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Modeling Crosslinking Polymerizationin Batch and Continuous Reactors

Ivan Kryven, Arjen Berkenbos, Priamo Melo, Dong-Min Kim, Piet D. Iedema*

A new pseudo-distribution approach is applied to the modeling of crosslinking copolymer-ization of vinyl and divinyl monomer and compared to Monte Carlo (MC) simulations. Withthe number of free pending double bonds as the main distribution variable, a rigoroussolution of the three leading moments of the molecularsize distribution becomes possible. Validation takes placewith data of methyl methacrylate with ethylene glycoldimethacrylate. Well within the sol regime perfect agree-ment is found, but near the gelpoint larger discrepanciesdo appear. This is probably due to the existence of multi-radicals that are not taken into account in the populationbalance approaches.

1. Introduction

Crosslinking polymerization occurs when a mixture of vinyl

and divinyl monomers polymerizes by radical polymeriza-

tion. The incorporation of divinyl units in the polymer chains

creates free pending double bonds (FPDB) that may react with

other growing chains to form crosslinks. The crosslinked poly-

mer has interesting industrial properties in view of its rheologi-

cal and processing characteristics. For instance, the influence of

a cross-linking agent (divinyl benzene) on new polystyrene–

polyethylene, interpenetrating-like networks has been inves-

tigated by Greco et al.[1] Higher divinyl content easily leads to

the formation of a gel network (Zhu and Hamielec[2]). However,

the present paper does not focus on the gel regime, but intends

to describe the process under pre-gelation conditions.

A key mathematical feature of the crosslinking polymer-

ization problem is the fact that there is no rigorous solution of

the chain length population balance equation in that

particular dimension only. This is caused by the fact that it

I. Kryven, D.-M. Kim, P. D. IedemaVan ’t Hoff Institute for Molecular Sciences, Universiteit vanAmsterdam, The NetherlandsE-mail: [email protected]. BerkenbosVattenfall-Nuon, Amsterdam, The NetherlandsP. MeloEscola de Quı́mica and Programa de Engenharia Quı́mica daCOPPE, Universidade Federal do Rio de Janeiro, Brazil

� 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim wileyonlin

Early View Publication; these are NOT

is a second dimension, the number of FPDB that determines

the reactivity of the chain backbones rather than the length of

the chains themselves. This is unlike some other non-linear

polymerizations as, for instance, radical polymerization with

long-chain branching by transfer to polymer, where basically

all monomer units on a chain backbone may in principle

react. The mathematical modeling of crosslinking polymer-

ization has received attention in several studies, among

which are the interesting series by Tobita et al.,[3–7] They treat

the polymerization under batch reactor conditions by Monte

Carlo sampling of primary polymers being coupled by

probability rules based on the ‘‘pseudo-kinetic’’ approach.

Cyclization and gel formation is taken into account by these

authors. More recently, Costas and Dias[8,9] have applied a

generating functions approach to the problem, both in the

pregel- and postgel-regimes, also for batch reactors only.

Their method is comprehensive and rigorous, but requires a

complex implementation. No explicit solutions for the chain

length distribution are given in their publications. Finally,

Kizilel et al.[10] present a model for crosslinking polymeriza-

tion with Methyl Methacrylate (MMA) and Ethylene Glycol

Dimethacrylate (EGDMA) for the sol and gel regime using the

‘‘numerical fractionation (NF)’’ technique. They check their

model with experimental data by Li et al.[11] We will apply our

method to MMA/EGDMA as well and compare the results to

those of these authors.

In the present paper, we aim at solving the crosslinking

polymerization problem for both batch reactor and

elibrary.com Macromol. React. Eng. 2013, DOI: 10.1002/mren.201200073 1

the final page numbers, use DOI for citation !! R

Rn;i

Rn;i

Rn;i

T� þ

T� þ

Rn;i

Rn;i

Rn;i

dRn;i;k

dt

1

2

REa

www.mre-journal.de

I. Kryven, A. Berkenbos, P. Melo, D.-M. Kim, P. D. Iedema

continuous stirred tank reactor (CSTR). We start with the full

population balance formulation in three dimensions: chain

length, number of FPDBs, and number of radicals per

molecule. However, while staying in the pre-gel regime,

we disregard the 3rd dimension, assuming that the

multiradical concentration is negligible. Whether this is

a valid assumption, will in this paper only be checked by

comparing the results of this solution to those of a

Monte Carlo simulation that does (implicitly) take multi-

radicals into account. An explicit check of the role of

multiradicals requires taking the number of radical sites

on molecules into account as an additional dimension.

This is left for future publication. In the present paper, the

population balance problem reduced to two dimensions

could be solved in a relatively simple and straightforward

manner using the ‘‘pseudo-distribution approach’’, first

introduced by us in Iedema et al.[12,13] This implies a further

reduction of the problem to one dimension only, but it still

allows us to solve a part of the problem in a fully rigorous

way. A Monte Carlo sampling method developed[3–7] for

crosslinking polymerization in batch reactors has now

been extended to the CSTR case by us. This will serve as a

reference for the pseudo-distribution results.

This paper is organized as follows. First, the population

balances are presented, still in terms of all the dimensions

involved, including the number of radical sites. Then

various versions of the pseudo-distribution approach to the

crosslinking polymerization problem will be formulated,

including one explicitly addressing multiradicals. The MC

simulation technique will then be explained as introduced

earlier[3–7] and extended to the CSTR case by us in this paper.

Finally, the results are presented for the deterministic

and MC models and comparisons are made, among others,

using experimentally validated kinetic parameters for

the MMA/EGDMA system.

2. Population Balance Equations

The indices in the following reaction equations, n, i, k, p

denote the four dimensions of the full problem, i.e. chain

length, number of FPDB, number of cross-linking points, and

number of radical sites, respectively. Note that a propagation

step involving the divinyl monomer, M2, leads to an increase

in the number of free pending vinyl groups in a living chain:

Initiator Decomposition

I2 �

I� þ

I� þ

þ2

kt

rly V

!kd2I� (1)

þ kp
Monovinyl initiation:
M1 �!ki1

R1;0;0;1 (2)

� Rn;
Divinyl initiation:
M2 �!ki2

R1;1;0;1 (3)

Macromol. React. Eng. 2013, DO

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Monovinyl propagation:

I: 10.100

H & Co

e nu

;k;p þM1 �!pkp1

Rnþ1;i;k;p (4)

Divinyl propagation:

;k;p þM2 �!pkp2

Rnþ1;iþ1;k;p (5)

Chain transfer to Chain Transfer Agent (CTA)

This is considered as a termination step generating a

transfer agent radical T� that in consecutive steps re-

initiates new monomers, M1 or M2 (see also Kizilel et al.[10]):

;k;p þ T �!pkf

Rn;i;k;p�1 þ T� (6)

M1 �!k0

i1R1;0;0;1 (7)

M2 �!k0

i2R1;1;0;1 (8)

A cross-linking step is a reaction between a growing

radical chain and a free vinyl on a living or dead chain,

leading to a combined living chain having one more cross-

linking point and one less free vinyl group:

Cross-linking:

;k;p þ Rm;j;l;q ��!ðiqÞkp�

Rnþm;iþj�1;kþlþ1;pþq (9)

Termination by disproportionation:

;k;p þ Rm;j;l;q ��!ðpqÞktd

Rn;i;k;p�1 þ Rm;j;l;q�1 (10)

Termination by recombination:

;k;p þ Rm;j;l;q ��!ðpqÞktc

Pnþm;iþj;kþl;pþq�2 (11)

