Monte Carlo simulation of the dispersion polymerization of styrene

Abstract

The Monte Carlo kinetic simulation method was performed to simulate the entire process of the dispersion polymerization of styrene stabilized by polyvinyl pyrrolidone (PVP) in ethanol. The equilibrium distributions of each component between the continuous phase and the particle phase were calculated by a thermodynamic model. In order to calculate the time course of the concentration of different species in each phase a kinetic model was proposed for the mechanism of dispersion polymerization which took into account the reactions both in the ethanol phase and particles. The role of the stabilizer and aggregation by Brownian diffusion and aggregation by the shear stress of the fluid was quantified for simulating the time course of particle concentration and particle size. The simulation results indicated that aggregation due to Brownian diffusion was the dominant mechanism of aggregation in dispersion polymerization. The effect of monomer concentration was investigated on the particle formation stage by simulating the monomer conversion, particle diameter and particle concentration. Subsequently, the particle growth stage was simulated after sufficient stabilizer was absorbed on the particle surface. The Monte Carlo simulation results for the monomer conversion, particle diameter and particle concentration were in agreement with experimental data.

---

Introduction

Dispersion polymerization (DP) is a unique technique for the preparation of monodisperse microspheres in a single batch process. These particles have found a wide range of applications and become a very hot topic recently.^1,2 These applications include spacers for liquid-crystal panels, electrophoretic displays, colored inks for electrostatic imaging systems, enzyme immobilization, biomedicine, drug delivery, ion-exchange beads, information technology and sensors for biochemical analysis.^3,4

Particles with selected surface properties and tunable size can be easily obtained from a wide variety of monomers via dispersion polymerization.⁵ Compared to the other methods of obtaining these particles such as the two-stage swelling method, successive seeded method, modified suspension polymerizations and dynamic monomer swelling method (DSM), which are more complex and less economical,^6,7 DP is operationally and economically superior and gives unparalleled results.

Several studies have been performed on the kinetics of dispersion polymerization. Dispersion polymerization can be divided into two main stages of particle formation and growth. In the first stage, the particles are formed by the nucleation and aggregation of primary particles. In the second stage, particles grow by absorbing polymer and monomer from the media followed by polymerization within the particles.^8,9 Lu et al.¹⁰ obtained a thermodynamic model for the monomer partitioning in dispersion polymerization of styrene in ethanol. Paine¹¹ developed a multibin kinetic model for the coalescence of unstabilized particles and quantified the role of stabilizer molecules in the particle formation step. Yasuda et al.^9,12 concluded that the aggregation of particles due to the shear stress of the fluid plays an important role in the particle formation stage. They also simulated the particle growth stage by using the experimental data after two hours reaction. Although, recent efforts have provided a deeper understanding of the dispersion polymerization process, the detailed mechanism of dispersion polymerization is not well understood as of yet.^6,13

Due to the statistical nature of chain propagation and termination reactions, mathematical methods that deal with estimating probabilities and specifically Monte Carlo have proven to be very effective and robust tools in simulating (co)polymerization systems. Monte Carlo simulation methods are capable to determine ingredients concentration, chain distributions and average molecular weights at any instant throughout the reaction. Moreover, the reaction rate constants are the only data required to simulate polymerization reactions.^14,15

Several studies have been performed on the kinetics of dispersion polymerization process but none of them are capable to simulate the entire process and predict the particle size without using experimental data throughout the reaction. Therefore, in the present study, the entire process of dispersion polymerization of styrene stabilized by polyvinyl pyrrolidone in ethanol, which is the most common dispersion polymerization system in the literature, was simulated using the Monte Carlo method.

Dispersion polymerization model

A mechanism has been proposed based on the literature^{9,11,12,16,17} to model the dispersion polymerization of styrene stabilized by polyvinylpyrrolidone in ethanol, which takes into account the reactions of styrene, 2,2′-azobisisobutyronitrile (AIBN) and PVP in the two phases involved namely the continuous and particles. The mechanism contains fifteen reactions as follows:

Polymerization in the continuous phase:

Primary radical formation:

Initiation:

Propagation:

Chain transfer to monomer:

Chain transfer to initiator:

Chain transfer to solvent:

Chain transfer to stabilizer:

Termination:

Polymerization in the particle phase:

Primary radical formation:

Initiation:

Propagation:

Chain transfer to monomer:

Chain transfer to initiator:

Chain transfer to solvent:

Termination:

Before polymerization, the monomer, stabilizer and initiator are dissolved in the continuous phase to form a homogenous solution. By commencement of the polymerization the free radicals obtained from the initiator decomposition propagate in the continuous phase. The molecules precipitate as unstable primary particles if they reach the critical chain length j_cr. The unstable particles in the ethanol phase aggregate by the Brownian diffusion and shear stress of the fluid. Simultaneously, stabilizer molecules and propagating polymer radicals form graft molecules (PVP-g-PS) that could stabilize the particles by absorbing on them. The aggregation of particles continues until sufficient stabilizer molecules absorb on them to form sterically stabilized particles.

