simulation of particle growth in the dispersion polymerization of styrene: the termination rate...

54

Simulation of Particle Growth in the Dispersion

Polymerization of Styrene: The Termination Rate

Constant in Particles

Masahiro Yasuda,* Hideki Yokoyama, Hidetoshi Seki, Hiroyasu Ogino, Kosaku Ishimi, Haruo Ishikawa

Department of Chemical Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka 599-8531, JapanFax: +81-722-54-9911; E-mail: [email protected]

Introduction

Monodisperse polymer particles of micron-size are used

as standard particles for calibrating instruments, spacers

of liquid-crystal panels, carrier particles for liquid chro-

matography columns, and particles for biomedical ana-

lyses.[1, 2] To obtain monodisperse particles whose di-

ameter is 1–50 lm is difficult by conventional methods

such as emulsion polymerization and suspension poly-

merization. Therefore, the seeding method,[3] the two-step

swelling method,[4] and dispersion polymerization[5, 6] has

been developed to obtain such monodisperse particles of

micron-size.

Since monodisperse polymer particles whose diameter

is 1–20 lm can be synthesized in a single-step, many

researchers have been interested in dispersion polymeri-

zation. In dispersion polymerization, particles are formed

in a reaction mixture which is initially homogenous in the

presence of a suitable steric stabilizer polymer. Poly(di-

methylsiloxane), polyisobutylene, poly(12-hydroxysteric

acid), and poly(2-ethylhexyl methacrylate) were

employed as the stabilizers for the polymerization of

methyl methacrylate.[7] Styrene was polymerized in alco-

hols in the presence of steric stabilizers such as hydroxy-

propyl cellulose, poly(acrylic acid), or poly(N-vinylpyr-

rolidone) (PVP).[7] When the polymerization conditions,

such as temperature, agitation speed, solvent type, mono-

mer composition and concentration, initiator type and

concentration, type and concentration of steric stabilizer

polymer, are favorable, the monodisperse particles can be

obtained. To synthesize the monodisperse particles it is

necessary to clarify the mechanism of the dispersion

polymerization. To control the particle diameter and

attain a narrow particle diameter distribution, several

studies concerning the effects of polymerization para-

meter were performed.[5, 6, 8–15] However, the mechanism

of dispersion polymerization is still not well understood.

Full Paper: A model is proposed for simulating the parti-cle growth in the dispersion polymerization of styrene inethanol. The model is based on the following assump-tions: (i) the termination reaction in an ethanol-phase andthe chain-transfer reactions in the ethanol-phase and parti-cles can be neglected, (ii) the mean volume of the radicalscaptured by particles is approximately equivalent to thatof monomeric radicals, and (iii) the termination rate con-stant in particles is bgel times that of the ethanol-phase.The experimental results of the conversion, the particlediameter and the particle number measured at the reactiontime of 2 h were used to determine the initial conditions.When the termination rate constant in particles was takento be about 1/130 of that of the ethanol phase, the calcula-tion results of the conversion and the particle diameterwere in good agreement with the experimental data.

Macromol. Theory Simul. 2001, 10, No. 1 i WILEY-VCH Verlag GmbH, D-69451 Weinheim 2001 1022-1344/2001/0101–0054$17.50+.50/0

Time course of the particle diameter.

Macromol. Theory Simul. 2001, 10, 54–62

Simulation of Particle Growth in the Dispersion Polymerization of Styrene ... 55

Teseng et al.[5] described qualitatively particle forma-

tion and growth in dispersion polymerization. According

to them, the reaction mixture is homogeneous at the start

of polymerization. When the reaction mixture is heated,

free radicals are formed by initiator decomposition and

grow in the continuous phase. Free radicals which attain

a sufficiently high degree of polymerization precipitate

and the stabilizer adsorbs on the resulting particles to

form stable particles. Once particles are formed, they

absorb the monomer from the continuous phase. After a

sufficient number of particles are formed, polymerization

mainly takes place within the monomer-swollen particles

until all of the monomer is consumed.