The 4-D formulation reads:

;p ¼ kp1M1pðRn�1;i;k;p � Rn;i;k;pÞ

þ kp2M2pðRn�1;i�1;k;p � Rn;i;k;pÞ� ktdl0001ðpþ 1ÞRn;i;k;pþ1 � kf Tðpþ 1ÞRn;i;k;pþ1

c

Xn�1

m¼1

Xi

j¼0

Xk

l¼0

Xp

q¼0

qRm;j;l;qðp� qþ 2ÞRn�m;i�j;k�l;p�qþ2

8<:

9=;

�(Xn�1

m¼1

Xi

j¼0

Xk

l¼0

Xp

q¼0

jRm;j;l;qðp� qÞRn�m;i�jþ1;k�l�1;p�q

i;k;pl0100

)

(12)

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Modeling Crosslinking Polymerization in Batch and Continuous Reactors

www.mre-journal.de

with moments defined as:

l0100

l0001

dRn;i

dt

þ kp �

dPn;i

dt

m01 ¼

l00 ¼

www.M

¼X1n¼1

X1i¼0

X1k¼0

X1p¼0

iRn;i;k;p (13)

¼X1n¼1

X1i¼0

X1k¼0

X1p¼0

pRn;i;k;p (14)

dRn

dt¼

Note that the moment as defined by Equation (13), l0100,

equals the total number of FPDB. It is obvious that the

reactivity of a polymer molecule for the cross-linking

reaction is proportional to its number of FPDB. This

formulation of the population balance does not take

eventual volume changes during polymerization into

account. Neither do the further population balances in

the remainder of this text do so.

Now, we simplify the population balance by dropping

two dimensions: the number of crosslinks and the

number of radical sites. The crosslinking is an essential

property in itself as it is important for processing, but it does

not directly influence the reactivity of the polymer

molecules. Therefore, it is possible to obtain a rigorous

solution of the population balance without considering

the crosslinks, which we will do here. In a further

publication, the number of crosslinks will be re-introduced

in the solution procedure. The kinetic parameters are

chosen in such a way that no gelation occurs. Note that

we have not removed the dimension number of radical

sites completely, but we rather represent it by

formulating the usual two polymer classes ‘‘dead’’ (P)

and ‘‘living’’ chains (R), possessing zero and one radical site,

respectively. The population balance in two dimensions

now is expressed as two equations, one for living and the

other for dead chains:

dPn

dt¼

¼ kp1M1ðRn�1;i � Rn;iÞ þ kp2M2ðRn�1;i�1 � Rn;iÞ

�ðktd þ ktcÞl00Rn;i � kf TRn;i

Xn�1

m¼1

Xi

j¼0

jPm;jRn�m;i�jþ1 � Rn;im01

8<:

9=; � 1

tRn;i ¼ 0

� �CSTR

(15)

¼ ktdl00Rn;i þ kf TRn;i þ1

2ktc

Xn�1

m¼1

Xi

j¼0

Rm;jRn�m;i�j

8<:

9=;

� kp � l00iPn;i �1

tPn;i ¼ 0

� �CSTR

(16)

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Macromol. React. Eng. 2013, DOI



with the moments:

: 10.100

H & Co

the

X1n¼1

X1i¼0

iPn;i (17)

X1n¼1

X1i¼0

Rn;i (18)

Note that we have presented the population balance

variants for both the batch reactor and CSTR, the latter

implying the RHS to be zero while at the same time an

outflow term containing the factor 1=t is added, where t is

the average residence time equal to the ratio of the CSTR’s

volume, V, to the volumetric flow, F, t¼V/F. Furthermore, we

have not accounted for eventual changes in density.

Finally, the equations describing the balances of the

low-molecular species are presented in Table 1.

3. Pseudo-Distribution Approach

3.1. Taking Moments Over the Number of FPDB

Now, we want to solve the 2-D population balances of

Equation(15) and (16) using the pseudo-distributionapproach

(Iedema et al.[12]). The method implies to reduce the

dimensionality by formulating expressions for the leading

moments of one dimension, while the other is preserved.

Thus, one obtains a multiple set of 1-D population balance

equations. Usually, the preserved dimension is chain length,

since one would most often be interested to compute the

chain length distribution. In the present case, this would

imply to take the moments over the number of FPDB:

dP1

i¼0 Rn;i

dt¼ kp1M1ðRn�1 � RnÞ þ kp2M2ðRn�1 � RnÞ

� ðktd þ ktcÞl00Rn � kf TRn

þ kp �Xn�1

m¼1

CmRn�m � Rnm01

( )� 1

tRn ¼ 0

� �CSTR

(19)

dP1

i¼0 Pn;i

dt¼ ktdl00Rn þ 1

2ktc

Xn�1

m¼1

RmRn�m

þ kf TRn � kp � l00Cn �1

tPn ¼ 0

� �CSTR

(20)

with the first FPDB moment:

Cn ¼X1i¼0

iPn;i (21)

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dR0

dt¼

�ðktd

dR1

Table 1. Balances of low-molecular species (subscript f refers to the feed stream in the CSTR case.

Species Equation Number

Initiator dI2

dt¼ �kdI2 þ

1

tðI2f � I2Þ ¼ 0

� �CSTR

T1

Initiator radical, I�dI�

dt¼ 2kdI2 � ðki1M1 þ ki2M2ÞI� �

1

tI� ¼ 0

� �CSTR

T2

Transfer agent, TdT

dt¼ �kf Tl00 þ

1

tðTf � TÞ ¼ 0

� �CSTR

T3

Transfer agent radical, T�dT�

dt¼ kf l00T � ðk0i1M1 þ k0i2M2ÞT� �

1

tI� ¼ 0

� �CSTR

T4

Vinyl monomer, M1dM1

dt¼ �kp1M1l00 �M1ðki1I� þ k0i1T�Þ þ 1

tðM1f �M1Þ ¼ 0

� �CSTR

T5

Divinyl monomer, M2dM2

dt¼ �kp2M2l00 �M2ðki2I� þ k0i2T�Þ þ 1

tðM2f �M2Þ ¼ 0

� �CSTR

T6

Macroradical, l00

dl00

dt¼ ki1M1I� þ ki2M2I� þ k0i1M1T� þ k0i2M2T�

�kf Tl00 � ðktd þ ktcÞðl00Þ2 �1

tl00 ¼ 0

� �CSTR

T7

Polymer, m00 ¼X1

n¼1

X1i¼0

Pn;i

dm00

dt¼ ktdðl00Þ2 þ kfTl00 � kp � l00m01 �

1

tm00 ¼ 0

� �CSTR

T8

Number of FPDB, h01 ¼X1n¼1

X1i¼0

iðPn;i þ Rn;iÞdh01

dt¼ kp2M2ðI� þ l00Þkp � l00m01 �

1

th01 ¼ 0

� �CSTR

T9

Number of crosslinks, xdx

dt¼ kp � l00h01 �

1

tx ¼ 0

� �CSTR

T10

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If it would be possible to solve these equations including

those for the higher FPDB moments, then the number,

weight, and higher averages of FPDB would be obtained as a

function of the chain length. For instance, the ratio Cn/Pn

represents the number average FPDB. However, one observes

the existence of the first moment Cn in the equations for the

zeroth moments of Pn and Rn, Equation (19) and (20). This

means that a closure problem exists in these moment

formulations. Obviously, the expressions for the higher

moments also possess this closure problem, since the

balances for the mth moment contain the unknown mþ 1th

1th moment term. Hence, we conclude that this pseudo-

distribution formulation cannot be solved withoutadditional

assumptions as those proposed by Hulburt and Katz.[14] The

existence of this closure problem is a logical consequence of

the fact that the number of FPDB on a chain is actively

involved in its reactivity.

