constant on chain length is calculated byusing two different equations, is defined. Finally, wherepossible, the simulation results are accompanied byexperimental data.

Simulating bimolecular termination rate constant

Bimolecular cross-termination rate coefficients, ki;jtc , (calcu-lated based on geometric mean) changes as a function ofchain length (Fig. 1). It is evident that ki;jtc decreases as thechain length rises and finally plateaus (see Fig. 2 for a

An exhaustive study of chain-length-dependent and diffusion-controlled 1543

1544 M. Najafi et al.

better comparison between ATRP and FRP); therefore, itcan be inferred that the dependency of bimoleculartermination rate constants on chain length is mainlycontrolled by a certain chain length beyond which themobility, and consequently reactivity, of radical chainsremain constant, and thus no more decrease in the reactionrate constants is seen. In this regard, a better representationcan be found in case of FRP, in which very long chains areproduced at each stage of the reaction. For instance, there isa negligible difference between k1000; jtc and k3000; jtc . Further-more, the drastic drop in ki; jtc (i>1) demonstrates that whenone of the two components involved in bimolecularterminations is a monomeric species, the termination ratecoefficient is higher than when both radicals are largesluggish macromolecules (k1; jtc � k3000; jtc ).

Figure 3 illustrates the dependence of average chain-length-dependent and diffusion-controlled termination rateconstant (ktc) on monomer conversion. In ATRP, ktcdecreases as the polymerization proceeds, whereas it isalmost constant throughout free radical polymerization;besides, it is approximately an order of magnitude smallerin FRP. Since the molecular weight of polymer chains riseslinearly as the ATRP reaction proceeds, ktc (because of adecrease in ki;jtc as indicated in Fig. 1) falls over the course ofthe reaction. Nevertheless, because polymers with highmolecular weight are available in the reactor of FRP systemsfrom the onset of the polymerization, ktc remains smaller andunchanged during the reaction. However, a very slightdecrease in ktc can be seen at the end of the polymerization,which is because of gel effect as a result of increasedviscosity of the system. It is also seen that ktc is higher inATRP systems even at the late stages of the reaction; this ismainly assigned to the fact that the molecular weight ofchains produced by “living” radical polymerizations is muchlower than that of chains free-radically synthesized.

Simulating change in the concentration of reactants

The variation of monomer conversion—in terms of ln(M0/M)versus reaction time—is demonstrated in Figs. 4 and 5. Inaccordance with the results, ln(M0/M) ascends linearly in theatom transfer radical polymerization of styrene; nonetheless,this is not the case with FRP. In fact, ln(M0/M) slowlyincreases at the initial stages of free radical polymerization,and finally soars at the late stages of the reaction. This can beattributed to the slow initiation of the polymerization at thebeginning of the reaction. In addition, due to higher viscosityof the polymerization medium, the termination reactions areeven more reduced at the final stages of the reaction, whichaccelerates propagation reaction and thereby steeply raising

ln(M0/M). It can additionally be concluded that the atomtransfer radical polymerization, rather than the free radicalpolymerization, takes much longer times to complete.Finally, it is obvious that the simulation results are inexcellent agreement with the experimental data, whichverifies the suitability of the composite model used to takeaccount of diffusion-controlled and chain-length-dependenttermination reactions.

In order to further evaluate the simulation results, thedependence of conversion and ln(M0/M) on reaction timefor FRP and ATRP has been compared with the dataavailable in the literature [24, 59]. It should be noted thatthe simulation program has been run in accordance with thekinetic parameters given in the mentioned references;Figs. 6 and 7 illustrate the dependence of conversion uponreaction time for ATRP (110 °C) and FRP (70 °C) systemsrespectively. It is obvious that the simulation results are ingood agreement with the experimental data in both ATRPand FRP cases.

