Ind. Eng. Chem. Res. 1997, 36, 2597-2604
2597
Kinetics of Dispersion Polymerization of Styrene in Ethanol. 1. Model Development S. Farid Ahmed and Gary W. Poehlein* School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0100
Dispersion polymerization in polar organic media is a heterogeneous polymerization process that combines solution polymerization in the continuous phase with microbulk polymerization in the disperse particle phase. The relative prevalence of the two polymerization modes is dependent on a number of reaction variables. A mechanistic picture of dispersion polymerization during the particle growth phase is presented and mathematical models of the rate of polymerization and particle growth during seeded batch and continuous polymerization is developed. 1. Introduction Dispersion polymerization in polar organic media is a unique process for the preparation of polymer dispersions with narrow particle size distributions and diameters in the 1-10 µm range. Batch dispersion polymerization starts as a homogeneous mixture of monomer, organic solvent, soluble polymeric stabilizer, and initiator. The solvent is selected on the basis of its miscibility with the monomer, nonsolvent nature for the polymer, and solubility for the stabilizer. Polymerization is induced in solution by an initiator that is soluble in both the solvent and monomer. Polymer particles are nucleated and form a separate phase, stabilized by the soluble polymeric stabilizer. These dispersions can be redispersed after settling by shaking or agitation. If the stabilizing polymer is soluble in water, the nonaqueous solvent can be replaced to form an aqueous dispersion. This two-part paper presents a mathematical model of seeded batch dispersion polymerization of styrene in ethanol. Simulation of the rate of polymerization and the competitive size evolution of two different seed particle populations has been adopted as a model validation technique and for estimation of adjustable parameters in the mathematical model. This batch model has been adapted to describe seeded continuous polymerization in a well-mixed reactor at steady state. Simulation of the polymerization rate and effluent particle size distribution was used as a model validation technique. Barrett and Thomas (1969) found the rate of dispersion polymerization and the average molecular weight of PMMA formed in n-dodecane to be much higher than that for solution polymerization in benzene at the same temperature and initiator level. The rate of polymerization and the average molecular weight of polystyrene dispersion formed in an ethanol-water mixture were also found to be significantly higher than those for polystyrene dispersion prepared in an ethanol-2-methoxyethanol mixture, indicating a more pronounced gel effect in the former case. In studies of dispersion polymerization of MMA in CCl4-isooctane mixtures, it was also observed that the rate of polymerization and the average molecular weight both decreased with an increasing fraction of CCl4, i.e., as solvency of the media for the polymer increased (Williamson et al., 1987). Another study found that the average molecular weight of polystyrene dispersion in ethanol increased as the initiator concentration was increased (Chen and Yang, 1992). The average molecular weight was found to decrease with increasing polymerization temperature. S0888-5885(96)00505-2 CCC: $14.00
All of these results indicate that polymerization in both the continuous and disperse phases is important and the relative prevalence of solution polymerization in the continuous phase or microbulk polymerization in the particles is determined by the process conditions and recipe. A kinetic competition between the two polymerization loci, which can be influenced by the appropriate choice of process conditions, is the crux of this process and needs to be well understood. Dispersion polymerization can be viewed as a hybrid process. During the initial phase, the reaction resembles solution polymerization in that the mixture is totally homogeneous. Particle formation is likely to be predominantly by homogeneous nucleation, possibly augmented by stabilizer graft formation and subsequent precipitation or coagulative nucleation. Solvency of the dispersion medium for the polymer, influenced by the choice of the solvent, monomer concentration, and temperature, is a critical parameter determining the outcome of the nucleation process and hence the particle number density. Particle nucleation in batch dispersion polymerization is, in general, substantially over within a short initial period, and the subsequent rate of monomer conversion is affected only by the kinetic processes taking place during the particle growth phase and the number of stable particles generated during the nucleation period. The particle growth phase in dispersion polymerization resembles the reaction environment present during Interval III in the emulsion polymerization of