Modeling of Differential Microemulsion Polymerization for

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Modeling of Differential Microemulsion Polymerization for Synthesizing Nanosized Poly(methyl methacrylate) Particles Guangwei He, Qinmin Pan,* and Garry L. Rempel* Department of Chemical Engineering, UniVersity of Waterloo, Waterloo, Ontario, Canada N2L 3G1

A mathematical model was developed for the differential microemulsion polymerization of methyl methacrylate to produce nanosized poly(methyl methacrylate) particles. The homogeneous nucleation mechanism is applied because of the operation characteristics of the differential microemulsion polymerization in which the concentration of the reaction monomer is low. An analysis has been carried out and it is confirmed that the model can predict the experimental data in reasonable agreement. Introduction Emulsion polymerization techniques have been widely used for synthesizing various polymers, such as synthetic rubbers and coating materials.1 Understanding and modeling the mechanism are essential for the analysis and design of the processes. The first mathematical model to describe emulsion polymerization was developed by Smith and Ewart more than half a century ago.2 Since then, a few more sophisticated models have been proposed, which can interpret the experimental phenomena of the reactions with monomers of low water solubility, such as styrene (St).3-5 It has been generally accepted in the concept of emulsion polymerization that the free radicals generated by the decomposition of an initiator inoculate the monomer-swollen micelles in the aqueous medium and initiate polymerization, which implies that the nucleation does not happen in the water phase; i.e., heterogeneous nucleation is assumed. However, for moderately water-soluble monomers, such as methyl methacrylate (MMA), it has been observed that emulsion polymerization can happen even in the absence of surfactants;6-8 i.e., monomerswollen micelles are not necessary to realize the polymerization. Thus, the concept of homogeneous nucleation for MMA polymerization was proposed,7,8 in which the free radicals formed in the aqueous phase react with the monomer dissolved in water and become oligomeric radicals. When these oligomeric radicals grow to a certain length so that their hydrophobicities become high, they precipitate from the aqueous phase to form particle precursors and further polymerization then takes place in these precursor particles. In modeling emulsion polymerization, efforts have been made to predict the reaction conversion, molecular weight, radical concentrations in the aqueous/particle phases, and the number of polymer particles in the reaction system as well as the particle size of the polymers.9,10 Microemulsion polymerization was first proposed and developed for methyl acrylate (W/O, water/oil systems, i.e., the dispersed phase is the aqueous phase surrounded by oil (continuous phase)) by Stoffer et al.11 Many investigations have been carried out in recent years, and most of them have dealt with the microemulsion polymerization of styrene.12-16 This method could be used to synthesize nanosized polymer particles. However, the surfactant used was high, sometimes even higher than the amount of monomer. A modified microemulsion polymerization, differential microemulsion polymerization method, was proposed,16,17 in * To whom correspondence should be addressed. Tel: (519)8884567 ext. 37111 (Q.P.) and 32702 (G.L.R.), Fax: (519)7464979. E-mail: [email protected], [email protected].

which a reaction system similar to emulsion polymerization is involved; i.e., it is composed of water, surfactant, a waterinsoluble monomer (e.g., St, MMA), and a water-soluble initiator (e.g., ammonium persulfate, APS) and requires a certain temperature to initiate polymerization and suitable agitation to form an emulsion. This method could be used to make similar particles as in traditional microemulsion polymerization; however, it uses a much smaller amount of surfactant. This method has been confirmed to be useful for synthesizing nanosized poly(methyl methacrylate) (PMMA) and polystyrene nanoparticles.17,18 However, the modeling and mechanism of differential microemulsion polymerization remains uninvestigated. To facilitate further understanding of this method for the synthesis of polymer nanoparticles, this paper is to exploit the mechanism and to develop a model for the differential microemulsion polymerization of MMA. Experimental Section The PMMA nanoparticles were synthesized through differential microemulsion polymerization17 by a semibatch operation, in which the monomer was added in a differential manner (continuously adding monomer in very small drops). The chemicals used in the experiments, the procedure for the differential microemulsion polymerization of MMA, and the techniques for characterization of the resultant polymer latexes are the same as reported previously.17 APS, sodium dodecyl sulfate (SDS), and deionized water were put into a three-necked, 250 mL flask equipped with a magnetic stirrer, a reflux condenser, and a thermometer. After the temperature was raised to 75 °C, MMA was added continuously in very small drops over a fixed period of time (monomer addition time). After the addition of MMA, the reaction mixture was kept at 80-85 °C for a certain time (aging time) before a cooling operation was applied. The solid content was determined by a weighing method, and the z-average particle size was measured using a dynamic light scattering method with a 90Plus particle size analyzer (Brookhaven Instrument Corp.). The number average particle size was measured with a Natrac 250 instrument (Microtrac Inc.). Physical Model Proposed for Differential Microemulsion Polymerization. In the reaction system for the differential microemulsion polymerization system, once the reaction is initiated, there are initiators, radicals (primary radicals, monomer radicals, and polymer radicals), and nonaggregated surfactants, as well as a very small amount of monomer and micelles in the

