Modeling of polymerization rate and microstructure in the anionic ...

J Polym Res (2012) 19:9909 DOI 10.1007/s10965-012-9909-2

ORIGINAL PAPER

Modeling of polymerization rate and microstructure in the anionic polymerization of isoprene using n-butyl lithium and N,N,N′,N′tetramethylethylenediamine considering different reactivities of the structural units José Alfredo Tenorio-López & Juan José Benvenuta-Tapia & Eduardo Vivaldo-Lima & Miguel Ángel Ríos-Enriquez & María Lourdes de Nieto-Peña

Received: 19 October 2011 / Accepted: 11 June 2012 / Published online: 30 June 2012 # Springer Science+Business Media B.V. 2012

Abstract A mathematical model for the anionic polymerization of isoprene using n-butyl lithium (n-BuLi) as initiator and N,N,N′,N′-tetramethylethylenediamine (TMEDA) as microstructure modifier, considering a system similar to a tetrapolymerization and a scheme of reaction that considers that the active sites are different in configuration, has been developed. Experimental data of conversion versus time and structure development (1,4-cis, 1,4-trans, vinyl or isopropenyl units) were taken from the literature. Since 1,4-cis structural units are difficult to measure, directly, we used reports based on indirect measurements for natural polyisoprene. The cis structural unit fraction was varied from 0.1 to 0.9 (referred to cis+trans content) in order to provide enough data for parameter estimation purposes. Rate expressions for monomer consumption as well as microstructure and dyad development were obtained from the proposed scheme of reaction. The fraction of active sites and dyad distribution were calculated

J. A. Tenorio-López : M. Á. Ríos-Enriquez : M. L. de Nieto-Peña Facultad de Ciencias Químicas, Universidad Veracruzana, Coatzacoalcos, Veracruz, Mexico J. J. Benvenuta-Tapia (*) Unidad de Desarrollo de Productos y Procesos, Resirene, S.A de C.V, Grupo Kuo. Carr. Federal Puebla-Tlaxcala Km. 15.5, Tlaxcala, Mexico e-mail: [email protected] E. Vivaldo-Lima Facultad de Química, Departamento de Ingeniería Química, Universidad Nacional Autónoma de México, México D.F., Mexico

using Markov chains theory, based on conditional probabilities. The kinetic model correctly describes the performance anionic polymerizations with and without TMEDA. Keywords Anionic polymerization . Polyisoprene . Kinetics . Microstructure Nomenclature (C) (T) (D) (V) FC FT FD FV FCC FCT FCD FCV FTC FTT FTD FTV FDC FDT FDD FDV FVC FVT FVD

Cis active sites Trans active sites Isopropenyl active sites Vinyl active sites Cis isomer formation Trans isomer formation Isopropenyl isomer formation Vinyl isomer formation Cis-cis dyad formation Cis-trans dyad formation Cis-isopropenyl dyad formation Cis-vinyl dyad formation Trans-cis dyad formation Trans-trans dyad formation Trans-isopropenyl dyad formation Trans-vinyl dyad formation Isopropenyl-cis dyad formation Isopropenyl-trans dyad formation Isopropenyl-isopropenyl dyad formation Isopropenyl-vinyl dyad formation Vinyl-cis dyad formation Vinyl-trans dyad formation Vinyl-isopropenyl dyad formation

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FVV Fij i [I] j kij kC, kT, kD, kV [M] MR Р

Pij [Pc], [PT] [PD], [PV] Pc 1,0,0,0, PT 0,1,0,0, PD 0,0,1,0 PV 0,0,0,1

PX

a,b,f,m

val R rij Wi W XCj, XTj, XDj, XVj

J Polym Res (2012) 19:9909

Vinyl-vinyl dyad formation Total quantity of the ij dyad Active site that reacts with a isoprene molecule Initiator concentration Terminal living isomer resulting from the propagation reaction Propagation rate constant Initiation rate constants Isoprene concentration [TMEDA]/[I]o ratio 2 3 PCC PCT PCD PCV 6 PTC PTT PTD PTV 7 7 matrix ¼6 4 PDC PDT PDD PDV 5 PVC PVT PVD PVV Conditional probability Concentration of active sites Active sites with one monomeric unit with the negative Charge located on the cis, trans, isopropenyl and vinyl isomer units, respectively Living polymer with the negative charge located on the isomer X, (X 0 C, T, D or V) with a, b, f and m units of cis, trans, isopropenyl and vinyl, respectively Valvassori equality: (rCT)(rTD)(rDC) 0 (rCD)(rDT)(rTC) Cis/(cis+trans) ratio Reactivity ratio Fraction of active sites Vector (Wc, WT, WD, WV) Isoprene conversion converted to the dyad Cj, Tj, Dj, Vj