watersoluble monomers using a nonionic polymeric stabilizer. Some similarity with suspension polymerization of water-soluble monomers is also observed, in that particles are large enough to accommodate many radicals, and polymerization rates in both phases can be important. However, since an oil-soluble initiator is used, radicals are initiated both in the continuous phase and inside the particles. In emulsion polymerization, radicals are generated in the aqueous phase while most of the polymerization takes place inside the particles. All three particle nucleation mechanisms, e.g., micellar entry, radical precipitation, and monomer droplet entry, may be prevalent in the formation of stable polymer particles (Sutterlin et al., 1976; Sutterlin, 1980; Hansen and Ugelstad, 1979b; Poehlein, 1982). Homogeneous nucleation through radical precipitation and subsequent homocoagulation of unstable nuclei and stabilization through surfactant adsorption are likely to be the more important particle formation processes in the emulsion © 1997 American Chemical Society
2598 Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997
polymerization of monomers with relatively high solubility in water (Song and Poehlein, 1989). Final particle concentration, monomer partitioning, radical interchange between phases, and polymerization kinetics in the particles determine the progress of the reaction after the nucleation phase. Use of nonionic polymeric stabilizer, in place of ionic surfactant, alters the particle nucleation mechanism but does not significantly affect the reaction kinetics. While stabilizer micellization and grafted macroradical precipitation may be present to some extent, homogeneous nucleation is the more prevalent particle formation process in the polymerization of water-soluble monomers, e.g., in vinyl acetate polymerization using poly(vinyl alcohol) as the stabilizer (Gilmore et al., 1993). The critical chain length for oligomer precipitation during homogeneous nucleation, the efficiency of irreversible absorption of aqueous-phase radicals by existing particles, and the stability factor associated with particle coagulation were found by Gilmore et al. to be important parameters in predicting the final particle number concentration. The average number of radicals per particle was still the most important parameter in predicting the polymerization rate, although the rate of polymerization in the aqueous phase can be significant and was accounted for in the kinetic model of vinyl acetate polymerization used by Gilmore et al. In aqueous suspension polymerization, because of their large size, polymerizing droplets contain a very large number of radicals. The polymerization kinetics generally follows that of bulk polymerization (Munzer and Trommsdorff, 1977). The same kind of dependence of polymerization rate on initiator concentration and temperature as that in bulk polymerization is observed. In the suspension polymerization of monomers with appreciable water solubility, quantitative agreement between experimental results and the prediction based on the bulk polymerization model were found to be poor (Taylor and Reichert, 1985). Kalfas and Ray (1993) obtained good fits of experimental conversion and molecular weight distributions in the aqueous suspension polymerization of styrene, vinyl acetate, and methyl methacrylate using a two-phase microbulk polymerization model that took into consideration the mutual solubility of monomer and water in each phase. Paine (1990) proposed a mathematical model of nucleation and growth of monodisperse polymer particles by dispersion polymerization in a polar media, stabilized by homopolymers such as HPC, PVP, or PAA. Formation of stable polymer particles was postulated to be a two-step process, with the initial nucleation of very large numbers of small and unstable precursor particles by the aggregation and precipitation of polymer chains in solution and subsequent homocoagulation of these precursor particles to form much larger particles. The mechanism of stabilization was thought to be by graft formation with the stabilizer homopolymer and adsorption of these grafted species on the particle surface. The number of polymer particles formed by dispersion polymerization is determined very early in the reaction (Lok and Ober, 1985), and the particle number remains relatively constant after nucleation. Particle growth occurs by absorption of radicals and polymer from the continuous phase and by radical generation and polymerization within the particles. Temperature is a key variable, as it governs both the polymer solubility and
