kinetic modeling of controlled living microemulsion

Kinetic Modeling of Controlled Living MicroemulsionPolymerizations That Use ReversibleAddition–Fragmentation Chain Transfer

KEVIN D. HERMANSON,* SHIYONG LIU,{ ERIC W. KALER

Center of Molecular Engineering and Thermodynamics, Department of Chemical Engineering,University of Delaware, Newark, Delaware 19716

Received 12 January 2006; accepted 23 June 2006DOI: 10.1002/pola.21652Published online in Wiley InterScience (www.interscience.wiley.com).

ABSTRACT: Reversible addition–fragmentation chain transfer (RAFT) polymerizationis a useful technique for the formation of polymers with controlled architectures andmolecular weights. However, when used in the polymerization of microemulsions,RAFT agents are only able to control the polymer molecular weight only at highRAFT concentrations. Here, a kinetic model describing RAFT microemulsion polymer-izations is derived that predicts the reaction rates, molecular weight polydispersities,and particle size. The model predicts that at low RAFT concentrations, the RAFTagent will be consumed early in the reaction and that this will result in uncontrolledpolymerization in particles nucleated late in the reaction. The higher molecularweight polydispersity that is observed in RAFT microemulsion polymerizations is theresult of this uncontrolled polymerization. The model also predicts a shift in the con-version at which the maximum reaction rate occurs and a decrease in the particlesize with increasing RAFT concentration. Both of these trends are also consistentwith those observed experimentally. VVC 2006 Wiley Periodicals, Inc. J Polym Sci Part A:

Polym Chem 44: 6055–6070, 2006

Keywords: controlled living polymerization; emulsion polymerization; microemul-sion polymerization; kinetics (polym.); reversible addition fragmentation chain trans-fer (RAFT)

INTRODUCTION

In recent years, controlled living radical poly-merization has become an important techniquefor the formation of polymers with controlledarchitectures and molecular weights.1,2 One ofthe more important techniques for controlled

free-radical polymerizations is reversible addi-tion–fragmentation transfer (RAFT) polymeriza-tion. Although the RAFT process has been app-lied to numerous bulk and solution polymeriza-tions, very few RAFT polymerizations have beensuccessfully conducted in a heterogeneous poly-merization environment.3–5 Recently, a RAFT po-lymerization was successfully completed in amicroemulsion.6 In that case, 2-cyanoprop-2-yldithiobenzoate (CPDB) was used in the polymer-ization of n-hexyl methacrylate (C6MA), solubi-lized in a dodecyltrimethylammonium bromide(DTAB) microemulsion, yielding polymers withlow polydispersities and low molecular weightsfor RAFT/initiator ratios above 3.0. At lowerRAFT/initiator ratios, a shoulder developed in the

*Present address: E22-Biophysics, Department of Physics,Technische Universitat Munchen, 85747 Garching, Germany

{Present address: Department of Polymer Science andEngineering, University of Science and Technology of China,Hefei, 230026, Anhui Province, People’s Republic of China

Correspondence to: E. W. Kaler (E-mail: [email protected])

Journal of Polymer Science: Part A: Polymer Chemistry, Vol. 44, 6055–6070 (2006)VVC 2006 Wiley Periodicals, Inc.

6055

final polymer molecular weight distribution, andthis resulted in a high molecular weight polydis-persity. An understanding of why this shoulderdevelops when RAFT is applied to microemulsionpolymerizations would aid in the proper selectionof RAFT agents for use in microemulsion formula-tions. To obtain a better understanding of themechanisms leading to this unusual behavior, thepolymerization kinetic model outlined here hasbeen developed.

Kinetic models that describe the homoge-neous RAFT process have previously been devel-oped,7–11 and some aspects of heterogeneousRAFT polymerizations have also been previouslymodeled.12–16 Previous reports of kinetic model-ing involve the use of rate constants that varyby as much as 7 orders of magnitude. The basisfor these discordant rate constants is the unset-tled debate over the inclusion of a combinationtermination mechanism.7,17–27 Fortunately, inmicroemulsion polymerizations, there is not sig-nificant termination through combination,28 sothe polymerization mechanism is less ambiguous

and the kinetics can be modeled with the mech-anism shown in Scheme 1, which is similar tothat used previously.7

The initiator molecules first decompose to givetwo initiator radicals (reaction I). The initiatorradicals are then free to propagate with the mon-omer to form a polymer chain (reaction II). Thispolymer chain can either continue to react withthe monomer or react with an unpolymerizedRAFT agent (XR) to form a macro-RAFT radical(PXR; reaction IIIa). This macro-RAFT radicalcan then fragment to give either the initial poly-mer chain or a new initiator radical (R) and apolymerized RAFT agent (XP). If fragmentationresults in an initiator radical, the newly formedinitiator radical can react with the monomer toform a second polymer chain (reaction II). Thissecond polymer chain will continue to propagateuntil it finds either an unpolymerized RAFTagent (XR) or polymerized RAFT agent (XP). Ifthe second polymer chain finds a polymerizedRAFT agent (XP), it can react to form the macro-RAFT radical (PXP; reaction IV). Once formed,

Scheme 1

6056 HERMANSON, LIU, AND KALER

Journal of Polymer Science: Part A: Polymer ChemistryDOI 10.1002/pola

this macro-RAFT radical can fragment to give ei-ther of the attached polymer chains. Throughreaction IV, an equilibrium is established betweenthe active polymer chains (P) and the dormantpolymer chains attached to the RAFT agent (XP).This equilibrium is the core mechanism andresults in a reduced molecular weight polydisper-sity.