In the first step of the simulation, the partitioning of the reaction components were calculated at different conversions. In the ternary system composed of styrene, ethanol and polystyrene the three components separate into two phases, i.e., an ethanol-rich continuous phase and a polystyrene-rich disperse phase. The thermodynamic model developed by Lu et al.¹⁰ was used to calculate the equilibrium distribution of each component between the continuous phase and the particle phase. X-ray photoelectron spectroscopy of the interior of the particles have indicated that there is no PVP inside the polymer particles.¹⁸ The initiator which is soluble in both the continuous phase and monomer will exist uniformly throughout the system.¹⁹ By considering that the partial molar free energy of ethanol and styrene in the continuous phase is equal to their partial molar free energy in the particle phase and the assumption that the continuous phase contains a negligible amount of polymer, the thermodynamic equations become:¹⁰

ϕ_ep + ϕ_mp + ϕ_pp = 1 (18)

ϕ_ec + ϕ_mc = 1 (19)

where

In the above equations, the subscripts e, m, c, and p represent ethanol, styrene, continuous phase, and polymer phase, respectively; ϕ_ij is the volume fraction of species i (m: styrene, p: polystyrene, e: ethanol) in phase j (c: ethanol-rich continuous phase, p: polystyrene-rich particle phase); M_ij is the ratio of molar volume of component i to that of component j; χ_ij is the Flory–Huggins interaction parameter between species i and j; γ is the interfacial tension; V̄_i is the molar volume of species i (m: styrene, e: ethanol); R is the gas constant and T is the absolute temperature.

By considering the thermodynamic equations and the material balances of styrene and polystyrene at different conversions, the volume fraction of different components in each phase were calculated. This was performed by solving the system of equations with appropriate initial conditions by using Polymath 6.1 software.

Using the thermodynamic model the volume fraction of different species in each phase could be calculated at different conversions. In order to obtain the particle diameter, the particle concentration should be calculated, which is one of the most difficult tasks in simulating dispersion polymerization. Furthermore, the time course of concentration of different ingredients in each phase is also required during the simulation. Therefore, the polymerization kinetics in both phases were simulated to attain the mentioned goals.

Monte Carlo method was used to simulate the kinetics of dispersion polymerization. Due to the statistical nature of Monte Carlo simulation a huge amount of computation is required for simulating polymerization on a molecular scale.¹⁴ To keep the simulation time reasonable a simulation volume containing 10¹² styrene molecules was chosen. Further increasing the initial number of styrene molecules did not change the simulation results. Therefore, it was concluded that a sample containing 10¹² styrene molecules determines the statistical properties of the whole reaction mixture effectively.

Monte Carlo simulation was performed in parallel in the continuous phase and the particle phase to consider the reactions in both phases. In order to perform this task, the reactions were performed 0.001 seconds in the continuous phase and then 0.001 seconds in the particle phase. The reaction time interval was calculated separately for each phase. The transfer of different species between the continuous phase and particle phase was also taken into account. The simulation was also performed in 0.01 and 0.0001 seconds time intervals instead of 0.001 seconds time intervals. In all three cases, the simulation results were the same and negligible differences were observed. It should be mentioned that the Monte Carlo simulation in each phase were performed based on the volume as well as the concentration of species in that phase. Furthermore, the formation of particles, the aggregation of particles, the partitioning of styrene, ethanol and AIBN between the continuous phase and particles and the entrance of radicals from the continuous phase to the particles were also considered at each moment of the simulation, which are explained in the subsequent sections.

At the beginning of the reaction, there were no particles in the system and all the reactions were performed in the continuous phase. Therefore, at the start of the reaction, the initial number of other ingredients in the continuous phase was determined based on initial number of styrene molecules:

N_X = [X]N_avV (22)

where V is the simulation volume, N_M number of monomers, [M] monomer concentration, N_X number of X molecules and [X] is the concentration of X species.²⁰

By formation of particles, the reactions are performed in both the continuous phase and particle phase. The following procedure is used to perform the reactions in each phase, by taking into account the number of reactants in each phase and the volume of each phase. The total volume of polymer particles (V_pt) is proportional to the polymer volume in the particles (V_pp), monomer volume in the particles (V_pm) and the ethanol volume in the particles (V_pe) as follows:

V_pt = V_pp + V_pm + V_pe (23)

V_pp, V_pm and V_pe were calculated by the following procedures:

The amount of polymer available in the particles increases by absorbing the chains in the continuous phase or polymerization of the monomer available in the particles. In order to calculate V_pp, the rate of entrance of radicals from the ethanol phase to the particles was also calculated by the following equation:^9,21

J_Rj = 2πd_pND_j[R_j]/V_et (24)

where d_p is the average particle diameter, N the total number of particles, D_j the diffusion constant of a molecule R_j with a degree of polymerization of j, [R_j] is the molar amount of a radical molecule R_j, and V_et is the total volume of the ethanol phase. The diffusion constant D_j could be estimated by the following equation:²²where η is the viscosity of ethanol, T the absolute temperature, U_m the molar volume of monomer and V_B the molar volume of ethanol.