monomer partitioning, as well as the rate of radical generation. The rate at which new polymer chains are initiated and terminated in the continuous phase, monomer and initiator partitioning between phases, and the rates of chain initiation, propagation, and termination in the particles are all expected to be important factors. Irreversible absorption of radicals by the polymer particles and desorption of radicals into the continuous phase affect the radical concentration in both phases. Polymerization rate and the relative contribution from the two reaction loci is thus also dependent on the radical interchange mechanisms. During batch polymerization, the monomer and initiator concentrations in both the phases change with time. Rates of radical generation and interphase transport also change with time. In addition, the nature of the continuous phase changes due to monomer depletion, while reaction conditions inside the particles also change as the polymer volume fraction increases with monomer conversion. Most polymers are soluble in their monomer, so that at the beginning of a batch reaction radicals can grow to a relatively large size before becoming insoluble in the media. As the reaction proceeds, however, the monomer concentration in the media is reduced by polymerization and radicals in the continuous phase cannot grow to their previous size in solution. A kinetic model has been developed by Lu (1988), based on the assumption that the mechanism of radical absorption by polymer particles is by precipitation of oligomer radicals to form primary particles, which are then instantaneously absorbed. Simulation of seeded dispersion polymerization based on this kinetic model was moderately successful in predicting the observed conversion-time profile at the lowest initiator concentration, although adjustment of parameters in the polymerization rate coefficient models was necessary. Prediction of monomer conversion profiles at higher initiator concentrations was not satisfactory. It was possible, however, to match the conversion profile at higher initiator concentrations by adjusting the radical entry rate coefficient downward at increasing initiator levels. No explanation for this phenomenon was advanced. 2. Model Development A mathematical model has been developed to predict the rate of polymerization and particle growth for seeded dispersion polymerization in polar organic media, and a model validation method employing competitive growth of two different sized and monodisperse seed particle populations has been utilized. Polymerization of monodisperse seed particles avoids the complicacy of the nucleation process and also simplifies the particle growth modeling. Competitive growth of bimodal seed dispersions has been employed as a tool for the qualitative understanding of kinetic mechanisms in emulsion polymerization (Vanderhoff et al., 1956). This technique has been adopted in this development to test model prediction and to improve the model parameter estimation results. Figure 1 is a schematic of seeded dispersion polymerization, illustrating the kinetic events taking place during the particle growth phase. Radicals initiated in the continuous phase continue to grow by chain propagation until they either undergo termination with other radicals or are captured by polymer particles. However, radicals that grow to a critical chain length at which
Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 2599
Figure 1. Schematic representation of seeded dispersion polymerization.
they become insoluble in the continuous phase precipitate to form unstable primary particles. These primary particles undergo homocoagulation with other primary particles before they are eventually captured by a seed particle. Radicals in solution in the continuous phase, before they reach the critical size, may be captured by seed particles or by primary particles. The primary particles are small enough, so that only one radical center may exist at any time. Polymerization Model. Rate of polymerization in the continuous phase is determined by accounting for polymerization taking place in solution as well as in the primary particles. Rate of polymerization in the disperse phase is accounted for separately for each of the particle populations in the bimodal seed dispersion. The rates of polymerization in the continuous phase, RpC, and in the two particle phases, RpD, are given by the following equations:
RpC ) VCkpC[M]C[MT*]C + kpD[M]DNP*n j
(1)
RpD ) RpD1 + RpD2 ) VD1kpD[M]D[MT*]D1 + VD2kpD[M]D[MT*]D1 (2) VC and VD are the total volumes of the continuous and disperse phases, in dm3 per dm3 of ethanol, kp is the rate coefficient for chain propagation, NP* is the molar concentration of primary particles per dm3 of ethanol, and n j is the average number of radicals per primary particle. [M] and [MT*] denote the concentrations of monomer and radicals, respectively. Subscripts C and D denote the continuous and disperse phases, while 1 and 2 refer to the two seed particle populations. The rate coefficient for propagation inside the primary particles and the monomer concentration inside these particles have been assumed to be similar to those inside the seed particles. The overall rate of polymerization is the sum of polymerization in the two reaction loci.