There are several models of dispersion polymerization.

It is convenient to divide the whole process into two

major stages, that is, an early stage (particle formation

stage) in which the formation of particles or nuclei and

aggregation between them are predominant and the latter

stage (particle growth stage) in which the particle growth

is predominant.

Paine[7] reported that an in situ stabilizer which is pro-

duced by the chain-transfer reaction of a radical with a

stabilizer molecule plays a critical role in the particle for-

mation stage. He developed a multibin kinetic model for

the aggregation of precipitated radicals or unstabilized

particles. It was assumed that stabilized particles of which

all the surface is covered with the in situ stabilizer mole-

cules do not aggregate each other. According to the

model, when particles which can capture radicals in the

continuous phase are sufficiently stabilized, no new

nuclei or particles are formed. This was the first model

that simulated quantitatively the role of the stabilizer

molecules in the particle formation stage. However, it is

still not sufficient for the particle growth stage because

the prediction of the particle diameter requires the experi-

mental time course of the conversion.

After the particle formation, there exist two places in

which polymerization proceeds, that is, polymer particles

and a continuous phase. A large part of monomer and

initiator exists in the continuous phase. Almost all the

initiation and the propagation reactions of oligomer radi-

cals take place in the continuous phase. The propagation

and the termination reactions of the polymer radicals

mainly take place in the particles when the number of sta-

bilized particles capturing oligomer radicals in the contin-

uous phase is sufficient. Lu et al.[16] studied the monomer

partition behavior in the dispersion polymerization of

styrene in ethanol. The volume fraction of polystyrene in

the particles was high from an early stage to the final

stage of the polymerization. Since the particles containing

a high volume fraction of polystyrene are viscous, the ter-

mination rate in polymer particles decreased due to the

gel effect. The termination rate constant in particles is

regarded to be smaller than that in the continuous phase.

Using the thermodynamic model of the monomer parti-

tion of the dispersion polymerization, Lu et al.[16] devel-

oped a kinetic model based on the assumption that the

stabilized particles are formed by precipitation of oligo-

mers. However, their model also cannot simulate the time

course of the particle diameter.

There are several models of dispersion polymerization

as mentioned above, but no model can simulate the whole

mechanism of the dispersion polymerization. This is

because the particle formation mechanism of the disper-

sion polymerization is too complicated. Therefore, as a

first step to develop a model which can simulate the

whole process of the dispersion polymerization, we pro-

pose a simple model for the particle growth stage in the

dispersion polymerization. Using this model, we simulate

the time courses of the particle diameter and the conver-

sion.

A Model for Polymer Particle Growth

The system under consideration consists of mature poly-

mer particles and an ethanol phase. The diameter of the

mature polymer particles are larger than 1 lm. The etha-

nol phase contains monomer and initiator. Our model was

based on the following three assumptions: (i) the termina-

tion reaction in the ethanol phase and the chain-transfer

reactions in the ethanol phase and particles could be

neglected, (ii) the mean volume of the radicals captured

by the particles was approximately equivalent to that of

the monomeric radicals, and (iii) the termination rate con-

stant in the particles was bgel times that of the ethanol

phase.

Number of Radical Molecules in Particles

At a steady state, the overall mass balance equation or the

equation for determining the number of radicals in the

ethanol phase is given by

rate qi of radical production

in the ethanol phase

� �

ÿ total rate JR of radical capture

by preexisting polymer particles

� �

ÿ rate of radical termination

in the ethanol phase

� �¼ 0 ð1Þ

When Fick’s first law of diffusion is applied, the total

diffusion rate JR of radical molecules in the ethanol phase

or the radical entry rate into particles is given by:

JR ¼ kRe p d2p NpðCRe ÿ CRsÞ ð2Þ

where Np is the total number of particles, dp the particle

diameter, kRe the mass transfer coefficient, CRe the molar

56 M. Yasuda, H. Yokoyama, H. Seki, H. Ogino, K. Ishimi, H. Ishikawa

concentration of the radical molecules in the ethanol

phase, and CRs the molar concentration of the radical

molecules at the particle surface. Since the particle di-

ameter is quite small, the following relation can be used

for estimating the mass transfer coefficient kRe .