V_pm and V_pe were calculated using the thermodynamic model of partitioning of species in dispersion polymerization.

The total volume of the ethanol phase V_pt was calculated based on the total volume of the reaction V_t and the total volume of the polymer particles V_pt by the following equation:

V_et = V_t − V_pt (26)

The volume contraction during polymerization was also taken into account by using the volume contraction factor ε.

Since the Monte Carlo simulation method deals with the number of molecules, the stochastic reaction rates must replace the macroscopic ones. The following procedure was applied for each phase separately:

Having considered L_i reactions in phase i, the probability of reaction l_i incidence (P_li) in phase i is given by:²³

where, a_li, the stochastic reaction rate for reaction l_i, is defined by

a_li = h_i × c_i (28)

in which h_i represents the number of reactants in phase i and c_i stands for stochastic rate constants in phase i and correlates with ordinary reaction rates:

First-order reactions

c_i = k_i (29)

Second-order reactions

The entrance of radicals from the continuous phase to the particles and the initiator decomposition were considered as first order reactions.

Considering r₁ as a random number generated during the simulation, based on the following criteria, a reaction is chosen to happen in phase i:

The probability of each polymer chain group reaction is calculated by dividing the number of chains in the group by all the chains available in phase i. Subsequently, expression (31) is utilized to select a reaction among different reactions in phase i. Eventually, another random number is generated to compute the reaction time interval in phase i:²⁴

Reaching random numbers to a suitable criterion required for uniformity and sequential correlation was achieved by applying an improved Mersenne twister random number generator.²⁵

The monomer is consumed by the propagation reaction in the ethanol and particle phase. The total number of monomers NM_t at time t is proportional to NM_p and NM_e, the number of monomers in the particles and ethanol phase respectively, by the following mass balance equation:

NM_t = NM_p + NM_e (33)

The number of monomers in the particles NM_p was calculated using the thermodynamic model of dispersion polymerization of styrene in ethanol.

The monomer conversion is defined by:

X = (NM_t0 − NM_t)/NM_t0 (34)

where NM_t0 is the total number of monomers in the start time of polymerization.

At each moment of the simulation the particle diameter was calculated simultaneously by the following procedure:

The diameter of the particles was assumed monodisperse based on the literature in order to use well-established theoretical equations for aggregation of particles.^12,26 The rate of decrease of N monodisperse particles due to aggregation by the Brownian diffusion is expressed by the following equation:²⁶

where k_B is the Boltzmann constant.

Using smolochowski's model, the rate of decrease of N monodisperse particles due to aggregation by the shear stress of the fluid is expressed by:²⁶

where [small gamma, Greek, dot above] is the shear rate and ϕ_p is the volume fraction of particles. The shear rate is proportional to the mixing speed n:²⁷

[small gamma, Greek, dot above] = Kn (37)

where K is a constant.

The particle diameter d_p is computed from V_pt as follows:

The stabilization mechanism by PVP-g-PS molecules suggested by Paine¹¹ was used in the present model. Paine assumed that the aggregation of particles stops when all the particle surface is covered by PVP-g-PS molecules, and after that the particles grow by capture of propagating radicals from the ethanol phase and polymerization within the particles. The chain transfer reactions between the propagating radicals and PVP molecules in the ethanol phase produce PVP-g-PS molecules. The amount of PVP-g-PS produced in the ethanol phase is called “graft available”. The minimum amount of PVP-g-PS required for preventing particle aggregation is called “graft required”. The rate of diffusion of graft molecules to the particle surface is by far larger than the production rate of graft molecules.¹² Therefore, an ideal situation where all the PVP-g-PS molecules are absorbed on the particle surface was assumed in the present study. Paine computed the minimum amount of PVP-g-PS molecules necessary for stabilizing the particles against aggregation.¹¹ Thus, the minimum amount of PVP-g-PS molecules was calculated in this model. The “graft available” was calculated by using the following equation:^11,12,28

