Nonhomogeneous Mixing Population Balance Model for the

Article pubs.acs.org/IECR

Nonhomogeneous Mixing Population Balance Model for the Prediction of Particle Size Distribution in Large Scale Emulsion Polymerization Reactors Aleck H. Alexopoulos,‡ Prokopis Pladis,‡ and Costas Kiparissides*,†,‡,§ †

Department of Chemical Engineering, Aristotle University of Thessaloniki, P.O. Box 472, 541 24 Thessaloniki, Greece Chemical Process and Energy Resources Institute, CERTH, Thessaloniki, Greece § Department of Chemical Engineering, The Petroleum Institute, Abu Dhabi, UAE ‡

ABSTRACT: The present study describes the development of a nonhomogeneous two-compartment model for the prediction of particle size distribution in a semibatch emulsion ter-polymerization reactor. The multicompartment model accounts for spatial variations of particle size distribution (PSD) in the reactor due to nonideal mixing conditions. A comprehensive emulsion polymerization model is applied to each compartment, which allows the calculation of the various species concentrations in the aqueous and particle phases in each compartment. Moreover, a particle population balance equation is solved for each compartment to determine the individual PSDs as well as the overall PSD in the reactor. The effects of the two-compartment nonhomogeneous model parameters, that is, the volume ratio of the two compartments, the compartment exchange flow rates, and the partitioning of the monomer and initiator feed streams into the two compartments, on the overall polymerization rate and PSD are analyzed in detail. It is shown that depending on the selected values of the two-compartment model parameters, the overall PSD in the reactor can significantly vary (i.e., from a narrow and/or broad unimodal distribution to a bi- and/or multimodal PSD). Small compartment exchange flow rates, uneven monomer and initiator feed partitioning, or unequal compartment volumes can result in very different PSDs in the two compartments. Moreover, it is shown that for a range of parameter values in the two-compartment model (i.e., reflecting the degree of reactor nonhomogeneity), the calculated overall PSD in the industrial-scale reactor can be unimodal but significantly broader than the respective PSD calculated by the homogeneous one-compartment model.

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INTRODUCTION Emulsion polymerization is an important industrial process commonly employed in the manufacture of particulate polymer dispersions.1−3 Emulsion polymerization is a highly complex process involving polymerization kinetics coupled with multiphase thermodynamic equilibrium calculations and heat/mass transfer phenomena occurring at different spatial and temporal scales.4−8 At the molecular scale, emulsion polymerization is characterized by the pertinent polymerization kinetics (i.e., initiation, propagation, termination reactions, etc.) and thermodynamic calculations of species concentrations in the various phases present in the system. At the particle scale, the process is dominated by particle nucleation and growth, particle stabilization via the adsorption of surfactant molecules, transfer of radicals between the particles and the continuous aqueous phase, and the associated particle heat and mass transfer processes. At the mesoscale, particle population dynamics and particle−particle interactions as manifested by the particle size distribution (PSD) and the polymer solids fraction largely affect the rheology of the dispersion which in turn influences the mixing conditions in the reactor as well as the particle aggregation rate. Finally, at the reactor scale, emulsion polymerization is influenced by macro-mixing and heat-transfer processes. In experimental investigations of emulsion polymerization carried out in well-stirred lab-scale reactors, homogeneous mixing conditions can be assumed.4,9 On the other hand, when © 2013 American Chemical Society

inadequate mixing occurs in industrial-scale polymerization reactors, the final latex product is typically of lower quality, that is, its PSD is broader than the corresponding PSD of the latex produced in lab-scale reactors even when an identical emulsion polymerization recipe is implemented. In general, these product differences have been attributed to nonideal mixing conditions in industrial-scale reactors.1 In high-solids emulsion polymerization reactors, the polymer volume fraction and the viscosity as well as the pseudoplastic nature of the dispersion increase during polymerization, resulting in a significant change in mixing efficiency and homogeneity.10 Specifically, the energy delivered by the impeller to the surrounding dispersion is dissipated in a smaller region around the impeller. In industrial-scale emulsion polymerization reactors a number of problems associated with nonhomogeneous mixing conditions have been identified, including formation of broad PSDs, particle settling, coagulum formation, and the appearance of temperature hot spots in the reactor.1,6,10 Consequently, there is a pressing need to understand, describe, and quantify the nonhomogeneous mixing phenomena appearing in large-scale emulsion polymerSpecial Issue: John MacGregor Festschrift Received: Revised: Accepted: Published: 12285

December 17, 2012 February 20, 2013 February 21, 2013 March 11, 2013 dx.doi.org/10.1021/ie303500k | Ind. Eng. Chem. Res. 2013, 52, 12285−12296

Industrial & Engineering Chemistry Research

Article

variables in the emulsion polymerization reactor, including the concentrations of the various species (i.e., monomer(s), initiator, surfactant(s), and radical concentrations in both the aqueous and the particulate polymer phase), were allowed to vary in each compartment. Consequently, the relevant particle nucleation, growth, and coagulation rates could vary in each compartment, which in turn led to different PSDs in the two compartments. Thus, the overall PSD of the latex in the industrial-scale reactor was in general broader (or even bimodal) than the respective PSD measured in a lab-scale reactor for the same emulsion polymerization recipe. Clearly additional compartments can be extracted from CFD simulation results to represent the complicated fluid flow in an emulsion polymerization reactor. The two-compartment approach is the simplest possible representation of imperfect mixing, and it provides useful information that demonstrates the effect of imperfect mixing on key reaction characteristics such as the PSD. Consequently, the two-compartment approach can also be employed as an initial simulation to detect possible effects of imperfect mixing and also as a guideline for development of more complicated multicompartment models. In what follows, the proposed two-compartment modeling approach is detailed. The key parameters of the twocompartment model (i.e., the compartment volume ratio, the two compartment exchange flow rates, and the monomer and initiator feed stream partitioning coefficients) are first defined. Subsequently, several monomer and initiator feed policies and nonhomogeneous reaction mixing conditions are identified and their effects on the overall PSD in an industrial-scale emulsion polymerization reactor are assessed using a computer simulated model of the real industrial process.