In this scheme, termination is assumed tooccur at the same rate, regardless of the polymerchain size. As a result, all the macro-RAFT radi-cals are classified as either one of two species,PXR or PXP. PXR refers to a radical that con-tains both a polymer chain (P) and a cyanopropylgroup (R), whereas PXP refers to a radical thatcontains two polymer chains. The reaction ofa cyanopropyl radical with an unpolymerizedRAFT agent is assumed to have a negligibleeffect on the kinetics. All radical species (R, I,and P) are assumed to propagate at the chainpropagation rate (kp). In this scheme, the valueof rate constant kfra,2 is controversial but isexpected7,18,20,22,27 to be within the very broadrange of 10�2–105 s�1.

A common variation of Scheme 1 assumes asteady-state equilibrium for reaction IIIa. In thiscase, reaction IIIa becomes either an irreversiblechain-transfer reaction with ktr ¼ kact,2kfra,1/(kfra,2 þ kfra,1) or a reversible chain-transfer reac-tion with both a forward rate constant, ktr, anda reverse rate constant, k�tr ¼ kact,1kfra,2/(kfra,2þ kfra,1) (reaction IIIb). For the microemulsionpolymerization simulations reported here, theuse of the steady-state assumption gave smallbut nonnegligible differences in the predictedresults.

THEORY

Nonliving microemulsion polymerizations havebeen studied extensively, and a general mechanismby which they occur has been developed.28–35 In anonliving microemulsion polymerization, an ini-tiator (typically water-soluble) is added to a one-phase and thermodynamically stable solution ofmonomer-swollen micelles. The added initiatorfirst begins to propagate in the aqueous domainuntil a critical degree of polymerization isreached, whereupon the radical becomes suffi-ciently hydrophobic to enter a monomer-swollenmicelle, thereby forming a polymer particle.Once entry has occurred, the radical continuesto propagate until either it undergoes chain

transfer to the monomer, by which exit may occur,or all the monomer in the microemulsion isconsumed. The monomer consumed in the poly-mer particle by the radical is replenished by mon-omer diffusion from the uninitiated micelles. Theconcentration of the monomer at the locus of thepolymerization is governed by thermodynam-ics.28,33 Because there are a large number ofmicelles in comparison with the final number ofparticles, new particles are nucleated throughoutthe entire reaction, and biradical termination isnegligible.

Kinetic Model for RAFT Polymerization

In the polymerization scheme used here, the ini-tiator first decomposes in the aqueous domain toproduce two radicals (reaction I in Scheme 1).The radical concentration in the aqueous phaseis assumed to be in the steady state, and so therate of radical entry (q) is equal to the rate ofinitiator decomposition:

qðtÞ ¼ ceff 2kd½I� expð�kdtÞ ð1Þ

where ceff is an efficiency constant that accountsfor radical termination in the aqueous domain,kd is the dissociation rate constant, [I] is the ini-tiator concentration, and t is the time.

Once the initiator radical enters a monomer-swollen micelle, it begins to propagate and therebyforms a new polymer particle. The radical canreact with either the monomer or the RAFT agentwithin the particle. It is assumed that the RAFTagent is free to exit a particle only if it has not yetreacted with a polymer chain (Pm). As a result,each polymerizing particle experiences a differentlocal concentration of the RAFT agent, and subse-quent reaction rates are then a function of both agiven time (t) and the time at which the particle isinitiated (t1). On the basis of Scheme 1, the equa-tions that describe the reaction rate within a parti-cle initiated at time t1 are

@PXRðt1; tÞ@t

¼ kact;2 CpartXR ðtÞP�ðt1; tÞ

þ kact;1 CXPðt1; tÞR�ðt1; tÞ� ðkfra;1 þ kfra;2ÞPXRðt1; tÞ ð2Þ

@PXPðt1; tÞ@t

¼ kact;2 CXPðt1; tÞP�ðt1; tÞ� 2kfra;2PXPðt1; tÞ ð3Þ

MICROEMULSION POLYMERIZATIONS 6057


@CXPðt1;tÞ/partðt1;tÞ@t

¼ 2kfra;2PXPðt1;tÞþkfra;1PXRðt1;tÞ� kact;1CXPðt1;tÞRðt1;tÞ

� kact;2CXPðt1;tÞP�ðt1;tÞ ð4Þ@R�ðt1;tÞ

@t¼�kpR

�ðt1;tÞCpartmonðtÞþkfra;1PXRðt1;tÞ

� kact;1CXPðt1;tÞR�ðt1;tÞ ð5Þ@P�ðt1;tÞ

@t¼ kpR

�ðt1;tÞCpartmonðtÞ�kact;2C

partXR ðtÞP�ðt1;tÞ

�kact;2CXPðt1;tÞP�ðt1;tÞþ2kfra;2PXPðt1;tÞþkfra;2PXRðt1;tÞ ð6Þ

where R*(t1,t), P*(t1,t), PXP(t1,t), and PXR(t1,t)are defined in Scheme 1 as radical concentra-tions per unit of volume of the microemulsion ina particle initiated between time t1 and t1 þ dt1;CXP and Cpart

XR are the concentrations of themacro-RAFT agent and unpolymerized RAFTagent per a unit of volume of the particle, re-spectively; Cpart

mon is the concentration of the mon-omer in the polymer particle; and /part is theunit of volume of the polymerizing domain perunit of volume of the microemulsion.

