Research ArticleModeling of the Penultimate Unit Effect in Chain-Growth Copolymerizations

Fabricio Machado 1,2

1Instituto de Química, Universidade de Brasília, Campus Universitário Darcy Ribeiro, CEP: 70910-900 Brasília, DF, Brazil2Programa de Pós-Graduação em Engenharia Química, Universidade Federal de Goiás, CEP: 74690-900 Goiânia, GO, Brazil

Correspondence should be addressed to Fabricio Machado; fmachado@unb.br

Received 17 January 2019; Accepted 26 March 2019; Published 28 April 2019

Academic Editor: Mehdi Salami-Kalajahi

Copyright © 2019 Fabricio Machado. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The present work addresses the modeling and simulation of the addition of copolymerizations of styrene and methyl methacrylatein batch mode, and the formation of tailored vinyl acetate/acrylic acid copolymers is evaluated through stochastic optimizationprocedures based on the Monte Carlo method. A kinetic model of the free-radical reaction was proposed in order to predict thebehavior of the reaction system taking into consideration the presence of the penultimate unit effect. The profiles of conversionand copolymer composition were also evaluated considering the effect of the medium viscosity (kinetic phenomena related togel and glass effects) on the reaction performance. It was shown that the proposed model for chain-growth copolymerization isable to describe strong nonlinear behaviors such as autoacceleration of the polymerization and drift of copolymer composition.It was also shown that copolymers with homogeneous composition can be successfully synthesized through manipulation of themonomer feed flow rate based on a stochastic optimization procedure.

1. Introduction

It is well-known that all reaction mechanism steps playan important role for the determination of copolymeriza-tion kinetic behavior. Despite this, the propagation step isfundamental for the proper description of the overall propa-gation rate coefficients, of polymer composition, and of thesequence distribution of final copolymer, important proper-ties that define the material polymeric application.

In chain-growth polymerizations, two propagationmodels are normally used to describe the polymerization.The first one is the well-known terminal model [1], whichassumes that the reactivity of the propagation reaction is gov-erned only by the nature of the monomer and of the terminalunit of the growing polymer chain. The second one is knownas the penultimate model [2], which considers the effectof both the terminal and penultimate monomeric unitsof the growing polymer chain. According to this assumption,electronic and geometric effects or steric interactionsbetween the penultimate monomeric unit and reactant spe-cies (e.g., monomers, transfer agents, and solvents) can be

incorporated into the propagation model [3, 4]. As a conse-quence, one has to keep in mind that the penultimate modelnormally leads to a considerable increase in the number ofmodel parameters, and for this reason, the terminal modelis the most popular propagation model and is the startingpoint for most studies on polymerization kinetics.

The penultimate model is generally employed when theterminal model (first-order Markovian model) is not able tosatisfactorily describe polymerization rates. Two types ofthe penultimate model have been proposed: (i) the first oneis named the explicit penultimate model which assumes thatboth the terminal and penultimate units of the radical canaffect both the reactivity and selectivity; (ii) the second onedenominated the implicit penultimate model which considersthat despite the terminal and penultimate units of the poly-meric radical affect the reactivity, only the terminal unitaffects the selectivity [5–13].

Coote and Davis [6] presented an extensive list of mono-meric pairs, where the terminal model is not suitable todescribe copolymerization kinetic, and the reader is referredto this publication for more detailed information. Although

HindawiInternational Journal of Polymer ScienceVolume 2019, Article ID 2912417, 12 pageshttps://doi.org/10.1155/2019/2912417

the penultimate effect is important to describe both the com-position and the polymerization kinetics in several polymersystems, only few kinetic parameters for the penultimatemodel are reported in the open literature. Recently, Deb[14] calculated the penultimate model reactivity ratios fromdifferent binary monomer mixtures employed in copolymer-ization systems.