ization reactors in order to control the reactor operating conditions and, thus, the end-product quality (i.e., PSD). A great number of segregated mixing and multicompartment models have been proposed in the literature11−20 to address the issues associated with nonhomogeneous mixing conditions in industrial-scale reactors. Baldyga et al.16 developed a multizone modeling approach to describe the nonhomogeneous mixing conditions in a stirred crystallizer and determined the effects of several operating parameters (i.e., addition feed rates, mixing intensity, compartment volume ratios) on the crystal size distribution. Alopaeus et al.17 used a multizone approach to determine the droplet size distribution in large-scale liquid− liquid dispersions. The individual compartment energy dissipation rates and exchange flow rates were determined from computational fluid dynamics (CFD) calculations.18 Generally, the relevant compartment parameters can be calculated from information on the fluid flow field or can be determined from more detailed CFD simulations (i.e., kinetic energy dissipation rates and exchange flow rates). Usually, these compartment parameters will be time-dependent due to timevarying process conditions (i.e., dispersion viscosity, solids concentration, etc.). CFD simulations have been also employed to describe nonhomogeneous mixing conditions in several systems19−26 and determine the number and individual compartment volumes. Despite the benefits of automatically generating a multicompartment model to describe nonideal mixing phenomena in large-scale reactors, multicompartment approaches have not been implemented to emulsion polymerization systems mostly due to the difficulty in calculating the dynamic evolution of the particle size distribution. It should be noted that the optimal number of required compartments to adequately resolve the nonideal mixing conditions in an industrial-scale reactor depends on many factors (e.g., polymerization process conditions, vessel and impeller design, timevarying viscosity of the dispersion, etc.). Moreover, the specific advantages of a multicompartment model over the more classical segregated mixing models are yet to be established.27 In some cases, CFD multicompartment models have been shown to result in similar or even inferior performance compared to the results obtained via the implementation of classical segregated mixing models.21 The present paper deals, for the first time, with the development of a two-compartment nonhomogeneous mixing model to calculate the dynamic evolution of PSD in large-scale emulsion polymerization reactors. The proposed two-compartment model provides a simple description of nonhomogeneous mixing conditions in an industrial-scale emulsion polymerization reactor by assuming that the nonhomogeneity can be approximated by two homogeneous compartments having different volumes, connected together by two exchange flow rates.28 Each compartment is characterized by its volume and a nonhomogeneous property of the flow field (e.g., the turbulent kinetic energy dissipation rate, the dispersion viscosity or the average shear rate), characteristic of the mixing conditions in each compartment. In particular, the two-compartment modeling approach developed in this study was employed to describe the dynamic behavior of an industrial-scale nonhomogeneous semibatch reactor for the emulsion copolymerization of butyl acrylate with methyl methacrylate (BuA/MMA). The reactor included two separate feed streams, namely, a monomer pre-emulsion feed stream as well as an initiator feed input. All the key process

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TWO-COMPARTMENT NONHOMOGENEOUS MIXING MODEL According to the proposed two-compartment nonhomogeneous mixing model (see Figure 1), the total reaction volume

Figure 1. Schematic representation of a nonhomogeneous, twocompartment emulsion copolymerization reactor. FI and FP denote the respective initiator and monomer pre-emulsion feed streams.

VT is assigned to two separate homogeneous reaction zones, namely V1 and V2, that are connected together via the two exchange flows, namely F12 and F21. Following the operational procedure of an industrial semibatch emulsion copolymerization reactor, the total initiator and monomer pre-emulsion feed streams are assumed to be partitioned into the two compartments according to the initiator and pre-emulsion feed coefficients, αI and αP, defined by eqs 1 and 2, respectively: αI = FI,1/(FI,1 + FI,2) = FI,1/FI 12286

(1)

dx.doi.org/10.1021/ie303500k | Ind. Eng. Chem. Res. 2013, 52, 12285−12296


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Figure 2. Possible monomer and initiator feeding policies to a semibatch emulsion copolymerization reactor (Compartment parameters: (a) λ < 0.5, αI = 1.0, aP = 1.0; (b) λ < 0.5, αI = 0, aP = 0; (c) λ < 0.5, αI = 1.0, aP = 0; (d) λ < 0.5, αI = 0, aP = 1.0).