Introduction Natural rubber and its products have long been used in a variety of applications, ranging from historical reports on rubber balls in 500 B.C. to reports from Spanish explorers in the 15th and 16th centuries of waterproof shoes in Mexico and Brazil [1]. Synthetic rubbers were developed as a replacement for naturally occurring rubber materials and are characterized as rubbers using the same general characteristics of natural rubber, such as high deformability, rapid recovery from deformation, and good mechanical strength. The most extensively used synthetic rubbers are polybutadiene, polyisoprene, and styrene/butadiene copolymers, which are used as direct replacements for natural rubber.

Synthetic polyisoprene represents one of the important types of polymers commercially produced worldwide. Currently, synthetic polyisoprene is used in a wide variety of industries, in applications requiring low water swell, high gum tensile strength, good resilience, and good tack. Compounds based on synthetic polyisoprene are used as rubber bands, cut thread, baby bottle nipples, extruded hoses, tires, motor mounts, pipe gaskets, as well as many other molded and mechanical products. Polyisoprene has been produced over the years using a variety of methodologies such as radical polymerization [2] emulsion polymerization [3], living anionic polymerization [4, 5], Ziegler-Natta catalyzed polymerization [6], and more recently, living free radical polymerization [7]. Living anionic polymerization makes it possible to produce tailor-made polymers. The desired properties of the polymers are varied and controlled by regulating important product variables, such as average molecular weight, molecular weight distribution, chain-end functionality, distribution of structural units, branching, and copolymer composition. In ionic polymerization, the negatively and positively charged free ions and ion pairs, which often are in equilibrium, are the active propagating species. These processes are complicated because of the presence of multiple propagating species and other complex reactions, such as reversible association of the growing polymer species and the initiator, exchange, and coupling reactions [8]. As a result, the correct modeling of ionic polymerization is still a challenging task. The available kinetic studies on the anionic polymerization of isoprene have been carried out only to obtain kinetic constants. In these studies, usually just one type of active site is considered [9, 10], mainly because of the experimental difficulty in distinguishing the active sites that belong to each of the isomers formed, although it is generally well known that the active sites are present in multiple associated states, and that they depend on the solvent used, initiator concentration, temperature, and active center modifiers. Several studies have shown that the microstructure of polyisoprene greatly influences the final properties of the polymer, ranging from highly crystalline trans-1,4-polyisoprene to amorphous cis-1,4-polyisoprene [11]. It is therefore important to control the polymerization of isoprene, in order to obtain the desired microstructures and, ultimately, the best material properties. Through the combination of living anionic polymerization methodologies and the use of polar additives, the microstructure of polyisoprene can be tuned, ranging in microstructure from ~97 % cis-1,4 to ~88 % vinyl and thus altering the final physical properties of the material. On the other hand, microstructure determination of polyisoprene has only been accomplished in experimental analyses, mainly because of the experimental difficulty in distinguishing the active sites that belong to each of the isomers formed. Not a single study on the possibility of obtaining the microstructure

J Polym Res (2012) 19:9909

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and dyad distribution of polyisoprene using a kinetic scheme has been reported so far. Recently, we used 13C NMR to obtain experimental data of microstructure (cis, trans and vinyl content) in the homopolymerization of butadiene with n-butyl lithium [12] and barium-based initiators [13], and microstructure and sequence length distribution in the anionic copolymerization of styrene and butadiene using Al/Li/Ba [14] as initiator and n-butyl lithium as initiator and tetramethylethylenediamine (TMEDA) as modifier [15]. With these experimental data, reliable estimates of the kinetic rate constants were obtained. In the present contribution, a kinetic model considering the interaction between initiator or living polymer molecules with different active sites, such as cis, trans, isopropenyl, and vinyl units, which have their own reactivities, independent of polymer chain length, is proposed. Polymerization rate and sequence length distribution in the anionic living polymerization of isoprene with n-butyl lithium and N,N,N′, N′-tetramethyl ethylenediamine as active center modifier can be calculated with our model.