Sh ¼ kRe dp

DRe

¼ 2 ð3Þ

where Sh is the dimensionless parameter concerning mass

transfer and DRe the diffusion coefficient of the radical

molecules in the ethanol phase. Using Equation (3),

Equation (2) is rewritten as:

JR ¼ 2 p dp Np DReðCRe ÿ CReÞ ð4Þ

CRe is regarded to be much larger than CRs. Equation (4)

reduces to:

JR ¼ 2 p dp NpDReCRe ð5Þ

The system under consideration consists of mature

polymer particles of which the particle diameter is larger

than 1 lm and the particle number is smaller than 1016

particles/m3. In such a system, the total rate of radical

entry into the particles is much greater than that of the

typical emulsion polymerization, in which the particle

diameter is 50–300 nm[17] and the particle number is

between 1019–1021 particles/m3.[18] Therefore, in the pre-

sent system, the average number np of radicals in a parti-

cle is much greater than 0.5 and the termination in parti-

cles is dominant. The termination of oligomer radicals in

the ethanol phase can be neglected, indicating that the

third term in the left hand side of Equation (1) can be

neglected. Therefore, the overall rate JR of radical capture

by preexisting polymer particles can be equated to the

rate qi of radical production in the ethanol phase as:

JR X qi ¼ 2f kd ½I�Ve ð6Þ

where f is the initiation efficiency, kd the initiator decom-

position rate constant in the ethanol phase, [I] the initiator

concentration and Ve the volume of the ethanol phase.

In dispersion polymerization, the unstabilized nuclei

aggregate until a sufficient amount of stabilizers are

adsorbed onto their surfaces. Therefore, the diameter of

the resulting mature particles is large and each polymer

particle contains some growing radicals. The change of the

number np of radicals in a particle is given by Equation (7).

dnp

dt¼ JR ÿ ktp

np=NA

Vp

� �2

Vpt Np

( )NA=Np ð7Þ

where ktp is the termination rate constant in particles, Vp

the volume of a polymer particle, Vpt the total volume of

polymer particles, and NA the Avogadro’s number.

Termination Rate Constant in the Particles

As mentioned above, there is a significant gel effect in

the particles.[16] Therefore, the termination rate constant

in the particles is probably smaller than that in the ethanol

phase. Since the volume fraction of polystyrene in the

particles is high from the early stage of polymerization,[16]

the termination rate constant in the particles is regarded

to be bgel times that of the ethanol phase. Therefore, the

termination rate constant ktp in the particles is expressed

by Equation (8).

ktp ¼ bgel kt ð8Þ

Total Volume of Polymer Particles

The total volume Vpt of polymer particles is related to the

polymer volume Vpp , the monomer volume Vpm , and the

ethanol volume Vpe in a particle as follows:

Vpt ¼ Np Vp ¼ NpðVpp þ Vpm þ VpeÞ ð9Þ

Vpm and Vpe are estimated using a thermodynamic

model of the dispersion polymerization of styrene in etha-

nol.[16] The change in the polymer volume of a particle is

caused by the propagation reaction in a particle and the

entry of growing radicals into a polymer particle. In gen-

eral, the volume of a monomer unit in a polymer mole-

cule can be equated to that of the monomer molecule.

Using the molar volume Um of the monomer and the aver-

age degree of polymerization je of the radicals in the etha-

nol-phase, the growth rate of the polymer volume in a

particle is given by Equation (10).

dVpp

dt¼ kp ½M�p

np=NA

Vp

� �VpUm þ

JRUm je

Np

ð10Þ

where kp is the propagation rate constant in the particles.