Knowledge of the monomer and radical concentrations and the propagation rate coefficients in each phase is necessary for calculation of the polymerization rates. Subsequent sections describe how the various concentration terms are evaluated from molar balances and thermodynamic equilibrium considerations. Seed particles are swollen with monomer and solvent, and thermodynamic equilibrium is assumed to be maintained between the continuous and particle phases during the reaction. Initiator is assumed to be equally distributed into both phases, and radical initiation occurs in both reaction loci. The total concentration of radical species in each reaction loci is obtained from balances for each radical species in that phase. Making the quasi-steady-state assumption with respect to the concentration of each radical species allows expressions to be derived for overall radical concentrations. The radical concentration in the continuous phase is augmented by radicals that desorb from polymer particles. In this development, it has been assumed that only monomeric radicals generated by chain transfer to monomer can undergo desorption and that after desorption these radicals have the same fates as the initiator-derived radicals with one monomer unit in the aqueous phase. Desorbed monomeric radicals are not distinguished from radicals that are generated by initiator dissociation and are included in the molar balance of radicals with one monomer unit. The radical concentration in the continuous phase is determined by a summation of balances for radicals of all chain lengths. A detailed development of radical and primary particle balances is given by Ahmed (1996). From a balance of primary radicals in the continuous phase and by application of the quasi-steady-state approximation, the monomeric radical concentration in the continuous phase is given by
[M1*]C )
j RiC + kdNP*n kpC[M]C + ktC[MT*]C + ka1SNS* + ka1PNP* (3)
where RiC is the rate of generation of initiator radicals in the continuous phase, kd is the rate coefficient for desorption of monomeric radicals from primary particles, and ka1S and ka1P are the rate coefficients for absorption of radicals of chain length one into seed and primary particles. ktC is the average rate coefficient for chain termination in the continuous phase. Solubility of the growing oligomer in the continuous phase starts to decrease as their chain lengths increase. These oligomers fall out of solution as their chain length reaches a critical value Z. Hence, the largest radical in solution is of chain length Z - 1. From a balance on the oligomer radicals of chain length Z - 1 and substituting eq 3, the following expression for the concentration of the largest radical in solution is obtained.
[MZ-1*]C ) {kpC[M]C}Z-2(RiC + kdNP*n j) {kpC[M]C + ktC[MT*]C + kaSNS* + kaPNP*}Z-1
(4)
A balance on the total radical concentration in the continuous phase is obtained by summing the balances
2600 Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997
for each radical species in solution.
d[MT*]C ) RiC + kdNP*n j - kpC[M]C[MZ-1*]C - 2ktC dt [MT*]C2 - kaSNS*[MT*]C - kaPNP*[MT*]C (5) Primary particles are generated as growing oligomers and become insoluble in the continuous phase and precipitate from solution. These collapsed coils form a separate microphase by absorbing some monomer and solvent molecules from the continuous medium. Primary particles are unstable and can undergo heterocoagulation with seed particles or homocoagulation with other primary particles. A primary particle may undergo homocoagulation with several primary particles before being absorbed by a seed particle. At the same time, a primary particle can adsorb grafted or ungrafted stabilizer chains while continuing to grow in size until it becomes a stable particle. However, when the stable polymer particle concentration is sufficiently high, most of the primary particles are captured by the much larger stable particles, and very few new stable particles are formed. Molar balance of primary particles in the continuous phase is given by eq 6, where the first term accounts for generation from growing radicals in solution and the second and third terms account for depletion by heterocoagulation with seed particles and homocoagulation with other primary particles. kfP is the rate coefficient
dNP* 1 ) VCkpC[M]C[MZ-1*]C - kcSNS*NP* - kfPNP*2 dt 2 (6) for homocoagulation of primary particles and kcS is the rate coefficient for heterocoagulation of primary and seed particles. Radicals initiated inside the particles or absorbed from the continuous phase can either grow by monomer addition, terminate with another radical, or chain transfer to monomer to form a monomeric radical. Monomeric radicals formed by initiation or chain transfer may either be desorbed from the particle or undergo propagation, termination, or further chain transfer. Desorption of radicals from the relatively very large seed particles has not been accounted for in the radical balance, as the time required for diffusion to the surface from the interior of a seed particle will be much larger than the average time required for addition of one or more monomer units. Balances on the total radical concentrations in each of the two seed particle populations result in the following expressions for the radical concentrations in each population:
[MT*]D1 )
{
}
VD1RiD + kcS1NS1*NP*n j + VCkaS1[MT*]CNS1* 2ktDVD1
[MT*]D2 )
{
}
VD2RiD + kcS2NS2*NP*n j + VCkaS2[MT*]CNS2* 2ktDVD2
1/2
(7)
1/2
(8)
Kinetic parameters controlling the polymerization behavior are influenced by the physical environment of the reacting system. Such molecular processes as