Typically, polymerization systems comprising styrene/a-crylonitrile [15–20], methyl methacrylate/n-butyl acrylate[21], ethylene/styrene [22], dimethyl itaconate/styrene [23],ethylene/4-methyl-1-pentene [24], α-alkylstyrenes/acryloni-trile [25], styrene/methyl methacrylate [26], p-chlorostyre-ne/methyl acrylate [27], styrene/isobutylene [28], methyl α-(trifluoromethyl)acrylate and α-(trifluoromethyl) acryloni-trile with styrene, p-chlorostyrene and methyl methacrylate[29], styrene/maleic anhydride [30], isobutyl methacrylate/-lauryl methacrylate [31], and styrene/fumaronitrile [32, 33]demonstrate a strong penultimate unit effect, representinggood examples of the successful implementation of the pen-ultimate model [9, 34].

The kinetic mechanism represented by the penulti-mate model (second-order Markovian model) is composedof four distinct radicals that can combine with monomericspecies, generating eight distinct stages of propagation(~M⋅

ij +Mnkijn ~M⋅

ijn with i, j, n = 1, 2, where kijn is the

reaction rate constant that characterizes the addition of themonomer unit n to a growing polymer chain containingthe monomeric units i and j). Given the complex nature ofpolymerization systems, the introduction of additional stagesof propagation with a consequent increase in the numberof kinetic parameters is the major practical limitation ofemploying the penultimate model in polymerization kineticmodels intended to be used to describe the reaction behavior.

In spite of the popularity of the terminal model, it is gen-erally agreed that the existence of the penultimate unit effectin important chain-growth polymerization systems seems tobe general rather than an exception, which clearly indicatesthat this polymerization kinetic based on the terminal modeloversimplifies actual polymerization reaction processes [8,14]. Initial studies on the influence of the penultimate uniteffect in free-radical copolymerizations date from 1940s.Among then, the pioneering works of Merz et al. [2], Barb[30], and Ham [33] must be highlighted.

The penultimate model was originally developed by Merzet al. [2] to predict the copolymer composition and sequencedistribution of the final copolymer. This propagation modelwas extensively explored by Fukuda and coworkers [5, 9,26, 27], some years later, in order to understand the devia-tions from the terminal model, showing that the penultimatemodel provides a good description of copolymer composi-tion, distribution sequences, and rate of propagation. Fukudaand coworkers have distinguished the penultimate unit effectbehavior, originating two distinct models: explicit penulti-mate and implicit penultimate models of copolymerization.Fukuda et al. [26] also demonstrated through measurementsof average propagation rate constant from copolymerizationsof styrene/methyl methacrylate that the terminal model wasnot suitable to appropriately describe the experimental data.

Li et al. [35, 36] have described the combined effect ofdepropagation and penultimate unit effect kinetic on theinstantaneous copolymer composition and average copoly-merization rate coefficients in styrene/butyl methacrylatereactions performed at elevated temperatures. According tothe authors, when depropagation appears combined withthe penultimate unit effect, the effective propagation ratecoefficient deviates significantly from terminal model predic-tions regarding the copolymer composition.

Recently, Nikitin and Hutchinson [37] have evaluated thepenultimate unit effect in homopolymerization of acrylates.According to the proposed kinetic mechanism, the differ-ences in reactivity of radicals formed by monomer additionto midchain radicals are taking into consideration. Modelcalculations have shown that deviation from the terminalmodel may depend on the radical reactivity ratio si andmonomer concentrations, which generally complicates thekinetic analysis of acrylate homopolymerizations.

In the present work, a phenomenological polymerizationmodel is proposed based on the kinetic mechanism that takesinto account the penultimate monomeric unit effect in chain-growth copolymerizations and simulations are performedbased on a typical polymerization system (styrene/methylmethacrylate) governed by the penultimate kinetic model.Additionally, the penultimate model is also employed todescribe the experimental data of vinyl acetate/acrylic acidsuspension copolymerization carried out in a batch mode.It is also proposed strategies for controlling the copolymercomposition based on stochastic optimization procedureswith the Monte Carlo method.

