j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 9 ( 2 0 0 8 ) 56–63

journa l homepage: www.e lsev ier .com/ locate / jmatprotec

Numerical simulation of reactive extrusion processes foractivated anionic polymerization

Lili Wua, Yuxi Jiaa,b,∗, Sheng Suna, Guofang Zhanga, Guoqun Zhaoa, Lijia Anb,∗

a School of Materials Science and Engineering, Shandong University, Jinan 250061, Chinab State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry,Chinese Academy of Sciences, Changchun 130022, China

a r t i c l e i n f o

Article history:

Received 2 February 2006

Received in revised form

20 December 2006

Accepted 27 July 2007

a b s t r a c t

In order to deal with the complicated relationships among the variables of the reactive extru-

sion process for activated anionic polymerization, a three-dimensional equivalent model of

closely intermeshing co-rotating twin screw extruders was established. Then the numeri-

cal computation expressions of the monomer concentration, the monomer conversion, the

average molecular weight and the fluid viscosity were deduced, and the numerical simu-

lation of the reactive extrusion process of styrene was carried out. At last, our simulated

results were compared with Michaeli’s simulated results and experimental results.

Keywords:

Activated anionic polymerization

Kinetics

Simulation

© 2007 Elsevier B.V. All rights reserved.

2000a, 2000b). Hyun and Kim made an engineering analysis of

Reactive extrusion

1. Introduction

The reactive extrusion for polymerization has been character-ized by means of a polymer synthesis process, in which theextruder is used as a reactor. However, the reactive extrusionprocess is more complex than traditional polymer synthesisprocesses because there are complex interactions among suchvariables as the fluid flow, heat transfer and chemical reac-tion (Janssen, 1998; Berghaus and Michaeli, 1991; Hoecker etal., 1996; Hornsby et al., 1994; Hornsby and Tung, 1994; Wollnyet al., 2003; Si et al., 2002; Michaeli et al., 1993a, 1993b; Siadatet al., 1979), which cause the researches on reactive extrusionprocesses to be very difficult.

To solve the problems, a lot of mathematical descriptions

and numerical simulations of reactive extrusion processes forpolymerization have been developed on the basis of experi-mental researches. Michaeli et al. used the ideal “cascade of

∗ Corresponding authors. Tel.: +86 431 5262137/5262206; fax: +86 431 52E-mail addresses: jia yuxi@sdu.edu.cn (Y. Jia), ljan@ciac.jl.cn (L. An)

0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.jmatprotec.2007.07.026

continuous stirred tank reactors” and “pipe reactor” modelsto develop the numerical simulation of the anionic polymer-ization of polystyrene and nylon 6 and the copolymerizationof styrene–isoprene in a closely intermeshing co-rotatingtwin screw extruder (Michaeli et al., 1995; Michaeli et al.,1993a, 1993b; Michaeli and Grefenstein, 1995). Janssen et al.carried out the numerical simulations of reactive extrusionprocesses for polymerization in a counter-rotating twin screwextruder (Ganzeveld and Janssen, 1992; Ganzeveld et al., 1994;de Graaf et al., 1997; Janssen et al., 2001). Gimenez et al.numerically simulated the anionic polymerization process of�-caprolactone in a co-rotating twin screw extruder (Gimenezet al., 2000a, 2000b; Poulesquen et al., 2001; Gimenez et al.,

62969..

the reactive extrusion process of thermoplastic polyurethanein a single screw extruder via the numerical simulation (Hyunand Kim, 1988). Kye and White conducted research into the

g t e c h n o l o g y 1 9 9 ( 2 0 0 8 ) 56–63 57

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2s

2

Uepscdtmno1

eom

in2

2< x ≤

2

j o u r n a l o f m a t e r i a l s p r o c e s s i n

olymerization of caprolactam in a co-rotating twin screwxtruder (Kye and White, 1996a, 1996b). Vergnes et al. proposedcomputation model for polymer flows in a self-wiping co-

otating twin screw extruder on the basis of one-dimensionalpproximated approach (Vergnes et al., 1998; Vergnes et al.,002; Berzin and Vergnes, 1998). Kim and White performedn engineering analysis of the increase in molecular weightnd the shear-induced reduction of the molecular weightfter the polymerization of �-caprolactone during the reac-ive extrusion process (Kim and White, 2004). Zhu and Jaluriaumerically investigated the flow of chemically reactive non-ewtonian fluids in a fully intermeshing co-rotating twin

crew extruder (Zhu and Jaluria, 2002). Fukuoka numericallynalyzed the reactive extrusion process in a self-wiping co-otating twin screw extruder (Fukuoka, 2000).