segmental diffusion of radical centers, the onset of chain entanglement, translational diffusion of the entangled chains, diffusion of radical centers by propagation, and diffusion of monomer to the propagation sites become important at various stages of the homopolymerization reaction. Due to the nature of the reaction environment during dispersion polymerization, most of these processes are relevant over the entire course of the particle growth phase. Polymerization in the continuous phase is approximated by low-conversion homopolymerization in solution. Values of rate coefficients for propagation and termination at the lower conversion region, as obtained from bulk or solution homopolymerization, were used for the continuous-phase reaction. Polymerization inside the particles resembles highconversion bulk polymerization when both translational diffusion and entanglement effects, as well as diffusioncontrolled propagation, may be important. The entanglement and chain length effect may be offset to some extent due to swelling of the particles by both monomer and solvent. The bimolecular termination reaction inside the particles is limited by the translational diffusion of the radical chains (gel effect), the rate of individual termination reactions being dependent on the chain lengths of the terminating radicals and on the polymer volume fraction and the average molecular weight of the polymer matrix (entanglement effect). Conventional kinetic rate equations must be modified by using a proper average termination rate constant, kt. The rate coefficient for chain propagation, kp, is independent of molecular weight and volume fraction of polymer formed until the polymer volume fraction of the reaction mixture attains a high value near the glass transition point of the mixture. In the low- and intermediate-conversion region, propagation by monomer addition is controlled by the activation barrier of the reaction. The measured propagation rate coefficient is assumed to be constant and equal to kp0, the rate coefficient at zero conversion. In the high-conversion region, however, propagation is thought to be controlled by monomer diffusion. Kinetic studies of bulk polymerization by the rotating sector method (Sack et al., 1988), of emulsion polymerization (Ballard et al., 1986) and bulk polymerization (Carswell et al., 1992) by ESR spectroscopy, indicate that the propagation rate coefficient starts to decrease with conversion well before the mixture glass transition point is approached. This has been attributed to the propagation reaction switching from being chemically rate determining to diffusion controlled. Arai and Saito (1976), Marten and Hamielec (1978), and Sundberg et al. (1980) proposed a semiempirical relation in terms of the fractional free volume, vf, of the system to account for the reduction in kp with increasing conversion.
[ {
kp ) kp0 exp -B
1 1 vf vfC2
}]
(9)
Both B and vfC2, the free volume at which kp begins to decrease, are treated as adjustable parameters. In the low-conversion region, the rate of termination is expected to be dominated by interactions involving at least one chain which is short, and undergoing rapid translational diffusion. Such interactions are likely to be controlled by segmental diffusion of the radical center, and they will depend only very weakly on chain
Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 2601
length, average molecular weight, or volume fraction of polymer. The corresponding value of the termination rate coefficient is denoted by kt0. During dispersion polymerization, polymer formed in the continuous phase is continually taken up by the particles. Polymerization in the continuous phase throughout the reaction resembles solution polymerization in the low-conversion region. In this polymerization model, ktC, the rate coefficient for termination in the continuous phase is taken to be equal to kt0. At intermediate conversions, the rapid acceleration of polymerization rate is generally attributed to a slowing down of the rate of termination due to increased viscosity of the polymerization medium. Rate of termination is controlled by the translational diffusion of the terminating chains in close proximity of each other. This process is influenced by chain lengths of the terminating radicals, volume fraction of polymer in the system, and their average molecular weight. The overall termination rate coefficient is the average of all the individual termination rate coefficients, ktij, where i and j are the chain lengths of the terminating radicals. Various kinetic models based on the free-volume theory have been put forward to account for diffusion control of termination reactions (Marten and Hamielec, 1982; Chiu et al., 1983; Soh and Sundberg, 1982a,b; Achilias and Kiparissides, 1988, 1992). Marten and Hamielec proposed the following relation for the rate coefficient of diffusion-controlled bimolecular termination reaction.
( ) [ { }]
0
kt ) kt
Mwc1
1.75
exp -A
Mw
1 1 vf vfC1
(11)
The parameter Crd has been defined by the authors in terms of the sweep out volume of a radical chain end. In a recent study of the performance of the freevolume kinetic models, Vivaldo-Lima et al. (1994) found that the “serial” approach to modeling of the termination rate coefficient was superior in monomer conversion and molecular weight prediction in the high-conversion region and for styrene polymerization where the gel effect is not so strong. In contrast to the “parallel” approach where the rate coefficient is assumed to be dominated by a single mechanism, the authors proposed additive contributions from both the mechanisms. Thus, the rate coefficient for termination in the intermediateand high-conversion region has been defined as
kt ) ktn + ktp
[ {
(12)
1 1 vf vfC1
ktn ) kt0 exp -A
}]
(13)
The rate coefficent for termination in the disperse phase throughout the dispersion polymerization of styrene, ktD, has been represented by the termination rate coeffcient model expressed by eqs 11-13. The rate of initiation is expressed in terms of a rate coefficient for initiator decomposition, kI, and an initiator efficiency factor, f. All initiator decomposition products do not initiate polymer chains, and the efficiency factor accounts for the radical loss process due to recombination or side reactions. Calculation of the efficiency factor from bulk polymerization data using kp, kt, and kI values obtained by independent methods indicates that f decreases exponentially at high polymer volume fraction. The extension of the Hamielec model proposed by Vivaldo-Lima et al. (1994) included an expression for f in terms of the free volume of the system.