2. Kinetic Mechanism andPolymerization Model

The kinetic mechanism proposed to describe the (e.g., bulkor suspension) copolymerizations comprises the followingfundamental steps: initiation, propagation, transfer to mono-mer, and termination by disproportionation and combina-tion, as follows:

Step 1. Initiation

IkD 2Z,

Z +M1k1 P1,0,

Z +M2k2 S0,1

1

Step 2. Propagation

Pi,j +M1kP111 Pi+1,j,

Pi,j +M2kP112 Qi,j+1,

Qi,j +M1kP121 Ri+1,j,

Qi,j +M2kP122 Si,j+1,

2 International Journal of Polymer Science

Ri,j +M1kP211 Pi+1,j,

Ri,j +M2kP212 Qi,j+1,

Si,j +M1kP221 Ri+1,j,

Si,j +M2kP222 Si,j+1

2

Step 3. Transfer to monomer

Pi,j +M1ktrM111 Θi,j + P1,0,

Pi,j +M2ktrM112 Θi,j + S0,1,

Qi,j +M1ktrM121 Θi,j + P1,0,

Qi,j +M2ktrM122 Θi,j + S0,1,

Ri,j +M1ktrM211 Θi,j + P1,0,

Ri,j +M2ktrM212 Θi,j + S0,1,

Si,j +M1ktrM221 Θi,j + P1,0,

Si,j +M2ktrM222 Θi,j + S0,1

3

Step 4. Termination by disproportionation

Pi,j + Pm,nkTD11 Θi,j +Θm,n,

Pi,j +Qm,nkTD12 Θi,j +Θm,n,

Pi,j + Rm,nkTD13 Θi,j +Θm,n,

Pi,j + Sm,nkTD14 Θi,j +Θm,n,

Qi,j +Qm,nkTD22 Θi,j +Θm,n,

Qi,j + Rm,nkTD23 Θi,j +Θm,n,

Qi,j + Sm,nkTD24 Θi,j +Θm,n,

Ri,j + Rm,nkTD33 Θi,j +Θm,n,

Ri,j + Sm,nkTD34 Θi,j +Θm,n,

Si,j + Sm,nkTD44 Θi,j +Θm,n

4

Step 5. Termination by combination

Pi,j + Pm,nkTC11 Θi+m,j+n,

Pi,j +Qm,nkTC12 Θi+m,j+n,

Pi,j + Rm,nkTC13 Θi+m,j+n,

Pi,j + Sm,nkTC14 Θi+m,j+n,

Qi,j +Qm,nkTC22 Θi+m,j+n,

Qi,j + Rm,nkTC23 Θi+m,j+n,

Qi,j + Sm,nkTC24 Θi+m,j+n,

Ri,j + Rm,nkTC33 Θi+m,j+n,

Ri,j + Sm,nkTC34 Θi+m,j+n,

Si,j + Sm,nkTC44 Θi+m,j+n,

P = 〠∞

i=1〠∞

j=0Pi,j,

Q = 〠∞

i=0〠∞

j=1Qi,j,

R = 〠∞

i=1〠∞

j=0Ri,j,

S = 〠∞

i=0〠∞

j=1Si,j,

5

where I is the concentration of the initiator; kD is thekinetic constant for initiator decomposition; Z is the con-centration of free radical of the initiator; Mi is the concen-tration of monomer i; ki is the kinetic constant for theformation of the first polymeric radical i; P1,0 is the firstpolymeric radical containing 1 mer of species 1; Q0,1 isthe first polymeric radical containing 1 mer of species 2;kPijl

is the kinetic constant for the propagation of polymeric

radical containing the monomeric units i (penultimate) andj (terminal) with monomer unit l; Pi,j is the growing poly-meric chain containing the meric units i and j and species 1at the active site (polymeric radical 1— P = ∼M1M1 ⋅ ); Qi,jis the growing polymeric chain containing the meric units iand j and species 2 at the active site (polymer radical2— Q = ∼M1M2 ⋅ ); Ri,j is the growing polymeric chain con-taining the meric units i and j and species 1 at the active site( R = ∼M2M1 ⋅ ); Si,j is the growing polymeric chain contain-ing the meric units i and j and species 2 at the active site( S = ∼M2M2 ⋅ ); Θi,j is the dead polymer chain containingmer units i and j; ktrMijl

is the kinetic constant of transfer of

3International Journal of Polymer Science

polymeric radical containing the monomeric units i and jto monomer l; kTDij

and kTCijare the kinetic constants for

termination by disproportionation and combination ofgrowing polymeric radicals i and j, respectively; P, Q, R,and S represent the total concentration of growing polymericradicals; kTij

is the kinetic constant that characterizes the

combined contribution of the termination by both dispro-portionation and combination (kTij

= kTDij+ kTCij

); and f is

the initiator efficiency.