Since the radical entry rate is greater than the propagation

rate in the ethanol phase, the average degree of polymeri-

zation je of radicals captured by the polymer particles is

assumed to be 1, that is, the radicals captured by the poly-

mer particles are monomeric. The particle diameter dp is

calculated from Vpt as

dp ¼6Vpt

pNp

� �1=3

ð11Þ

Monomer Concentration in the Ethanol Phase and

Particles

Monomer is consumed by the propagation reaction in the

ethanol phase and polymer particles. Since the radical

entry rate is greater than the propagation rate in the etha-

nol phase, almost all the monomer is consumed by the


propagation reaction in the polymer particles. The con-

sumption rate of monomer is given by Equation (12).

d½M�tdt

¼ ÿkp ½M�pnp=NA

Vp

� �ðVp=VÞNp ð12Þ

The total monomer concentration [M]t is related to

[M]p and [M]e , the monomer concentrations in the parti-

cles and ethanol phase, respectively, by the following

mass balance equation:

½M�tV ¼ ½M�pVpt þ ½M�eVe ð13Þ

where V is the volume of the present system. The mono-

mer concentration [M]p in the particles can be estimated

using the thermodynamic model of the dispersion poly-

merization of styrene in ethanol.[15]

The conversion of monomer is given by Equation (14).

X ¼ ð½M�t0 ÿ ½M�tÞ=ð½M�t0 ð14Þ

where [M]t0 is the monomer concentration at the start of

polymerization.

Experimental Part

Materials

Styrene and 2,29-azoisobutyronitrile (AIBN) were purchasedfrom Wako Pure Chemicals Co. Ltd. (Osaka, Japan). Styrenewas washed with a 10% sodium hydroxide solution andpassed through a column packed with activated aluminumoxide to remove an inhibitor before use. AIBN was recrystal-lized twice in methanol. PVP with a nominal molecular massof 40000, cetyl alcohol, ethanol and tetrahydrofuran (THF)were purchased from Nacalai Tesque (Kyoto, Japan). Thesereagents were used without purification.

Measurements

The molecular masses of the polymers were measured usinga Shimadzu liquid chromatograph LC-5A (Shimadzu, Kyoto,Japan) with a Wakobeads G-40 column (Wako; 7.8 mm indiameter and 300 mm in length) using THF as a carrierliquid (8.3610–9 m3 N s–1). A Shimadzu SPD-2A UV detec-tor was calibrated at 265 nm using the polystyrene molecularmass standards from GL-Science (Tokyo, Japan) and wasused to measure the polymer concentration. The conversionwas measured by a gravimetric method. Particle diameterdistribution was measured using a laser particle diameteranalyzer MICROTRAC FRA (LEEDS & NORTHRUP, Sum-neytown Pike, USA). Scanning electron micrographs weretaken using a HITACHI S-2150 SEM (Tokyo, Japan). Theparticle number was counted using a Burker Turk hemacyto-meter (Elma, Tokyo, Japan).

Synthesis of Particles

To study the experimental particle growth and to use it forsimulation, the dispersion polymerization of styrene in etha-

nol was performed in a glass batch reactor. The standardrecipe is shown in Table 1. A clean and dry 300 ml separa-ble-flask was charged with ethanol, PVP, and cetyl alcohol,and then covered with a 3-neck separable-cover attachedwith a Dimroth condenser. The flask was heated to 708Cwith mild shaking for 20 min. AIBN was weighed and thenput into styrene and the resulting solution was quicklypoured into the flask. The reaction was proceeded for 24 hwith agitation at 30 rpm. Then, the flask was cooled in aniced-bath and the reaction mixture was transferred to four 50ml test tubes. The particles were washed by repeated centri-fugation and suspension in ethanol and distilled water.

Results and Discussion

Dispersion Polymerization

To solve Equation (7), (10) and (12) simultaneously, three

initial conditions are required, that is, the number np of

radical molecules in a particle, the total monomer con-

centration [M]t and the particle volume Vpt at a specified

time. In the present study we decided to use these experi-

mental data as initial conditions. To determine the initial

conditions and to test the validity of the present model,

the experiments of the dispersion polymerization of sty-

rene in ethanol were performed in an isothermal batch

reactor. To compare the polymerization rate with that in

the dispersion polymerization, the solution polymeriza-

tion of styrene in cyclohexane was also carried out under

the same conditions as the dispersion polymerization

except that the stabilizer was not used.