{ (
f ) f0 exp -D
(10)
where vfC1 is the free volume at the onset of the gel effect and A is an adjustable parameter. This model requires simultaneous modeling of the molecular weight of the polymer matrix. At a high-conversion level, the rate of polymerization levels off and then starts decreasing. The region where the rate essentially remains constant at a high value has been attributed to a leveling off of the termination rate coefficient. This leveling off of the termination rate coefficient occurs as the radical chains become essentially immobile, and termination occurs mainly due to diffusion of the chain ends by propagation. An expression for the termination rate coefficient due to propagation has been proposed by Soh and Sundberg
ktp ) Crdkp[M]
As noted by Zhu and Hamielec (1989), the use of termination rate coefficients based on free-volume considerations alone will make adequate predictions of reaction rate and number-average molecular weight but not of weight-average molecular weight evolution. In the dispersion polymerization rate model used in this study, a simplified version of the Marten and Hamielec expression (eq 10) based on free-volume changes alone has been used.
1 1 vf vfo
)}
(14)
f0 is the efficiency factor at low conversion, while vfo is the free volume at zero conversion and D is an adjustable parameter. The volume fractions of monomer and solvent in both phases and the total volumes of the continuous and particle phases are obtained from free-energy relations and volume balances developed for dispersion polymerization of styrene in ethanol, using PVP as the stabilizing species (Lu et al., 1988). In this development, it was assumed that the monomer and solvent partitionings in both the particle populations and the primary particles are identical. The free-energy contribution due to the interfacial tension term for particle sizes in the 1-5 µm range in the ethanol-styrene system is small compared to contributions from the other terms. The rates of radical absorption by the two seed particle populations and by the primary particles, the rate of radical desorption from primary particles, and the rates of homocoagulation of primary particles and heterocoagulation of primary and seed particles are needed in order to calculate the radical and primary particle concentrations during competitive growth polymerization. Hansen and Ugelstad (1976) applied Fick’s law of diffusion to the absorption process, taking into account the concentration gradient of radicals at the particle surface, and found that, for large particles, the rate of absorption of a radical species of length j is given by the following expression:
FAj ) kajNS[Mj*]C
(15)
kaj ) 4πDCjrSFj
(16)
where
2602 Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997
kaj is the absorption rate coefficient expressed in dm3 per particle per second, rS is the swollen radius of the polymer particle, and DCj is the diffusion coefficient of a radical of length j in the surrounding medium. Fj is a correction factor to account for the reversibility of radical absorption and potential barrier to absorption. The total absorption rate is expressed in terms of an average absorption coefficient, kaS,
FA ) kaSNS[MT*]C
DPC )
Z-1k
(17)
∑ j)1
aj[Mj*]C
[MT*]C
The absorption coefficient is, in turn, expressed in terms of an effective mean diffusion coefficient for radical capture, DC, which includes the correction factor Fj.
kaS ) 4πDCrS
dvP 1 dMP ) dt NSdP dt
kd )
3DwδCm mdrP2
(19)
Dw is the diffusion coefficient of monomeric radical in the continuous phase, Cm is the ratio of the rate coefficient for chain transfer to monomer to the propagation rate coefficient, and md is the partition coefficient of monomeric radicals between the particle surface and continuous phase; δ is the ratio of mass-transfer resistance in the continuous phase to that inside the particle. During seeded dispersion polymerization, the unstable primary particles undergo homocoagulation with other primary particles but are eventually taken up by the stable polymer particles by heterocoagulation. Coagulation of primary particles is likely to be diffusion controlled, and the rate coefficient is given by the Smoluchowski relations (Hansen and Ugelstad, 1979a,b).
kfP )
kCS )
16πDPCrPNA WPP
(20)
4πDPC〈rS〉NA WPS
(21)
WPP and WPS are the Fuchs stability factors, while DPC is the diffusion coefficient of primary particles in the continuous phase and is given by the following relation.