dt¼

3.2. Taking Moments over the Chain Length

In view of the aforementioned closure problem, we

decided to solve differently formulated pseudo-distribution



rly View Publication; these are NOT the final pag

balances. One would not expect a closure problem, if

the moment over the chain length were taken instead of the

number of FPDB, since the former is not involved in the

chain reactivity. We thus formulate the pseudo-distribu-

tion balances as follows by taking the first three leading

moments over the chain length and express them in terms

of the number of FPDB distribution variable, i:

0th chain length moment:

I: 10.100

H & Co

e nu

dP1n¼1

Rn;0

dt¼ M1ðki1I� þ k0i1T�Þ � kf TR0 � kp2M2R0

þ ktcÞl00R0 þ kp � P1R0 � kp � m01R0 �1

tR0 ¼ 0

� �CSTR

(22)

dP1

n¼1Rn;1

dt¼ M2ðki2I� þ k0i1T� þ kp2R1Þ

� kp2M2R1 � kf TR1 � ðktd þ ktcÞl00R1

þ kp �X2

j¼0

jPjR2�j � kp � m01R1 �1

tR1 ¼ 0

� �CSTR

(23)

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dRdP1

Rn;i
i
dt¼

dPi

dt¼

þ

dF0

dt¼

dF1

dt¼

dFi

dt¼

dCi

dt¼

dV0

dt¼

dV1

dt¼

www.M

n¼1

dt¼ kp2M2ðRi�1 � RiÞ � kf TRi � ðktd þ ktcÞl00Ri

þkp �Xiþ1

j¼0

jPjRi�jþ1 � kp � m01Ri �1

tRi ¼ 0

� �CSTR

(24)

dP1n¼1

Pn;i

dt¼ ktdl00Ri þ 1

2ktc

Xi

j¼0

RjRi�j

kf TRi � kp � l00iPi �1

tPi ¼ 0

� �CSTR

(25)

1st chain length moment:

dFi

dt¼

dVi

dt¼

dP1n¼1

nRn;0

dt¼ M1ðki1I� þ k0i1T� þ kp1R0Þ � kf TF0

� kp2M2F0 � ðktd þ ktcÞl00F0 þ kp � ðC1R0 þ P1F0Þ

� kp � m01F0 �1

tF0 ¼ 0

� �CSTR

(26)

dP1

n¼1nRn;1

dt¼ M2ðki2I� þ k0i2T� þ kp2R1 þF1Þ

þkp1M1R1 � kp2M2F1 � kf TF1 � ðktd þ ktcÞl00F1

þkp �X2

j¼0

ðjCjR2�j þ jRjF2�jÞ

� kp � m01F1 �1

tF1 ¼ 0

� �CSTR

(27)

dP1

nR

dLi

dt¼

n¼1n;i

dt¼ kp2M2ðRi�1þFi�1 �FiÞ þ kp1M1Ri � kf TFi

�ðktd þ ktcÞl00Fi þ kp �Xiþ1

j¼0

jðCjRi�jþ1 þ PjFi�jþ1Þ

�kp � m01Fi �1

tFi ¼ 0

� �CSTR

(28)

dP1

n¼1nPn;i

dt¼ ktdl00Fi þ ktc

Xi

j¼0

RjFi�j þ kf TFi

� kp � l00iCi �1

tCi ¼ 0

� �CSTR

(29)

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2nd chain length moment

: 10.100

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the

dP1n¼1

n2Rn;0

dt¼ M1fki1I�k0i1T� þ kp1ðR0 þ 2F0Þg

� kf TV0 � kp2M2V0 � ðktd þ ktcÞl00V0

þ kp � ðL1R0 þ 2C1F0 þ P1V0Þ

� kp � m01V0 �1

tV0 ¼ 0

� �CSTR

(30)

dP1

n2R
n¼1
n;1

dt¼ kp1M1ðR1 þ 2F1Þ þM2ðki2I� þ k0i2T�Þ

þ kp2M2ðR0 þ 2F0 þV0 �V1Þ � kf TV1

� ðktd þ ktcÞl00V1þ kp �X2

j¼0

jðLjR2�j þ 2CjF2�j þ PjV2�jÞ

� kp � m01V1 �1

tV1 ¼ 0

� �CSTR

(31)

dP1n¼1

nRn;i

dt¼ kp2M2ðRi�1 þFi�1 �FiÞ þ kp1M1Ri

� kf TFi � ðktd þ ktcÞl00Fi

þ kp �Xiþ1

j¼0

jðCjRi�jþ1 þ PjFi�jþ1Þ

� kp � m01Fi �1

tFi ¼ 0

� �CSTR

(32)

dP1n¼1

nRn;i

dt¼ kp2M2ðRi�1 þ 2Fi�1 þVi�1 �ViÞ

þ kp1M1ðRi þ 2FiÞ � kf TVi � ðktd þ ktcÞl00Vi

þ kp �Xiþ1

j¼0

jðLjRi�jþ1 þ 2CjFi�jþ1 þ PjVi�jþ1Þþ

� kp � m01Vi �1

tVi ¼ 0

� �CSTR

(33)

dP1n¼1

n2Pn;i

dt¼ ktdl00Vi þ ktc

Xi

j¼0

ðRjVi�j þFjFi�jÞ

þ kf TVi � kp � l00iLi �1

tLi ¼ 0

� �CSTR

(34)

The definitions of all the chain length moments

are given at the LHS parts of these expressions. Note

that the minimum of the FPDB distribution variable i equals

2/mren.201200073

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0, denoting the sum of chains with various length and

zero FPDB. The minimum of the moment summations in the

previous equations is chain length n¼ i, since a chain with a

number of FPDB equal to i has minimum length i. Note that

the balance equations for i¼ 0 and 1 FPDB contain an

initiation term with I�; the former refers to the initiation of

radical chains with 0 FPDB by a reaction between I� and M1,

the latter to the initiation of chains with 1 FPDB, by a

reaction between I� and M2. The main assumption of the

scheme expressed by the system of equations presented is

that only FPDB on dead chains undergo crosslinking

reactions. This is obviously consistent with the underlying

assumption that chains with more than one radical need

not to be considered under pre-gel conditions. Finally, one

may see that this system of balance equations does not

contain closure problems and may readily be solved.

From the solution of the moment equations presented

above, we may infer the number and weight average chain

lengths as a function of the number of FPDB, i, by:

Dead chains

nnPi

nnRi

Pn;

rly V

¼ Ci

PinwP

i ¼Li

Ci(35)

Living chains

¼ Fi

RinwR

i ¼Vi

Fi(36)

An approximation of the chain length distribution at

each i-value may be constructed from these moments using

a two-parameter distribution (Iedema[15]):

i;Rn;i ¼ mP;R0 ðiÞ ð1� bP;RðiÞ

� �1þaP;RðiÞ

�n� 1þ aP;R

n� 1

!ðbP;RÞn�1

(37)

The two parameters in this distribution, a(i) and b(i), are

associated to the three moments as follows:

a ¼ 2m21 � m1m0 � m2m0

m0m2 � m21 � m1m0 þ m2

0

b ¼ m0m2 � m21 � m1m0 þ m2

0

m0m2 � m21

(38)

With

mP0;i ¼ Pi; mP

1;i ¼ Ci; mP2;i ¼ Li

mR0;i ¼ Ri; mR

1;i ¼ Fi; mR2;i ¼ Vi

(39)




Thus, we obtain approximations for the two-

dimensional distributions of chain length and number of

FPDB for dead and living chains, Pn,i and Rn,i.