The investigations have been mainly confined to one-imensional models, and have generally simplified theomplicated relationships among the fluid flow, the chemicaleaction, the material structures and physicochemical prop-rties, and heat transfer.

In this paper, the mathematical model of screw reactors,he kinetic model of activated anionic polymerization, theverage molecular weight and the fluid viscosity are studied,nd then the numerical computation expressions of the vari-bles are established. At last, by means of an example, thevolvements of the monomer concentration, the monomeronversion, the average molecular weight and the fluid viscos-ty are shown and compared with Michaeli’s simulated resultsnd experimental results.

. Construction of the equivalent model ofcrew reactors

.1. Construction of the physical model

sually, a closely intermeshing co-rotating twin screwxtruder is used as the reactor of the reactive extrusionrocess for anionic polymerization, which often consists ofeveral forward conveying screw elements, several reverseonveying screw elements, several kneading elements and aie. And in the reactor, the fluid flows forward by means ofhe drag of screws and the pressure. Therefore, the above-

entioned complexity makes it very difficult to perform theumerical simulation of reactive extrusion processes basedn the real reactor model (David et al., 2000; Goffart et al.,996; Van Der Wal et al., 1996; Kalyon et al., 1999), and the

h(x) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Rs

[1 + cos

2�(x − Wb/2)

Ts · cos �̄

]−

√C2

L − R2s s

2Rs − CL + ıf

ıf

quivalent reactor model should be constructed. At present,ne-dimensional equivalent reactor models are adopted inany investigations.

Fig. 1 – Cross section of screw channels (Geng, 2003).

If the extruder is assumed to be fully filled, the melt leak-ages are neglected in the intermeshing zone and the distortionof the fluid from one screw to another screw is not considered,the fluid flow space can be unfolded and equivalently consid-ered as an axisymmetrical model along the axial direction ofthe screw channel, in which the effect of the axial motion ofthe wall of the model on the fluid flow is equivalent to the com-prehensive effect of the rotary screws and the static barrel onthe fluid flow.

2.2. Calculation of the length of the model

In the model, the fluid is unfolded along the axial direction ofthe screw channel, and the average unfolded length, Z̄, of everylead, Ts, of screws is Ts/ sin �̄ (Zhu, 2001), where �̄ representsthe average helix angle of screws. So the equivalent length,Lm, of the model can be calculated

Lm = LZ̄

Ts= L

sin[arctan Ts/�(Rs + Rb)](1)

where L denotes the axial length of the screw and Rs and Rb

denote the radius of the excircle and the radius of the radicalcircle, respectively.

2.3. Calculation of the radius of the model

As shown in Fig. 1, the screw channel profile is made up of fivecurves, which are expressed as IJ, JK, KL, LM and MN. Accordingto the geometry of normal screws, the depth of screw channelprofile, h(x), can be obtained (Geng, 2003)

2�(x − Wb/2)

Ts · cos �̄+ ıf

Wb

2< x ≤ Ws

2

0 ≤ x ≤ Wb

2Ws (Ws + Wb)

(2)

where CL represents the centerline distance of two screws.For a double-thread screw, there are three parallel fluids.

For every fluid, the equivalent radius of the model, Rm, can beobtained

i n g

58 j o u r n a l o f m a t e r i a l s p r o c e s s

Rm =(

2�

∫ Ws/2

Wb/2

{Rs

[1 + cos

2�(x − Wb/2)

Ts · cos �̄

]

−√

C2L − R2

s sin2 2�(x − Wb/2)

Ts · cos �̄+ ıf

}dx

+ 1�

(2Rs − CL + ıf) · Wb + 1�

ıf · Wb

)1/2(3)

3. Construction of the kinetic model ofanionic polymerization

3.1. Calculation of monomer concentration

The following assumptions are made to simplify the mecha-nisms of activated anionic polymerizations (Pan, 1997):

(1) Initiator molecules all transform to active centers veryquickly.