αP = FP,1/(FP,1 + FP,2) = FP,1/FP

compartment volume ratio will vary with time. In general, the dynamic compartment volume ratio, λ(t), will depend on the two inflow feed rates (i.e., FI and FP), the two exchange flow rates (i.e., F12 and F21), the dynamic evolution of the density of the polymerization mixture, and the spatial variation of the local shear rate. By neglecting the time variation of αI, αP, F12, and F21 (i.e., assuming that they are constant), one can easily show that the time-dependent compartment volume ratio will be given by

(2)

where FI,i is the volumetric flow rate of the initiator feed stream into the “i” compartment and FI denotes the total initiator volumetric flow rate. Similarly, Fp and FP,i indicate the total flow rate of the pre-emulsion feed stream and the flow rate into the “i” compartment, respectively. The volume fraction of compartment “1” with respect to the total reaction volume will be given by λ = V1/VT = V1/(V1 + V2)

λ(t) = [λ(0)VT(0) + (αIFI + αPFP + F21 − F12)t ]/VT(t )

(3)

(4)

Thus, the parameters αI, αP, F12, F21, and λ, are the key design/ operational parameters in the two-compartment model. In general, the volumes of the two compartments (V1 and V2), the two exchange flow rates (F12 and F21), and the two feed partitioning coefficients (αI and αP) will not be equal (i.e., λ(t) ≠ 0.5, αI(t) ≠ αP(t), and F12(t) ≠ F21(t). In fact, the twocompartment model parameters will depend on the hydrodynamic flow conditions in the emulsion copolymerization reactor and will vary with time. For example, if the compartment volumes are identified from local shear-rate data in the reaction vessel, based on a characteristic shear rate, γ*, then the volume of the high shear-rate impeller compartment (i.e., γ > γ*) will change with the polymerization time due to the increase of the zero-shear viscosity. In principle, the time -dependence of the two-compartment model parameters (i.e., αI, αP, λ, F12 and F21) can be extracted from time-dependent CFD simulations of the fluid flow field in the nonhomogeneous emulsion polymerization system or, at least, from steady-state CFD simulations conducted at different rheological conditions corresponding to different stages of polymerization.28 In the present work, a parametric investigation with respect to the two-compartment model parameters on the emulsion polymerization kinetics and PSD developments was carried out. Note that the time-variation of the compartment volume ratio, λ, will depend not only on changes in the internal flow field but also on the values of the inflow feed streams and compartment exchange flow rates. Thus, assuming that the monomer and initiator feed streams partitioning parameters (i.e., αI and αP) are constant, one can identify the following two limiting cases. Constant Exchange Flow Rates F12 and F21. In this case, the volumes of the two compartments as well as the

where λ(0) is the initial value of the compartment volume ratio and VT(t) is the total reaction volume given by VT(t ) = V1(t ) + V2(t ) = VT(0) + (FI + FP)t

(5)

Constant Compartment Volume Ratio λ. In this case, the two exchange flow rates will be different (i.e., F21 ≠ F12). Accordingly, the value of F21 can be expressed in terms of the value of F12 using the following equation: F21 = F12 + λ(FI + FP) − αIFI − αPFP

(6)

Partitioning of Initiator and Pre-Emulsion Feed Streams. The actual inlet points (i.e., geometric positions in the vessel) of the pre-emulsion and initiator feed streams are important design parameters since they can affect the nonhomogeneous, time-varying mixing conditions in the reactor. Thus, for unequal monomer and initiator feed partitioning coefficients (i.e., αI ≠ αP), the initiator and monomer concentrations in the two compartments can be significantly different. This can in turn influence the local polymerization rate and, thus, the local heat generation rate, which can lead to the appearance of “hot-spots” and/or thermal run-away conditions. Moreover, variations in the compartment initiator and monomer concentrations can result in nonuniform particle nucleation and particle growth rates leading to nonuniform particle number concentrations and particle size distributions. Specifically, in monomer-rich regions monomer-swollen particles will tend to exhibit larger growth rates. On the other hand, initiator-rich regions will tend to exhibit larger particle nucleation rates and, thus, larger particle number concentrations. Consequently, these two limiting cases can lead to 12287



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significantly different PSDs (i.e., low particle number concentrations can result in larger size particles while large particle number concentrations can result in smaller size particles). From Figure 2 one can identify four possible partitioning scenarios for the monomer and initiator feed streams. In particular, in the cases shown in Figure 2a,b both monomer and initiator feed streams are added to the same compartment (i.e., αI = αP, see Figure 2a,b). On the other hand, in the cases shown in Figure 2c,d, the monomer and initiator streams are fed separately to the two compartments (i.e., αI ≠ αP, see Figure 2c,d). For example, in Figure 2a, a nonhomogeneous feeding policy is depicted in which both initiator and monomer streams are being fed to the smaller compartment. Note that reactor nonhomogeneity is increased when the monomer and initiator inflow streams are separately fed to the two compartments (e.g., cases shown in Figure 2c,d). It should be noted that irrespectively of the values of the parameters λ, αI, and αP, the two-compartment model resulted in identical predictions (i.e., for the overall monomers conversion and PSD) with those obtained by the singlecompartment homogeneous model for sufficiently high values of the exchange flow rates (e.g., F12, F21 ≫ FI, FP). Moreover, when the monomer and initiator feed streams were partitioned into the two compartments proportionally to the compartment volume ratio (i.e., λ = αI = αP) and the two exchange flow rates were equal (i.e., F12 = F21), then the predictions of the twocompartment model were identical to those of the singlecompartment homogeneous model even for small values of F12 and F21 (e.g., F12, F21 ≪ FI, FP). Development of a Nonhomogeneous Emulsion Copolymerization Reactor Model. In a previous study,29 a comprehensive mathematical model was developed to describe the dynamic behavior of (St/2EHA) emulsion copolymerization in a homogeneous (single-compartment) semibatch reactor. The model included a detailed kinetic mechanism, equilibrium thermodynamic equations for the calculation of molecular species concentrations in the various phases, monomer swelling, diffusion-controlled models for the termination and propagation reactions, detailed dynamic balances for all the molecular species of interest, in the polymer and aqueous phases, and a particle population balance model. Commonly, the kinetic mechanism for an emulsion copolymerization system includes the following elementary reactions:

Chain Transfer to CTA: k tsi

R ip,q + X → D=p,q + R i2 − i , i − 1

Chain Transfer to Polymer: k tpij

R ip,q + Dx , y ⎯→ ⎯ Dp,q + R xj , y

k bi

R ip,q → R ip,q

Termination by Combination: k tcij

R ip,q + R xj , y ⎯→ ⎯ Dp + x ,q + y

k tdij

R ip,q + R xj , y ⎯→ ⎯ D=p,q + Dx , y

The symbols I, R , Mj, and X denote the initiator, the primary initiator radicals, the monomers (j = 1, 2) and the chain transfer agent, respectively. Rix,y and Dx,y denote the corresponding “live” and “dead” polymer chains with degrees of polymerization x and y with respect to the two monomers (1 and 2). On the basis of the above kinetic mechanism, a set of ordinary differential equations can be derived to describe the conservation of the various molecular species in the aqueous, polymer, and emulsion phases in a semibatch reactor.29 Initiator concentration in the aqueous phase: dC Iw C dV F = I − Iw w − kIC Iw dt Vw Vw dt

⎛ Vw ⎞ m dC ie C ie dVe Fi = − − ⎜ ⎟ ∑ (k pji + k tmji)C •j wCi w dt Ve Ve dt ⎝ Ve ⎠ j = 1 ⎛ Vp ⎞ m − ⎜ ⎟ ∑ (k pji + k tmji)C •j pCi p ⎝ Ve ⎠ j = 1

(17)

CTA concentration in the emulsion phase: ⎛ Vp ⎞ m dCxe C dV F = x − xe e − ⎜ ⎟ ∑ k tsjC •j pCxp dt Ve Ve dt ⎝ Ve ⎠ j = 1

dCyw

(18)

=

Fy Vw

−

Cyw dVw Vw dt

(19)

Surfactant concentration in the emulsion phase: j = 1, 2

dCse F C dV = s − se e dt Ve Ve dt

(8)

Propagation Reactions: + Mj ⎯→ ⎯

(16)

Monomer concentration in the emulsion phase:

dt

k pij

(15)

•

Chain Initiation Reactions:

R ip,q

(14)

Termination by Disproportionation:

(7)

I → 2R*

R* + Mj ⎯→ ⎯

(13)

Electrolyte concentration in the aqueous phase:

kI

R 2j − j , j − 1,

(12)

Intramolecular Chain Transfer:

Primary Radical Formation:

k pjj

(11)

R pj + 2 − j ,q + j − 1

Monomer concentration in the “dead” polymer chains: dCiqe

(9)

dt

Chain Transfer to Monomer(s):

=−

⎛V ⎞ m Ciqe dVe + ⎜ w ⎟ ∑ (k pji + k tmji)C •j wCi w Ve dt ⎝ Ve ⎠ j = 1

⎛ Vp ⎞ m + ⎜ ⎟ ∑ (k pji + k tmji)C •j pCi p ⎝ Ve ⎠ j = 1

k tmij

R ip,q + Mj ⎯⎯⎯→ D=p,q + R 2j − j , j − 1

(20)

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(21)



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“Live” polymer moments (i = 0,1,2) in the emulsion phase: dλ i e λ dV = − ie e + rλie dt Ve dt

each compartment. In the Appendix, the particle nucleation and particle growth rate functions as well as the particle aggregation kernel employed in the present study are summarized. It should be noted that due to differences in the surfactant concentration, ionic strength, temperature, etc., in the two compartments, the numerical values of the aggregation rate kernels (i.e., β1(U,V) ≠ β2(U,V)) will be different. Moreover, the dominant particle aggregation mechanism in each compartment will not be in general the same. For example, in the impeller region of the reactor, shear-induced particle aggregation will be more important than Brownian particle aggregation. Finally, the particle nucleation rates (i.e., S1(t) ≠ S2(t)) as well the particle growth rates (i.e., G1(V) ≠ G2(V)) in the two compartments can also differ, because of the differences in the molecular species concentrations (e.g., initiator, surfactant, etc.). Consequently, due to differences in the particle nucleation, growth, and aggregation rates, the calculated number density functions in the two compartments will not be in general the same (i.e., n1(V,t) ≠ n2(V,t)). Implementation of the Two-Compartment Model. In general, during polymerization, variations in the viscosity and polymer solids content can lead to changes in the reaction flow field and, thus, in the mixing conditions and the values of the two-compartment exchange flow rates. In the present work, the volumes of the two compartments (i.e., V1 and V2) were assumed to vary with time. The compartment exchange flow rates, F12 and F21, were considered to remain constant while the compartment volume ratio, λ, was assumed to be either constant or varied according to eq 4. In the proposed nonhomogeneous two-compartment model, the concentrations of the various molecular species (i.e., initiator, surfactants, monomers, and radicals in both the aqueous and particle phases) in the two compartments will be in general different. Thus, the overall concentration of species “i” in the reactor, Ci,t(t), will be given by the following equation:

(22)