3.3. Numerical Methods

The population balance equations for the batch reactor

have been solved using a direct method for up to i¼ 1 000

with the help of the stiff MATLAB ordinary differential

equation solver ode15s.m. For the CSTR, we employed a

Galerkin-hp method (e.g., PREDICI[16]) in a simple explicit

Euler scheme solving the equations in a dynamic form until

steady-state is achieved.

4. Monte Carlo Sampling

The Monte Carlo sampling scheme (Tobita et al.[3–7]) has

been employed here to generate a comparison basis for

the results of the pseudo-distribution approach described

in the previous paragraph. The algorithm for the batch

reactor is briefly summarized, followed by our extension to

the CSTR case. Note that all the variables employed in the

equations below follow by solving the balance equations of

Table 1. With respect to the balance equations of the radical

species, I� and l00, the quasi-steady state approximation

has to be applied.

4.1. Batch Reactor

As usual the basic building blocks of the Monte Carlo

scheme are the linear primary polymers being formed by

propagation and termination. This is considered as an

instantaneous process, taking place under the prevailing

conditions at a certain time instant in the batch reactor. For

the first primary polymer, both the time instant of its

creation and its length have to be determined by MC

sampling. The first should follow from a conversion (x)

‘‘intensity’’ distribution over the batch time:

I: 10.100

H & Co

e nu

dx

dt¼ kp1M1l00 þ kp2M2l00 þ kp � l00m01

M1ð0Þ þM2ð0Þ(40)

The denominator of RHS denotes the total monomer

concentration at t¼ 0. From the conversion rate, distribu-

tion of Equation (40) the ‘‘birth’’ time, u, is sampled. Now,

rather than using real time as the time variable, the

conversion time is employed to find the instants of creation

of primary polymers. For the first primary polymer, it

implies that its ‘‘conversion birth’’ time, u, is sampled by

selecting a random number between 0 and end conversion,

w. Furthermore, it requires expressing the rate intensity

distribution of the other reactions to be still discussed

as amount of production/consumption per conversion

2/mren.201200073




www.mre-journal.de

interval, dx, instead of amount per time interval, dt. The

relation between these is obviously given by Equation (40).

The length of the primary polymer is sampled from the

weighted length distribution of Flory shape, with nðuÞas the

conversion (time)-dependent number average primary

polymer length:

1

Figme‘‘threcis ‘

www.M

pwn ðuÞ ¼

n 1� 1=nðuÞf gn�1

nðuÞ2

nðuÞ ¼ ðkp1M1l00 þ kp2M2l00 þ kp � l00m01Þðktd þ ktcÞl00

2 þ kf Tl00

� � (41)

In the case of recombination termination, a primary

polymer may be connected with another primary polymer

with the same birth time. This happens with a probability Pc

that follows as the ratio of the rate of combination to the

overall termination rate (denominator of Equation (41)):

PcðuÞ ¼ktcl00

ðktd þ ktcÞl00 þ kf T� � (42)

If a connection occurs, the first sampled primary polymer

is connected to a second one, whose length is sampled from

the number distribution:

dra

d

pnnðuÞ ¼

1� 1=nðuÞf gn�1

nðuÞ (43)

The lengths are added and the two primary polymers,

having identical birth time, henceforth act as one in the MC

algorithm.

Now, the key feature of the Monte Carlo scheme for

crosslinking polymerization is, for the linear primary

polymers, to view the crosslinking process from two

perspectives, as shown in Figure 1. From the perspective

of the still growing primary polymer 1 an instantaneous

crosslink is formed by a propagation reaction with an

already formed FPDB at primary polymer 2. Since this

reaction is in competition with the propagation by reaction

with M1 and M2, the instantaneous crosslink fraction, ri,

equals the following ratio between instantaneous rates

2

ure 1. Instantaneous and additional crosslinking. Primary poly-r 1 gets an instantaneous crosslink as its radical site propagatesrough’’ an FPDB on primary polymer 2. Primary polymer 2eives an additional crosslink (arrow at RHS) as one of its FPDB‘consumed’’ by a growing radical chain.

aterialsViews.com




at conversion u (0< u<w, where w is the batch end

conversion):

: 10.100

H & Co

the

riðuÞ ¼kp � l00m01

ðkp1M1l00 þ kp2M2l00 þ kp � l00m01Þ(44)

Since all the concentrations in Equation (44) are functions

of conversion (time), ri is a function of conversion (time).

Thus, we see that the primary polymer formed at

conversion birth time u possessing instantaneous cross-

links right from the start and FPDB between u and batch

end conversion w may undergo a crosslinking reaction with

radical chains, thus producing additional crosslinks. The

fraction of divinyl inserted at u is given by a similar

expression as Equation (40):

F2ðuÞ ¼kp2M2l00

kp1M1l00 þ kp2M2l00 þ kp � l00m01

(45)

The part of this fraction F2(u) being converted into

crosslinks is called ‘‘additional crosslink density’’, ra. The

rate of this reaction is proportional to the decaying fraction

of unconverted FPDB as expressed by F2ðuÞ � raðu; tÞ. This

raðu; tÞ increases from 0 at t¼ u to raðu;’Þ at the end of

the batch, w, which is calculated by integration between u

and w of the differential equation:

ðu; tÞt¼ d F2ðuÞ � raðu; tÞf g

dt¼ kp � l00ðtÞ F2ðuÞ � raðu; tÞf g; raðu; uÞ ¼ 0

(46)

The actual numbers of instantaneous and additional

crosslinks then follow by a sampling procedure using riðuÞand raðu; tÞ, as calculated from Equation (44) and (46), and the

length n, sampled from Equation (41) and, evt., Equation (43).

The number of instantaneous crosslinks, ni, is directly

sampled from a binomial distribution, representing the

probability distribution of the number of instantaneous

crosslinks on a primary polymer of length n:

pðnijn; riÞ ¼nni

� �ri

nið1� riÞn�ni (47)

Sampling the number of additional crosslinks, na,

proceeds in two steps. First, the sum of unconverted and

converted FPDB on a chain of sampled length n, n2,

follows from F2(u) as given by Equation (45), also by

sampling from a binomial distribution:

pðn2jn; F2Þ ¼nn2

� �F2

n2ð1� F2Þn�n2 (48)

The sampling of the number of additional crosslinks,

na, proceeds by a binomial distribution, containing

2/mren.201200073

. KGaA, Weinheim7


8

REa

www.mre-journal.de


n2, and the fraction FPDB turned into crosslinks,

r0a¼ ra/F2(u):

rly V

pðnajn2; r0aÞ ¼

n2

na

� �ðr0aÞ

nað1� r0aÞn2�na (49)

The number of unconverted FPDB then finally becomes

i¼n2–na. Note that the total number of these FPDB on

all primary polymers in a molecule generated with MC

sampling represents the distribution variable in the

pseudo-distribution approach, i.