Kinetic model The main feature of our kinetic model is that it takes into account the interaction between the different isoprene active sites with initiator or living polymer molecules, thus producing geometric active sites in cis, trans, isopropenyl, or vinyl configurations, which have their own reactivities, independent of polymer chain length. Consequently, in the reaction scheme proposed, the negative charge is located on some of the terminal units (Fig. 1). The reaction scheme includes four initiation reactions namely the formation of four different active sites and the

CH2 C

CH2

Li

CH2 C

C H

CH3

cis active site

H CH2

Li

(1,4 addition product)

H CH2

Reaction scheme Four active sites: cis (C), trans (T), isopropenyl (D), and vinyl (V) were considered in the reaction scheme. Termination and chain transfer reactions were neglected. The initiator reacts with isoprene, producing one living molecule. Initiation kC I þ M ! Pc 1;0;0;0

ð1Þ

kT I þ M ! PT

0;1;0;0

ð2Þ

kD I þ M ! PD 0;0;1;0

ð3Þ

kV I þ M ! PV

ð4Þ

0;0;0;1

C

CH3

trans active site

(1,4 addition product)

corresponding propagation reactions. So, the kinetic model has sixteen propagation kinetic rate constants. From the reaction scheme, the corresponding kinetic equations were derived, considering the reactions to be irreversible and first order with respect to isoprene concentration. The equations for rate of monomer consumption, isomer formation (cis FC, trans FT, 3,4- FD, and vinyl FV), and dyad formation (FCC, FCT, FCD, FCV, FTC, FTT, FTD, FTV, FDC, FDT, FDD, FDV, FVC, FVT, FVD, FVV) were derived from the propagation reactions, and depend on isoprene and active site concentrations. The sixteen kinetic rate constants were estimated from conversion and microstructure experimental data, considering a Bernoulli dyad distribution and the proper definitions of conditional probabilities and active site fractions.

I, M, and P in Eqs. (1) to (4) are the initiator, monomer, and living polymer, respectively. In Px a,b,f,m, the subscripts are as follows: a for cis units, b for trans units, f for 3,4units, and m for vinyl units, with a living anionic terminal unit in x (x0C, T, D, V).

CH3

C

Li

C

CH3

CH2

Isopropenyl active site (3,4 addition product)

CH2

C

Li

CH CH2

vinyl active site (1,2 addition product)

Fig. 1 Terminal units in the anionic polymerization of isoprene

Propagation In the propagation reactions, a living polymer molecule reacts with isoprene, producing a new molecule of living polymer, with the negative charge located in the same or a different type of active site. kij is the propagation kinetic constant, where i represents the active site that reacts with an isoprene molecule, and j is the terminal live unit resulting from this reaction.

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J Polym Res (2012) 19:9909

kTT ! PT PT a;b;f ;m þ M

a;bþ1;f ;m

ð10Þ

the active sites of polyisoprene is four [10]. The effect of TMEDA is to promote dissociation of the living polymeric aggregates. It was also assumed that all the species are dissociated according to [n-BuLi]/[TMEDA]04 or greater [10]. As mentioned before, the rate equations were derived, considering that all reactions are bimolecular, irreversible, and of first order with respect to initiator, isoprene, and active sites [10]. The rate of initiator consumption is given by Eq. (21), where [I] is initiator concentration and kC, kT, kD, and kV are kinetic rate constants for the different structural units. It was assumed that the initiation rate is very fast, compared to the propagation rate. The polymerization process is controlled by the propagation rate. Therefore, the active sites are instantaneously generated, and their total concentration remains constant, which makes Eq. (21) unnecessary.

kTD ! PD PT a;b;f ;m þ M

a;b;f þ1;m

ð11Þ



kTV PT a;b;f ;m þ M ! PV

a;b;f ;mþ1

ð12Þ

The rate of production of living polymer with terminal units of cis, trans, vinyl, isopropenyl or vinyl structural units, is given by Eqs. (22)–(25), respectively.