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the polymer volume; (6) the reactivity of radicals is independent of their degree of polymerization; and (7) the effect of collisions is not considered in this work. Based on these assumptions, a mechanism as well as the mathematical model for the differential microemulsion polymerization of MMA is proposed as follows: Initiation. The initiator (I) used in the reaction system decomposes to generate free radicals, R·, which attack monomer (M), at suitable reaction conditions.

I f 2R•

(symmetric I)

R• + M f RM•

Figure 1. Proposed mechanism: differential microemulsion polymerization of MMA.

water phase. The initiator decomposes in the water phase to form primary radicals. Some of the primary radicals will attack monomers to form polymer radicals, as illustrated in Figure 1. These polymer radicals will propagate in the water phase until they reach the critical chain length and precipitate to form polymer particles (homogeneous nucleation) or enter into monomer-swollen micelles to generate polymer particles (heterogeneous nucleation). The radicals in the water phase also can terminate by combination or disproportion or be captured by either active (polymer particles having free radicals in them) or dead polymer particles. A quasi-steady-state hypothesis with respect to the free radicals will be used to calculate the concentration of the radicals in the water phase since the mainstream of radicals comes from the initiator decomposition and the half-life time of the initiator is at least as long as the reaction time. Although radical termination also occurs in the particles, the number of radicals consumed in the particles is much smaller compared with the radical termination in the water phase. The number of polymer particles is much smaller than the micelle number,19 and thus, the probability of a free radical in the water phase entering an active particle for termination is so small that the termination in the water phase is the main termination reaction. Therefore, the steady-state hypothesis for the radicals in the water phase is reasonable. After the oligomeric radicals precipitate or enter micelles to form the polymer particles, these radicals will continue to propagate and the polymer chain will end after chain transfer to a monomer molecule or be terminated by another radical coming from the water phase. The monomeric radicals generated by chain transfer can continue to start polymerization or diffuse out into the water phase. It is well accepted that desorption of monomeric radicals from the active particles is unavoidable for emulsion polymerization since monomers have, more or less, some degree of water solubility.20 For microemulsion polymerization, monomeric radicals may diffuse more readily because of the much smaller particle sizes than those encountered in emulsion polymerization. Mathematical Model. The reaction system for differential microemulsion polymerization is so complex that some reasonable assumptions are necessary, which include the following: (1) the polymer particles are spherical; (2) the polymer nanoparticles contain 0 or 1 free radical; (3) the monomer in the polymer particles is in equilibrium with that in the water phase; (4) a quasi-steady state of free radicals is adopted in this process; (5) the particle volume is equal to the monomer volume plus

The end products of the initiator decomposition process (R• and RM•) can combine to become initiator molecules again, called geminate recombination species, or other inert products. They can also transfer to an initiator molecule, causing the waste of the initiator, but without the loss of free-radical activity. Therefore, it is common to employ an efficiency factor f in the initiation rate:

RI )

d[R•] ) 2f kd[I] dt

(1)

The rate coefficient kd (s-1) is temperature dependent. This is usually represented by the Arrhenius equation:

kd ) Ae-Ea/RT

(2)

where A is the frequency factor and Ea is the activation energy. For a persulfate-type initiator, kd ) 8 × 1015e-135000/RT.20 Fractional Conversion. The fractional conversion is calculated according to its definition:

x)

1 Vw

∫0t Mcharge dt - [M]s 1 Vw

∫0t Mcharge dt

(3)

where x is the fractional conversion of monomer to polymer, Mcharge is the monomer feed rate (mol/s), Vw is the volume of water (L), and [M]s is the apparent monomer concentration in the system based on water (in water and the particles) (mol L-1).