Graft available = C_sSN_avX (39)

where C_s is the chain transfer constant for transfer to stabilizer (per monomer unit), S is the concentration of the monomer unit of PVP in the ethanol phase and N_av is the Avogadoro's number. Chain transfer to PVP stabilizer can be approximated by the chain-transfer constant for 4-hydroxybutyric acid lactone 0.4 × 10⁻⁴.^11,29

The radius of gyration of a PVP chain molecule in the ethanol phase R_g can be calculated by AM_m-PVP^b where A and b are constants and M_m-PVP is the molecular weight of a PVP molecule. For estimating R_g, the average length of the long chain from the grafting point is required. The average length of the long chain from the grafting point was assumed 75% of that of PVP.¹² Therefore, the coverage surface Q of a PVP-g-PS molecule is given by:¹²

Q = πR_g² = πA²(0.75M_m-PVP)^2b (40)

The “graft required” is given by:¹²

The simulation program was written in C++ and executed on a high-memory system with 32 GB main memory and 16-core 64-bit 2.4 GHz processors. The various parameters used for the simulation of dispersion polymerization of styrene are demonstrated in Table 1.

Table 1 Parameters used for the simulation of dispersion polymerization of styrene

Parameter	Unit	Value	Reference
kd	s^−1	3.77 × 10^−5	32
kini	L mol^−1 s^−1	10kp	33
kp	L mol^−1 s^−1	3.52 × 10^2	32
kt,ethanol	L mol^−1 s^−1	6.1 × 10^7	32
kt,particle	L mol^−1 s^−1	4.58 × 10^5	9
Ctr,M		1 × 10^−4	32
Ctr,I		0.02	Estimated from ref. 32
Ctr,S		2 × 10^−4	32
Ctr,PVP		0.4 × 10^−4	11 and 29
f		0.58	17
ε		−0.147	34
K		13	27
jcr		120	12
Um	m^3 mol^−1	1.33 × 10^−4	12
n	rpm	30
Cs		0.4 × 10^−4	11 and 29
	cm^−3 mol^−1	120.7	10
	cm^−3 mol^−1	61.8	10
Rg	m	1.4 × 10^−8	11
A	m mol kg^−1	5.3 × 10^−7	35
b		0.32	35
χme		2.05	10
χem		1.05	10
χmp		0.25	10
χep		2.1	10

---

Results and discussion

The simulations were performed according to the experimental conditions and recipes of Yasuda et al.,^9,12 where the experiments were performed in a glass batch reactor equipped with a motor-driven stirrer with agitation at 30 rpm at 70 °C. The recipes of the reaction mixtures are presented in Table 2. The experimental results of these papers were used for investigating the accuracy of our simulation. The simulations were performed without using any experimental data during the reaction.

Table 2 Recipes of the reaction mixtures (concentrations based on wt%)

Reaction no.	Styrene	AIBN	PVP K-30	Cetyl alcohol	Ethanol	Reference
1	10.00	0.10	1.80	0.50	87.60	12
2	15.00	0.15	1.80	0.50	82.55	12
3	20.00	0.20	1.80	0.50	77.50	12
4	25.00	0.25	1.80	0.50	72.50	12
5	20.00	0.20	1.80	0.57	77.43	9

---

After the formation of particles, there are two locations in which the polymerization proceeds, which are the particles and the continuous phase. Initially, almost all the initiation reactions and propagation reactions of oligomeric radicals occur in the continuous phase. When the number of stabilized particles absorbing oligomeric radicals in the continuous phase are sufficient, the propagation and termination reactions of polymeric radicals mainly occur in the particles. Fig. 1 shows the simulation results of the volume fraction of styrene in particles at different conversions for reactions (1)–(4). The styrene concentration in the polymer particles increases with increasing initial styrene concentration. The volume fraction of styrene in the particles is quite low and decreases as the polymerization proceeds.

Fig. 1 Simulation of volume fraction of styrene in particles for different monomer concentrations.

In the proposed model for the stabilization mechanism in dispersion polymerization, the unstable particles aggregate until sufficient amount of stabilizer absorbs on their surfaces. When “graft available” reaches “graft required”, the entire surface of particles is covered by PVP-g-PS molecules and particles do not aggregate anymore. This time is defined as t_stb. The graft available and graft required for reactions (1)–(4) are presented at Fig. 2. At early stages of the reaction, the graft available is less than graft required. After t_stb, the graft available exceeds the graft required and the whole particle surface is covered with PVP-g-PS molecules. The time which the graft available attains the graft required (t_stb) determines the end of the particle formation stage. By increasing the monomer concentration, the rate of polymerization in the continuous phase and subsequently the particle formation rate increases. Therefore, the total area of particles that should be covered with PVP-g-PS molecules increases which tends to increase the graft required and prolong the particle formation stage. One the other hand, the rate of particle aggregation also increases with increasing the particle concentration. Furthermore, the production rate of PVP-g-PS increases as a result of higher polymerization rate in the continuous phase. The two latter phenomena tend to increase the graft required and shorten the particle formation stage. The simulation results at Fig. 2 indicate that the particle formation stage was prolonged under higher monomer concentrations, since the increase in graft required was more than the increase in graft available.