When the size of the polymerizing domain islarger than a swollen micelle (ca. 3 nm), the vol-ume of the polymerizing domain is equal to thatof a monomer-swollen polymer chain of lengthL. L(t1,t) is the length of a chain initiated attime t1 at a latter time t. L can be found fromthe average propagation rate of a radical. Theaverage propagation rate is equal to the fractionof active particles multiplied by the propagationrate of those particles:

@L

@tðt1;tÞ¼kpC

partmonðtÞ

� P�ðt1;tÞþR�ðt1;tÞP�ðt1;tÞþR�ðt1;tÞþPXPðt1;tÞþPXRðt1;tÞ ð7Þ

The quotient on the right-hand side of eq 7 is thefraction of radicals initiated at t1 that are stillpropagating at time t. The volume of a monomer-swollen polymer particle at time t is then given by

rðt1;tÞ¼ 3MWLðt1;tÞ4pqpolyNA 1�Cpart

monðtÞMWqmon

� �0B@

1CA

13

ð8Þ

and

/partðt1;tÞ¼4

3prðt1;tÞ3NAqðt1Þ ð9Þ

where MW is the monomer molecular weight, NA

is Avogadro’s number, qmon is the bulk monomerdensity, and qpoly is the bulk polymer density.

At very small particle sizes, the volume of thepolymerizing domain is dominated by the vol-ume of the surfactant tails. In this case, the poly-merizing domain resembles a micelle more thana polymer particle, and it is assumed that par-ticles less than the size of the micelles (r ¼ 3nm)36 have a radius equal to that of the mono-mer-swollen micelle.

Cpartmon and Cpart

XR are not the same as the micel-lar concentrations because of thermodynamicpartitioning. The partitioning of the monomer innonliving polymerizations has been investigatedpreviously for the same monomer and surfactantconcentrations used in this study.33 Thoseresults show that the concentration at the react-ing site decreases linearly with the polymer con-version. Thus

CpartmonðtÞ ¼ C0ð1� f ðtÞÞ ð10Þ

where f is the monomer conversion and C0 is theinitial concentration of the monomer in the poly-mer particle. The partitioning of the unpolymer-ized RAFT agent (XR) is not known, but approxi-mate values can be estimated under the assump-tion that the agent partitions similarly to themonomer (see the appendix). Equations 2–6assume that the initiator-derived radicals (I inScheme 1) behave identically to the RAFT-agent-derived radicals (R* in Scheme 1). A furtherassumption is that the initiator and polymer rad-icals react with the monomer at the same rate. Ithas been argued by some37 that the inhibition pe-riod observed in some bulk RAFT polymeriza-tions may be due to the slow reaction rate of theRAFT leaving group (R) with the monomer. How-ever, 2-cyanopropyl is a highly reactive leavinggroup, and in polymerizations of methyl acrylate,the rate of its reaction with the methyl acrylatemonomer is larger than that of a methyl acrylateradical.38 This is also true for the case modeledhere because 2-cyanopropyl reacts with metha-crylates faster than acrylates,39 and acrylate rad-icals react faster with the monomer than methac-rylate.40,41 Therefore, in the modeling of CPDB/methacrylate RAFT polymerizations, the initia-tion rate can safely be taken to equal the poly-mer/monomer propagation rate.

To solve this set of equations, the concentra-tion of initiator radicals that enter a particlebetween time t1 and t1 þ dt1 must be known. In



the case of no radical exit, this concentration isequal to the rate of initiator decomposition:

R�ðt1; tÞ ¼ q at t ¼ t1 ð11Þ

The concentrations found by the solution of eqs2–6 are thus for the particles initiated at timesbetween t1 and t1 þ dt1, and the total concentra-tion of each species can be found by summationover all t1 values. The rate of monomer con-sumption is calculated by the multiplication ofthe total number of propagating radicals by theradical propagation rate:

@fmon

@t¼ kp C

partmonðtÞM0

Z t

0

ðR�ðt1; tÞ þ P�ðt1; tÞÞdt1ð12Þ

Similarly, the rate of unpolymerized RAFT agent(XR) consumption is given by

@fXR@t

¼ 1

XR0

Z t

0

ðkact;2 CpartXR ðtÞP�ðt1; tÞ� kfra;2PXRðt1; tÞÞdt1 ð13Þ

where fmon and fXR are the conversions of themonomer and RAFT agent, respectively, and M0

and XR0 are the concentrations at the start ofthe reaction of the monomer and RAFT agent

per liter of the microemulsion, respectively.The predicted reaction kinetics of a RAFT

microemulsion polymerization are given by thesimultaneous solutions of eq A5 and eqs 1–13.With the method of moments, a similar set ofequations can be derived that describes the evo-lution of the molecular weight and polydisper-sity (see the appendix). The discrete particle sizedistribution can be determined by the solutionof eqs 1, 7 and 8 for all t1 values and by thecounting of the number of particles between sizer and larger size r þ Dr. The limit of this distri-bution as Dr approaches 0 yields the distributionnr(r, t), where nr(r, t)dr is the number of particleswith a size between r and r þ dr. The predictedz-average size that would be measured by qua-sielastic light scattering (QLS), neglecting theeffect of the particle form factor, is

hrðtÞiQLS ¼R10 r6nrðr; tÞdrR10 r5nrðr; tÞdr

ð14Þ

This kinetic model addresses two issues that arenot a concern in nonliving microemulsion poly-merization. First, because of the compartmental-