According to the proposed kinetic mechanism andassuming that the long-chain and quasi-steady-state hypoth-eses are valid for the polymer radicals and admitting that thepropagation terms are much larger than the initiation,chain transfer, and termination terms, it is possible to writethe following set of mass balance equations for the copoly-merization process:

dIdt

= −kDI,

dM1dt

= − kP111P 1 +Ω 1r11

+r12r11

M2M1

+M2

r11M1M1,

dM2dt

= − kP111P1r11

+Ω r12r11

+r22r12r11

M2M1

+M2

r21r11M1M2,

d℘1dt

= kP111P 1 +Ω 1r11

+r12r11

M2M1

+M2

r11M1M1,

d℘2dt

= kP111P1r11

+Ω r12r11

+r22r12r11

M2M1

+M2

r21r11M1M2,

6

where

Ω =M2 + 1/r21 M2

2/M1M1 + r12M2

,

 rii =kPiiikPii j

, rij =kPij j

kPiji

, si =kPjii

kPiii

, i = 1, 2 i ≠ j ,

kP111P = 2f kDI

1ψ11

+1ψ22

Ω r12s2r11

2+

1ψ33

M2s1r11M1

2

+1ψ44

Ω r12r22M2r11M1

2+

2Ωζ12ψ11ψ22

r12s2r11

+2ζ13ψ11ψ33

M2s1r11M1

+2Ωζ14ψ11ψ44

r12r22M2r11M1

+2Ωζ23ψ22ψ33

r12M2s1s2r

211M1

+2Ω2ζ24ψ22ψ44

r212r22M2s2r

211M1

+2Ωζ34ψ33ψ44

r12r22s1r

211

M2M1

2 −1 1/2

,

ζij =kTij

kTiikT jj

, i = 1, 2, j = 2, 4 i ≠ j ,

ψii =kPiii

2

kTii

, i = 1, 2 i ≠ j ,

ψjj =kPiii

2

kT jj

, i = 1, 2, j = 3, 4,

7

where rij is the reactivity ratio monomers i and j, si is theradical reactivity ratios for growing polymer chain i, ζij isthe cross-termination constant between polymer radicals iand j, and ℘i is the moles of monomer i incorporated intopolymer chains.

3. Results and Discussion

3.1. Styrene/Methyl Methacrylate Copolymerization. Figure 1shows the simulation result for the styrene (S)/methylmethacrylate (MMA) system, which is considered a typicalpolymerization where the penultimate effect has to be takeninto account. All model parameters (see equations (6) and(7) and Table 1) used for simulations were obtained fromSchmidt and Ray [38], Kalfas et al. [39], and Burke et al.[34], and the reader is referred to these works for moredetailed information. Copolymerization model equationswere implemented in FORTRAN and solved numericallywith the integration package DDASSL [40]. A maximumreaction conversion of approximately 92% at the end of thereaction with a molar fraction of styrene in the copolymerequal to 17%, which is kept almost constant along the reactiontime (Figure 1(b)) and reaction conversion (Figure 1(a)),is observed.

It is generally agreed that viscosity effects play a funda-mental role in free-radical polymerization reactions, leadingto strong diffusion limitations in the reaction medium. Thisparticular and relevant nonlinear kinetic phenomenon ischaracterized by the glass and gel effects.

Additionally, for simulations of polymerization systemswhere glass and gel effects play a significant role, it is veryimportant to consider correlations adopted to correct the

propagation kf vPiii= kPiiigPi T and termination kf vTiii

= kTiii

gTiT constant deviations due to the diffusion limitations

of both the monomer and growth polymer chain species. Inthis scenario, correlations based on the free volume theoryare of fundamental importance [41].