Cis active site: kCC ! PC PC a;b;f ;m þ M

aþ1;b;f ;m

ð5Þ

kCT PC a;b;f ;m þ M ! PT

a;bþ1;f ;m

ð6Þ

kCD ! PD PC a;b;f ;m þ M

aþb;f þ1;m

ð7Þ

kCV PC a;b;f ;m þ M ! PV

aþb;f ;mþ1

ð8Þ

aþ1;b;f ;m

ð9Þ

Trans active site: kTC ! PC PT a;b;f ;m þ M

Isopropenyl active site: kDC ! PC PD a;b;f ;m þ M

aþ1;b;f ;m

ð13Þ

kDT ! PT PD a;b;f ;m þ M

a;bþ1;f ;m

ð14Þ

kDD ! PV PD a;b;f ;m þ M

a;b;f þ1;m

ð15Þ

kDV PD a;b;f ;m þ M ! PD

a;b;f ;mþ1

ð16Þ

kVC ! PC PV a;b;f ;m þ M

aþ1;b;f ;m

ð17Þ

kVT ! PT PV a;b;f ;m þ M

a;bþ1;f ;m

ð18Þ

d½I ¼ððkC þkT þkD þkV Þ½I½M dt

d½PC a;b;f ;m  dt

    ¼kCC Pc a1;b;f;m ½M  kCC Pc a;b;f;m ½M kCT Pc a;b;f ;m ½M    kCD PC a;b;f ;m ½M  þkTC PT a1;b;f;m ½M þ kDC  PD a1;b;f;m ½M kCV Pc a;b;f ;m ½M þ kVC Pv a1;b;f ;m ½M

ð22Þ

    ¼kCT Pc a;b1;f;m ½M þ kTT PT a;b1;f  ;m ½M kTT PT a;b;f ;m ½M  kTC PT a;b;f ;m ½M  kTD PT a;b;f ;m ½M þ kDT PD a;b1;f ;m ½M kTV PT a;b;f ;m ½M þ kVT PV a;b1;f ;m ½M

ð23Þ

    ¼kCD PT a;b;f 1;m  ½M þ kTD  PT a;b;f1;m ½M kDC PD a;b;f ;m ½M    kDT PD a;b;f ;m ½M  ½ M   k P þkDD PD a;b;f 1;m DD D a;b;f ;m   ½M kDV PD a;b;f ;m ½M þ kVD PV a;b;f 1;m ½M

ð24Þ

    ¼kVV Pv a;b;f ;m1 ½M  kVV Pv a;b;f ;m ½M þkCV Pc a;b;f ;m1 ½M þ kTV PT a;b;f ;m1  ½M þkDV PD a;b;f ;m1  ½M  kVC Pv a;b;f;m ½M kVT Pv a;b;f ;m ½M  kVD Pv a;b;f ;m ½M

ð25Þ

d½PT a;b;f ;m  dt

Vinyl active site:

d½ PD

a;b;f ;m



dt

PV a;b;f ;m þ M! PD

a;b;f þ1;m

ð19Þ

kVV PV a;b;f ;m þ M ! PV

a;b;f ;mþ1

ð20Þ

kVD

It was assumed that the associated species are not reactive to polymerization and that they are in balance with the dissociated species, given that the degree of association of

d½PV

a;b;f ;m

dt

ð21Þ



J Polym Res (2012) 19:9909

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Assumed

Microestructure data

data rij

Bernoulli dyads calculated Optimization A Conditional probability (39-41)

d½IC  dt

¼ððkCT þkCD þkCV Þ½IC þkTC ½IT þkDC ½ID þkVC ½IV Þ½M

ð27Þ

d½IT  dt

¼ðkCT ½IC ðkTC þkTD þkTV Þ½IT þkDT ½ID þkVT ½IV Þ½M

ð28Þ

d½ID  dt

¼ðkCD ½IC þkTD ½IT ðkDC þkDT þkDV Þ½ID þkVD ½IV Þ½M

ð29Þ

d½IV  dt

¼ðkCV ½IC þkTV ½IT þkDV ½ID ðkVC þkVT þkVD Þ½IV Þ½M

ð30Þ

Fraction of active site (46) First-order dyads (47) rij Assumed data

Data

kii

Isoprene conversion

Optimization B

Microstructure formation

Isoprene conversion*dyads distribution Conditional probability (39-41)

The rates of formation of cis (FC), trans (FT), isopropenyl (FD), and vinyl (FV) polyisoprene, independent of the position of the structural units within the chains, are given by Eqs. (31)–(34),

Fraction of active site (46) Dyads formation (35-37)

kCC, kTT, kDD

d ½ FC  ¼ðkCC ½IC þkTC ½IT þkDC ½ID þkVC ½IV Þ½M dt

ð31Þ

d½FT  ¼ðkCT ½IC þkTT ½IT þkDT ½ID þkVT ½IV Þ½M dt

ð32Þ

d ½ FD  ¼ðkCD ½IC þkTD ½IT þkDD ½ID þkVD ½IV Þ½M dt

ð33Þ

d ½ FV  ¼ðkCV ½IC þkTV ½IT þkDV ½ID þkVV ½IV Þ½M dt

ð34Þ

kij = kii/rij

Arrhenius parameters kij = Aij exp(-Eij/RT)