d[M]s ) -Rp + Mcharge/Vw dt

(4)

Rp ) kp[M]pN1 + kpw[M]w[R•]w

(5)

where Rp is the apparent polymerization rate (mol L-1 s-1), kp is the propagation rate constant (L mol-1 s-1), [M]p is the monomer concentration in the polymer particles, (mol L-1), [M]w is the monomer concentration in the aqueous phase (mol L-1), N1 is the concentration of free radicals in the polymer particles per unit volume of water (mol L-1), and [R•]w is the concentration of total free radicals in the aqueous phase (mol L-1). To determine the monomer concentration in a polymer particle, [M]p, the following empirical equation is used:20

[M]w [M]w* For MMA, a ) 0.72.21

)

( ) [M]p

[M]p*

a

(6)

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Generation Rate of Dead Polymer Particles. The dead polymer particles will be generated if the free radicals diffuse out of the polymer particles containing free radicals or if free radials in water enter the particles containing free radicals. The dead polymer particles will disappear if free radicals from the water phase enter them.

dN0 ) -kcp[R•]wN0 + (kdes + kcp[R•]w)N1 dt

(7)

where N0 is the concentration of dead polymer particles (mol L-1) and kcp is the apparent rate constant for radical capture by polymer particles (L mol-1 s-1). The first term on the right side of eq 7 is the active particle generation rate for capturing free radicals in the aqueous phase by inert particles. The second term is the diminishing rate of active particles through capturing free radicals from the water phase or through desorption of a monomer radical into the aqueous phase, where kdes is the rate coefficient of monomer radical desorption from a polymer particle (s-1).

dN ) kpw[Rc-1•]w[M]w + kcm[R•]wNd dt

Table 1. Parameters Used for Simulation of MMA Differential Microemulsion Polymerization at 80 °C

(8)

where kpw is the propagation rate constant in water (L mol-1 s-1), kcm is the apparent rate constant of radical capture by monomer-swollen micelles (L mol-1 s-1), and [Rc-1‚]w is the concentration of radicals with critical length in the water phase (mol L-1). At quasi-steady state, [R•]w can be obtained as eq 9.

a

parameter

value

ref

kcm, L mol-1 s-1 kp, L mol-1 s-1 kpw, L mol-1 s-1 Mm, g mol-1 Fm, g cm-3 Fp, g cm-3 Dw, dm2 s-1 Dp, dm2 s-1 NA ktrM, L mol-1 s-1 ktw, L mol-1 s-1 kd, L mol-1 s-1 f jcra [M]p*, mol L-1 [M]w*, mol L-1 md [S]CMC, mol L-1 Sa, dm2

5.5 × 102 1307 4kp 100 0.943 1.178 1.7 × 10-7 1.7 × 10-7 6.02 × 1026 3.24 × 10-2 3.7 × 109 8.6 × 10-5 0.5 10 6.6 0.15 44 0.008 3 × 10-17

12 23 20

23 23 24 20 20 25 20 20 25 9 9

jcr is the critical length of oligomeric radicals in the aqueous phase.

is the contribution of radicals terminated by reacting with chain transfer agents, surfactants, or other impurities. Since there are some unknown factors in the calculation of β, it is treated as an empirical parameter in the range of 0-1. Volume Growth of Polymer Particles. The total volume growth of all polymer particles is determined by the conversion of monomer into polymer via homogeneous nucleation.

-A1 + xA12 + 4ktw(RI + kdesN1) [R•]w ) 2ktw

(9)

dVp (kp[M]pN1 + kpw[Rc-1•]w[M]w)VwMm ) dt FpΦp

A1 ) kcpN + kcmNd

(10)

Rate of Micelle Depletion. The micellar concentration is calculated as follows:

where ktw is the water phase termination rate constant (L mol-1 s-1). The first term on the right side of eq 8 is the homogeneous particle nucleation rate. The primary free radicals are watersoluble and they mainly stay in the water phase. When a primary radical attaches a monomer unit, its hydrophobicity increases and it is easier to enter the monomer droplets or particles within the monomer-polymer mixture. When more monomer units are attached to a primary radical, it becomes more and more hydrophobic, and finally it cannot remain in the aqueous phase and precipitates as a primary polymer particle. This is the socalled homogeneous nucleation. The second term is the polymer particle formation rate whereby free radicals enter the monomerswollen micelles. The radical desorption rate coefficient of a monomer radical from a polymer particle, kdes (s-1), is as follows:22