Once the numbers of instantaneous and additional

crosslinks has been determined, the primary polymer

containing them is known to be connected to the same

number of other primary polymers. The next step is then to

determine the conversion birth times of these primary

polymers. In the case of instantaneous crosslinking,

the sampled primary polymer becomes attached, at

conversion time u, to the FPDB on primary polymers that

were already present before that time, in the conversion

interval between 0 and u. Now, the probability to find it at

a certain conversion time depends on the distribution

over time of the production rate of FPDB, but – since the

FPDB should still be available at u – also on the consumption

rate of FPDB. This implies that the desired distribution over

time of the crosslinking rate rcl is given by:

rclðu; tÞ ¼ kp � l00ðtÞf1� r0aðu; tÞg (50)

The probability distribution of finding an FPDB created at

conversion birth time u in an infinitesimal conversion

interval Dz at z thus becomes:

piðzjuÞDz ¼ r0clðu; zÞDzR u

0 r0clðu; zÞdz(51)

0
Here, r cl is the conversion based rate, which is related to
the time based rate of Equation (50) by rcl¼ (dx/dt)r0cl,

where dx/dt is given by Equation (40).

In the case of additional crosslinking, the primary

polymer becomes attached to further primary polymers

that were reacting with the FPDB formed at u at the

conversion interval between u and w. Therefore, the birth

times of those primary polymers should again follow

from the crosslinking intensity distribution given by

Equation (50), but now in the interval between u and w:

paðujuÞDu ¼ r0clðuÞDuR f

ur0clðxÞdx

(52)

0

dra

dt

The conversion based rate rclðu;uÞ is again related to the

time based rate in Equation (50), rcl, by rcl¼ (dx/dt)r0cl,

where dx/dt is given by Equation (40).




Thus, the conversion birth times are determined of

the primary polymers instantaneously and additionally

becoming attached to the first primary polymer, in

generation 0. Then the algorithm continues with the

determination of the properties of these primary polymers

in generation 1: lengths and further connectivity. Note that

the lengths of the new primary polymers, both instanta-

neously and additionally crosslinked, are sampled from

the weighted distribution. As regards instantaneously

coupled chains, this is the case since they are connected

by a FPDB along their backbones and more FPDB are

found on longer backbones. This holds also for additionally

crosslinked primary polymers, since the probability of

being connected is proportional to their length. The

algorithm proceeds to determine the further connectivity

for every new primary polymer with identified length and

conversion birth time until no new primary polymers are

found.

A flow chart of the MC algorithm is shown in Figure 2.

4.2. CSTR

The sampling procedure for the CSTR in steady state

resembles that for the batch reactor. The main difference is

the fact that the instantaneous properties in the CSTR are

constant, whereas the properties of the primary polymers

depend on their residence time in the CSTR, t. In the batch

reactor, we had to sample the birth conversion time of

primary polymers, while in the CSTR, we need to sample

their residence time. Consequently, the rate expressions

that determine the distribution over residence time of the

crosslinking rate at individual primary polymers remain

time-dependent. We assume that the CSTR follows the ideal

exponential residence time distribution with average t.

The weighted length distribution of the primary polymer

is now Flory distributed around a constant number average

n and is given by exactly the same expressions as in

Equation (41) and, if recombination termination is occur-

ring, Equation (43). Similarly, the instantaneous cross-

linking density, ri, as well as the instantaneous fraction

divinyl incorporated in chains, F2, are given by the same

equations as for the batch case, Equation (44) and (45),

respectively, but now with constant concentrations. The

situation is different for the additional crosslinking, which

depends on the residence time, t, of the primary polymer in

the CSTR. Every primary polymer possesses the same

fraction F2 and FPDB upon its creation. However, a part of

these FPDB is converted into crosslinks and evidently this

part increases with residence time. Therefore, a similar

differential equation as for the batch reactor, Equation (46),

holds for this case:

I: 10.100

H & Co

e nu

¼ �dðF2 � raÞdt

¼ kp � l00ðF2 � raÞ; rað0Þ ¼ 0 (53)

2/mren.201200073



i+1

i+1

τi+1 > τi (batch)

τi+1 < τi (CSTR)

i+1

i+1

i+1

i+1

i+1

i

i

rand(1) < Pc

rand(1) > Pc

i+1

i+1

i

τi+1 < τi (batch)

τi+1 > τi (CSTR)

Sampling of:• Length pp’s gen. i, Eq. (41)• Connections, Pc, Eq. (42)• Nr of instantaneous cl’s, ni• Nr of FPDB, nFPDB• Birth times of new pp’s, τi+1

Sampling of:• Nr of additional cl’s, ni• Birth times of new pp’s, τi+1

Instantaneous cl loop Additional cl loop

nFPDB

stopni

na

= 0

> 0

= 0ni + na

Generation i = 01 pp

start

> 0

i = i + 1

Figure 2. Flow chart of the Monte Carlo simulations. It starts in generation i¼0 with one primary polymer (pp). In the instantaneouscrosslinking loop the numbers of FPDBs (nFPDB) and instantaneous crosslinks (ni) is determined. If nFPDB>0, the additional cl loop is entered,eventually producing na additional crosslinks. By sampling a random number, rand1, and requiring that it is< Pc, a recombination point isfound present (left hand structures). In subsequent cycles, the instantaneous loop is entered with niþ na pp0s. The algorithm stops if no newcrosslinks are formed.


www.mre-journal.de

Since F2 and l00 are constant in the CSTR this is a simple

first-order differential equation with an exponential

function in residence time, t, as the solution for ra:

www.M

raðtÞ ¼ F2 1� ekp�l00t� �

(54)

In order to determine the residence time for the first

sampled primary polymer, we select randomly from the

exponential residence distribution:

pðtÞ ¼ e�t=t (55)

piðz

Once the fraction FPDB, F2, and the instantaneous and

addition crosslink densities, ri and ra are known, the

actual numbers of crosslinks and unconverted FPDB are

obtained from the sampled primary polymer chain length

(Equation 41 and 43) and the binomial distributions as

given for the batch case, Equation (47)–(49).

For the new series of primary polymers connected to the

first one at the identical number of crosslink points, the

residence time has to be determined as well, in order to

find the additional crosslinking density, ra. As similar as

to the batch case this proceeds differently for primary

polymers connected by instantaneous crosslinks than

for those connected by additional crosslinks. An instanta-

neous crosslink forming at a primary polymer with

residence time t does so by reacting with an FPDB on

another primary polymer. The latter primary polymer

aterialsViews.com




exits the reactor at the same time as the first one, so the

residence time of the latter should be equal or longer than t.

Now, the probability of finding an unreacted FPDB on a

new chain in a certain residence time interval depends on:

a. T

: 10

H &

t

he residence time distribution of the chain,

Equation (55).

b. T
he decaying amount of FPDB on the chain. This is
expressed by Equation (53) describing the decay of the

fraction non-converted FPDB, F2–ra(t).

The probability distribution of finding a primary polymer

with residence time z is thus given by:

.100

Co

he

jtÞDz ¼ F2 � raðzÞf gexpð�z=tÞDzR1t

F2 � raðzÞf gexpð�z=tÞdz

¼ exp � kp � l00 þ1

t

� �zþ kp � l00t

� Dz

(56)

The second equality in Equation (51) easily follows from

the expression for ra of Equation (54).