ization in microemulsion polymerizations, eachRAFT agent is not accessible to each polymeriz-ing radical. Once a macro-RAFT agent (XP inScheme 1) reaches a critical degree of hydropho-bicity, it is no longer able to exit the particle,and this situation results in a local polymerenvironment that is different from the averagepolymer environment. Therefore, the polymer-ization rate within a particle not only is a func-tion of the overall time of reaction but alsodepends on the time at which the particle isnucleated. Here for simplification we assumethat the critical degree of polymerization thatinhibits the exit of a macro-RAFT agent or radi-cal (XP, PXR, or PXP) is equal to 1 (Pn ¼ P1).Second, the concentration of the unpolymerizedRAFT agent (XR in Scheme 1) in the particlewill not equal the concentration in the sur-rounding micelles. The concentration will eitherbe controlled by mass transfer (diffusion) fromsurrounding micelles or be set by a thermody-namic partition coefficient. Because CPDB has asolubility (0.12 mM)6 close to that of C6MA (0.4mM)35 and because the diffusion rate of C6MAis high enough to enable thermodynamic parti-tioning,28,33 it is assumed that the CPDB con-centration in the particle will also be set by apartitioning coefficient, as discussed further inthe appendix.

In this model, the small amount of the RAFTagent soluble in the aqueous domain is assumedto have a negligible effect on the initiation effi-ciency. The effect of the RAFT agent on initia-tion has been investigated with a modified Max-well–Morrison model proposed by Smulderset al.15 that assumes that reaction III in Scheme1 can be modeled with only the irreversiblechain-transfer constant, ktr ¼ kact,2kfra,1/(kfra,2 þkfra,1). Using values for the polymerization stud-ied here in the proposed Maxwell–Morrisonequations, we have found that CPDB is pre-dicted to have a negligible impact on the initia-tion efficiency for polymerizations in which theinitial local micelle RAFT concentrations are lessthan 0.2 mM and ktr/kp is less than 1000.

This model also assumes that radical exitdoes not affect the polymerization kinetics. Theexits of both cyanopropyl radicals and mono-meric radicals produced by chain-transfer wereinitially included in the model. Preliminarilyresults showed that radical exit had a negligibleeffect on the polymerization kinetics, molecularweight, and particle size. The inclusion of theradical exit terms also often leads to numerical



instabilities, so they were omitted for the finalcalculations.

SOLUTION METHOD

We solved equations A5 and 1–13 by first ass-uming monomer and RAFT concentrations fortime Dt. As an initial guess, the monomer andRAFT concentrations were assumed to decreaselinearly from time zero to time Dt. With theseassumed monomer and unpolymerized RAFTconcentrations, eqs 1–9 and 11 were then solvedfor all t1 values between time zero and Dt withthe Fortran subroutine DDASSL.42 The result-ing species concentrations at time Dt were thenknown for all particle initiation times (t1). Thetotal concentrations of P*, R*, and PXR at Dtwere found by integration over all t1 values.With these total concentrations, and under theassumption of a linear change in the total con-centration from time zero to Dt, new monomerand unpolymerized RAFT concentrations,Cpart

mon(t) and CpartRAFT(t), were determined by the so-

lution of eqs 12, 13, and A5. If these newly cal-culated monomer and unpolymerized RAFT con-centrations were not equal to the initially ass-umed concentrations, the new monomer andunpolymerized RAFT concentrations were thenused as the assumed concentrations, and eqs 1–9 and 11 were once again solved. These stepswere repeated until the assumed and calculatedconcentrations were equal. This process wasrepeated for subsequent time steps (Dt) until atime was reached at which the polymerizationwas near completion. A step size of Dt ¼ 2 s orless yielded accurate numerical solutions.

RESULTS AND DISCUSSION

Figures 1 and 2(a,b) show the predicted reactionkinetics with appropriate values for the RAFTmicroemulsion polymerization of C6MA underthe experimental conditions used by Liu et al.6

The propagation rate constant was approxi-mated to be kp ¼ 1500 M�1 s�1 by the interpola-tion of pulsed-laser polymerization data for otherlinear alkyl methacrylates.41 kd of the initiator,V50, was used as supplied by the manufacturer(kd ¼ 2.03 � 10�4 s�1), and ceff was assumed to be0.25. The initial concentration of the monomer inthe polymer particle, C0, was found through the

fitting of the Morgan–Kaler kinetic model,28

which accurately describes the polymerizationof DTAB/C6MA/H2O microemulsions, to kineticdata of polymerizations performed without aRAFT agent. The kinetic constants for thereaction of CPDB with C6MA are unknown,but values for kact2, ktr, and k�tr have beenmeasured for the polymerization of CPDB withmethyl methacrylate.43,44 For the purpose ofthese simulations, similar chain-transfer con-stants, Ctr ¼ ktr/kp, C�tr ¼ k�tr/kp (reaction IIIbScheme 1), and Cact,2 ¼ kact,2/kp, were used aswell as an assumed kfra,2 value of 30 s�1. Val-ues of kfra,2 in the range of 100–103 s�1 give ki-netic rates of the appropriate order of magni-tude and demonstrate similar kinetic trends.Figure 3 shows the effects of different kfra,2values on the predicted kinetics for a [RAFT]/[Initiator] ratio of 1.5. Because few other simu-lations have been conducted on CPDB polymer-ized under either bulk or solution conditions,the accuracy of the kfra,2 value used is un-known. However, multiple simulations of sty-rene polymerizations with cumyl dithioben-zoate as the RAFT agent have been con-

Figure 1. Model-predicted reaction kinetics at sevendifferent RAFT concentrations with the following pa-rameter values: M0 ¼ 0.183 M, [I] ¼ 6.1 � 10�4 M, C0

¼ 4.5 M, ceff ¼ 0.25, kd ¼ 2.03 � 10�4 s�1, kp ¼ 1500M�1 s�1, kfra,1 ¼ 2.9 s�1, kfra,2 ¼ 30 s�1, kact,1 ¼ 7.5 �105 M�1 s�1, and kact,2 ¼ 4.2 � 105 M�1 s�1.



ducted7,18,20–22 and have given values of kfra,2over the enormous range of 10�2–105 s�1.