(23)
where vP is the volume of polymer in a particle, NS is the seed particle number density in number of particles per dm3 of solvent, MP is the total polymer mass in grams per dm3 of solvent, and dP is the polymer density. Any polymer generated in the continuous phase is eventually transferred into the particle phase. The amount of polymer in the continuous phase is small and has been neglected. The rate of generation of polymer mass is given by
(18)
In this study the rate coefficients for absorption of radicals from the continuous phase by each of the seed populations were defined in molar terms, following eq 18. A similar equation was used to define the rate coefficient for radical absorption by primary particles, kaP. Nomura and Harada (1981) derived the following expression for the rate coefficient for radical desorption by taking into account the rate of generation of monomeric radicals by chain transfer in polymer particles containing only one radical at a time and the diffusion of this monomeric radical into the continuous phase.
(22)
kb is the Boltzmann constant and η is the viscosity of the continuous phase. Particle Growth Model. The rate of volume growth of polymer with time t in a monodisperse seed particle during heterogeneous polymerization can be expressed as
where
kaS )
kbT 6πηrP
dMP d[M] ) -MM ) MMRP dt dt
(24)
For a bimodal particle population, eq 24 can be rewritten as
NS1
( )
dvP1 dvP2 MM + NS2 ) [RPD + RA] dt dt dP
(25)
NS1 and NS2 are the number densities of the two seed particle populations. The rate of polymer absorption by the seed particles, RA, is assumed to be proportional to the radius of the seed particles and is given by the following relation. Since all the polymer formed in the continuous phase is assumed to be taken up by the particles, RA is equal to RPC.
RA )
[
]
RPC NS1v1y NS2v2y + NS 〈vy〉 〈vy〉
(26)
where
NS1v1y + NS2v2y 〈v 〉 ) NS1 + NS2 y
and NS ) NS1 + NS2. v1 and v2 are the swollen volumes of the seed particles, and exponent y is equal to 1/3. Therefore, the rate of particle growth for each particle population can be written as follows:
( )[
]
(27)
( )[
]
(28)
dvP1 MM RPC NS1v1y ) RPD1 + dt NS1dP NS 〈vy〉 and
MM RPC NS2v2y dvP2 ) RPD2 + dt NS2dP NS 〈vy〉
Continuous Polymerization. Steady-state polymerization in a well-mixed continuous reactor allows the modeling of reaction kinetics in a nondynamic reaction environment and has been extensively employed for the
Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 2603
study of emulsion polymerization kinetics and for model validation (Degraff and Poehlein, 1971; Greene et al., 1976; Kirillov and Ray, 1978; Nomura and Harada, 1981). Poehlein (1982b) presented a general mathematical framework for simulation of the polymerization kinetics and latex particle size distribution in seeded emulsion polymerization in a steady-state, isothermal CSTR. This mathematical model adapted the theories of Ugelstad and co-workers (Ugelstad and Hansen, 1976; Ugelstad et al., 1967) to the particle age distribution and accounted for the absorption and desorption of free radicals from latex particles and termination in the aqueous phase. Lee and Poehlein (1987) and Mead and Poehlein (1988, 1989) obtained values for radical absorption and desorption rate coefficients for emulsion polymerization of various monomers. The distribution of particle volumes in a CSTR is broadened due to the broad exit age distribution and is related to the age distribution and the rate of particle growth by the following relation:
U(vP) )
f(τ) (dvP/dτ)
(29)
τ is the dimensionless residence time of particles in the reactor. The rate of radical absorption by particles and, as such, the rate of volume growth are dependent on the particle size. The effluent particles are distributed over a range of size classes, depending on the measuring technique employed. An expression for polymer particle volume growth can be obtained from a polymer mass balance in the CSTR, based on the assumption of constant particle number density and equal inflow and outflow rates. The rate of particle volume growth of a particle in size class q is given by eq 30.