It is widely adopted that the free volume of the poly-merization system vf can be expressed as the sum of theindividual contribution of each species vf i , as follows [38]:

vf =〠i

vf iϕi 8

4 International Journal of Polymer Science

The free volume contribution of methyl methacrylate(vfMMA

) and its homopolymer (vf PMMA) is proposed by

Schmidt and Ray [38] in the following form:

vfMMA= 0 025 + 0 0010 T − 167 15 ,

vf PMMA= 0 025 + 0 00048 T − 387 15 ,

ϕi =ρi/mi

∑iρi/mi,

9

where ρi is the pure density of species i and mi is themass of species i. It is also assumed that gTMMA

T can beexpressed as [38, 39, 42]

0 25 50 75 100 125 150 175 200 2250.00

0.05

0.10

0.15

0.20

0.25

0.30

Styr

ene m

olar

frac

tion

0.35

0.40

Polymer compositionConversion

Time (min)

0

10

20

30

40

50

60

70

80

90

100

Reaction conversion (%)

(a)

0 10 20 30 40 50 60 70 80 90 1000.00

0.05

0.10

0.15

0.20

0.25

Styr

ene m

olar

frac

tion

0.30

0.35

0.40

Reaction conversion (%)

(b)

Figure 1: Conversion and copolymer composition profiles. I = 0 17 mol, M1 = 1 54 mol, M2 = 5 00 mol, T = 85°C, r11 = 0 472, r12 = 0 454,r21 = 0 472, r22 = 0 454, s1 = 0 412, and s2 = 0 170 [34].

gTMMAT =

0 10575 exp 17 15vf − 0 01715 T − 273 15 , for vf > vTf c ,

2 3 ⋅ 10−6 exp 75vf , for vf ≤ vTf c ,10

Table 1: Model parameters used for simulation of styrene/methyl methacrylate system.

Parameter Unit Reference

ρ1 = 0 9193 − 6 65 ⋅ 10−4 T − 273 15 g/cm3 [39]

ρ2 = 0 9654 − 1 09 ⋅ 10−3 T − 273 15 − 0 970 ⋅ 10−6 T − 273 15 2 g/cm3 [39]

ρ3 = 0 9926 − 2 65 ⋅ 10−4 T − 273 15 g/cm3 [39]

ρ4 = 1 18 − 0 10 ⋅ 10−2 T − 273 15 g/cm3 [39]

kP111 = 1 090 × 107 exp −7051/RT L/(mol·s) [34]

kT111= 1 703 × 109 exp −2268/RT L/(mol·s) [34]

kP222 = 5 366 × 105 exp −4353/RT L/(mol·s) [34]

kT222= 9 800 × 107 exp −701/RT L/(mol·s) [34]

kD = 1 7 × 1014 exp −30000/RT s-1 [39]

f = 0 80 — [48]

Subscript 1 for styrene and 2 for methyl methacrylate; subscripts 3 and 4 for the homopolymers based on styrene and methyl methacrylate, respectively.

5International Journal of Polymer Science

where the critical free volume for termination is given by

vTf c = 0 1856 − 0 2965 ⋅ 10−3 T − 273 15 11

In order to correct the propagation constant, gPMMAT is

expressed in the following form [38, 39, 42]:

gPMMAT =

1, for vf > 0 05,

7 1 ⋅ 10−5 exp 171 53vf , for vf ≤ 0 0512

For the styrene species, the gel effect correlation isexpressed as a function of the styrene conversion (xS), asfollows [39]:

gTST = exp −0 4404xS − 6 362xS2 − 0 1704xS3 ,

gPST = 1 0

13

Figure 2 illustrates the simulation results for theS/MMA system. The copolymerization behavior is stronglyaffected when glass and gel effects are considered, exhibit-ing autoacceleration of the polymerization and copolymercomposition drift. According to Figure 2(a), a maximumconversion limit equal to 98% is achieved and the molarfraction of styrene in the copolymer is lying in the intervalfrom 17% to 22%. It is also observed that the copolymercomposition slightly changes at the beginning of the reac-tion when the conversion is below 40% and the viscosityeffects are not pronounced.

Figure 3 illustrates the effect of the reaction temperatureand the initiator concentration on the global conversionbehavior. According to Figure 3(a) the conversion is

significantly affected by the temperature, exhibiting a stronggel effect as the medium temperature is increased from70°C to 90°C. Simulations presented in Figure 3(b) were per-formed at 80°C and show that the global conversion isslightly affected by the initiator (benzoyl peroxide (BPO))concentration, lying in the range from 85% to 91% whenthe BPO concentration is increased from 0.10mol to 0.2mol.