In accordance with the experimentally meas-ured rates of Liu et al.6 [Figs. 4 and 5(a,b)], thekinetics simulated here show a large decrease inthe reaction rate with increasing RAFT concen-

trations [Figs. 1 and 2(a,b)]. However, the exper-imental kinetic rates could not be simulatedquantitatively; this was most likely a result ofthe uncertainty in the values of the kinetic con-stants or because of constraints imposed by ass-umptions about the RAFT agent partitioning. Inthe simulated polymerizations, two kinetic rateintervals can be observed. In the first interval,the concentration of the unpolymerized RAFTagent available is high enough to mediate thepolymerization effectively. As the polymerizationprogresses, the RAFT agent is consumed, andthis results in a low concentration of the unpoly-merized RAFT agent late in the reaction. As aresult, newly nucleated particles can polymerizerapidly and without control. This situationcauses the observed rapid increase in the reac-tion rate at higher conversions [Fig. 2(a,b)]. Asthe initial RAFT concentration increases, thelength of the first interval is extended, and theamount of uncontrolled polymerization at theend of the reaction decreases.

When the RAFT concentration is high enough([RAFT]/[Initiator] ¼ 4.5 or 6.0), the RAFT agentis present throughout the entire reaction, andthe full reaction proceeds under control. As a

Figure 3. Model-predicted reaction kinetics at differ-ent kfra,2 values for [RAFT]/[Initiator] ¼ 1.5 with thefollowing parameter values: M0 ¼ 0.183 M, [I] ¼ 6.1� 10�4 M, C0 ¼ 4.5 M, ceff ¼ 0.25, kd ¼ 2.03 � 10�4 s�1,kp ¼ 1500 M�1 s�1, kfra,1 ¼ 2.9 s�1, kact,1 ¼ 7.5 � 105

M�1 s�1, and kact,2 ¼ 4.2 � 105 M�1 s�1.

Figure 2. (a) Model-predicted reaction rates at sevendifferent RAFT concentrations and (b) model-predictedreaction rates for five RAFT concentrations above[RAFT]/[Initiator] ¼ 1.5 plotted with an expandedscale. For the parameter values, see Figure 1.



result, a shift in the maximum rate with inc-reasing RAFT concentration is observed, andthis is consistent with the shift observed in theexperimental reaction rates [Fig. 5(a,b)]. Whenno RAFT agent is present, both the modeled andexperimental reaction rate maxima occur at aconversion of 39% [Figs. 2(a) and 5(a)]. With theaddition of the RAFT agent, the maximum rateshifts to higher conversions until a RAFT con-centration is reached, at which the entire reac-tion is mediated by the RAFT agent. At thehighest RAFT concentrations ([RAFT]/[Initiator]¼ 4.5 or 6.0), the maximum rate once againshifts to lower conversions [Figs. 2(a,b) and5(a,b)].

The simulated trends in the final molecularweight polydispersity also resemble those meas-ured6 (Fig. 6). At low RAFT concentrations, theRAFT agent increases the molecular weightpolydispersity. The experimental molecularweight distributions show that this increase re-sults from the formation of a high-molecular-weight shoulder at low RAFT concentrations(Fig. 7). The simulation of these experimentsshows that this increase in the polydispersityresults from the early consumption of the RAFTagent and the subsequent uncontrolled polymer-ization. As the concentration of the RAFT agentincreases, the effects of the RAFT agent become

more beneficial, and beyond a critical concentra-tion, the RAFT agent acts to decreases the poly-dispersity. The critical concentration is the con-centration beyond which the unpolymerized

Figure 5. (a) Experimentally measured reaction rates[cf. Fig. 2(a)] at seven different RAFT concentrationsand (b) experimentally measured reaction rates [cf. Fig.2(b)] at five RAFT concentrations above [RAFT]/[Ini-tiator] ¼ 1.5 plotted with an expanded scale (the experi-mental results were taken from ref. 6).

Figure 4. Experimentally measured reaction kinetics(cf. Fig. 1) at seven different RAFT concentrations (theexperimental results were taken from ref. 6).



RAFT agent is present throughout the entirereaction, which is here equal to a [RAFT]/[Ini-tiator] ratio slightly higher than 3.0.

Figure 8 shows the simulated evolution of themolecular weight polydispersity. Early in thereactions, a low polydispersity is predicted at allRAFT concentrations, but late in the reaction,only the simulations at a high RAFT concentra-tion predict a low polydispersity. Like the rapidincrease in the polymerization rate observed atlow RAFT concentrations and high conversions[Fig. 2(a,b)], the rapid increase in the polydisper-sity results from the early consumption of theRAFT agent and the uncontrolled polymerizationof particles nucleated late in the reaction.

The critical RAFT concentration at which un-controlled polymerization commences is deter-mined by the consumption rate of the unpolymer-ized RAFT agent (III in Scheme 1). If the RAFTagent is consumed too quickly, the concentrationof the unpolymerized RAFT agent decreases rap-idly, and uncontrolled polymerization occurs latein the reaction. The rate of the RAFT agent con-sumption is determined by the kinetic rate ktr. Ifthis rate is too fast in comparison with the totalreaction time, all the RAFT agent will be con-sumed too early in the reaction. In contrast, inbulk and solution polymerizations, there are nosimilar problems associated with the rapid con-sumption of the RAFT agent, although slow RAFTagent consumption and the resulting slow estab-

Figure 6. Gel permeation chromatography traces offinal polymer samples at different RAFT concentra-tions. At low RAFT concentrations, a shoulder devel-oped that corresponded to the formation of a high-mo-lecular-weight polymer.