“Dead” polymer moments (i = 0,1,2) in the emulsion phase: dμie dt

=−

μie dVe Ve dt

+ rμie

(23)

Concentration of LCB in the emulsion phase: d[LCB]e [LCB]e dVe =− + rLCBe dt Ve dt

(24)

where Ciw and Cip are the concentrations of the “i” monomer in the aqueous and polymer phase, respectively. Similarly, C•jw and C•jp are the respective concentrations of radicals with a “j” terminal unit in the aqueous and polymer phase. Finally, rλie, rμie, and rLCBe are the net production rates for the leading moments of NCLDs (λie and μie) and long chain branches, respectively. Detailed information on the modeling of emulsion copolymerization in a semibatch reactor can be found in the original publication by Kammona et al.29 Development of a Two-Compartment Population Balance Model. In order to determine the time evolution of the PSD in each compartment in the nonhomogeneous emulsion copolymerization reactor, a population balance approach was employed. Accordingly, a number density function, ni(V,t), representing the number of particles in a differential volume size range, V to V + dV, per unit compartment volume was introduced to describe the dynamic evolution of the PSD in the “i” compartment.30,31 According to previous developments,32 one can write the following PBE for the dynamic evolution of the number density function n1(V,t) in the first compartment. ϑn1(V , t ) ϑ[G1(V ) n1(V , t )] + ϑt ϑV = f (V ) S1(t ) +

∫V

Ci , t(t ) = λ(t ) Ci1(t ) + (1 − λ(t ))Ci2(t )

V /2

Moreover, Ci,t will be different from the value of Ci(t) calculated by the homogeneous single-compartment model. Because of the differences in the molecular species concentrations in the two compartments, the polymerization rates in the aqueous and polymer phases in the two compartments will not be equal (e.g., Rpp,1 ≠ Rpp,2, Rpw,1 ≠ Rpw,2, etc.). It should be noted that the various molecular species concentrations (i.e., monomers, surfactant(s), initiator, chain transfer agent, electrolyte, radicals) in each compartment can determined by the following general dynamic molar species conservation equation:

β1(V − U , U ) n1(V − U , t )

min

n1(U , t ) dU −

∫V

Vmax

β1(V , U ) n1(V , t ) n1(U , t ) dU

min

−

n1(V , t ) dV1(t ) dt V1(t )

+

F21(t ) n2(V , t ) − F12(t ) n1(V , t ) V1(t )

(25)

The last two terms on the right-hand side of eq 25 represent the contributions of the compartment’s volume time derivative and of the two exchange flow rates to PBE (eq 25). Note that the overall number density function nT(V,t) will be given by the volume weighted average of the calculated number density functions in the two compartments: n T(V , t ) = λ(t ) n1(V , t ) + (1 − λ(t ))n2(V , t )

(27)

dC Iw, j dt

dCie, j dt

(26)

=

=

Fi , j Vj

Fpi , j Vj

j = 1, 2

In general, nT(V,t) will be different from the number density function, n(V,t), calculated using a single-compartment model (i.e., homogeneous case). The particle growth, nucleation, and aggregation rates in eq 25 ultimately depend on the particle size and various species concentrations (e.g., monomer(s), initiator, surfactant, etc.) in

−

−

C Iw, j dVj Vj dt

Cie, j dVj Vj dt

− RIw, j

− R pie, j ;

(28)

i = 1, 2

and (29)

where CIw,j and Cie,j denote the initiator and monomer concentrations in the aqueous and emulsion phases of compartment “j”, respectively. Consequently, the various species reaction rates in each compartment need to be calculated separately. For example, RIw,j is the initiator 12289



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decomposition rate in the aqueous phase in the “j” compartment and Rpie,j is the propagation rate of monomer “i” in the emulsion phase in the “j” compartment.

Table 2. Kinetic Rate Constants for the MMA/BuA/AA Polymerization Systema

SIMULATION RESULTS AND DISCUSSION The proposed nonhomogeneous, two-compartment model was employed to describe the emulsion copolymerization of BuA/ MMA in a semibatch reactor. The polymerization was conducted under monomer-starved conditions in the presence of both anionic and nonionic surfactants (see emulsion polymerization recipe in Table 1). The total polymerization time, tadd, was controlled by the monomer addition rate.

kp11 = 7.34 × 105 exp(−1154/T) kp11/kp12 = 0.498, kp11/kp13 = 1.080 ktc220 = 1.51 × 107 exp(−352.79/T) ktm11 = 4.2 × 104 exp(−1657.27/T) potassium persulfate (KPS)

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1. butyl acrylate

2. methyl methacrylate

3. acrylic acid

kp22 = 4.92 × 105 exp(−2190.7/T)

kp33 = 65000

kp21/kp22 = 1.789, kp23/kp22 = 0.697 ktd220 = 9.8 × 107 exp(−352/T)

kp33/kp31 = 0.590, kp33/kp32 = 1.138 ktd330 = 2.6 × 106

ktm22 = 2.0 exp(−1657.27/T)

ktij = (ktiiktjj)1/2

kI = 4.56 × 1016 exp(−16860/T)

a Kinetic rate coefficients in L/(mol·s); initiator decomposition rate in s−1, T in K.