The change in the concentration of initiator over thecourse of the polymerization reactions is delineated inFig. 8. Thanks to fast initiation, initiator is entirelydecomposed at the very early stages of the polymerizationin ATRP. Therefore, all chains are initiated at the beginningof the reaction therein, which provides the basic need forproducing polymers with narrow molecular weight distri-bution. However, as expected, initiator is exponentiallyconsumed in free radical polymerization and since thetermination reactions are highly reduced, there is a smallchange in its concentration. This means that initiation ofchains continues throughout the reaction up to very highconversion, and therefore it results in creation of chainswith different residence times.

Figure 9 shows the dependence of the concentration ofthe catalysts both in lower oxidation state, namely CuBr,and in higher oxidation state, that is CuBr2, in styreneATRP. According to Fig. 9, the concentration of CuBrlessens as the reaction progresses and finally reaches aplateau. Since the progress of the polymerization gives riseto radical termination, albeit so little in ATRP, theconcentration of growing radicals abates, which corre-spondingly shifts the equilibrium toward the formation ofnew growing radicals and thereby consuming more CuBr.On the other hand, the concentration of CuBr2 increases asthe polymerization reaction advances to higher conversionand eventually levels out. By reason of irreversibletermination of growing radicals, the equilibrium reactionin ATRP tends to move toward the activation of dormantchains and the production of new radicals; hence, contraryto CuBr, the concentration of CuBr2 increases. It is alsoevident that the control over the polymerization begins atthe initial stages of the reaction and both CuBr and CuBr2plateau at conversion of around 40%.

Scheme 3 The Monte Carlo simulation algorithm of styrene FRPR

An exhaustive study of chain-length-dependent and diffusion-controlled 1545

Scheme 4 The Monte Carlo simulation algorithm of styrene ATRP

1546 M. Najafi et al.

Figure 10 portrays the variation of growing radicalsduring the reaction. The concentration of free radicals isnearly two orders of magnitude smaller in ATRP comparedto FRP. It remains approximately unchanged during thereaction too. In contrast to ATRP, the concentration ofgrowing radicals is constant only up to conversion of about50% in FRP and steeply rises at the final stages of thereaction. This can largely be imputed with gel effect andconsequently a smaller amount of termination at high

conversion; actually, there is an accumulation of radicals atthe end of the polymerization, which correspondingly leadsto an acceleration in the rate of propagation reaction too.

Figure 11 demonstrates the dependence of the concentra-tion of dormant chains on monomer conversion for ATRP.Dormant chains are quickly generated upon the decomposi-tion of initiator at the early stages of the polymerization andnearly reach the initial concentration of initiator. However, theconcentration of dormant chains drops as the polymerizationproceeds since a few irreversible termination reactions—

Fig. 4 Illustration of the dependence of ln M0M1

� �on reaction time in

ATRP

Fig. 3 ktc as a function of monomer conversion

Fig. 2 Comparing dependence of ki;jtc on chain length in FRP andATRP

Fig. 1 Cross-termination rate coefficients as a function of chainlength j

An exhaustive study of chain-length-dependent and diffusion-controlled 1547

which are unavoidable—change dormant chains into deadpolymers. Nevertheless, the diminishment of the concentra-tion of dormant chains is not so large, and thus they keepcomposing a high (about 80%) portion of the whole chains upto very high conversion; in other words, the system keeps itslivingness up to high conversion.

The concentration of initiated chains increases sheer andreaches the concentration of initiator, that is to say 0.01 (mol.L−1), at the initial stages of the atom transfer radicalpolymerization (Fig. 12). This means that the initiator iscompletely decomposed at the beginning of the reaction and

all chains are simultaneously and quickly initiated. On thecontrary, in case of free radical polymerization, the concen-tration of initiated chains gradually augments during thepolymerization up to high conversion. In effect, chains areslowly initiated over the course of the reaction and hence aregiven different residence times. On the other hand, theconcentration of dead polymers gradually ascends during thereaction and it reaches about 20% of the initiated chains atthe end of the ATRP reaction. However, in free radicalpolymerization, dead polymers accumulate more rapidlyduring the reaction and eventually make up about 50% ofthe chains at the end of the reaction. Therefore, it is deduced

Fig. 7 Illustration of the dependence of conversion on reaction timein FRP at 70 °C; conditions and simulation parameters as in Ref. [59]

Fig. 8 Change in the concentration of initiator

Fig. 6 Illustration of the dependence of conversion and ln M0M

� �on

reaction time in ATRP; Conditions and simulation parameters as inRef. [24]

Fig. 5 Illustration of the dependence of ln M0M1

� �on reaction time in

FRP

1548 M. Najafi et al.

that the irreversible termination reactions are highly sup-pressed by reducing the concentration of growing radicals inatom transfer radical polymerization, and hence the greatmajority of polymer chains remain alive at the end of thereaction.