K0 kdes ) ktrM[M]p βK0 + kp[M]p K0 )

β)

12Dw/mddv2 1 + 2Dw/mdDp

kpw[M]w + ktw[R•]w kpw[M]w + ktw[R•]w + kcpN + kcmNd + φ

(11)

Nd )

[S]total - [S]CMC Ap Nagr SaNAVwNagr

(14)

(if Mmic > 0) (15)

where Nagr is the average aggregation number of SDS in a

Ap ) (36πVwNA)1/3N1/3Vp2/3

(16)

micelle, Ap is the surface area of the polymer particles, [S]CMC is the concentration of surfactant (mol L-1), Mmic is the monomer amount in the micelles (mol), and [S]total is the total concentration of surfactant (mol L-1). Monomer Balance. In microemulsion polymerization, monomer is distributed among three possible locations: the water phase, the monomer droplets (monomer-swollen micelles), and the polymer particles.

Munreacted ) [M]pVp + [M]wVw + Mmic Munreacted ) (1 - x)

∫0t Mcharge dt

(17) (18)

(12)

Mmic ) NdVwKeq[M]w

(13)

where Munreacted is the amount of unreacted monomer in the reaction system (mol) and Keq is the equilibrium constant of monomer between the aqueous phase and the micelles.

where N0 is the concentration of dead polymer particles (mol L-1), N is the total concentration of polymer particles (mol L-1), Nd is the concentration of monomer droplets (mol L-1), and φ

(19)

Results and Discussion Nucleation Mechanism. The experimental phenomenon for MMA polymerization via the differential microemulsion method

Ind. Eng. Chem. Res., Vol. 46, No. 6, 2007 1685 Table 2. Recipe for Differential Microemulsion Polymerization of MMA MMA, mL

water, mL

SDS, g

APS, g

T, °C

addition time, s

aging time,a s

14

84

1.4

0.08

80

3600

3600

a

The operation time after the monomer addition period was finished and before the reaction operation was terminated.

Figure 4. Prediction of particle size with respect to kcp for differential microemulsion polymerization of MMA at 80 °C (MMA 14 mL, APS 0.08 g, water 84 mL, addition time 3600 s, aging time 3600 s).

Figure 2. Prediction of particle size for differential microemulsion polymerization of MMA at 80 °C (MMA 14 mL, APS 0.08 g, SDS 1.4 g, water 84 mL, addition time 4500 s, aging time 6600 s).

Figure 5. Prediction of fractional conversions with respect to β for differential microemulsion polymerization of MMA at 80 °C (MMA 14 mL, APS 0.08 g, water 84 mL, addition time 3600 s, aging time 3600 s).

Figure 3. Prediction of particle sizes with respect to kcp for differential microemulsion polymerization of MMA at 80 °C (MMA 14 mL, APS 0.08 g, water 84 mL, addition time 3600 s, aging time 3600 s).

shows that the instantaneous fractional conversion is higher than 0.9 when the monomer addition time period is more than 10 min, which will be shown in the later section (Figure 8). More than 90% of the monomer was polymerized and ∼10% or less of the monomer may be left in the aqueous phase and polymer particles. Thus, most surfactant micelles in the water phase are not monomer-swollen and could not form polymer particles. Therefore, it is safer to consider only homogeneous nucleation for this moderately water-soluble monomer MMA and that no heterogeneous nucleation exists, i.e., Nd ) 0. Determination of Model Parameters. Most parameters employed here are from the literature, except kcp and β, and they are listed in Table 1, and the recipe for the polymerization is given in Table 2. Parameters kcp and β are not available from the literature. Therefore, estimation about them is performed as follows:

Figure 6. Prediction of particle size with respect to β for differential microemulsion polymerization of MMA at 80 °C (MMA 14 mL, APS 0.08 g, water 84 mL, addition time 3600 s, aging time 3600 s).

kcp is the apparent rate constant of radical capture by polymer particles. A higher value of kcp means that, over a period of time, a larger amount of free radicals will enter into the existing polymer particles (particles with or without free radicals). Since

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Table 3. Particle Sizes Predicted and Obtained from Experiments at the End of Polymerization