Fig. 2 Simulation of graft available and graft required for different monomer concentrations.

The aggregation due to the Brownian diffusion and shear stress of the fluid were both taken into account. Fig. 3 shows the time course of rate of aggregation of particles due to the Brownian diffusion for reactions (1)–(4). The rate of particle aggregation due to the Brownian diffusion quickly decreased with time, in accordance with eqn (35) that indicates the particle aggregation due to Brownian diffusion is proportional to the square of the particle concentration N. Fig. 4 shows the time course of rate of aggregation of particles particle due to the shear stress of the fluid for reactions (1)–(4). According to eqn (36) the aggregation of particles due to the shear stress of the fluid is proportional to the particle concentration N and the volume fraction of the particles ϕ_p. The results indicate that rate of aggregation due to Brownian diffusion is by far larger than that of the shear stress of the fluid. Hence, it was concluded that the aggregation due to the Brownian diffusion is the dominant mechanism of aggregation in dispersion polymerization.

Fig. 3 Simulation of the time course of rate of particle aggregation due to the Brownian diffusion for different monomer concentrations.

Fig. 4 Simulation of the time course of rate of particle aggregation due to the shear stress of the fluid for different monomer concentrations.

The effect of monomer concentration on the particle formation stage in dispersion polymerization was investigated. Fig. 5 shows the time course of conversion under different concentrations of monomers for recipes 1–4. The graft available and graft required both increased by increasing the monomer concentration and the simulation was continued until the graft available attained the graft required. According to the simulation results of monomer partitioning in particles at Fig. 1, the volume fraction of styrene in the particles increases at higher monomer concentrations; therefore due to the gel effect in particles, the monomer consumption rate and conversion increases.

Fig. 5 Simulation of the time course of conversion for different monomer concentrations.

The time course of particle diameter for different concentrations of monomer is presented at Fig. 6. At higher monomer concentrations t_stb was prolonged and therefore the particles have more time for aggregation. Therefore, the number of particles decreases and subsequently the particle diameter increases. Furthermore, by increasing the monomer concentration, the volume fraction of monomer in the particles increases. Therefore, the polymerization rate in the particles increases, which results in increasing the particle diameter. At the end of the particle formation stage, the particle diameter is bigger than the typical latex particles obtained by emulsion polymerization.

Fig. 6 Simulation of the time course of particle diameter for different monomer concentrations.

The time course of particle concentration under different monomer concentrations are shown at Fig. 7. After the very early stages of the reaction the particle concentration decreased until t_stb and after that became constant. The particle formation rate increased at higher monomer concentrations, therefore by increasing the monomer concentration the rate of decrease of particle concentration was smaller. The relatively low monomer concentration in polymer particles which was shown in Fig. 1 and the low particle concentration in this system which was shown in Fig. 7, explain the slow polymerization rates for dispersion polymerization compared with emulsion polymerization. At higher monomer concentrations there are more radical molecules available in a particle. This is a result of increased monomer and initiator concentrations in the particles according to the simulation results of the partitioning of species. Subsequently the rate of polymerization in the particles increases by increasing the monomer concentration. Furthermore due to the solution polymerization mechanism in the continuous phase the rate of polymerization also increases by increasing the monomer concentration in the continuous phase.

Fig. 7 Simulation of the time course of particle concentration for different monomer concentrations.

Fig. 8 compares the theoretical predictions of particle concentration under various concentrations of monomer with experimental data. The theoretical predication of particle concentration at t_stb under various monomer concentrations is in a good agreement with experimental results. Therefore, the model can simulate the particle formation stage in dispersion polymerization.

Fig. 8 Particle concentration at t_stb for different monomer concentrations.

The two-stage dispersion polymerization method was developed by Winnik et al. which is a type of seeded dispersion polymerization that the particles are formed in situ.⁷ Various research groups have found that if additives such as comonomers and crosslinking agents were added at the beginning of the reaction the size distribution became much broader.^30,31 Due to the high sensitivity of the particle formation step, Winnik et al. added the additives at the end of the particle formation step, thereby obtained particles with narrow size distributions.³¹ A method for determining the end of the particle formation step, is monitoring the number of particles during the dispersion polymerization process; however, this type of experiment remains too difficult for dispersion polymerization.⁷ One of the applications of Monte Carlo simulation of dispersion polymerization is the ability to compute the reaction time and conversion at the end of the particle formation step. Therefore, the point for adding the problematic additives could be calculated. The particle concentration, monomer conversion and particle diameter at the end of the particle formation stage for different monomer concentrations are presented at Table 3.