Figure 7. Effects of the RAFT concentration on themodel-predicted polydispersities at 85% conversion andthe final experimentally measured polydispersities (theexperimental results were taken from ref. 6). For theparameter values, see Figure 1.

Figure 8. Model-predicted polydispersity as a func-tion of the conversion at eight different RAFT concen-trations. For the parameter values, see Figure 1.



lishment of equilibrium are often cited as causesfor high molecular weight polydispersities.2

Even though the RAFT agent is continuouslyconsumed throughout the polymerization, thesimulated number-average molecular weight (Mn)

increases linearly and agrees well with the exper-imentally measured molecular weight measuredby Liu et al.6 [Fig. 9(a,b)]. A linear increase in Mn

is usually observed in bulk polymerizations2 andresults from RAFT agent consumption that is fastin comparison with the consumption of the mono-mer. If all the RAFT agent is consumed early inthe reaction, Mn can be determined by the divi-sion of the total amount of the polymer formed bythe total amount of the RAFT agent, and it isgiven by the following linear equation:

Mn ¼ FWRAFT þM0FWmonf

XR0ð15Þ

where FWRAFT and FWmon are the RAFT and mon-omer molecular weights, respectively. Because thepolymerization mechanism is much different inmicroemulsion polymerization, such linear behav-ior might be unexpected. The major differencebetween the Mn increase in microemulsion poly-merization and that in solution and bulk polymer-izations is that in microemulsion polymerization aline fit to the linear portion of the experimentaldata does not pass through the origin.

Figure 9. (a) Model-predicted and experimentallymeasured Mn as a function of the conversion at eightdifferent RAFT concentrations and (b) model-pre-dicted and experimentally measured Mn as a functionof the conversion at five RAFT concentrations above[RAFT]/[Initiator] ¼ 1.5 plotted with an expandedscale (the experimental results were taken from ref.6). For the parameter values, see Figure 1.

Figure 10. Effects of the RAFT concentration on themodel-predicted particle size at 85% conversion, andthe particle size experimentally measured by QLS(the experimental results were taken from ref. 6). Forthe parameter values, see Figure 1.



The predicted particle size also shows a dra-matic decrease with increasing RAFT concentra-tion (Fig. 10) that is similar to the decrease ob-served experimentally.6 However, unlike theexperimentally measured particle size, at highRAFT concentrations, the simulated particle sizecontinues to decrease. This difference betweenthe experimental and simulated particle sizes isprobably the result of particle aggregation andcoagulation processes late in the reaction. Thesecolloidal effects are not included in the model.

Figure 11 shows the simulated evolution of theparticle size for different RAFT concentrations. Atall RAFT concentrations, the simulated particlesizes are depressed in the initial stage of the reac-tion; however, for the low RAFT concentrationsimulations, the particle size increases rapidly athigh conversions. The onset of this particle sizeincrease corresponds to the onset of the uncon-trolled polymerization, as observed in the poly-merization kinetics [Fig. 2(a,b)].

CONCLUSIONS

The proposed model demonstrates that in poly-merizations at low RAFT concentrations, thelarge molecular weight polydispersity, particlesize, and reaction rates observed experimentally

at high conversions likely result from the earlyconsumption of the RAFT agent and the subse-quent uncontrolled polymerization. The generaltrends predicted by the model mirror the experi-mental measurements, but uncertainties in thevalues of the kinetic parameters have prohibitedquantitative predictions. Despite these limita-tions, the model is useful for the investigation ofRAFT microemulsion polymerization mecha-nisms. The predicted kinetics also demonstratea shift in the maximum rate that is consistentwith that observed experimentally. The properorders of magnitude in the particle size, molecu-lar weight, and reaction rate have been obtainedwith kfra,2 values in the range of 100–103.

The authors are grateful to D. G. Vlachos for his guid-ance in appropriate approaches to the numerical cal-culations herein.

APPENDIX

Raft Partitioning

The partitioning of a monomer or RAFT agentbetween polymer particles and swollen micellescan be written in terms of a thermodynamic par-titioning constant (K):

K ¼ Cpart

CmicðA1Þ

where Cpart is the monomer or RAFT concentra-tion in the polymer particle and Cmic is the mono-mer or RAFT concentration in the micelles. Equa-tion 10 can be written in terms of the monomerpartitioning constant:

KmonðtÞ ¼ C0

Cmicmon

ð1� f ðtÞÞ ðA2Þ

where Kmon is equal to Cpartmon/C

micmon. Cmic

mon is themolar density of the monomer in the micelles:

Cmicmon ¼

ð1� f ÞM0 � CpartmonfM0MW

qpoly 1�CpartmonMW

qmon

� �

ð1�f ÞM0MWqmon

� CpartmonfM0MW2

qmonqpoly 1�CpartmonMW

qmon

� �þ /surf

0@

1A

ðA3Þ

/surf is the volume fraction of the surfactant inthe microemulsion.

Figure 11. Model-predicted particle size as a func-tion of the conversion at RAFT concentrations of 0,0.5, 0.7, 1.0, 1.5, 2.25, 3.0, 4.5, and 6.0 ([RAFT]/[Ini-tiator]). For the parameter values, see Figure 1.