Figure 1 compares the experimental time course of the

conversion in the dispersion polymerization of styrene in

ethanol with that of the solution polymerization of sty-

rene in cyclohexane at 708C. The rate of the dispersion

polymerization is higher than that of the solution poly-

merization, indicating that there is a significant gel effect

in the dispersion polymerization. To further investigate

the gel effect in the dispersion polymerization, the molec-

ular mass distribution in the dispersion polymerization

was measured as a function of the reaction time. As

shown in Figure 2, the experimental mass average molec-

ular mass increases with the increase in the reaction time.

The mass average molecular mass in the solution poly-

merization is almost the same as that of the dispersion

Table 1. Standard recipe for the dispersion polymerization ofstyrene at 70 8C in a 300 ml reactor.

Materials wt.-%

Ethyl alcohol 77.43Poly(N-vinylpyrrolidone) (K-30)a) 1.80Cetyl alcohol 0.57Styrene 20.002,29-Azoisobutyronitrile 0.20

a) PVP K-30: average molecular mass is 46104.


polymerization at t = 0.5. These results suggest that the

termination rate constant in particles decreases due to the

gel effect.

To investigate whether the particle number varies with

the reaction time or not, the total particle number was

measured at various reaction times. Since particles were

very small until a reaction time of 2 h, the particle num-

ber could not be counted correctly until then. At the reac-

tion time from 2 h to 24 h, the total particle number was

constant at 1.3761012 within the experimental error. Fig-

ure 3 shows SEM micrographs of polystyrene particles

obtained at the reaction times of 2 h and 24 h. Monodis-

perse polystyrene particles of which the average diameter

was 1.62 lm were obtained at the reaction time of 2 h,

and thereafter they grew with time. If particle nucleation

and particle aggregation are balanced, the particle dia-

meter distribution will be broad.[7] Therefore, it was con-

cluded that new particle formation could be neglected in

the time range of 2–24 h. We decided to use the above

experimental value of the total particle number Np

(1.3761012) at the reaction time of 2 h as the initial par-

ticle number.

Simulation of the Particle Growth

In order to study the particle growth in dispersion poly-

merization, the differential equations, Equation (7), (10)

and (12), were solved numerically using the Runge-

Kutta-Gill method. The rate constants kd , kt and kp were

obtained from literature.[7] The values of the initiator effi-

ciency f was also taken from literature.[19] Um was esti-

Figure 1. Comparison of the polymerization rate in dispersionpolymerization with that of solution polymerization.

Figure 2. Time courses of molecular mass distribution in dis-persion polymerization.

Figure 3. SEM micrographs of particles.


mated by the method of Le Bas.[20] The rate constants and

the physical properties of monomer and polymer used in

the calculation are listed in Table 2.

As the initial conditions to solve the differential equa-

tions, three initial values of np , Vpp , and [M]t are required.

The Vpp value and the [M]t value were calculated using the

experimental particle diameter and the conversion obtained

at the reaction time of 2 h, respectively. These values are

also shown in Table 2. However, the number np of the radi-

cals in a particle could not be experimentally determined.

Therefore, the effect of the initial np value on the simulation

was first studied. As shown in Figure 4, when the initial np

value was changed from 0 to 106, the np value converged to

about 7100 within 1.5 s irrespective of the initial np value.

If a psudo-steady-state assumption is applicable for the np

value at the reaction time of 2 h, the equation which gives

the np value is derived by taking dnp /dt = 0 in Equation (7).

The equation is given by Equation (15).

np ¼NA

Np

2kd f ½I�VptVe

ktp

� �1=2

ð15Þ

The np value estimated using this equation was 6167.

This value was not so different from the above asymptoti-

cal value of 7100. Therefore, the value of 6167 was taken

as the initial np value.