Table 1. Emulsion Polymerization Recipea total amount (g) total water total MMA total BuA total AA initiator

940 525 455 20 3

temperature initiator anionic surfactant (Disponil FES 32 IS) nonionic surfactant (Disponil A 3065)

82 °C KPS: 0.3 wt % SA = 0.65 wt % SB = 1.0 wt %

were attributed to nonideal mixing phenomena present in the industrial-scale reactor. In what follows, the effects of nonhomogeneous mixing conditions on the overall monomers conversion, the average particle size, and the overall PSD are analyzed in terms of the two-compartment model parameters. The values of the parameters were selected to reflect some probable nonhomogeneous mixing scenarios in industrial-scale emulsion polymerization reactors. In some simulations, the value of the exchange flow rate has been on purpose taken to be very small (e.g., F12 ≪ FP) to enhance the effect of nonhomogeneity. The PBEs in the two compartments were solved simultaneously using a discretized numerical approach.35−37 The discretization of the volume domain was based on a geometric rule. Typically, 60−90 discrete elements were employed over the total particle volume domain. The PBEs together with the comprehensive emulsion polymerization model were solved for both compartments simultaneously, using the IMSL program DIVPAG.29 The resulting system of DAEs was very stiff and typically required a CPU time 2−30 times greater than that of the homogeneous single-compartment model. Comparison of Homogeneous and Nonhomogeneous Mixing Models. The homogeneous (i.e., one-compartment) model was first solved for the semibatch emulsion copolymerization of BuA/MMA. In Figures 3 and 4, model predictions are compared with experimental measurements on the overall monomers conversion and mean particle diameter. Apparently, there is a good agreement between model predictions and experimental results obtained from a lab-scale reactor.33 The two-compartment model was then solved using the same emulsion polymerization recipe with the one employed in the lab-scale reactor experiments. It was found that under certain nonhomogeneous mixing conditions (i.e., as realized by the selection of the two-compartment model parameters), the nonhomogeneous model could exhibit different particle nucleation profiles (even a secondary particle nucleation) in the two compartments that resulted in significantly different PSDs. Thus, in order to show the effect of nonhomogeneous mixing conditions on the PSD in an industrial-scale emulsion polymerization reactor, the following values of the parameters in the two-compartment model (i.e., αI = 0.25, αP = 0.75, and λ = 0.5 corresponding to the feeding policy of Figure 2d) were selected. Accordingly, the exchange flow rates F12 and F21 were taken to be equal to 10−9 m3/s; that is, they were significantly

a

The reported weight percentages are based on the total monomers mass.

A series of semibatch kinetic experiments were carried out using a 5 L stainless-steel fully automated pilot-scale reactor system.33 The polymerization procedure consisted of the following steps. A predetermined amount of the anionic surfactant (Disponil FES 32 IS) dissolved in distilled water was first introduced to the reactor. Subsequently, the reactor was evacuated (e.g., up to 0.4 bar) and the reactor content was heated up to the final polymerization temperature. Accordingly, small amounts of Disponil FES 32 IS and nonionic surfactant Disponil A 3065 dissolved in water were added to the reactor, followed by the addition of a small amount of the three monomers and a specified quantity of the initiator solution. The remaining amounts of the pre-emulsion (e.g., monomers, surfactants, and water) and initiator solutions were kept under a nitrogen blanket in two separate stirred tanks. The polymerization was initially carried out in a batch mode to control the particle nucleation period and the total number of generated particles. After particle nucleation, the remaining amounts of the two solutions were fed at constant rates to the reactor with the aid of two piston pumps. The semibatch polymerization was carried under monomer starved conditions while samples were withdrawn from the reactor at regular time intervals for further analysis. The total monomer addition time could vary from 1.5 to 4.5 h. The kinetic rate constants and the values of the physical and transport properties employed in the numerical simulations are reported in Tables 2 and 3. It should be pointed out that the homogeneous polymerization model (i.e., for a single compartment) was fully validated using a comprehensive series of experimental data obtained from a lab-scale reactor.33 However, experimental results obtained from an industrial-scale reactor showed a broader PSD than the corresponding PSD in the labscale reactor despite the fact that the same polymerization recipe was employed.34 The observed differences in the PSDs 12290



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Table 3. Physical, Thermodynamic, and Transport Properties of the MMA/BuA/AA Polymerization System Sa: anionic surfactant

Sb: nonionic surfactant

Ccmc = 7.8 × 10−5 kmol/m3 ρs = 1.05 kg/m3 ρs = 1.07 kg/m3 MWs = 348 MWs = 500 rm = 2.5 nm rm = 2.5 nm KPS ρI = 2.477 kg/m3 MWI = 270.33

1 BuA

2 MMA

3 AA

Dw1 = 1.7 × 10−9 m2 s−1 Dp1 = 10−11 m2 s−1 ρ1 = 0.898 kg/m3 K1wp = 5.12 × 10−3 MW1 = 128.2 C1sat = 0.013 mol/L ρq1 = 1.17 kg/m3

Dw2 = 1.7 × 10−9 m2 s−1 Dp2 = 10−11 m2 s−1 ρ2 = 0.93 kg/m3 K2wp = 3.2 × 10−2 MW2 = 100.1 C2sat = 0.156 mol/L ρq2 = 1.03 kg/m3

Dw3 = 1.7 × 10−9 m2 s−1 Dp3 = 10−11 m2 s−1 ρ3 = 1.046 kg/m3 K3wp = 1.0 × 10−4 MW3 = 72 ρq3 = 1.17 kg/m3

= 0.5, αI = 0.25, and αP = 0.75 and F12 = 10−9 m3/s) on the individual compartment and overall PSDs is shown in Figure 5,

Figure 3. Time evolution of the overall monomers conversion. Comparison of one- and two-compartment model predictions with experimental measurements (αI = 0.25, αP = 0.75, λ = 0.5, F12 = 10−9 m3/s, tadd = 2.5 h).