If an additional crosslink is formed at a primary

polymer with residence time t by a reaction of a FPDB on

that chain with a newly growing radical chain, then the

latter chain must have been growing and reacting during

the residence time of the first chain, while it leaves at the

same time. This implies that the residence time of the

new primary polymer attached by an additional crosslink

2/mren.201200073

. KGaA, Weinheim9


0.8

1

1.2 x 10−3

Fraction divinyl in chains, F2 = 1.12 × 10-3

10

REa

www.mre-journal.de


should be between 0 and t. The probability distribution of

finding this residence time, u, 0<u� t, is again given by the

function F2–ra(t) of Equation (54), describing the decay of

the FPDB on the first chain during that interval. Thus,

we have, similar to Equation (56):

p

Tab

Rat

con

Ini

Ini

Vin

Div

Ma

Bat

Av

Con

Vin

Div

Cha

Dis

Rec

Cro

Av

len

A con

coeffic

0.2

0.4

0.6Additional crosslinking density, ρa(τ)

Instantaneous crosslinking density, ρi = 0.078 × 10-3

Frac

tion

rly V

aðujtÞDu ¼ F2 � raðuÞf gDuR t

0 F2 � raðuÞf gdu¼ expð�kp � l00uÞDu

1� expð�kp � l00tÞ(57)

10−1 100 101 102 1030

Residence time, τ (s)

Figure 3. Instantaneous and additional crosslinking density andfraction divinyl in polymer chains, CSTR case. Initiator radicalconcentration, I� ¼ 1.73� 10�9, conversion x¼0.429; further dataas in Table 2.

The CSTR algorithm proceeds, similarly to that for the

batch reactor, in determining the further connectivity for

every new primary polymer with identified length and

residence time until no new primary polymers are found.

The values of the divinyl content, F2 (Equation 45), and

the instantaneous and additional crosslinking densities

(Equation 44 and 54), are shown for a representative case

(data in Table 2) in Figure 3, the latter as a function of

residence time, t. In Figure 4 are shown the probability

distributions for by instantaneous and additional cross-

linking, according to Equation (56) and (57), to a primary

polymer of average residence time, t¼ t¼ 30 s. One may

observe, from the strongly declining distribution for

instantaneous crosslinking, that the residence times for

such chains are close to that of the primary polymer in the

earlier generation. In contrast, the distribution for addi-

le 2. Kinetic data and concentrations in batch reactor and CSTR.

e coefficient/species

centration

Batch

(MMA/EGDMA[10,11])

tiator, I2 0.0171

tiator radical, I�yl start/feed, M1(0), M1f 9.2553

inyl start/feed, M2(0), M2f 0.0530 (0.1 wt%)

croradical, l00

ch end time, tend

erage res. time CSTR, t

version, x

yl propagation, kp1 462

inyl propagation, kp2 689

in transfer, kf 0

proportionation, k1Þtd

1.054� 107

ombination, k1Þtc

1.013� 107

sslinking, kp� 232

erage primary polymer

gth (Equation 42)

1140–1180

version (x) dependent gel effect was modeled using the empiri

ients Ai (dependent on CTA-concentration) taken from ref.[11].




tional crosslinking is rather flat, implying a mild preference

for the smaller times.

5. Results

We started simulations to validate the pseudo-distribution

model and MC simulations with data for the crosslinking

Batch[3–7] CSTR Units

0.015

1.73� 10�9 kmole �m�3

9.9 9.995 kmole �m�3

0.1 0.005 kmole �m�3

5.45� 10�6 kmole �m�3

60 s

30 s

0.45 –

5000 5000 m3 � kmole�1 � s�1

10 000 10 000 m3 � kmole�1 � s�1

0 0

1.97� 106 108 m3 � kmole�1 � s�1

0 0

500 500 m3 � kmole�1 � s�1

140–320 2900 –

cal formula (Li et al.[11]): ktðxÞ ¼ ktð0Þexp � A1xþ A2x2 þ A3x3ð Þf g;

I: 10.1002/mren.201200073

H & Co. KGaA, Weinheim www.MaterialsViews.com

e numbers, use DOI for citation !!

Figure 4. Residence time probability distributions for primarypolymers in CSTR connected by instantaneous and additionalcrosslinks to a primary polymer with average residence timet ¼ t ¼ 30 s. Residence time of instantaneously coupled chains,u, between t and 1, but with higher probability close to t.Residence time probability of additionally coupled chains moreevenly distributed over time between 0 and t.


www.mre-journal.de

polymerization of MMA and EGDMA from Li et al.[11] and

Kizilel et al.[10] All the kinetic data, based on experiments,[11]

have been listed in Table 2. Note that a small gel effect

reducing the termination rate has been accounted for.[11]

The population size for the MC simulations was always

500 000, requiring typically between 1 min and 1 h CPU-

time. For a batch end time of 632 s and a conversion of

h¼ 0.054 and a weight percentage of divinyl (EGDMA)

of 0.1 the system is still in a sol regime. The rigorous

pseudo-distribution solution for the concentration distri-

bution of FPDBs is compared to MC simulation results

in Figure 5. The concentration is plotted as d log(w)/di,

which scales as i2Pi, to better highlight the tail of the

Figure 5. Distribution of number of FPDBs scaling with i2Pi (GPC-distriution) from MC simulations and pseudo-distributions, forbatch end time 632 s, conversion x¼0.054, 0.1 weight fractionEGDMA and kinetic data from Table 2.

www.MaterialsViews.com




distribution. In order to obtain the FPDB distribution, Pi,

from the MC simulations, the following procedure has

been applied. First, one should realize that the MC

simulations generate chain length weighted populations.

When collecting the numbers of molecules with a certain

number of FPDB in certain FPDB-bins, those numbers have

to be corrected for the number average weight of all the

molecules collected in a bin. This is achieved by construct-

ing the chain length distribution for the molecules of each

FPDB-bin. Regarding Figure 5, it is obvious that perfect

agreement exists between the new pseudo-distribution

model and the MC simulations. A second set of results is

obtained for a batch end time of 875 s and a conversion of

h¼ 0.075 (0.1 wt% EGDMA) and the resulting chain length

weight fraction distributions are shown in the double-log

plot of Figure 6. At these conditions our MC simulations

reveal a small gel weight fraction of 0.00019 (number of

molecules having more than 10 000 generations[3–7]), hence

we assume that the distribution shown represents weight

distribution at the gelpoint of the sol. The chain length

distribution based on the pseudo-distribution model is

approximated using the two-parameter distribution,

Equation (37–39). We compare the pseudo-distribution

(Piþ Ri) and MC results at conversion h¼ 0.075 to a

distribution obtained by NF by Kizilel et al.,[10] also at the

gelpoint. However, latter authors find the gelpoint at

h¼ 0.09. There is global agreement between the three

curves, but more closely regarding one observes small

differences. The pseudo-distribution runs until n � 106

since the maximum number of FPDB in the model was set

to 2� 104. At high n the pseudo-distribution curve is

slightly higher than the MC. This is probably due to the

fact that the pseudo-distribution model does not account

for multiradicals that may become important already near

Figure 6. Weight fraction distribution of chain lengths, n, forbatch end time 875 s, conversion x¼0.075, 0.1 weight fractionEGDMA and further kinetic data from Table 2. From pseudo-distributions, MC simulations and NF-method.[10] At the tail of thedistributions small discrepancies are seen.

: 10.1002/mren.201200073

H & Co. KGaA, Weinheim11


10-3 10-2

10-1 100

101 102

100 101

102 Number of FPDB, i

x 10-6

0

2

4

6

Con

cent

ratio

n, R

i( t) (km

ole/

m3 )

Figure 8. Development of living chain FPDB distribution over timeas computed for the batch reactor from the pseudo-distributionmodel. Kinetic data according to Table 2 and Figure 7.

100 101

102

100

0.03

Con

cent

ratio

n, P

i(t) (km

ole/

m3 )

0.02

0.01

0

12

REa

www.mre-journal.de


the gel-regime. Apart from predicting the gelpoint at a

slightly higher conversion, h¼ 0.09 than the MC simula-

tions (�0.075), the NF-model shows higher concentrations

in the tail of the distribution. These differences may be

attributed to the inaccuracy introduced by taking moments

in the NF-method and/or to multi-radicals, also not taken

into account in the NF-method. Summarizing this part we

conclude that good agreement exists between the earlier

results, based on experiments,[11] and our models.