The bgel value was evaluated by fitting the experimental

conversion to the theoretical conversion curves. In Fig-

ure 5, the experimental time course of the conversion was

compared with the theoretical time courses calculated

using the bgel values of 1.0, 0.1, 0.01, 0.0075 and 0.001.

The experimental data agreed well with the theoretical

line calculated using a bgel value of 0.0075. This result

shows that the termination rate constant in particles was

about 1/130 of that in the ethanol phase. In the dispersion

polymerization, the volume fraction of polystyrene in

particles is higher than 0.7 from the beginning of the

polymerization.[16] Scheren et al.[21] simulated the termina-

tion rate constants of styrene polymerization at high poly-

mer fractions in the seeded emulsion polymerization of

styrene. They reported that the kt values in particles of the

seeded emulsion polymerization ranged from 60 to 600

m3 N mol–1 N s–1, which were from 1/1000 to 1/100 of the

termination rate constant in the continuous phase. There-

fore, the bgel value evaluated above is reasonable.

Table 2. Rate constants and physical properties used in the cal-culation (70 8C).

Rate constants:kd 3.77610–5 s–1

kp 3.52610–1 m3 N mol–1 N s–1

kt 6.106104 m3 N mol–1 N s–1

f 0.58

Physical properties:Np 1.3761012

Um 1.33610–4 m3 N mol–1

Initial condition of simulation at t = 7200 s[I] 1.01610 mol N m–3

[M]t 1.416103 mol N m–3

X 0.12Vpp 1.68610–18 m3

Vpt 3.07610–6 m3

Figure 4. Effect of initial np value on the time course of np .

Figure 5. Effect of bgel on conversion.


Figure 6 compares the experimental particle diameters

with the theoretical line. The theoretical line calculated

with the bgel value of 0.0075 is in good agreement with

the experimental data. From this result and the result of

the time course of conversion, the termination rate con-

stant in particles is regarded to be 1/130 of that in the

ethanol-phase due to the gel effect.

Polymerization in the Ethanol Phase

In the above simulation, the termination reaction in the

ethanol phase and the chain-transfer reactions in the etha-

nol phase and particles were not taken into account and

the mean volume of the radicals captured by particles

was approximately equivalent to that of the monomeric

radicals. The chain-transfer reactions in the ethanol phase

and particles can be neglected in the simulation because

their rates are about 1/1000-1/10000 of those of the prop-

agation reactions. However, the termination of the radi-

cals in the ethanol phase, if any, decreases the radical

entry rate into particles. Furthermore, the volume of the

radicals captured by particles increases the particle

volume. Therefore, the propriety of these simplifications

must be verified.

In order to evaluate the contribution of the termination

of radicals in the ethanol phase, the termination rate and

the radical entry rate into particles were estimated. The ter-

mination rate in the ethanol phase is given by kt CRe2 and the

radical entry rate JR into particles is given by Equation (5).

To compare the radical entry rate with the termination rate

in the ethanol phase, a parameter n defined by JR /ktCRe2 was

introduced. The diffusion coefficient of monomeric radi-

cal which was required to calculate the JR value was esti-

mated using the equation given by Lusis and Ratcliff.[22]

Figure 7 shows the effect of the particle diameter on the

parameter n. In the range of the particle diameter which

was dealt with in the present simulation, n is much larger

than 1, indicating that the rate of the radical entry into par-

ticles is much larger than that of the termination in the

ethanol phase. Therefore, it is reasonable to neglect the ter-

mination in the ethanol phase.

Next, the second simplification or assumption will be

discussed. In the present simulation, the mean volume of

Figure 6. Time course of the particle diameter.

Figure 7. Effect of the particle diameter on the ratio n.

Figure 8. Average chain length of radicals in the ethanolphase.


the radicals captured by particles was regarded to be

equivalent to that of the monomeric radicals. It means

that the average degree je of polymerization is 1. The

average degree je of polymerization of the radicals in the

ethanol phase was calculated kinetically from the ratio of

the radical entry rate into the particles to the propagation

rate in the ethanol phase. Figure 8 shows the calculation

result of the average degree of polymerization of the radi-

cals in the ethanol phase. In the present simulation, the

degree of polymerization of the initiator radicals formed

by the decomposition of the initiator was defined as 0.