Figure 4 illustrates the ability of the penultimate model topredict the experimental data of the bulk copolymerization ofstyrene and methyl methacrylate as provided by Jalili et al.[43]. According to Figure 4, the reaction conversion profileis satisfactory predicted by the penultimate model, which isable to represent typical monomer mass-transfer limitationthat is responsible for a strong nonlinear kinetic behaviorclosely related to gel and glass effects.

3.2. Vinyl Acetate/Acrylic Acid Copolymerization. Theeffect of the penultimate monomeric unit of the growingpolymer chains might also be important in vinyl acetate(VAc)/acrylic acid (AA) copolymerizations performed indispersed medium such as suspension polymerization pro-cess. In a series of articles, Silva and coworkers [44–46]evaluated the suspension copolymerization of VAc/AA.According to the authors, the role of the penultimate effectin the kinetic mechanism of VAc/AA must be considered.

In order to evaluate the ability of equations (6) and (7) topredict the kinetic behavior of VAc/AA, copolymerization isfundamental to describe the partition of AA between theorganic phase and the aqueous phase. For this reason, theAA partition coefficient must be included into the copoly-merization model to determine the AA concentrationthroughout the reaction. Equations (14) and (15) representthe solubility of AA in water. It is very important to keep inmind that the amount of AA available for polymerizationinside the organic droplets (dispersed into the aqueousphase) is different from the total amount of monomer addedinto the reactor (M2 =M2

aqueous +M2organic). The amounts of

0 20 40 60 80 100 120 140 160 180

0.05

0.10

0.15

0.20

0.25

0.30

Styr

ene m

olar

frac

tion

0.35

0.40

Polymer compositionConversion

Time (min)

0

10

20

30

40

50

60

70

80

90

100

Reaction conversion (%)

(a)

0 10 20 30 40 50 60 70 80 90 1000.00

0.05

0.10

0.15

Styr

ene m

olar

frac

tion

0.20

0.25

0.30

0.35

0.40

Reaction conversion (%)

(b)

Figure 2: Conversion and composition profiles: gel and glass effects. I = 0 17 mol, M1 = 1 54 mol, M2 = 5 00 mol, T = 85°C, r11 = 0 472,r12 = 0 454, r21 = 0 472, r22 = 0 454, s1 = 0 412, and s2 = 0 170 [34].

6 International Journal of Polymer Science

AA in the organic (M2organic) and aqueous (M2

aqueous) phasesis given as follows [46]:

M2organic =

M21 + K

,

M2aqueous =

K1 + K

M2,14

where

K = αVaqueous

Vorganic , 15

where Vorganic is the volume of the organic phase and Vaqueous

corresponds to the volume of the aqueous phase.The partition coefficient of AA considering the system in

the system AA/VAc/water is represented as a function ofboth the aqueous AA composition and temperature in thefollowing form [46]:

α T , M2II = A + B M2

II +C

M2II 2

−1

, 16

where coefficients A, B, and C are temperature-dependentadjustable parameters, expressed as follows [46]:

A = −16 67 + 0 455 T − 273 15 − 2 92 ⋅ 10−3 T − 273 15 2,

B = 23 02 − 0 594 T − 273 15 + 3 96 ⋅ 10−3 T − 273 15 2,

C = 0 317 − 9 02 ⋅ 10−3 T − 273 15 + 6 04 ⋅ 10−5 T − 273 15 2

17

Table 2 presents the model parameters, and Figure 5shows the prediction of the penultimate model (equations(6) and (7)) in comparison to experimental data providedby Silva et al. [45]. As depicted in Figure 5, the proposedmodel is able to describe experimental data of conversionand AA composition. Figure 6 illustrates the effect of theinitiator amount on the kinetic behavior, when BPO insidethe organic phase was varied from 0.004mol to 0.021mol.The copolymerization conversion is significantly affected bythe BPO concentration. According to simulation results, this

0 50 100 150 200 250 300 3500

10

20

30

40

50

60

70

80

90

100

Jalili et al. (2011)Penultimate model

Reac

tion

conv

ersio

n (%

)

Time (min)

Figure 4: Penultimate model prediction of the styrene/methylmethacrylate bulk copolymerization carried out with BPO [43].