Simulating dependence of molecular weightand polydispersity index on monomer conversion

The living nature of ATRP has successfully fulfilled therequirements of producing tailor-made polymers withnarrow molecular weight distribution. It also provides theprerequisites for obtaining molecular weight that linearlyaugments over the course of the polymerization. AMonte Carlo simulation based on diffusion-controlled

and chain-length-dependent bimolecular terminations isdeveloped to calculate the change in number- andweight-average molecular weights in ATRP and FRP. Inaddition, the variation of polydispersity index (PDI) isalso computed during the polymerization. Furthermore,molecular weight distributions (comprising number andweight distributions) have been calculated at differentstages of the reaction. Lastly, the simulation results arecompared with experimental data to confirm the validityof our simulation.

Number-average molecular weight starts from lowvalues and linearly increases over the course of theATRP (Fig. 13). It is also clear that the molecular weightfollows the trend one can expect according to itstheoretical values. For example, it is equal to about9000 g.mol−1—which is in good agreement with thetheoretical value, i.e. 9373.51 g.mol−1—at conversion of90%. Differently, in FRP, Mn takes its initial value as largeas that of commercial polymers, namely about 70000 g.mol−1, and continues with the same value up to conversionequal to 40%. Nonetheless, from this point on, number-average molecular weight increases as the reactionprogresses and finally it plateaus at the end of thepolymerization. This trend can be ascribed to the suppres-sion of termination reactions in the highly viscous mediumof the polymerization at high conversion, which helpsgrowing radicals live longer and add more monomers.However, due to a remarkable reduction in the concentra-tion of monomer, Mn levels out at the very late stages ofthe polymerization.

Fig. 11 Change in the concentration of dormant chains

1 Mn ¼ ½M�½I�

� ��M0 � P ¼ 100� 104:15� 0:90 ¼ 9373:5 g:mol�1

Fig. 10 Change in the concentration of growing radicals

Fig. 9 Change in the concentration of CuBr and CuBr2

An exhaustive study of chain-length-dependent and diffusion-controlled 1549

The dependence of weight-average molecular weight onmonomer conversion shows trends similar to number-averagemolecular weight (Fig. 14). In ATRP, Mw linearly augmentsover the course of the polymerization; to the contrary, it isalmost constant and equal to about 105000 g.mol−1 up toconversion of 40% in free radical polymerization. Also, itrises in the rest of the free radical polymerization and finally,owing to monomer shortage, plateaus. It is also evident inboth Figs. 13 and 14 that polymers produced by the freeradical polymerization system have much larger molecularweight. Lastly, excellent agreement can be seen between thesimulation results and the experimental data.

As it is shown in Fig. 15, polydispersity index of polymerproduced in ATRP soars at the beginning of the reaction and

then greatly drops over the course of the polymerization; infact, as the reaction proceeds, chains become more similar inlength. This decrease in polydispersity index continues to thevery end of the polymerization and it finally amounts tovalues nearly equal to 1.0. However, in case of free radicalpolymerization, PDI is well-nigh equal to 1.5 throughout thepolymerization, although it becomes slightly larger at thevery late stages of the reaction. The latter is attributed to theunsteadiness of the polymerization at very high conversionas a result of a considerable reduction in the concentration ofmonomer.

Fig. 13 Illustration of the dependence of number-average molecularweight on monomer conversion

Fig. 14 Illustration of the dependence of weight-average molecularweight on monomer conversion

Fig. 15 Illustration of the dependence of polydispersity index onmonomer conversion

Fig. 12 Change in the concentration of initiated chains and dead polymers

1550 M. Najafi et al.

Similar to Mn and Mw, the simulation results whollyconcur with the experimental data.