Fig. 2 Kinetic rate constants calculation procedure

Rate of monomer consumption It was assumed that the amount of isoprene consumed in the initiation reaction is negligible; thus, isoprene is consumed by propagation only, and its consumption rate is given by Eq. (26). where [IC], [IT], [ID], and [IV] are the concentrations of the active sites.  d½dtM ¼ððkCC þkCT þkCD þkCV Þ½IC þðkTC þkTT þkTD þkTV Þ½IT  þðkDC þkDT þkDD þkDV Þ½ID þðkVC þkVT þkVD þkVV Þ½IV Þ½M

ð26Þ

The overall concentrations of these active sites are given by: ½I C  ¼ ½I D  ¼

1 P 1 P 1 P 1  P PC

a¼1 b¼0 f ¼0 m¼0 1 P 1 P 1 P 1  P a¼0 b¼0 f ¼1 m¼0

PD

a;b;f ;m



a;b;f ;m



½I T  ¼ ½I V  ¼

1 P 1 P 1 P 1  P PT

a¼0 b¼0 f ¼0 m¼0 1 P 1 P 1 P 1 P

a¼0 b¼0 f ¼0 m¼1



PV

a;b;f ;m



a;b;f ;m



Balance of active sites The concentration of active sites of each type are given by Eqs. (27)–(30).

Table 1 Initial Conditions for the anionic polymerization of isoprene Experiment

T (°C)

MR ¼½TMEDA=½Io ¼ 0 1 40 2 50 3 60 70 4a 80 5a MR ¼½TMEDA=½Io ¼ 4 6 40 7 50 8 60 70 9a 80 10a a

Extrapolated

[M]o mol/L

[I]o mmol/L

1.827 1.779 1.589 1.589 1.589

0.6317 1.228 0.3662 0.3662 0.3622

1.583 1.583 1.583 1.583 1.583

0.3591 0.3591 0.3591 0.3591 0.3591

Page 6 of 12

J Polym Res (2012) 19:9909 r1

Table 2 Microstructure distribution in the polymerization of isoprene at different MR ([TMEDA]/[n-BuLi]o) ratios and R00.30

PCC ¼ 1þr1 þr11 þr1

CV

PCT ¼ 1þr1 þrCT1 þr1

MR

PCD ¼ 1þr1 þrCD1 þr1

PCV ¼ 1þr1 þrCV1 þr1

0

exp calc error (%)

4

exp calc error (%)

Cis

Trans

Isopropenyl

28.0800 28.1531

65.5200 65.4382

6.4000 6.4087

0.2602

0.1248

0.1359

15.1350 15.0904

35.3150 35.3062

49.5500 49.6034

0.2947

0.0249

0.1077

CT

CD

r1

CT

CD

CT

CV

r1

CD

CV

r1

CT

CD

CV

r1

PTC ¼ r1 þ1þrTC1 þr1

PTT ¼ r1 þ1þrTV1 þr1

PTD ¼

PTV ¼ r1 þ1þrTV1 þr1

TD TV r1 TD 1 r1 þ1þr1 TD þrTV TC TC

TC

TC

r1

TD

r1

TD

TV

r1

PDT ¼ r1 þr1DTþ1þr1

Dyad formation

PDD ¼ r1 þr11þ1þr1

PDV ¼

The rates of dyad formation are given by Eqs. (35)–(38), where Fij represents the total amount of the ij dyad.

PVC ¼ r1 þr1VCþr1 þ1

DC

DV

DT

r1

VC

d½FCC  dt ¼kCC ½IC ½M d½FCD  dt ¼kCD ½IC ½M

DV

DT

VT VD 1

r

d½FCT  dt ¼kCT ½IC ½M d½FCV  dt ¼kCV ½IC ½M

ð35Þ

PVD ¼ r1 þr1VDþr1 þ1 VC

VT

ð40Þ

TV

PDC ¼ r1 þr1DCþ1þr1 DC

ð39Þ

VD

DV DT r1 DV 1 r1 þr1 DT þ1þr DV DC DC

ð41Þ

r1

PVT ¼ r1 þr1VTþr1 þ1 VC

VT

VD

PVV ¼ r1 þr11þr1 þ1 VC

VT

ð42Þ

VD

The conditional probabilities are subjected to the following restrictions:

d½FTC  dt ¼kTC ½IT ½M d½FTD  dt ¼kTD ½IT ½M

d½FTT  dt ¼kTT ½IT ½M d½FTV  dt ¼kTV ½IT ½M

ð36Þ

d½FDC  dt ¼kDC ½ID ½M d½FDD  dt ¼kDD ½IT ½M

d½FDT  dt ¼kDT ½ID ½M d½FDV  dt ¼kDV ½ID ½M

ð37Þ

d½FVC  dt ¼kVC ½IV ½M d½FVD  dt ¼kVD ½IV ½M

d½FVT  dt ¼kVT ½IV ½M d½FVV  dt ¼kVV ½IV ½M

ð38Þ

PCC þPCT þPCD þPCV ¼ 1 PTC þPTT þPTD þPTV ¼ 1 PDC þPDT þPDD þPDV ¼ 1 PVC þPVT þPVD þPVV ¼ 1 Besides, the conditional probabilities and the fraction of active sites are related by Eq. (43) [16, 17], where W is the vector (WC, WT, WD, WV) and P is the matrix defined in Eq. (44). W  P ¼ lW 2

Fraction of active sites The conditional probabilities can be expressed in terms of reactivity ratios (rij0kii/kij) as shown in Eqs. (39)–(42).

PCC 6 PTC P ¼6 4 PDC PVC

ð43Þ

PCT PTT PDT PVT

PCD PTD PDD PVD

3 PCV PTV 7 7 PDV 5 PVV

ð44Þ

Table 3 Sequence distribution in the polymerization of isoprene at different MR ([TMEDA]/[n-BuLi]o) ratios and R00.30

MR00 Exp Calc Error(%) MR04 Exp Calc Error(%)

CC

CT

CD

TC

TT

TD

DC

DT

DD

7.8849 7.8875 0.0340

18.3980 18.4541 0.3047

1.7971 1.7955 0.0900

18.3980 18.4700 0.3913

42.9287 42.7833 0.3387

4.1953 4.2045 0.2669

1.7971 1.7955 0.0900

4.1933 4.2008 0.1804

0.4096 0.4087 0.2138

2.2907 2.2772 0.5884

5.3449 5.3279 0.3195

7.4994 7.4854 0.1872

5.3449 5.3279 0.3195

12.4715 12.4653 0.0497

17.4986 17.5131 0.0828

7.4994 7.4854 0.1872

17.4986 17.5131 0.0828

24.5520 24.6050 0.2156

J Polym Res (2012) 19:9909

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Fig. 3 Isoprene conversion as function of polymerization time at MR00 and 4, at 50, 60, 70 and 80 °C

So, the relationship between the fraction of active sites and the conditional probabilities is: WC WT WD WT WV WT

TT 1Þ½ðPDD 1ÞðPVV 1ÞPVD PDV PDT ½PTD ðPVV 1ÞPVD PTV þPVT ½PTD PDV PTV ðPDD 1Þ ¼ ðPP CT ½ðPDD 1ÞðPVV 1ÞPVD PDV þPDT ½PCD ðPVV 1ÞPVD PCV PVT ½PCD PDV ðPDD 1ÞPCV  TD ðPVV 1ÞPVVD PTV ðPTT 1Þ½PCD ðPVV 1ÞPVD PCV þPVT ½PCD PTV PTD PCV  ¼ PPCTCT½ð½PPDD 1ÞðPVV 1ÞPVD PDV þPDT ½PCD ðPVV 1ÞPVD PCV PVT ½PCD PDV ðPDD 1ÞPCV 

¼

ð45Þ

PCT ½ðPTD PDV ðPDD 1ÞPTV þðPTT 1Þ½PCD PTV ðPDD 1ÞPCV PDT ½PCD PTV PTV PCV  PCT ½ðPDD 1ÞðPVV 1ÞPVD PDV þPDT ½PCD ðPVV 1ÞPVD PCV PVT ½PCD PDV ðPDD 1ÞPCV 

As mentionated before, for the polymerization conditions studied in this system, it can be considered that no vinyl units are present, moreover; peaks indicating 1,2- addition were not observed in the 1H NMR, thus 1,2 units were found to be negligible. If only the cis, trans and isopropenyl actives sites are present, then Eq. (45) reduces to Eq. (46). WC WT WD WT