The average degree of polymerization of the radicals in

the ethanol phase ranges from 0 to 2. Since the initiator

radicals with the degree of polymerization of 0 were pre-

sent in the ethanol-phase, the degree of polymerization of

the radicals captured by the particles were regarded to be

approximately 1. Therefore, the assumption that the

volume of the radical molecules captured by particles

was approximately equal to that of the monomeric radi-

cals is reasonable.

From the results shown in Figures 7 and 8, it is con-

cluded that the vast majority of the oligomeric radicals

are captured by preexisting polymer particles before the

termination in the ethanol phase. Therefore, the above

assumptions that the termination of the radicals in the

ethanol phase can be neglected and the volume of radicals

captured by particles is equivalent to that of the mono-

meric radicals, are reasonable in the particle growth stage

of the dispersion polymerization.

Conclusions

A simple model which simulates the particle growth in

the dispersion polymerization of styrene in ethanol is pro-

posed. In the present model the following assumptions

are made: (i) the termination reaction in the ethanol phase

and the chain-transfer reactions in the ethanol phase and

particles can be neglected, (ii) the mean volume of the

radicals captured by particles is approximately equivalent

to that of the monomeric radicals, and (iii) the termina-

tion rate constant in particles is bgel times that of the etha-

nol phase. The theoretical time courses of the conversion

and the particle diameter calculated using a bgel value of

0.0075 were in good agreement with the experimental

data. This result indicates that the termination rate con-

stant in particles was about 1/130 of that of the ethanol

phase. By comparing the radical entry rate into the parti-

cles with the termination rate and propagation rate in the

ethanol phase, we illustrated that the assumptions of (ii)

and (iii) were found to be reasonable. We believe that the

present results are useful for developing a model that can

describe quantitatively the whole process of the disper-

sion polymerization and useful for industrial application,

such as designing and operating reactors which produce

micron-size monodisperse polymer particles.

CRe radical concentration in the ethanol phase,

[mol N m–3]

CRs radical concentration on particle surface,

[mol N m–3]

DRe diffusion coefficient of a radical in the ethanol

phase, [m2 N s–1]

dp particle diameter, [m]

f initiation efficiency

[I] initiator concentration in the ethanol phase,

[mol N m–3]

JR total rate of radical capture by polymer particles,

[mol N s–1]

je average degree of polymerization of radicals in

the ethanol phase

kd decomposition rate constant of initiator, [s–1]

kp propagation rate constant, [m3 N mol–1 N s–1]

kRe mass transfer coefficient of radical in the ethanol

phase, [m N s–1]

kt termination rate constant in the ethanol phase,

[m3 N mol–1 N s–1]

ktp termination rate constant in particles,

[m3 N mol–1 N s–1]

[M]e monomer concentration in the ethanol phase,

[mol N m–3]

[M]p monomer concentration in particles, [mol N m–3]

[M]t total monomer concentration, [mol N m–3]

[M]t0 initial monomer concentration, [mol N m–3]

NA Avogadoro’s number, [mol–1]

Np total number of polymer particles

np number of radical molecules in a particle

Sh Sherwood number

Um volume of repeating unit, [m3 N mol–1]

V total reaction volume, [m3]

Vp volume of a particle, [m3]

Vpe ethanol volume in a particle, [m3]

Vpm monomer volume of a particle, [m3]

Vpp polymer volume of a particle, [m3]

Vpt total volume of particles, [m3]

X conversion

bgel parameter of termination rate constant

n ratio of the rate of radical entry and termination

rate

qi rate of radical production in the ethanol phase,

[mol N s–1]

Received: February 16, 2000Revised: April 17, 2000

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simulation of particle growth in the dispersion polymerization of styrene: the termination rate...

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