Table 3 t_stb, monomer concentration and particle diameter at different monomer concentrations

Reaction no.	tstb (s)	Particle concentration (m^−3)	Monomer conversion (%)	Particle diameter (μm)
1	891	1.82 × 10^16	0.36	0.34
2	1336	1.64 × 10^16	0.83	0.55
3	1659	1.50 × 10^16	1.54	0.80
4	1896	1.40 × 10^16	2.54	1.09

---

The simulation for the entire process of dispersion polymerization was performed according to the experimental recipe and conditions of Yasuda et al.⁹ (reaction (5)). By using a multibin kinetic model, Paine¹¹ proved that the monodispersity of the particles is lost when the aggregation of particles occurs after t_stb. Therefore, the total concentrations of particles at the particle growth stage was considered constant.⁹ The volume fraction of polystyrene in the particles is high from the early stages of dispersion polymerization and there is a significant gel effect in the particles.¹⁰ Therefore the termination rate constant in the particles is lower than that of in the ethanol phase.

The volume fraction of styrene, ethanol and polystyrene in the particles are calculated based on the thermodynamic model and are presented at Fig. 9. Most of the ethanol and styrene remain in the continuous phase during the reaction. Furthermore, the partition coefficient of styrene, which is the ratio of volume fraction of styrene in the particles to the volume fraction of styrene in the continuous phase, was calculated at different conversions and is presented at Fig. 10. The styrene partitioning coefficient is not constant and increases slightly as the polymerization proceeds.

Fig. 9 Simulation of the volume fraction of styrene, ethanol and polystyrene in the particles.

Fig. 10 Simulation of partition coefficient of styrene at different conversions.

The aggregation due to the Brownian diffusion and the aggregation due to the shear stress of the fluid were both taken into account. In the early stages of the reaction the graft available was smaller than the graft required and after 1659 seconds the graft available exceeded the graft required and the entire surface of particles was covered by PVP-g-PS molecules. The particle formation step ends at t_stb and is followed by the particle growth stage. The simulation results of the time courses of conversion and particle diameter are plotted at Fig. 11 and 12 respectively. In both cases the simulation results match the experimental data with good accuracy.

Fig. 11 Simulation of the time course of monomer conversion.

Fig. 12 Simulation of the time course of particle diameter.

The simulation result for the time course of rate of polymerization is depicted in Fig. 13. A quick rise in rate occurs by initiating the polymerization, followed by a rise to a maximum and then a decrease. At the early stages of the reaction the continuous phase is the main locus of polymerization. As the polymerization proceeds the particle size increases and subsequently the adsorption rate of radicals to the particles increases. Furthermore in the first few hours of the reaction, the number of monomers in the particles increase according to the theoretical modeling of monomer partitioning behavior. Moreover, a significant gel effect occurs during dispersion polymerization due to the high concentration of polymer chains in the particles which results in decrease of the termination rate and increase in the rate of polymerization. The time course of number of monomers and number of radical molecules in a particle are depicted in Fig. 14 and 15 respectively.

Fig. 13 Simulation of the time course of rate of polymerization.

Fig. 14 Simulation of the time course of number of monomers in a particle.

Fig. 15 Simulation of the time course of number of radical molecules in a particle.

Solution polymerization mechanism occurs in the continuous phase. Therefore, the monomer concentration and subsequently the rate polymerization in the continuous phase decreases as the reaction proceeds. Two main factors affect the time course of monomer concentration in the continuous phase: the depletion of monomer due to solution polymerization and the decrease in monomer concentration due to the partitioning of monomer between the continuous phase and the particles. Therefore, the rate of decrease of monomer concentration in the continuous phase is more than solution polymerization.

Conclusion

The entire process of dispersion polymerization of styrene stabilized by polyvinyl pyrrolidone in ethanol was simulated using the Monte Carlo method. Subsequently, the validity of the simulation procedure was confirmed by employing different experimental systems. The partitioning of different components between the continuous phase and particle phase were calculated by a thermodynamic model. The volume fraction of styrene in the particles increased by increasing the initial monomer concentration. A kinetic model was proposed to take into account the reactions in the ethanol phase and particles. Initially the particle formation step was simulated by taking into account the role of stabilizer and aggregation due to the Brownian diffusion and aggregation due to the shear stress of the fluid. The aggregation due to the Brownian diffusion was the dominant mechanism of particle aggregation. Subsequently, the particle growth stage was simulated after sufficient stabilizer was absorbed on the particle surface. The simulation results indicated a quick rise in the rate of polymerization by initiating the reaction, followed by a rise to a maximum and then a decrease. The simulations were performed without using any experimental data throughout the reaction and by only using the initial reaction conditions. The simulation results enabled to predict monomer conversion, particle diameter and particle concentration at any stage of the reaction for various experimental systems. The results could be used to calculate the point for adding the problematic additives in two-stage dispersion polymerization.