Fig. 5 Microstructure formation as function polymerization time with [I]o00.3591 mmol/L, [M]o01.583 mol/L, MR04, R00.30, and T0 60 °C

ð1PDD ÞðPTC þPTD ÞPTD PVT ¼ ð1P DD ÞPCT þPDT ð1PCC PCT Þ

PTD þðPTC þPTD Þð1PCC PCT Þ ¼ PðCT1P DD ÞPCT þPDT ð1PCC PCT Þ

ð46Þ

Fig. 4 Microstructure conversion as function of polymerization time with [I]o00.3591 mmol/L, [M]o01.583 mol/L, MR04, R00.30, and T060 °C

Using a first-order Markov model, the dyad fractions can be calculated by the product of the fraction of active sites (Wi) times the conditional probability of active sites (Pij), as shown in Eq. (47).

CC ¼WC PCC TC ¼WT PTC DC ¼WD PDC VC ¼WV PVC

CT ¼WC PCT TT ¼WT PTT DT ¼WD PDT VT ¼WV PVT

CD ¼WC PCD TD ¼WT PTD DD ¼WD PDD VD ¼WV PVD

CV ¼WC PCV TV ¼WT PTV DV ¼WD PDV VV ¼WV PVV

ð47Þ

Fig. 6 Dyad conversion for the cis active sites. [I]o00.3591 mmol/L, [M]o01.583 mol/L, MR04, R00.30, and T060 °C

Page 8 of 12

J Polym Res (2012) 19:9909 Table 4 Reactivity ratios as function of R at MR00

Fig. 7 Dyad conversion for the trans active sites. [I]o00.3591 mmol/ L, [M]o01.583 mol/L, con MR04, R00.30, and T060 °C

The Bernoulli dyads are calculated as shown in Eq. (48). CC ¼FC FC TC ¼FT FC DC ¼FD FC VC ¼FV FC

CT ¼FC FT TT ¼FT FT DT ¼FD FT VT ¼FV FT

CD ¼FC FD TD ¼FT FD DD ¼FD FD VD ¼FV FD

CV ¼FC FV TV ¼FT FV DV ¼FD FV VV ¼FV FV

ð48Þ

CC ¼XC XC TC ¼XT XC DC ¼XD XC VC ¼XV XC

CT ¼XC XT TT ¼XT XT DT ¼XD XT VT ¼XV XT

CD ¼XC XD TD ¼XT XD DD ¼XD XD VD ¼XV XD

CV ¼XC XV TV ¼XT XV DV ¼XD XV VV ¼XV XV

ð48Þ

where XC,XT, XV and XD, are the fraction of isomer cis, trans, vinyl and isopropenyl respectively.

Cis active site

Trans active site

D active site

R

rCT

rTC

rTD

rDC

rDT

0.10

0.1111

1.4625

9.0000

13.1625

0.6837

0.0759

0.20 0.30

0.2500 0.4285

2.9250 4.3875

4.0000 2.3333

11.7000 10.2375

0.3418 0.2279

0.0854 0.0976

0.40

0.6666

5.8500

1.5000

8.7749

0.1709

0.1139

0.50

1.0000

7.3125

1.0000

7.3125

0.1367

0.1367

0.60

1.5000

8.7750

0.6666

5.8500

0.1139

0.1709

0.70

2.3333

10.2375

0.4285

4.3875

0.0976

0.2279

0.80 0.90

4.0000 9.0000

11.7000 13.1625

0.2500 0.1111

2.9250 1.4625

0.0854 0.0759

0.3418 0.6837

rCD

Calculation procedure Figure 2 shows the procedure for calculation of the kinetic rate constants. The Fortran code developed using our model was fed with initial data of isoprene and initiator initial concentrations, [M]o and [I]o, and with experimental data of isoprene conversion, microstructure, and Bernoulli dyads. The conversion data were obtained from the kinetic equations proposed by Chang et al. [10] using temperatures of 40, 50, 60, 70, and 80 °C; n-butyl-lithium as initiator; and N, N,N′,N′-tetramethylethylenediamine as microstructure modifier, with a molar ratio [TMEDA]/[I]o of 0 and 4. The initial conditions were as follows (see Table 1): ½Px ¼ 0; x ¼ 2; 3; 4;    ½IC  ¼ WC ½I ½ IT  ¼ WT ½ I   ½ID  ¼ WD ½I Dyads were calculated considering a random distribution (Bernoulli) [16] from the values of reported microstructure [19]. It was also assumed that the microstructure does not Table 5 Reactivity ratios as function of R at MR04