Simulating complete molecular weight distributionthroughout polymerization

Monte Carlo method, contrary to many analytical models, isable to easily calculate the molecular weight distributionsduring the polymerization as it is not dependent on solvingsome stiff, time-related differential equations. As mentionedabove, the number and weight distributions of polymer chainsare plotted as a function of polymerization conversion (seeFigs. 16, 17, 18, and 19). It can be noticed that the locationof the peaks of number and weight distributions nearlyremains intact throughout the free radical polymerization. Inaddition, the broadness of these distributions is almost

similar at different stages of the polymerization. Moreover,a detectable portion of chains produced by FRP is also inoligomeric forms as indicated by the low-molecular-weighttail of the number molecular weight distribution. Quitedifferently, the peak of molecular weight distributions shiftsto higher molecular weight while the atom transfer radicalpolymerization proceeds; in fact, this additionally affirms theliving nature of the polymerization. Besides, the contributionof low molecular weight polymers in ATRP can beneglected. In addition, the molecular weight distribution ofthe chains synthesized by ATRP is much narrower than thatof free-radically synthesized polymers. The agreementbetween the simulation results and the experimental data isquite acceptable, although a slight difference can be seen athigh conversion in ATRP. Indeed, thanks to couplingtermination reactions, the molecular weight distributions

Fig. 16 Evolution of number molecular weight distribution through-out free radical polymerization

Fig. 17 Evolution of number molecular weight distribution through-out atom transfer radical polymerization

Fig. 18 Evolution of weight molecular weight distribution throughoutfree radical polymerization

Fig. 19 Evolution of weight molecular weight distribution throughoutatom transfer radical polymerization

An exhaustive study of chain-length-dependent and diffusion-controlled 1551

widen at the final stages of the atom transfer radicalpolymerization as more clearly represented by theexperimental data (see curves plotted at 80% conversionin Figs. 17 and 19).

A better presentation of molecular weight distribu-tions is shown in two dimensional plots (Figs. 20 and21). The broadness of molecular weight distribution offree-radically generated chains is completely noticeablein comparison with that of chains synthesized by ATRP.Also, as stated before, the portion of chains with verylow molecular weight is much higher in FRP. This islargely attributed to the premature termination of chainsat the early stages of the polymerization, which produceschains composed of only a few monomers. However, asa result of reversible capping of growing radicals inATRP and thereby reducing their concentration, prema-ture terminations are highly suppressed after a short time

at the outset of the polymerization, and therefore thecontribution of the low-molecular-weight chains is unnotice-able. Finally, a remarked and rapid fall in PDI values at theinitial stages of the ATRP reaction verifies the above.

Experimental part

Materials

Copper(I) bromide (CuBr, Aldrich, 98%), Copper(II)bromide (CuBr2, Aldrich, 99%), N,N,N′,N″,N″-pentame-thyldiethylenetriamine (PMDETA, Aldrich, 99%), ethylalpha-bromoisobutyrate (EBiB, Aldrich, 97%), styrene(Aldrich, 99%), anisole (Aldrich, 99%), benzoyl peroxide(BPO, Aldrich, 97.0%), and neutral aluminum oxide(Aldrich) were used as received.

Fig. 20 Evolution of number molecular weight distribution throughout polymerization reaction