Using the same kinetic parameter set as Tobita et al.[3–7]

(see Table 2 and caption of Figure 7) further calculations

were made for the batch reactor. The results from both

the pseudo-distribution model and the Monte Carlo

simulations are presented in Figures 7 through 12.

Figure 7 presents the time profiles of the monomer

concentrations, M1 and M2, the total radical concentration,

l00, and the total number of FPDB, m01. The development

over time of the FPDB distributions for living and dead

chains is shown in Figure 8 and 9. The former plot, Ri,

reveals a relatively narrow distribution at short times that

gradually changes and then broadens after appreciable

conversion. It remains at a relatively high concentration

over the whole time interval, which reflects the profile of

l00 in Figure 7. The plot of the dead chains, Pi, is more

gradually increasing over time, as is the case with the

overall FPDB in Figure 7.

Figure 10 and 11 show good agreement between the

number of FPDB distributions from Monte Carlo simula-

tions and pseudo-distributions approach. In the double-log

0

5

10

0

0.05

0.1

0

5 x 10−6

10−3 10−2 10−1 100 1010

0.05

0.1

M1

M2

λ00 (radical concentration)

μ01 (number of FPDB)

kmole/m3

time, t (s)

Figure 7. Development of various overall concentrations overtime in the batch reactor. M1(0)¼9.9; M2(0)¼0.1; I2(0)¼0.015;ktd¼ 108 m3/(kmole � s); kd¼0.05 1/s; final conversion x¼0.454;further data according to Table 2.

10-3 10-2

10-1 101

102 Number of FPDB, i

Figure 9. Development of dead chain FPDB distribution over timeas computed for the batch reactor from the pseudo-distributionmodel. Kinetic data according to Table 2 and Figure 7.

0 50 100 150

10−6

10−4

10−2

100

Number of Free Pending Double Bonds

Rel

ativ

e co

ncen

tratio

n

Monte Carlo

Deterministic solution

Figure 10. Relative concentration distribution of FPDB accordingto deterministic model (Equation 22–34) at batch end time andMonte Carlo simulation (500 000 molecules) for the batch reac-tor. Kinetic data according to Table 2 and Figure 7.

Macromol. React. Eng. 2013, DOI: 10.1002/mren.201200073

� 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.MaterialsViews.com

rly View Publication; these are NOT the final page numbers, use DOI for citation !!

100 101 102

102

103

104

Number of Free Pending Double Bonds, i

Ave

rage

cha

in le

ngth

=

=

Monte Carlo

Deterministic solutions

Λψ

ψ

Figure 11. Number and weight average chain length as a functionof FPDB from deterministic pseudo-distribution model (Equation22–34) and Monte Carlo simulations for batch reactor (500 000molecules). Kinetic data according to Table 2 and Figure 7.

100 101 102 103 104 10510−15

10−10

10−5

100

Number of Free Pending Double Bonds, i

Rel

ativ

e co

ncen

tratio

n

Monte Carlo Deterministic solution

Figure 13. Relative concentration distribution of FPDB accordingto deterministic model (Equation 22–34) and Monte Carlo simu-lation (12 500 000 molecules) for CSTR. Kinetic data according toTable 2.


www.mre-journal.de

plot of Figure 11 the values of the average chain lengths

for i¼ 0 FPDB is not shown, but they also agree. The value

of the number average length is 50, which is equal to

the ratio of kp1 M1, and kp2 M2, which is consistent with the

fact that chains start to grow by propagation with on

average 50 steps with vinyl monomer before inserting a

divinyl. Figure 12 shows that even for an approximation of

the chain length distribution by three moments a good

agreement is achieved with the rigorous, though scattered

Monte Carlo result.

The results for the CSTR case from both the pseudo-

distribution model and the Monte Carlo simulations are

presented in Figure 13 and 14. The kinetic data used have

been listed in Table 2. They differ from the batch data as

101 102 103 1040

0.2

0.4

0.6

0.8

1

Chain length, n

dlog

(w)/d

n

Monte Carlo

Deterministic solution

Figure 12. Chain length distribution for batch reactor from deter-ministic pseudo-distribution model (Equation 22–34) and two-parameter distribution (Equation 37–39) and Monte Carlo simu-lations (500 000 molecules). Kinetic data according to Table 2 andFigure 7.





regards the disproportionation coefficient (lower) and the

divinyl fraction (also lower). Also the Monte Carlo sample

taken is much larger: 12 500 000 molecules, resulting in

fairly smooth distributions. The FPDB distributions shown

in the double-log plot of Figure 13 reveal good agreement,

but also undeniable differences for molecules over 100

FPDB. The distribution from the pseudo-distribution model

falls sharply for i> 1000 FPDB, whereas the MC distribution

continues to decrease at constant slope. Furthermore, the

deterministically obtained distribution features a small

upward deviation from the constant slope from MC directly

before the sharp decline. Since the conditions chosen in this

CSTR case are much closer to the gel regime than in the

batch case, we tend to believe that the special shape of the

deterministic curve over is an artifact due to the neglecting

the multiradicals. Since the MC method implicitly takes

102 103 104 105 106 1070

0.2

0.4

0.6

0.8

Chain length, n

dlog

(w)/d

n

Monte Carlo Deterministic solution

Figure 14. Chain length distribution for CSTR from deterministicpseudo-distribution model (Equation 22–34) and two-parameterdistribution (Equation 37–39) and Monte Carlo simulations(12 500 000 molecules). Kinetic data according to Table 2.

: 10.1002/mren.201200073

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multiradicals into account, it is expected that a straight

form of the FPDB distribution over 100 FPDB is closer to the

truth. We will discuss this phenomenon in a publication to

follow, where the number of radical sites is explicitly taken

into account as an additional dimension. Figure 14 shows

the molecular size distribution directly obtained from the

MC simulations and indirectly, using the two-parameter

distribution shape approximation after Equation (37)–(39),

from the pseudo-distribution approach. Being represented

as weighted distributions, in terms of n2Pn, the distribu-

tions feature quite good although no perfect agreement.

The aforementioned reasons may cause the small dis-

crepancy, apart from the approximate character of the

deterministic distribution.

6. Conclusion

A new pseudo-distribution approach has been applied to

the modeling problem of crosslinking copolymerization of

vinyl and divinyl monomer, both for batch and continuous

reactors. The usual formulation of the pseudo-distribution

balance equations in terms of molecular size, while taking

the moment over the number of FPDBs as the second

distribution variable, was shown to lead to a closure

problem. This is caused by the fact that it is rather the

number of FPDB that determines the reactivity of a

molecule than its number of monomer units. Therefore,

it was decided to construct a pseudo-distribution model

with the number of FPDB as the main distribution variable

and taking moments over the chain length distribution.

This model does not contain a closure problem and it allows

a fully rigorous solution of the three leading molecular size

moments. A Monte Carlo simulation model developed by

Tobita et al.[3–7] has been employed as a reference for our

model. This MC model is based on primary polymers that

are crosslinked by a reaction between the radical sites on

one molecule and the FPDB on another. It utilizes concepts

of instantaneous and addition crosslinking to connect the

primary polymers.