Assuming the unpolymerized RAFT agentbehaves in away thermodynamically similar tothe monomer [Kmon(t) ¼ KRAFT(t)], the RAFTpartitioning can be described by

KRAFTðtÞ ¼ C0

Cmicmon

ð1� f ðtÞÞ ðA4Þ

Combined with the RAFT mass balanceand under the assumption that the RAFT agentcontributes a negligible volume to both theparticle and micelles, the concentration of theunpolymerized RAFT agent in the polymer par-ticle is

CpartXR ¼ KRAFTð1� fXRÞXR0

ð1�f ÞM0

qmon� Cpart

monfM0MW

qmonqpoly 1�CpartmonMW

qmon

� �24

35MW þ /surf þ KRAFT

fM0MW

qpoly 1�CpartmonMW

qmon

� �0@

1A ðA5Þ

Molecular Weight Model

Recently, Wang and Zhu8 developed a polymer-ization model based on the method of momentsthat describes the evolution of the molecularweight and polydispersity of a bulk RAFT poly-merization in which the macro-RAFT radicals,PXP and PXR, do not terminate through biradi-cal combination. Because a microemulsion poly-merization contains negligible amounts of birad-ical termination, the evolution of the molecularweight and polydispersity can be calculated in asimilar way.

The method of moments is a discrete trans-formation method that allows for the first threemoments of the molecular weight distributionto be determined without the calculation of theentire distribution. In the method of moments,a generating function transform is applied tothe concentration-based kinetic rate equations,transforming these equations into a set ofmoment-based kinetic rate equations. A goodreview of the method of moments was pub-lished by Dotson et al.45 Here, the method ofmoments is used to calculate the evolution ofthe molecular weight and polydispersity of aRAFT microemulsion polymerization.

The goal of this derivation is to determine themoments of the molecular weight distribution forthe four types of chain species (Pr, CXPr, PrXP0,and

Pr�1s¼1 1=2Pr�sXPs). In the notation used in

this derivation, subscript r denotes the chainlength, and the summation over Pr�sXPs givesthe number of macro-RAFT radicals (PXP) of totalchain length r. The unpolymerized cyanopropylradicals, which are usually represented by R*,are now represented by P0 in this symbolism.Similarly, macro-RAFT radicals (PXR) are repre-sented by PrXP0.

A generating function transform operator isapplied to four species in the concentration-based rate equations to give a set of moment-based rate equations. For example, the transformoperator Yi �

P1r¼1 r

i will be used to transformPr in the concentration-based rate equations tomoment-based rate equations that are in termsof Yi, where Yi is defined as the ith moment ofPr:

Yi ¼X1r¼1

riPr ðA6Þ

The transformed rate equations for the first threemoments (Y1, Y2, and Y3) can then easily besolved. Similarly, transforms are also applied tospecies PrXPs, CXPr, and PrXP0 in the concentra-tion-based rate equations, giving moment-basedrate equations with moments that are defined asfollows:

Gi ¼X1r¼1

riCX Pr ðA7Þ

Hi ¼X1r¼1

riPrXP0 ðA8Þ

Qi ¼ 1

2

X1r¼1

riXr

s¼1

Pr�s XPs � 1

2

X1r¼1

riPrXP0 ðA9Þ

Qi;j ¼X1r¼1

X1s¼1

rjsiPrXPs ðA10Þ

Equation A10 is required for closure in themoment-based rate equations. Equation A10 can



be related to eq A9 through the following proper-ties:

Q0 ¼ 1

2

X1r¼1

Xr

s¼1

Pr�sXPs � 1

2

X1r¼1

PrXP0

¼ 1

2

X1r¼1

X1s¼1

PrXPs ¼ 1

2Q0;0 ðA11Þ

Q1 ¼ 1

2

X1r¼1

rXrs¼1

Pr�sXPs � 1

2

X1r¼1

rPrXP0

¼X1r¼1

X1s¼1

rPrXPs ¼ Q1;0 ðA12Þ

Q2 ¼ 1

2

X1r¼1

r2Xr

s¼1

Pr�sXPs � 1

2

X1r¼1

r2PrXP0

¼X1r¼1

X1s¼1

r2PrXPs þX1r¼1

X1s¼1

rsPrXPs

¼ Q2;0 þQ1;1 ðA13Þ

The concentration-based rate equations thatmust be transformed are

@CXPrðt1; tÞ/partðt1; tÞ@t

¼X1s¼1

kfra2PrXPsðt1; tÞ

þ kfra1PrXP0ðt1; tÞ � kact1CXPrðt1; tÞP0ðt1; tÞ

�X1s¼1

kact2Psðt1; tÞCXPrðt1; tÞ ðA14Þ

@PrXPsðt1; tÞ@t

¼ kact2Prðt1; tÞCXPsðt1; tÞþ kact2Psðt1; tÞCXPrðt1; tÞ

� 2kfra2PrXPsðt1; tÞ ðA15Þ

@PrXP0ðt1; tÞ@t

¼ kact2Prðt1; tÞCpartXR ðtÞ

þ kact1P0ðt1; tÞCXPrðt1; tÞ� ðkfra1 þ kfra2ÞPrXP0ðt1; tÞ ðA16Þ

@Prðt1; tÞ@t

¼ kpCpartmonðtÞPr�1ðt1; tÞ

þX1s¼1

kfra2PrXPsðt1; tÞ

�X1s¼1

kact2Prðt1; tÞCXPsðt1; tÞ

� kact2 CpartXR ðtÞPrðt1; tÞ

þ kfra2PrXP0ðt1; tÞ� kpC

partmonðtÞPrðt1; tÞ ðA17Þ

where each species concentration is for a particleat time t that was previously initiated at time t1.Using these concentration-based rate equationswith the defined transformations (eqs A6–A10)gives the zero moment rate equations