(2) The reactant is stirred well, and the distribution ofmonomer is uniform.

(3) All growing chains form at the same time and have theequal growth probability.

(4) No chain transfer and chain termination take place.(5) Depolymerization can be neglected.

Therefore, the reaction rate of activated anionic polymer-ization, rp, can be expressed as (Allcock et al., 2004)

rp = −dcm

dt= kpcn

i cm (4)

where kp denotes the rate constant for chain propagation, ci

the initiator concentration, cm the monomer concentration,and n is the kinetic order for initiator initiation.

The correlation between kp and temperature T can beobtained according to the Arrhenius equation

kp = Ap e−Ep/RT (5)

where Ap denotes the frequency factor for chain propagation,Ep the activation energy for chain propagation, and R is thegeneral gas constant.

Then the monomer concentration can be obtained

cm = cm,0 e−kt = cm,0 e−Ape−Ep/RTcni

t (6)

where cm,0 denotes the initial monomer concentration.

3.2. Numerical calculation of monomer conversion

In the process of activated anionic polymerization, themonomer conversion X can be expressed as

X = 1 − e−Apcni

e−Ep/RTt (7)

For an isothermal process, the monomer conversion can becalculated directly with Eq. (7). But in practical processes, the

t e c h n o l o g y 1 9 9 ( 2 0 0 8 ) 56–63

reaction temperature is variable, and therefore the monomerconversion cannot be calculated directly via Eq. (7).

Upon using the partial differential calculus Eq. (7) becomes

∂X

∂t

∣∣∣I,J

= Apcni e−Ep/RT(I,J)[1 − X(I, J)] (8)

Based on the incremental theory, the following equation canbe obtained (Jia et al., 2003)

∂X

∂t

∣∣∣I,J

= �X(I, J)�t

+ O(�t) = X(I, J) − X(I − 1, J)�t

+ O(�t) (9)

where X(I,J) denotes the monomer conversion in the Ith timestep on the Jth space point, X(I−1,J) the monomer conversionin the (I−1)th time step on the Jth space point, �X(I,J) themonomer conversion increment in the Ith time step on theJth space point, O(�t) the truncation error, and (�t) is the timestep.

If the truncation error is neglected, combining Eqs. (8) and(9), the numerical calculation equation of the increment ofmonomer conversion can be obtained

�X(I, J) = Apcni e−Ep/RT(I,J)�t(I, J)[1 − X(I − 1, J)]

1 + Apcni e−Ep/RT(I,J)�t(I, J)

(10)

Then the numerical calculation equation of monomer conver-sion can be derived

X(I, J) = X(I − 1, J) + Apcni e−Ep/RT(I,J)�t(I, J)

1 + Apcni e−Ep/RT(I,J)�t(I, J)

(11)

Because the initial monomer conversion X (0,J) is zero, themonomer conversion in any time step can be calculated.

4. Numerical calculation of averagemolecular weight

4.1. Numerical calculation of the average molecularweight of chains

When the monomer conversion is 100%, the average degreeof polymerization, x̄n, should be equal to the number ofmonomer molecules that have been added to every activechain end, which can be calculated as follows (Pan, 1997)

x̄n = lcm,0

ci(12)

where l denotes the number of initiator molecules in a poly-mer chain. For an anionic polymerization initiated by singleinitiator molecule, l = 1.

Therefore, the number-average degree of polymerization at

the node (I,J) can be obtained

x̄n(I, J) = cm,0X(I, J)ci

(13)

g t e c h n o l o g y 1 9 9 ( 2 0 0 8 ) 56–63 59

Tc

M

w

ta

o

m

M

4m

Wuc

M

5m

TeteC

�

wvs

utP

�

wmmc

a

j o u r n a l o f m a t e r i a l s p r o c e s s i n

hen the number-average molecular weight at the node (I,J)an be acquired

¯ n(I, J) = Mmcm,0X(I, J)ci

(14)

here Mm denotes the molecular weight of monomer.For an activated anionic polymerization without chain

ermination reaction, the distribution of molecular weightpproximately fulfils the Poisson distribution (Pan, 1997)

x̄w

x̄n= 1 + x̄n

[x̄n + 1]2≈ 1

x̄n+ 1 (15)

r is even in a monodisperse state x̄w/x̄n = 1.Therefore, for a monodisperse state, the weight-average

olecular weight can be expressed as follows

¯ w(I, J) = Mmcm,0X(I, J)ci

(16)