The pseudo-distribution model has been validated with,

in part experimental, data from literature[10,11] for the

crosslinking copolymerization of MMA with EGDMA in a

Nomenclature

(Note that the terms ‘‘chain length’’, ‘‘number of monomer un

Symbol Description

F Volumetric flow rate

F2 Fraction divinyl monomer in primary polym

I2; I� Concentration Initiator, Initiator radical




batch reactor. Good agreement with the weight fraction

distribution obtained by NF[10] was observed. Under

conditions near the gelpoint a small discrepancy was

found at longer chain lengths between the deterministic

methods (NF and pseudo-distribution) on the one hand and

the MC simulations at the other. This may be attributed to

the fact that multiradicals in the deterministic methods are

not taken into account, whereas in the MC simulations they

(implicitly) are. In the case of a batch reactor under non-

gelling conditions and relatively short primary polymer

lengths (�140), perfect agreement was found (Figure 9

and 11) between the model outcomes for the leading

moments (FPDB distributions) and those from the Monte

Carlo simulation model. Even a complete molecular size

distribution obtained by an assumed two-parameter shape

distribution based on the three leading moment features

very good agreement with the MC simulations (Figure 12).

In the CSTR case, we again ventured closer to a gel-

forming regime by choosing different conditions like a

longer linear chain length. The original pseudo-distribution

model, Equation (22)–(34), was formulated for the living

and dead molecules only, as usual in radical polymerization

modeling. The outcomes from this model under these CSTR

conditions have been compared with a MC simulation

model that we developed for this purpose. This MC model is

based on the same principles as the original batch

version,[3–7] but it allows for residence time dependent

properties of the linear primary polymers, assuming the

usual exponential residence time distribution shape of a

CSTR. Comparing the results of the deterministic and MC

model showed fair but not perfect agreement (Figure 13

and 14). Again, one should realize that the MC simulation

implicitly takes multiradicals into account, whereas the

pseudo-distribution model does not. From these results, we

conclude that the pseudo-distribution model is well

applicable within the sol regime. The small differences

observed under conditions near the gelpoint, however,

should be explored further with respect to the role of

multiradicals. This is especially so, if one desires the models

to be valid until or even beyond the gelpoint. In a future

publication, we will address the multiradical issue in

population balance modeling by explicitly taking this into

account as an additional dimension.

its’’, and ‘‘molecular size’’ have been used as synonyms.)

Units Location

m3 � s�1 Par. 2

er – Equation (45)

kmol �m�3 1

I: 10.1002/mren.201200073



Table . (Continued)

Symbol Description Units Location

M1, M2 Concentration Monovinyl, Divinyl monomer kmol �m�3 Par. 2

Pc Recombination probability primary polymer – Equation (42)

Rn,i,k,p Concentration of polymer species with length n,

number of FPDB i, number of crosslinking points

k, number of radical sites p

kmol �m�3 Par. 2

Rn,i,Pn,i Concentration of living and dead chains with

length n, number of FPDB i

kmol �m�3 Equation (15) and (16)

Rn,Pn Concentration of living and dead chains with

length n

kmol/m�3 Equation (19)–(21)

Ri,Pi Concentration of living and dead chains with i

FPDB

kmol �m�3 Equation (22)–(34)

T; T� Concentration CTA, CTA radical kmol �m�3 7

f Functionality –

fi, fi0 Number fraction fragments with i connection

(interchange reaction) points

Table A1

kd Initiator decomposition rate coefficient s�1 Equation (1)

kf Chain Transfer to CTA rate coefficient m3 � (kmol�1 � s�1) Equation (6)

ki1, ki2 Monovinyl, divinyl initiation rate coefficient

(by I�)

m3 � (kmol�1 � s�1) Equation (2) and (3)

ki10, ki2

0 Monovinyl, divinyl initiation rate coefficient

(by T�)

m3/(kmol�1.s�1) Equation (7) and (8)

kp1, kp2 Monovinyl, divinyl propagation rate coefficient m3 � (kmol�1.s�1) Equation (4) (5)

kp� Crosslinking rate coefficient m3 � (kmol�1 � s�1) Equation (9)

ktd Disproportionation termination rate coefficient m3 � (kmol�1 � s�1) Equation (10)

ktc Recombination termination rate coefficient m3 � (kmol�1 � s�1) Equation (11)

nnPi ;nwP

i

nnRi ;nwR

i

Number and weight average chain lengths as a

function of number of FPDB.

Equation (35) and (36)

nðuÞ Number average length primary polymer – Equation (41)

ni, na Number of instantaneous and additional

crosslinks

Equation (47) and (49)

pw (u), pw (u) Number and weighted length distribution

primary polymer

– Equation (41) and (43)

pi, pa Probability distributions of birth times – Equation (51), (52), (56), and (57)

rcl, r0cl Crosslinking rate (conversion-based) Equation (50)–(52)

t Time s

u Conversion birth time – Equation (52)

x Conversion – Equation (40)

z Conversion birth time – Equation (51)


Macromol. React. Eng. 2013, DOI: 10.1002/mren.201200073

� 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim15

Early View Publication; these are NOT the final page numbers, use DOI for citation !! R


www.mre-journal.de

Greek letters

Symbol Description Units Location

Li 2nd chain length moment dead chains

(i number of FPDB)

m3 � (kmol�1 � s�1) Equation (22)–(34)

Fi 1st chain length moment living chains

(i number of FPDB)

m3 � (kmol�1 � s�1) Equation (22)–(34)

Cn 1st FPDB moment dead chains

(n chain length)

m3 � (kmol�1 � s�1) Equation (19)–(21)

Ci 1st chain length moment dead chains

(i number of FPDB)

m3 � (kmol�1 � s�1) Equation 22–34

Vi 2nd chain length moment living chains

(i number of FPDB)

m3 � (kmol�1 � s�1) Equation (22)–(34)

a(i), b(i) Parameters in two-parameter distribution – Equation (37)–(39)

l0100, l0001 Moments in 4D population balance

(l0100 number of FPDB)


m01, l00 Moments in 2D population balance

(m01 number of FPDB)


mP0;i;m

P1;i;m

P2;i

mR0;i:m

R1;i;m

R2;i

Moments used in two-parameter distribution kmol �m�3 Equation (37)–(39)

u Conversion birth time primary polymer – Par. 4.1

w End conversion batch reactor – Par. 4.1

ri, ra Instantaneous and additional crosslink fraction – Equation (44)–(46)

t, t Residence time CSTR (average) s Population balances

x Number of crosslinks Table 1

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Received: October 23, 2012; Revised: November 29, 2012;Published online: DOI: 10.1002/mren.201200073

Keywords: copolymerization; crosslinking; modeling; molarmass distribution; Monte Carlo simulation

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[2] S. P. Zhu, A. E. Hamielec, Makromol. Chem. Macromol. Symp.1992, 63, 135.

[3] H. Tobita, A. E. Hamielec, Macromolecules 1989, 22, 3098.[4] H. Tobita, A. E. Hamielec, Polymer 1990, 31, 1546.[5] H. Tobita, Macromolecules 1993, 26, 836.




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19, 323.[10] S. Kizilel, G. Papavasiliou, J. Gossage, F. Teymour, Macromol.

React. Eng. 2007, 1, 587.[11] W. H. Li, A. E. Hamielec, C. M. Crowe, Polymer 1989, 30, 1518.[12] P. D. Iedema, M. Wulkow, H. C. J. Hoefsloot, Macromolecules

2000, 33, 7173.[13] P. D. Iedema, H. C. J. Hoefsloot, Macromol. Theory Simul. 2001,

10, 855.[14] H. M. Hulburt, S. Katz, Chem. Eng. 1964, 19, 555.[15] P. D. Iedema, Macromol. Theory Simul. 2012, 21, 166.[16] M. Wulkow, Macromol. Theory Simul. 1996, 5, 393.

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