SDS used in the experiments, g particle size (experimental) dz (dn), nm SDS needed for fully covering the surface of particles, g particle size (predicted), dz, nm aExperimental

no. 1

no. 2

no. 3

no. 4

no. 5

2.1 13.4 7.6

1.4 14.5 (11) 7.1

1 15.4 6.7

0.7 16.1 6.4

0.35 22.6 4.6

13.3

conditions: MMA 14 mL, APS 0.08 g, water 84 mL, addition time 3600 s, aging time 3600 s, and T ) 80 °C.

the number of dead polymer particles is much higher than that of the polymer particles with free radicals, more active particles will be formed after the flux of free radicals enter into the polymer particles, and therefore, the active particles will continue to grow in volume until these active particles become dead ones. This results in larger particle sizes, as predicted in Figures 3 and 4 that, over the range of (0-5.5) × 106 L mol-1 s-1 kcp, the particle size could increase from 13.1 to 41 nm. The particle size dn in our system is ∼14.5 nm. To satisfy dn ) 14.5 nm, by a trial-and-error method, kcp ) 2.2 × 104 L mol-1 s-1 was determined. The common value of kcp for emulsion polymerization is ∼106 L mol-1 s-1, where the size of particles is usually larger than 100 nm. In our system, the particle size is ∼14.5 nm; thus, the kcp should be less than its normal level of emulsion polymerization systems by 1-2 orders of magnitude. Thus, a smaller kcp is reasonable and a value of kcp ) 2.2 × 104 L mol-1 s-1 is used in all the predictions in this paper. β is a combination parameter, having a value of 0-1 as defined by eq 13. To understand the effect of β, simulation analysis was carried out. The fractional conversion and the average diameter of the particles with different β are shown in Figures 5 and 6. Compared to the experimental data, the preferable value of β is from 0.1 to 0.2. Hereafter, β ) 0.15 is used in this paper. From eq 11, higher β makes kdes smaller, and thus, the oligomeric radicals will stay in the particle for a longer time to continue the polymerization and to make the particle bigger. The model predicts a trend similar to that shown in Figure 6. Particle Size. The particle size predicted by the aboveproposed model is very close to the experimental result; see Table 3 and Figure 2. It should be mentioned here that the particle size predicated by the above model is the numberaverage diameter (dn) while the experimental one is the z-average diameter (dz) in this paper except for those otherwise specifically noted. A comparison analysis between dz and dn was carried out for the sample obtained when 1.4 g of SDS was used; it indicated dz ) 14.5 nm was obtained by a Brookhaven 90 Plus while dn ) 11 nm was obtained by a Philips CM20 Super Twin high-resolution transmission electron microscope. The results of particle sizes from experiments with different amounts of surfactant used in the polymerization mixture are shown in Table 3. As indicated in Figure 2 and Table 3, except for the result obtained when the SDS concentration is particularly low (no. 5 in Table 3), the model prediction agrees relatively well with the experimental results. From a calculation of the coverage on the particle surface by SDS molecules, it could be seen that even when the largest amount of SDS (2.1 g) was used, it still cannot cover the whole surface of PMMA particles with the assumption of only one molecular adsorption layer (see Table 3). This low coverage of SDS should cause the particle size to become larger during the polymerization process due to particle collisions. However, the differential microemulsion polymerization operation fortunately shows different characteristics that over a broad range of SDS (SDS > 0.7 g) this phenomenon was not observed and the particle size kept at a low level.

Figure 7. Prediction of particle size with respect to monomer addition times for differential microemulsion polymerization of MMA at 80 °C (MMA 14 mL, APS 0.08 g, water 84 mL; addition time varied, total reaction time 7200 s).

Figure 8. Simulation of MMA differential microemulsion polymerization (80 °C, MMA 14 mL, APS 0.08 g, water 84 mL, addition time 3600 s, aging time 3600 s).

Fractional Conversion. The fractional conversions obtained from the experimental results and the model prediction results are shown in Figure 8. The model predictions are in good agreement with the experimental results. Only during the first 1000 s and during the aging time, the model predicts slightly higher fractional conversions. In the differential polymerization operation of MMA, both theoretical and experimental results reveal that, after the addition of MMA begins, the reaction proceeds very fast until the fractional conversion reaches the plateau region, where the polymerization rate is close to the addition rate. After finishing the addition of MMA, the reaction continues and the fractional conversion becomes slightly higher. Monomer Concentration. The prediction of the monomer concentrations in the aqueous phase and in the particles ([M]w and [M]p, respectively) during polymerization is shown in Figure

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Figure 9. Model prediction for the time evolution of monomer concentration in polymer particles (80 °C, MMA 14 mL, APS 0.08 g, water 84 mL, addition time 3600 s, aging time 3600 s).