0 20 40 60 80 100 120 140 160 1800

10

20

30

40

50

60

70

80

90

100

Time (min)

T = 70 °CT = 80 °CT = 90 °C

Reac

tion

conv

ersio

n (%

)

(a)

0 20 40 60 80 100 120 140 1600

10

20

30

40

50

60

70

80

90

100

Time (min)I = 0.103 mol I = 0.154 molI = 0.206 mol

Reac

tion

conv

ersio

n (%

)

(b)

Figure 3: Effect of (a) temperature and (b) initiator concentration on the conversion profiles. M1 = 2 4 mol, M2 = 9 25 mol, T = 80°C,r11 = 0 472, r12 = 0 454, r21 = 0 472, r22 = 0 454, s1 = 0 412, and s2 = 0 170 [34].

7International Journal of Polymer Science

kind of reaction is characterized by presenting an autoacce-leration of the polymerization and significant drift of copoly-mer composition.

The synthesis of copolymer materials presenting homo-geneous chemical composition throughout the reactionmay be regarded as an important challenge in the polymeri-zation field mainly when the copolymerization is carried outwith from monomers that present very different reactivities.In order to avoid composition drift control, compositionstrategies should be implemented. In the particular case ofVAc/AA copolymerization because of the complete solubility

of AA in the aqueous phase (AA is partitioned between theorganic and the aqueous phases during the reaction course),semibatch operation mode can be successfully performed, asthe aqueous phase can be used as a reservoir, supplying AAfor the organic phase.

Based on the previous assumptions, semibatch copo-lymerization can be performed to keep the copolymer com-position constant at the desired setpoint (yAA

d) valuethroughout reaction through manipulation of the feed flowrate of AA (FAA). AA feed rate profiles required to maintainAA molar fraction constant into the copolymer chains were

Table 2: Model parameters used for simulation of vinyl acetate/acrylic acid system.

Parameter Unit Reference

ρ1 = 0 9584 − 1 3276 ⋅ 10−3 T − 273 15 g/cm3 [48]

ρ2 = 1 0821 − 1 1969 ⋅ 10−3 T − 273 15 g/cm3 [45]

ρ3 = 1 211 − 8 496 ⋅ 10−4 T − 273 15 g/cm3 [48]

ρ4 = 1 22 g/cm3 [49]

ρw = 1 00066 − 7 35 ⋅ 10−5 T − 273 15 − 3 5 ⋅ 10−6 T − 273 15 g/cm3 [50]

kP11 = 3 2 × 107 exp −6300/RT L/(mol·s) [48]

kT11= 3 7 × 109 exp −3200/RT L/(mol·s) [48]

kP22 = 8 4 ⋅ 102 L/(mol·s) In this work

kT22= 3 2 ⋅ 107 L/(mol·s) In this work

kD = 1 7 × 1014 exp −30000/RT s-1 [51]

f = 0 8 — [48]

Subscript 1 is reserved for vinyl acetate, and subscript 2 is for acrylic acid. Subscripts 3 and 4 for the homopolymers based on vinyl acetate and acrylic acid,respectively.

0 20 40 60 80 100 120 140 160 180 2000.0

0.2

0.4

0.6

0.8

1.0

Silva et al. (2004)Penultimate model

AA

mol

ar fr

actio

n

Time (min)

0 20 40 60 80 100 120 140 160 180 200

0

20

40

60

80

100

Time (min)

Conv

ersio

n (%

)

Figure 5: Penultimate model prediction for the vinyl acetate/acrylicacid copolymerization. M1 = 1 67 mol, M2 = 0 50 mol, T = 70°C,r11 = 0 054, r12 = 2 927, r21 = 0 032, r22 = 1 902, s1 = 0 412, ands2 = 1 0 [52].

0 20 40 60 80 100 120 140 160 180 2000.0

0.2

0.4

0.6

0.8

1.0

AA

mol

ar fr

actio

n

Time (min)

0 20 40 60 80 100 120 140 160 180 200

0

20

40

60

80

100

I = 0.004 molI = 0.008 molI = 0.012 mol

I = 0.016 molI = 0.021 mol

Time (min)Co

nver

sion

(%)

Figure 6: Effect of the initiator amount on the conversion andcomposition profiles. M1 = 1 67 mol, M2 = 0 50 mol, T = 70°C,r11 = 0 054, r12 = 2 927, r21 = 0 032, r22 = 1 902, s1 = 0 412, ands2 = 1 0 [52].