Glossary notation

A	Proportionality constant for the relation Rg = AMm-PVP^b (mmol kg^−1)
al	Stochastic reaction rate for reaction l (s^−1)
b	Exponent, relating radius of gyration to molecular weight
c	Stochastic rate constant (s^−1)
Cs	Chain-transfer constant for transfer to stabilizer (per monomer unit)
Dj	Diffusion coefficient of a radical whose degree of polymerization is j in the ethanol phase (m^2 s^−1)
dp	Particle diameter (m)
f	Fraction of initiator fragments forming radicals
h	Number of reactants
JRj	Rate of radical entry into polymer particles (mol s^−1)
jcr	Critical degree of polymerization of radicals in the ethanol phase
K	Constant in eqn (37) (N s m^−2)
kB	Boltzmann constant (m^2 kg s^−2 K^−1)
kd	Rate constant for initiator decomposition (s^−1)
kini	Propagation rate constant of primary radicals (m^3 mol^−1 s^−1)
kp	Propagation rate constant (m^3 mol^−1 s^−1)
kt,ethanol	Termination rate constant in the ethanol phase (m^3 mol^−1 s^−1)
kt,particles	Termination rate constant in the particles (m^3 mol^−1 s^−1)
ktr,M	Rate constant for chain transfer to monomer (m^3 mol^−1 s^−1)
ktr,I	Rate constant for chain transfer to initiator (m^3 mol^−1 s^−1)
ktr,S	Rate constant for chain transfer to solvent (m^3 mol^−1 s^−1)
ktr,PVP	Rate constant for chain transfer to stabilizer (m^3 mol^−1 s^−1)
[M]	Monomer concentration (mol m^−3)
Mm-PVP	Average molecular mass of PVP (kg mol^−1)
n	Stirring speed (s^−1)
N	Total number of polymer particles
Nav	Avogadoro's number (mol^−1)
NB	Rate of decrease of number of particles due to the aggregation by the Brownian diffusion (s^−1)
NM	Number of monomers
NMe	Number of monomers in the ethanol phase
NMp	Number of monomers in particles
NMt	Total number of monomers
NMt0	Initial total number of monomers
NR	Molar amount of radical molecules in particles (mol)
NS	Rate of decrease of number of particles due to the aggregation by the shear stress of the fluid (s^−1)
NX	Number of X molecules
Pl	Probability of incidence of reaction l
Q	Coverage surface of a PVP-g-PS molecule (m^2)
r	Random number generated during the simulation
Rg	radius of gyration of stabilizer chains in the ethanol phase (m)
Rj	Radical whose degree of polymerization is j
[Rj]	Molar amount of radical Rj (mol)
S	Concentration of monomer unit of stabilizer in the ethanol phase (mol m^−3) × PVP monomer unit
T	Temperature (K)
t	Time (s)
tstb	Time required to stabilize particles or the time when the graft available attains the graft required (s)
Um	Molar volume of repeating unit in a polymer chain (m^3 mol^−1)
V	Simulation volume (m^3)
	Molar volume of ethanol diluent (m^3 mol^−1)
Vet	Total volume of the ethanol phase (m^3)
	Molar volume of styrene monomer (m^3 mol^−1)
	Molar volume of polystyrene (m^3 mol^−1)
Vpt	Total particle volume (m^3)
Vpe	Ethanol volume in particles (m^3)
Vpm	Monomer volume in particles (m^3)
Vpp	Polymer volume in particles (m^3)
Vt	Total reaction volume (m^3)
X	Conversion
[X]	Concentration of species X (mol m^−3)
ϕec	Volume fraction of ethanol in continuous phase
ϕmc	Volume fraction of styrene in continuous phase
ϕpc	Volume fraction of polystyrene in continuous phase
ϕep	Volume fraction of ethanol in polymer phase
ϕmp	Volume fraction of styrene in polymer phase
ϕpp	Volume fraction of polystyrene in polymer phase
χem	Interaction parameter of ethanol and styrene
χep	Interaction parameter of ethanol and polystyrene
χme	Interaction parameter of styrene and ethanol
χmp	Interaction parameter of styrene and polystyrene
ϕp	Volume fraction of particles
ε	Volume contraction factor
η	Viscosity of ethanol (Pa s)
[small gamma, Greek, dot above]	Shear rate (s^−1)
τ	Reaction time interval (s)

Acknowledgements

The High Performance Computing Research Center (HPCRC) of Amirkabir University of Technology is acknowledged for providing computational facilities to perform this simulation.