Figure 10. Time evolution of number of active polymer particles for MMA differential microemulsion polymerization at 80 °C (MMA 14 mL, APS 0.08 g, water 84 mL, addition time 3600 s, aging time 3600 s).

9. At the beginning of the polymerization, due to the smaller number of active particles in the system (see Figure 10), the polymerization rate is low and the added monomer is accumulated, which results in the increase in [M]w and [M]p. However, with the increase of the active particles in the reaction mixture, the reaction rate increases, which consumes monomer quicker. When the monomer addition rate is low, a monomer starving phenomenon occurs, which results in the decrease of the monomer concentration in the water and in the particles. Average Number of Radicals per Particle. The average number of radicals per particle was calculated based on the model established in this work, and the results are illustrated in Figures 11 and 12, in which three zones are observed, the quick decrease zone, the steady zone, and the quick increase zone. Although the number of active particles increases during the polymerization process (see Figure 10), the number of free radicals per polymer particle (N1/N) decreases quickly at the very beginning of the reaction because of the rapid nucleation and then keeps at almost a constant level, although a slight decrease is observed, until the monomer addition is finished. During the aging period (after the monomer addition period), the number of free radicals per particle increases quickly because no more or less new particles could be formed to share the free radicals in this period.

Figure 11. Time evolution of number of free radicals per particle for MMA differential microemulsion polymerization at 80 °C (MMA 14 mL, APS 0.08 g, water 84 mL, addition time 3600 s, aging time 3600 s).

Figure 12. Prediction of the average number of radicals per polymer particle for MMA differential microemulsion polymerization at 80 °C (MMA 14 mL, APS 0.08 g, water 84 mL, addition time 3600 s, aging time 3600 s).

Effect of Monomer Addition Speed. The effect of the monomer addition speed has been numerically analyzed on the particle size based on the model and is shown in Figure 7. At the beginning, the particle sizes for faster addition methods are larger than for slower ones. However, during the last stage, the particle size for the slowest addition is the highest. This is due to the entry of the radicals into the existing polymer particles. If the addition rate is so fast, there will be monomer droplets in the reaction mixture. Furthermore, the behavior of this method will be like that of a traditional emulsion polymerization method, in which the particle size will be much larger as reported by other researchers.9 Applications of the Model to Other Reaction Systems To Predict the Particle Size. Differential microemulsion polymerization of three other systems, vinyl acetate (VAc), methyl acrylate (MA), and ethyl acrylate (EA), which have moderate water solubilities, are also discussed based on the model developed above. The parameters used for the simulation of these systems are provided in Table 1, except for those listed in Table 4. Since the polymers resulting from these three monomers have much lower Tg, the resultant particles might be very viscous at the reaction temperature (80 °C) and the particles might easily collide with each other and adhere together to form larger particles. Therefore, the real particle sizes would be probably higher than predicted. The simulation of the particle sizes with respect to time is given in Figure 13.

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Figure 13. Predictions of particle sizes of polymers synthesized by differential microemulsion polymerization of MA, EA, and VAc at 80 °C (monomer 14 mL, APS 0.08 g, water 84 mL, addition time 3600 s, aging time 3600 s). Table 4. Parameters for Simulations of the Differential Microemulsion Polymerization of VAc, MA, and EA parameter mol-1 s-1

kp, L kpw, L mol-1 s-1 Mm, g mol-1 Fm, g cm-3 Fp, g cm-3 ktrM, L mol-1 s-1 [M]p*, mol L-1 [M]w*, mol L-1 jcr β

VAc 25800 4kp 86 0.950 1.15 6.192 7.7 20 0.3 20 20 27 1

MA 26

3651 4kp 86 0.956 1.22 0.0818 (75 °C) 24 6.6 (assumed) 0.61 (50 °C) 20 20 (assumed) 1