( )
dvPq θMM ) (R + RAq) dτ dPNSq PDq
(30)
where RPDq is the total rate of polymerization in all particles in the qth size class, and RAq is the rate of polymer absorption from the continuous phase by all the particles in the qth range. On the basis of the continuum diffusion theory, the rate of absorption is assumed to be proportional to v1/3. The particle growth equation is rewritten in the form of eq 31.
( )(
)
qMM RPC NSqvq1/3 dvPq ) RPDq + dt dPNSq NS 〈v1/3〉
(31)
where k
U(vP) )
sIU′(vPi) ∑ i)1
(32)
The radical concentration within particles in any size class is assumed to be the same for all particles in this class but different from the radical concentration inside particles in other size classes. The particle size distribution and particle growth equations presented above have been derived on the basis of a uniform seed particle size. However, seed dispersions with narrow size distributions, even though monodisperse by definition (PDI of 1.01 or less), are composed of particles whose diameters may span across several size classes of the measuring technique em-
ployed. The seed distribution is considered to be comprised of several monodisperse populations corresponding to the mean size of the size classes. The particle size distribution and growth equations (eqs 29 and 31) are reformulated as follows: k
U(vP) )
siU′(vPi) ∑ i)1
(33)
where
U′(vPi) )
f(τi) (dvPq/dτi)
(34)
Here si denotes the particle number density fraction of seed size class i, and k is the total number of seed size classes. The volume growth rates of particles in the qth class are the same irrespective of the seed size class i that they are arising from. However, the age distribution of particles in the qth class will vary for particles that arise from different seed classes. The models presented in this paper have been used to simulate bimodal seeded batch and seeded continuous polymerization experiments. A comparison of model predictions and experimental results is presented in part 2 of this paper. Nomenclature DCj: diffusion coefficient of radicals of chain length j in the continuous phase, dm2/s DC: mean effective diffusion coefficient for radical absorption by particles, dm2/s DPC: diffusion coefficient of primary particles in the continuous phase, dm2/s DW: diffusion coefficient of monomeric radicals in the continuous phase, dm2/s dP: density of polymer, g/dm3 f: initiation efficiency [I]: initiator concentration, mol/dm3 kaj: rate coefficient for absorption of radical of chain length j, dm3/mol‚s ka: average rate coefficient for radical absorption, dm3/ mol‚s kd: rate coefficient for radical desorption, s-1 kI: rate coefficient for initiation, s-1 kP: rate coefficient for chain propagation, dm3/mol‚s kt: average rate coefficient for chain termination, dm3/ mol‚s ktp: rate coefficient for termination by propagation, dm3/ mol‚s kCS: rate coefficient for heterocoagulation of primary and seed particles, dm3/mol‚s kfP: rate coefficient for primary particle homocoagulation, dm3/mol‚s MM: molecular weight of monomer, g/mol Mp: total mass of polymer, g/dm3 M0: mass concentration of monomer at zero conversion, g/dm3 [M]: monomer concentration, mol/dm3 [Mj*]: concentration of radicals of chain length j, mol/dm3 [MT*]: total radical concentration, mol/dm3 n j : average number of radicals in a primary particle NP: number density of primary particles, particles/dm3 NP*: molar density of primary particles, mol/dm3 NS: number density of seed particles, particles/dm3 NS*: molar density of seed particles, mol/dm3(ethanol) Ri: rate of radical initiation, mol/dm3‚s RP: rate of polymerization, mol/dm3‚s
2604 Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 rP: radius of primary particle, dm rS: radius of stable polymer particle, dm VC: total volume of continuous phase, dm3/dm3(ethanol) VD: total volume of disperse phase, dm3/dm3(ethanol) vP: volume of polymer in a seed particle, dm3/particle v: volume of swollen seed particle, dm3/particle vf: free volume inside a polymer particle WPP: stability factor for primary-primary interaction WPS: stability factor for primary-seed interaction Greek Symbols θ: mean residence time, min τ: dimensionless residence time, θ/t Fa: rate of radical absorption, mol/dm3‚s Subscripts C: D: F: 1: 2:
continuous phase disperse phase feed seed population 1 seed population 2
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Received for review August 14, 1996 Revised manuscript received April 10, 1997 Accepted April 11, 1997X IE960505R
X Abstract published in Advance ACS Abstracts, June 1, 1997.