References

1. Z. Jia, V. A. Bobrin, N. P. Truong, M. Gillard and M. J. Monteiro, J. Am. Chem. Soc., 2014, 136, 5824–5827.
2. J. Tan, X. Rao, J. Yang and Z. Zeng, RSC Adv., 2015, 5, 18922–18931.
3. A. P. Richez, H. N. Yow, S. Biggs and O. J. Cayre, Prog. Polym. Sci., 2013, 38, 897–931.
4. S. Hosseinzadeh and Y. Saadat, RSC Adv., 2015, 5, 35325–35337.
5. W. Li and K. Matyjaszewski, Macromol. Chem. Phys., 2011, 212, 1582–1589.
6. S. Jiang, E. D. Sudol, V. L. Dimonie and M. S. El-Aasser, Macromolecules, 2007, 40, 4910–4916.
7. J.-S. Song, F. Tronc and M. A. Winnik, J. Am. Chem. Soc., 2004, 126, 6562–6563.
8. J. Tan, X. Rao, X. Wu, H. Deng, J. Yang and Z. Zeng, Macromolecules, 2012, 45, 8790–8795.
9. M. Yasuda, H. Yokoyama, H. Seki, H. Ogino, K. Ishimi and H. Ishikawa, Macromol. Theory Simul., 2001, 10, 54–62.
10. Y.-Y. Lu, M. El-Aasser and J. Vanderhoff, J. Polym. Sci., Part B: Polym. Phys., 1988, 26, 1187–1203.
11. A. J. Paine, Macromolecules, 1990, 23, 3109–3117.
12. M. Yasuda, H. Seki, H. Yokoyama, H. Ogino, K. Ishimi and H. Ishikawa, Macromolecules, 2001, 34, 3261–3270.
13. N. M. Smeets, R. A. Hutchinson and T. F. McKenna, ACS Macro Lett., 2011, 1, 171–174.
14. A. Mahjub, S. H. Jafari, H. A. Khonakdar, U. Wagenknecht and G. Heinrich, Macromol. Theory Simul., 2013, 22, 207–216.
15. W. Bruns, I. Motoc and K. F. O'Driscoll, Monte Carlo applications in polymer science, Springer-Verlag, 1981.
16. S. Kawaguchi and K. Ito, in Polymer Particles, Springer, 2005, pp. 299–328.
17. S. F. Ahmed and G. W. Poehlein, Ind. Eng. Chem. Res., 1997, 36, 2597–2604.
18. Y. Deslandes, D. F. Mitchell and A. J. Paine, Langmuir, 1993, 9, 1468–1472.
19. M. Croucher and M. Winnik, in An Introduction to Polymer Colloids, Springer, 1990, pp. 35–72.
20. A. Mahjub, H. Mohammadi, M. Salami-Kalajahi and M. T. Angaji, J. Polym. Res., 2012, 19, 1–9.
21. R. G. Gilbert, Emulsion polymerization: a mechanistic approach, Academic Press, 1995.
22. M. Lusis and C. Ratcliff, Can. J. Chem. Eng., 1968, 46, 385–387.
23. M. Salami-Kalajahi, M. Najafi and V. Haddadi-Asl, Int. J. Chem. Kinet., 2009, 41, 45–56.
24. H. M. Jalali, RSC Adv., 2014, 4, 32928–32933.
25. M. Matsumoto and T. Nishimura, ACM Transactions on Modeling and Computer Simulation (TOMACS), 1998, 8, 3–30.
26. D. L. Swift and S. K. Friedlander, J. Colloid Sci., 1964, 19, 621–647.
27. A. Metzner, R. Feehs, H. L. Ramos, R. Otto and J. Tuthill, AIChE J., 1961, 7, 3–9.
28. P. J. Flory, Principles of Polymer Chemistry, Cornell University Press, 1953.
29. C. H. Bamford and C. F. H. Tipper, Free-radical polymerisation, Elsevier Pub. Co., 1976.
30. J. Kim, B. Kim and K. Suh, Colloid Polym. Sci., 2000, 278, 591–594.
31. J.-S. Song and M. A. Winnik, Macromolecules, 2006, 39, 8318–8325.
32. J. Brandrup, E. H. Immergut and E. A. Grulke, Polymer Handbook, Wiley, 1999.
33. P. Vana and A. Goto, Macromol. Theory Simul., 2010, 19, 24–35.
34. M. Benbachir and D. Benjelloun, Polymer, 2001, 42, 7727–7738.
35. S. Ali and N. Ahmad, Br. Polym. J., 1982, 14, 113–116.

---