EA 26

1412 4kp 100 0.924 1.12 0.134 (70 °C) 24 6.6 (assumed) 0.15 10 (assumed) 1

Conclusions A mathematical model has been developed for the differential microemulsion polymerization of the monomer MMA, which has moderate water solubility. In this model, only homogeneous nucleation is considered since the conversion of monomer is so high such that the monomer concentrations in the aqueous and particle phase are always very low. The evolution of the fractional conversion and the particle size of PMMA nanoparticles agree very well with the experimental data. The particle size does not change much with the addition rate. The particle size is determined from the kinetic parameters. The prediction of the particle sizes of other moderately water-soluble monomers has also been carried out using this model. Acknowledgment The support from Natural Sciences and Engineering Research Council of Canada (NSERC) is greatly acknowledged. Nomenclature A ) frequency factor, s-1 Ap ) surface area of the polymer particles, dm2 dn ) number-average diameter, nm Dw ) diffusion coefficient for monomer radicals in the water phase, dm2 s-1 Dp ) diffusion coefficient for monomer, dm2 s-1 dv ) volume average diameter, dm dz ) z-average diameter, dm Ea ) activation energy, J mol-1 f ) initiator efficiency [I] ) initiator concentration. mol L-1

jcr ) critical length of monomeric radicals in the aqueous phase kdes ) rate coefficient of monomer radical desorption from a polymer particle, s-1 Keq ) equilibrium constant of monomer between the aqueous phase and micelles kcm ) apparent rate constant of radical capture by monomerswollen micelles, L mol-1 s-1 kcp ) apparent rate constant of radical capture by polymer particles, L mol-1 s-1 kd ) initiator decomposition rate constant, L mol-1 s-1 kp ) propagation rate constant, L mol-1 s-1 kpw ) propagation rate constant in water, L mol-1 s-1 ktrM ) chain transfer rate constant to monomers, L mol-1 s-1 ktw ) water-phase termination rate constant, L mol-1 s-1 Mcharge ) monomer charge rate mol s-1 md ) partition coefficient for monomer radicals between the polymer particles and water phase Mm ) molecular weight of monomer, mol L-1 Mmic ) monomer amount in the micles, mol [M]p* ) monomer concentration in the polymer particles at saturation, mol L-1 [M]p ) monomer concentration in polymer particles, mol L-1 [M]s ) monomer concentration in the system based on water, mol L-1 [M]w* ) monomer concentration in water at saturation, mol L-1 [M]w ) monomer concentration in water phase, mol L-1 [M]unreacted ) amount of unreacted monomer in the reation system, mol Nagr ) average aggregation number of SDS in micelle N1 ) concentration of active polymer particles, mol L-1 N0 ) concentration of dead polymer particles, mol L-1 N ) total concentration of monomer droplets, mol L-1 Nd ) concentration of monomer droplets, mol L-1 NA ) Avogadro’s number R ) gas constant, J mol-1 K-1 RI ) initiator decomposition rate, mol L-1 s-1 Rp ) polymerization rate, mol L-1 s-1 [Rc-1•]w ) concentration of radicals with critical length in water phase, mol L-1 [R•]w ) concentration of radicals in water phase, mol L-1 [S]total ) total concentration of surfactant, mol L-1 [S]CMC ) concentration of surfactant, mol L-1 Sa ) area covered by a monolayer of surfactant molecules, dm2 t ) time, s T ) reaction temperature, K Vp ) particle volume, L Vw ) volume of water, L Vm ) molar volume of monomer, L mol-1 x ) monomer fractional conversion Fp ) polymer density, g cm-1 Fm ) monomer density, g cm-1 Φm ) volume fraction of monomer in the particle Φp ) volume fraction of polymer in the particle Literature Cited (1) Athey, R. D., Jr. Emulsion Polymer Technology; Marcel Dekker Inc.: New York, 1991. (2) Smith, W. V; Ewart, R. H. Kinetics of Emulsion Polymerization. J. Chem. Phys. 1948, 16, 592. (3) Gardon, J. L. Emulsion polymerization. I. Recalculation and extension of the Smith-Ewart theory. J. Polym. Sci. Chem. 1968, 6, 623. (4) Gardon, J. L. Emulsion polymerization. II. Review of experimental data in the context of the revised Smith-Ewart theory. J. Polym. Sci. Chem. 1968, 6, 643.

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ReceiVed for reView November 1, 2005 ReVised manuscript receiVed January 8, 2007 Accepted January 8, 2007 IE051216I