Figure 5. Time evolution of the individual compartment and overall PSDs. (a) t = 15 min, (b) t = 30 min, (c) t = 150 min (αI = 0.25, αP = 0.75, λ = 0.5, F12 = 10−9 m3/s, tadd = 2.5 h).

Figure 4. Time evolution of the average particle diameter. Comparison of one- and two-compartment model predictions with experimental measurements (αI = 0.25, αP = 0.75, λ = 0.5, F12 = 10−9 m3/s, tadd = 2.5 h).

at three different polymerization times (t = 15, 30, and 150 min). It is evident that, due to the uneven initiator concentrations in the two compartments (i.e., αI ≠ λ and the low value of F12), particle nucleation conditions will be different, resulting in different PSDs at the end of the nucleation period, i.e., t = 15 min (see Figure 5a). Note that as the polymerization time increases the separation distance of the two peaks in the overall bimodal PSD increases (see Figure 5b,c), due to the nonhomogeneous monomer concentrations in the two compartments (i.e., αP ≠ λ), leading to different monomer particle growth rates, and, thus, different PSDs. In Figure 5c, the PSDs calculated by the single- and the twocompartment model are shown. As can be seen the PSD predicted by the nonhomogeneous model is broader than that

smaller than the respective monomer and initiator feed rates of FP = 1.4 × 10−7 m3/s and FI = 1.5 × 10−8 m3/s, giving rise to nonhomogeneous mixing conditions. In Figures 3 and 4, the overall monomers conversion and the mean particle diameter determined by the nonhomogeneous two-compartment models are shown, respectively. As can be seen, the overall monomers conversion predicted by the two-compartment model is nearly identical to that calculated by the single compartment model. This is an expected outcome since in both cases the emulsion polymerization was carried out under monomer starved conditions. The effect of the nonhomogeneous mixing conditions as manifested by the low compartment exchange flow rate (i.e., λ 12291



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calculated by the homogeneous model, mainly due to the differences in the particle growth rates caused by the unequal partitioning of monomer in the two compartments. The effect of the exchange flow rates, F12 and F21, on the calculated compartment PSDs as well as on the overall PSD was next examined for moderately nonhomogeneous conditions (i.e., λ = 0.5, αI = 0.25, and αP = 0.75). In Figure 6, the

Figure 7. Effect of the compartment volume ratio λ, on the overall PSD in the reactor. (αI = 0.25, αP = 0.75, F12 = 10−9 m3/s, tadd = 2.5 h).

to 100 nm. In this moderately nonhomogeneous case, the differences in the compartment PSDs are primarily due to the uneven monomer partitioning, resulting in different particle growth rates. It should be noted that as the value of the monomer partitioning coefficient (αP) approaches the value of λ (i.e., αP ∼λ), the mixing conditions in the reactor become almost homogeneous. On the other hand, when the value of the compartment volume ratio is significantly different from the monomer partitioning coefficient (i.e., αP ≠ λ), the two peaks of the final bimodal PSD are moving apart due to the significant difference in the compartment monomer concentrations and, thus, in the particle growth rates. As was noted in the introduction, the value of the compartment volume ratio can vary with polymerization time due to changes in the dispersion rheology and mixing efficiency. Thus, the variation of the compartment volumes during the reaction is an additional important factor concerning the homogeneity of the semibatch emulsion polymerization reactor. It should be noted that the variations in the compartment volumes can be determined from steady-state CFD simulations conducted at different polymerization times if the time-varying viscosity of the dispersion is known.28 Effect of Monomer Addition Time. The effect of the monomer and initiator addition time, tadd, on the final PSD was next examined for a moderately nonhomogeneous system (i.e., λ = 0.25, αI = 0.25, αP = 0.75, and F12 = 10−7 m3 s−1). In Figure 8, the final compartment and overall PSDs are shown for two different addition times, i.e., tadd = 2.5 and 4.5 h. Note for an addition time of 2.5 h the respective values of Fp and FI will be 1.4 × 10−7 and 1.5 × 10−8 m3 s−1. On the other hand, for an addition time of 4.5 h the corresponding values of Fp and FI will be 0.78 × 10−7 and 0.83 × 10−8 m3 s−1. This means that as the addition time increases (i.e., from 2.5 to 4.5 h) the mixing conditions in the reactor become more homogeneous due to the high value of F12 = 10−7 m3 s−1. Moreover, as the reactor is operating under starved conditions, when the monomer addition time increases, the monomer addition rate, Fp, decreases, resulting in a decrease in the particle growth rate due to the lower monomer concentration. This reduced particle growth rate can result in an increase of the surfactant concentration in a compartment (e.g., above the CMC), causing the generation of new particles. It is important to note that although the compartment PSDs in Figure 8b are nearly identical to each other, they are both substantially different from the homogeneous single-compartment PSD shown in Figure 5c. This means that nonhomogeneous mixing conditions in a polymerization system

Figure 6. Effect of the exchange flow rate on the individual compartment PSDs and the overall PSD in the reactor at the final polymerization time, t = 150 min (F12 = (a) 10−9 m3/s, (b) 10−8 m3/s, (c) 10−7 m3/s ; αI = 0.25, αP = 0.75, λ = 0.5, tadd = 2.5 h).