@Y0ðt1; tÞ@t

¼ 2kfra2Q0ðt1; tÞ � kact2Y0ðt1; tÞG0ðt1; tÞþ kpC

partmonðtÞP0ðt1; tÞ � kact2 C

partXR ðtÞY0ðt1; tÞ

þ kfra2H0ðt1; tÞ ðA18Þ

@G0ðt1; tÞ/partðt1; tÞ@t

¼ 2kfra2Q0ðt1; tÞ þ kfra1H0ðt1; tÞ� kact1P0ðt1; tÞG0ðt1; tÞ

� kact2Y0ðt1; tÞG0ðt1; tÞ ðA19Þ

@Q0ðt1; tÞ@t

¼ kact2Y0ðt1; tÞG0ðt1; tÞ � 2kfra2 Q0ðt1; tÞðA20Þ

@H0ðt1; tÞ@t

¼ kact2 CpartXR ðtÞY0ðt1; tÞ

þ kact1 P0ðt1; tÞG0ðt1; tÞ� ðkfra1 þ kfra2ÞH0ðt1; tÞ ðA21Þ

the first moment rate equations

@Y1ðt1; tÞ@t

¼ kP CpartmonðtÞY0ðt1; tÞ þ kp C

partmonðtÞP0ðt1; tÞ

þ kfra2 Q1ðt1; tÞ � kact2 Y1ðt1; tÞG0ðt1; tÞ� kact2 C

partXR ðtÞY1ðt1; tÞ þ kfra2 H1ðt1; tÞ

ðA22Þ




¼ kfra2 Q1ðt1; tÞ þ kfra1 H1ðt1; tÞ� kact1 P0ðt1; tÞG1ðt1; tÞ

� kact2 G1ðt1; tÞY0ðt1; tÞ ðA23Þ

@Q1ðt1; tÞ@t

¼ kact2 Y1ðt1; tÞG0ðt1; tÞþ kact2 G1ðt1; tÞY0ðt1; tÞ

� 2kfra2 Q1ðt1; tÞ ðA24Þ

@H1ðt1; tÞ@t



and the second moment rate equations

@Y2ðt1; tÞ@t

¼ kP CpartmonðtÞð2Y1ðt1; tÞ þ Y0ðt1; tÞÞ

þ kp CpartmonðtÞP0ðt1; tÞ þ kfra2 Q2;0ðt1; tÞ

� kact2Y2ðt1; tÞG0ðt1; tÞ� kact2 C

partXR ðtÞY2ðt1; tÞ

þ kfra2 H2ðt1; tÞ ðA26Þ


¼ kfra2 Q2;0ðt1; tÞ þ kfra1 H2ðt1; tÞ� kact1 P0ðt1; tÞG2ðt1; tÞ

� kact2 G2ðt1; tÞY0ðt1; tÞ ðA27Þ

@Q2;0ðt1; tÞ@t

¼ kact2 Y2ðt1; tÞG0ðt1; tÞþ kact2 G2ðt1; tÞY0ðt1; tÞ

� 2kfra2 Q2;0ðt1; tÞ ðA28Þ

@Q2ðt1; tÞ@t

¼ kact2 Y2ðt1; tÞG0ðt1; tÞþ 2kact2 Y1ðt1; tÞG1ðt1; tÞþ kact2 G2ðt1; tÞY0ðt1; tÞ

� 2kfra2 Q2ðt1; tÞ ðA29Þ

@H2ðt1; tÞ@t



The number of initiator radicals present is givenby

@P0ðt1; tÞ@t

¼ �kp CpartmonðtÞP0ðt1; tÞ

þX1r¼1

kfra1 PrXP0ðt1; tÞ

�X1r¼1

kact1 CX Prðt1; tÞP0ðt1; tÞ ðA31Þ

This expression can also be transformed into anexpression in terms of the defined moment equa-tions (eqs A6–A10):

@P0ðt1; tÞ@t

¼ �kp CpartmonðtÞP0ðt1; tÞ þ kfra1 H0ðt1; tÞ� kact1 P0ðt1; tÞG0ðt1; tÞ ðA32Þ

These expressions are consistent with those usedin the kinetic analysis because the zero momentrate equations (eqs A18–A21) are identical to eqs2–6. With the 14 initial conditions at t ¼ t1

Y0 ¼ 0;P0 ¼ q;Q0 ¼ 0;G0 ¼ 0;H0 ¼ 0

Y1 ¼ 0;Q1 ¼ 0;G1 ¼ 0;H1 ¼ 0

Y2 ¼ 0;Q2 ¼ 0;Q2;0 ¼ 0;G2 ¼ 0;H2 ¼ 0 ðA33Þ

and the monomer and RAFT profiles determinedfrom the solution to the polymerization kinetics(eqs 10 and A5), eqs A18–A30 and A32 can besolved. The number-average molecular weight(Mn) and weight-average molecular weight (Mw)for the total population can then be determinedfrom



Mn ¼

Rt0

Y1ðt1; tÞ þQ1ðt1; tÞ þG1ðt1; tÞ/partðt1; tÞ þH1ðt1; tÞ� �

dt1

Rt0


dt1

ðA34Þ

Mw ¼

Rt0


dt1

Rt0


dt1

ðA35Þ

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kinetic modeling of controlled living microemulsion

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