.2. Numerical calculation of the weight-averageolecular weight of fluids

hen M̄eqw(I, J) is used to express the weight-average molec-lar weight of the fluid at the node (I,J), M̄eqw(I, J) can bealculated (Michaeli et al., 1995)

¯ eqw(I, J) = M̄w(I, J)X(I, J) + Mm[1 − X(I, J)] (17)

. Construction of chemorheologicalodels

he viscosity change during polymerization reaction is influ-nced by molecular weight, temperature, and shear rate. Andhen, to describe the rheological property of the fluid in anxtruder, the Carreau formulation is adopted (Rubinstein andolby, 2003)

= �∞ + (�0 − �∞) · [1 + (��̇)2]n−1/2

(18)

here � denotes the apparent viscosity, �∞ the infinite sheariscosity, �0 the zero shear viscosity, � the time constant, �̇ thehear rate, and n is the non-Newtonian index.

To describe the concentration and weight-average molec-lar weight dependence of the zero shear viscosity �0 ofhe fluid, the empirical formula is adopted (Richards andrud’homme, 1986)

0 ={

K1cM̄w, M̄eqw ≤ Mc

K2c5.4M̄3.4w , M̄eqw > Mc

(19)

here Mc denotes the critical molecular weight for entangle-ent effects in viscosity, c denotes the mass concentration of

acromolecular chains, and K1 and K2 denote the material

onstants which are related to temperature.To describe the dependence of fluid viscosity on temper-

ture, the thermally activated Arrhenius equation is adopted

Fig. 2 – Flow chart of the numerical simulation.

(Rubinstein and Colby, 2003)

�0(T) = K eE�/RT (20)

where K denotes the material constant and E� the activationenergy for fluid flow.

6. Simulation procedures

In order to deal with the complicated relationships amongthe variables such as reaction rate, average molecular weight,fluid viscosity, pressure, temperature and flow velocity, a semi-implicit iterative algorithm is proposed, and the flow chart ofthe numerical simulation is shown in Fig. 2.

7. Example and verification

The reactive extrusion process for the activated anionic poly-merization of styrene, with sec-butyl lithium as the initiatorand cyclohexane as the solvent in a closely intermeshing co-rotating twin screw extruder was investigated on the basis offinite volume analysis (Michaeli et al., 1995). The input keydata related to the second type of extruder, the material prop-erties, and the processing conditions are shown in Tables 1–3,respectively (Michaeli et al., 1995; Michaeli et al., 1993a, 1993b;Geng, 2003; Wu and Wu, 2002; Liang et al., 2003). And the num-ber of the nodes in the axial direction of the extruder is 8807,and the number of the nodes in the radial direction is 60.

According to the experimental work, the evolvement oftemperature along the axial direction of the extruder is shownin Fig. 3 ((�) 0.003357 m represents the nodes whose distanceto the centerline is 0.003357 m, which is close to the model

60 j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 9 ( 2 0 0 8 ) 56–63

Table 1 – The input key data related to the twin screwextruder

Parameters Numerical values

Nominal diameter of screws 30 mmCenterline distance of screws 26 mmSlenderness ratio of screws 29Number of thread starts 2Lead of screws 20 mmAxial screw position of initiator feeding 375 mm

Table 2 – The input key data related to the materialproperties

Parameters Numerical values

Density of monomer 0.9059 × 103 kg m−3

Molecular weight of monomer 104.15 g mol−1

Molecular weight of initiator 64.09 g mol−1

Critical molecular weight forentanglement effects

38,000 g mol−1

The infinite shear viscosity inCarreau equation

100 Pa s

The time constant in Carreauequation

3 s

The non-Newtonian index inCarreau equation

0.4

Activation energy for fluid flow 95,000 J mol−1

Frequency factor for chainpropagation reaction

1.13842 × 106 (m3 mol−1)1/2 s−1

Activation energy for chainpropagation reaction

5.9 × 104 J mol−1

Table 3 – The input key data related to the processingconditions

Parameters Numerical values

Initial concentration of initiator 12.723 mol m−3

Initial concentration of monomer 8698.99 mol m−3

Flow speed on the wall of the model 0.0152 m/sFlow speed in the entrance of the model 0.0101 m/s

Fig. 3 – Evolvement of temperature along the axial directionof the extruder (Michaeli et al., 1995).