Fig. 8 Dyad conversion for isopropenyl active sites. [I]o 0 0.3591 mmol/L, [M]o01.583 mol/L, con MR04, R00.30, and T0 60 °C

Cis active site

Trans active site

D active site

R

rCT

rCD

rTC

rTD

rDC

rDT

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

0.1111 0.2500 0.4528 0.6666 1.0000 1.5000 2.3333 4.0000 9.0000

0.1018 0.2036 0.3054 0.4072 0.5090 0.6108 0.7127 0.8145 0.9163

9.0000 4.0000 2.3333 1.5000 1.0000 0.6666 0.4285 0.2500 0.1111

0.9163 0.8145 0.7127 0.6108 0.5090 0.4072 0.3054 0.2036 0.1018

9.8216 4.9108 3.2738 2.4554 1.9643 1.6369 1.4030 1.2277 1.0912

1.0912 1.2277 1.2277 1.6369 1.9643 2.4554 3.2738 4.9108 9.8216

J Polym Res (2012) 19:9909

Page 9 of 12

Table 6 Kinetic parameters in the anionic polymerization of isoprene with MR00 and R00.30

kCC

a

kCT kCD kTC kTT kTD kDC kDT kDD kp [7]

At 60 °C

ln A

E (kJmol−1)

Δ≠G (kJ/mol)a

Δ≠S (J/molK)a

Δ≠H (kJ/mol)a

22.63 23.48 21.15 23.27 24.11 21.79 21.52 22.37 20.04

70.648 70.648 70.648 70.648 70.648 70.648 70.648 70.648 70.648

89.8 87.5 93.9 88.0 85.7 92.2 92.9 90.6 97.0

−65.9 −58.9 −78.3 −60.6 −53.6 −72.9 −75.2 −68.1 −87.5

67.9 67.9 67.9 67.9 67.9 67.9 67.9 67.9 67.9

24.86

70.615

83.7

−47.4

67.9

change with monomer conversion [18]. Since 1,4-cis structural units are difficult to calculate, we obtained these from reports of indirect measurements, based on natural polyisoprene [19, 20]. The fraction of cis structural units (R0cis/ (cis+trans)) was varied from 0.10 to 0.90. For the polymerization conditions studied this system, it was assumed that no vinyl units are present [19, 20]. The kinetic rate constants were estimated using a multivariate nonlinear regression procedure, which takes into consideration the error in all variables [21]. In the code for parameter estimation purposes, experimental data of isoprene conversion, microstructure, and Bernoulli dyad distribution were used. The six reactivity ratios were determined from conditional probabilities (Eqs. 39–41), fraction of active sites (Eq. 46), and first-order dyads (Eqs. 35– 37). The model solution involves an iterative process that requires the supposition of initial values of kii. The functionality with respect to time was solved with a fourth-order Runge–Kutta method. The calculation was repeated with different initial values with the aim of proving that uniqueness exists. The comparison of experimental and calculated data XCj X X ; ΣXTjTj ; ΣXDjDj where XCj, was realized in terms of the quotient ΣX Cj XTj, and XDj represent the isoprene converted to the dyad Cj, Tj, Dj, respectively. Table 7 Kinetic parameters in the anionic polymerization of isoprene with MR04 and R00.30

kCC

a

kCT kCD kTC kTT kTD kDC kDT kDD kp [7]

At 60 °C

Results and discussion The conditional probabilities of the active sites are based on the reactivity ratios. Therefore, the monomer sequence distribution is determined by the propagation step. Nuclear magnetic resonance studies do not allow the determination of the distribution of the nine dyads in isoprene polymerization. However, considering that the dyad distribution follows a Bernoulli distribution, these values can then be calculated. Therefore, the monomer sequence distribution can be inferred from microstructure data. The predictive power of the model was tested by simulating isoprene polymerizations at the conditions used in the experimental studies. Aij, Eij, and T were the only data fed to the simulation program, and then monomer conversion, microstructure, and the monmer sequence distribution were calculated. Table 2 shows experimental and calculated microstructures, whereas Table 3 shows experimental and calculated monomer sequence distributions. The agreement is good in both cases. With respect to microstructure, the experimental error is