8 International Journal of Polymer Science

determined for discretized time intervals of 10 minutes. Theoptimization problem can be defined as follows:

minFAA

 f = 〠ND

i=1yi − ydi

2,

subject to li ≤ xi ≤ ui, i = 1, 2,… ,m,

18

where yi corresponds to the cumulative copolymer composi-tion at each discretized sampling time i, yi

d is the desiredcopolymer composition at each discretized sampling time i,ND is the number of discretized sampling times in the con-trol window of 10 minutes, xi is the manipulated variable ateach discretized sampling time i, f is the objective function

that must be minimized, ui is the upper limit of the con-straints, and li is the lower limit of the constraints.

AA feed rate profiles were estimated based on stochasticoptimization procedures with the Monte Carlo method[47]. Optimum candidates were estimated by generating ran-dom numbers in the following form:

xi,j = xi,j−1 + ri,j Δi,j, i = 1,… , ND, j = 1,… , NITER, 19

where ri,j are pseudorandom numbers uniformly distributedin the interval (-1,1) and Δi,j defines a search interval for var-iable i at iteration j. Dynamic trajectories of conversion andcopolymer composition were computed after the generation

0 20 40 60 80 100 120 140 160 1800.000

0.002

0.004

0.006

0.008

AA flow rateCopolymer composition95% confidence intervals

Time (min)

AA

feed

rate

(mol

/h)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

AA

copolymer com

position

(a)

AA flow rateCopolymer composition95% confidence intervals

0 20 40 60 80 100 120 140 160 1800.000

0.004

0.008

0.012

0.016

0.020

Time (min)

AA

feed

rate

(mol

/h)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

AA

copolymer com

position

(b)

0 20 40 60 80 100 120 140 160 1800

10

20

30

40

50

60

70

80

90

100

Conv

ersio

n (%

)

Time (min)

AA 30 mol%AA 60 mol%

(c)

Figure 7: Dynamic behavior of vinyl acetate/acrylic acid suspension copolymerizations in semibatch operation mode. AA feed flow rate andAA copolymer composition profile for yAA = 30 mol% (a) and yAA = 60 mol% (b) and conversion profiles (c).

9International Journal of Polymer Science

of the candidate optimum solutions at iteration j, and theobjective function was minimized.

Figure 7 shows the optimum AA feed rate profiles gener-ated to keep the AA copolymer composition equal to30mol% and 60mol% as well as global conversions predictedby the penultimate polymerization model. According toFigures 6(a) and 6(b), copolymers with homogeneous chem-ical composition at the desired setpoint values are easilyobtained based on the proposed optimization procedure.The copolymer composition is kept within the upper control(UCL) and lower control (LCL) limits considering a 95%confidence interval and standard deviation equal to 1%related to the setpoint value of the AA molar fraction. It isimportant to note that because of the high consumption ratesof the AA, the conversion trajectories are not affected by fluc-tuation of the AA feed operation, as shown in Figure 7(c).

4. Conclusion

A kinetic model of free-radical copolymerization of styreneand methyl methacrylate was proposed, taking into accountthe penultimate unit effect. It was shown that the additioncopolymerization reactions can be properly simulated byusing the proposed penultimate polymerization model,being suitable to evaluate strong nonlinear behavior thattakes place due to the presence of viscosity effects relatedto important kinetic phenomena such as autoaccelerationof the polymerization, copolymer composition drift, diffu-sion, and heat-transfer limitations. Copolymer with homoge-neous composition along the whole polymerization time canbe successfully obtained through manipulation of the AAfeed flow rate based on stochastic optimization procedures.

Data Availability

The data used to support the findings of this study areincluded within the article.

Conflicts of Interest

The author declares no conflict of interest.

Acknowledgments

The authors thank the Coordenação de Aperfeiçoamento dePessoal de Nível Superior (CAPES) and Conselho Nacionalde Desenvolvimento Científico e Tecnológico (CNPq) forproviding scholarships and financial support.

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