Fig. 4 – Evolvement of monomer concentration along the

axial direction of the extruder.

wall; (�) 0.001831 m represents the nodes whose distance tothe centerline of the geometrical model is 0.001831 m; (�)0.000366 m represents the nodes whose distance to the cen-terline of the geometrical model is 0.000366 m, which is closeto the centerline of the model).

The evolvements of such variables as the monomer con-centration, the monomer conversion, the weight-averagemolecular weight of the chains, the weight-average molecularweight of the fluid and the apparent viscosity along the axialdirection of the extruder are shown in Figs. 4–8, respectively.

It can be seen that with the increase of the fluid flowlength, the monomer concentration gradually decreases; themonomer conversion, the weight-average molecular weightof the chains and the weight-average molecular weight of thefluid gradually increase. But the change of the apparent viscos-ity is complex, which piecewise increases firstly, and piecewisedecreases secondly. It can be seen from Eqs. (18) to (20) thatthe apparent viscosity of the fluid is the increasing functionof the weight-average molecular weight, and the decreasing

function of the temperature and the shear rate. And so thecomplex change is attributed to the comprehensive influenceof the above three factors.

Fig. 5 – Evolvement of monomer conversion along the axialdirection of the extruder.

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 9 ( 2 0 0 8 ) 56–63 61

Fig. 6 – Evolvement of the weight-average molecular weightof chains along the axial direction of the extruder.

Fo

seIoa

Fd

Fig. 9 – Comparison of our simulated results with

ig. 7 – Evolvement of the weight-average molecular weightf the fluid along the axial direction of the extruder.

The comparisons of monomer conversion between ourimulated results and Michaeli’s simulated results as well as

xperimental results are shown in Fig. 9 (Michaeli et al., 1995).t can be seen that the evolvement trends of the two typesf simulated results are uniform, and our simulated resultsre in good agreement with Michaeli’s experimental results.

ig. 8 – Evolvement of apparent viscosity along the axialirection of the extruder.

Michaeli’s simulated results and experimental results(Michaeli et al., 1995).

The number-average molecular weight in the experiment is74,000 g mol−1 (Michaeli et al., 1993a, 1993b), and our simu-lated result is 71215 g mol−1. Obviously, the difference betweenthem is small.

However, for the same screw length, the monomer con-version in our simulation is less than Michaeli’s simulatedresults.

The main reasons for the above phenomena are as follows.

(1) In the three-dimensional model adopted in this paper, thecomprehensive effect of the rotary screws and the staticbarrel on the fluid flow is equivalent to the effect of theaxial motion of the model wall on the fluid flow. And thenthe velocity of the fluid near the wall is bigger than thevelocity in the inner, therefore the residence time of thefluid near the wall is less than that in the inner.

(2) The incremental theory is adopted to solve the complexnon-isothermal problem in this paper. But the one-dimensional model is subdivided into isothermal balancesections in Ref. (Michaeli et al., 1995).

8. Summary

Based on the geometry of the closely intermeshing co-rotatingtwin screw extruder, the equivalent reactor model has beenestablished, and the parameters of the model have beencalculated. The numerical computation expressions of themonomer concentration, the monomer conversion, the aver-age molecular weight and the fluid viscosity have beendeduced in the reactive extrusion process for activated anionicpolymerization. The numerical analysis of the reactive extru-sion process of polystyrene has been successfully made bymeans of the finite volume method.

Twin screw extruders often consist of different elements

like forward conveying screw elements, reverse conveyingscrew elements, kneading elements and die, and they are notalways fully filled actually, so more precise three-dimensionalreactor models should be built. Eq. (4) is better fit to low

i n g

r

62 j o u r n a l o f m a t e r i a l s p r o c e s s

monomer conversion phase; but at high monomer conversionphase, the polymerization kinetic equation should be morecomplex because of the influence of diffusion on reactionrate.

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (50390096, 50573079, 50425517,50340420392) and the Special Funds for Major State BasicResearch Projects (2003CB615601).

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