1552 M. Najafi et al.

Polymerization

The ATRP polymerizations were performed in a 250 mllab reactor which was placed in an oil bath thermostatedat the desired temperature. A number of batch polymer-izations were run at 110 °C in bulk and with the molarratio of 100:1:1:1 for [M]:[EBiB]:[CuBr]:[PMDETA],giving a theoretical polymer molecular weight of10415 g.mol−1 at 100% conversion. The reactor wasdegassed and back-filled with nitrogen gas 3 times, andthen left under N2. The batch experiments were run byadding deoxygenated monomer (styrene), catalyst (CuBr),ligand (PMDETA), and deoxygenated anisole as aninternal standard to the reactor and then increasing thereaction temperature to 110 °C (within about 15 min.). Thesolution turned light green as the CuBr/PMDETA complexformed. Finally, after the majority of the metal complexhad formed, initiator (EBiB) was added to the system to

start the styrene ATRP. In case of FRP, the polymerizationstarted by adding initiator (BPO) to the reactor, which waspurged by N2 and contained deoxygenated styrene andanisole as an internal standard at 110 °C. The molar ratioof monomer to initiator was 100. In both polymerizations,a sample was taken before the reaction started and used asa comparison reference for the samples later taken at thedifferent stages of the reaction so as to measure themonomer conversion.

Analysis

Gas chromatography (GC) is simple and highly sensitivecharacterization method and does not require removal of themetal catalyst particles. GC was performed on an Agilent-6890N with a split/splitless injector and flame ionizationdetector (FID) detector, using a 60mHP-INNOWAX capillarycolumn for the separation. The GC temperature profile

Fig. 21 Evolution of weight molecular weight distribution throughout polymerization reaction

An exhaustive study of chain-length-dependent and diffusion-controlled 1553

included an initial steady 60 °C heating for 10 min. anda 10 °C/min. ramp from 60 to 160 °C. The sampleswere also diluted with acetone. The ratio of monomer toanisole at different stages of the reaction was measuredby GC to calculate monomer conversion throughout thereaction. The average molecular weight and molecularweight distributions were measured by gel permeationchromatography (GPC) technique. Polymer sampleswere dissolved in THF and passed through a neutralaluminum oxide column to remove catalyst particles. AWaters 2000 ALLIQNCE with a set of three columnswas utilized to determine polymer average molecularweight and polydispersity index (PDI) using toluene asan internal standard against linear polystyrene standards.

Conclusion

An optimized and high-performance Monte Carlo simulationbased on diffusion-controlled and chain-length-dependentbimolecular termination reactions is developed to thoroughlysimulate free radical and atom transfer radical polymerizationof styrene. It is found out that as the chain length of radicalsincreases, the termination rate constant drops and finallylevels out beyond a certain chain length. In addition, thesimulation results show that average termination rate constantgreatly decreases over the course of the atom transfer radicalpolymerization, while it remains almost unchanged in FRP.The linear plot of ln(M0/M) against reaction time attests thelivingness of the atom transfer radical polymerization;however, because of slow initiation, the FRP progressesslowly at the initial stages and then speeds up, which resultsin a steep slope in ln(M0/M) curve. In addition, initiator iscompletely decomposed at the early stages of the polymer-ization in ATRP, whilst it is consumed slightly in anexponential manner in FRP. The concentration of CuBrfirstly decreases and then levels out. On the contrary, that ofCuBr2 rises and finally reaches a plateau. The concentrationof growing radicals is very low and constant in ATRP,whereas it is more noticeable in FRP and rises at the latestages of the reaction. Furthermore, the concentration ofdormant chains in ATRP is high enough throughout thereaction, although it shows a little reduction. Besides,molecular weight linearly increases with conversion inATRP; however, high-molecular-weight chains are availablein the FRP system from the beginning of the reaction; Mn

remains nearly the same up to conversion equal to 50%and then increases up to the end of the reaction. Also,polymers produced through FRP have higher molecularweight. In addition, the peaks of molecular weightdistributions shift toward higher molecular weightthroughout the ATRP, whilst it nearly stays at the sameplace in FRP. Finally, the molecular weight distributions

become narrower at the end of the atom transfer radicalpolymerization. However, as expected, the broadness ofmolecular weight distribution of the chains produced byfree radical polymerization is well-nigh similar at differentstages of the reaction.

Acknowledgements We would like to acknowledge Ms. GhazalAssadi-pour for her invaluable cooperation in the optimization andtune-up of the simulation program. Also, the financial and technicalsupport of Petrochemical Research and Technology Company (NPC-RT)is gratefully acknowledged.

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