compartment PSDs and the overall PSD are shown at the final polymerization time t = 150 min, for three different values of the exchange flow rate F12, (i.e., 10−9, 10−8, and 10−7 m3/s). It is apparent that as the value of F12 increases from 10−9 to 10−7 m3/s, the overall PSD in the reactor changes from bimodal to unimodal. Thus, it can be concluded that for moderately nonhomogeneous mixing conditions by increasing the exchange flow rate between the two compartments the overall PSD in the reactor becomes identical to that calculated by the one-compartment homogeneous model. Effect of Compartment Volume Ratio. In Figure 7, the effect of the compartment volume ratio, λ, on the overall PSD is examined for a case of unequal monomer and initiator feed partitioning in the two compartments (i.e., αI = 0.25, αP = 0.75, F12 = 10−9 m3/s). It can be seen that the final overall PSD in the reactor is bimodal for all the examined values of λ. Moreover, as the value of λ decreases from 0.6 to 0.3 (i.e., the reaction mixing conditions become less homogeneous), the separation distance between the two peaks of the bimodal PSD increases from 40 12292



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secondary particle nucleation in the homogeneous well-mixed case. Figure 9a,b shows the compartment and total PSDs

Figure 8. Effect of monomer addition time, tadd, on the final compartment and total reactor PSDs. tadd = (a) 2.5 h, (b) 4.5 h (αI = 0.25, αP = 0.75, λ = 0.25, F12 = 10−7 m3/s). Figure 9. Effect of the anionic surfactant concentration on the compartment and total reactor PSDs. SA = 0.8%. (a) t = 15 min, (b) 150 min (αI = 0.25, αP = 0.75, λ = 0.5, F12 =10−7 m3/s, tadd =2.5 h).

can affect the overall PSD despite the fact that the two compartment PSDs are similar. Effect of Surfactant Concentration in the TwoCompartment Model. The effect of the anionic surfactant concentration on the compartment PSDs and the overall PSD in the reactor was also investigated using the nonhomogeneous two-compartment model. According to the polymerization recipe employed in this study (see Table 1), the surfactants are added both at time zero and continuously via the monomer pre-emulsion feed stream. Note that the surfactant concentrations in the initial batch emulsion polymerization recipe and in the subsequent pre-emulsion monomer feed stream are chosen so that particle nucleation does take place during the initial batch polymerization period.33 That is, the surfactant concentration in the continuous monomer feed stream can ensure sufficient particle stabilization while, at the same time, secondary particle nucleation is excluded during the semibatch operation, provided that mixing conditions in the reactor remain homogeneous. Thus, during the subsequent monomer fed batch policy, the originally generated particles undergo a continuous growth due to the continuous monomer addition. However, under nonhomogeneous mixing conditions, the surfactant concentrations in the two compartments can be significantly different which can lead to secondary participle nucleation in a compartment (i.e., when the surfactant concentration in the aqueous phase exceeds the critical micelle concentration, CMC). The two-compartment model was applied to a moderately nonhomogeneous case characterized by the following compartment parameters: λ = 0.5, αI = 0.25, αP = 0.75, and F12 = 10−7 m3 s−1. In order to assess the effect of nonhomogeneous mixing conditions with respect to the surfactant concentration on particle nucleation, the anionic surfactant concentration in the emulsion polymerization recipe was increased to a value of 0.80 wt % on monomers (compared to its nominal value of 0.65 wt % reported in Table 1). Note that the increased surfactant concentration is just below the critical value required for

immediately after the initial nucleation phase (i.e., t = 15 min) and at t = 150 min, respectively. Due to the nonhomogeneous partitioning of the surfactant (i.e., αP ≠ λ), after a polymerization time of about 70 min, the free surfactant concentration in the aqueous phase of compartment “1” exceeds its CMC value leading to a secondary particle nucleation. As a result, a bimodal PSD is formed with a distinct narrow second peak at ∼50 nm corresponding to the secondary nucleated particles (Figure 9b). It should be noted that in this case the compartment PSDs and the overall PSD are similar (i.e., due to the large exchange flow rate of F12 = 10−7 m3 s−1) but different from the PSD calculated by the homogeneous singlecompartment model.

■

CONCLUSIONS A two-compartment model was developed and applied to a nonhomogeneous semibatch industrial-scale reactor for the emulsion copolymerization of BuA/MMA. A parametric study was conducted to determine the effects of the two-compartment model parameters, i.e., λ, αI αP FI, FP, F12, F21, on the compartment and overall particle size distributions. Depending on the values of the two-compartment model, homogeneous, moderately or severely nonhomogeneous mixing conditions could be observed. It was found that, under well-stirred mixing conditions, the final PSDs in the industrial-scale reactor were narrow and almost identical to the PSDs obtained in a lab-scale homogeneous emulsion polymerization reactor. However, for nonhomogeneous mixing conditions the overall PSD in the reactor was broader and even bimodal and significantly varied from the PSD calculated by the single-compartment homogeneous model. In particular it was shown that low values of the compartment exchange flow rates and uneven partitioning of the monomer and initiator feed streams as well as unequal 12293



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compartment volumes increased the spatial nonhomogeneity in the reactor, leading to quite different PSDs in the two compartments. Thus, by increasing the exchange flow rate between the two compartments, the overall PSD in the reactor, as calculated by the two-compartment model, was found to converge to the PSD calculated by the single-compartment homogeneous model. However, for some nonhomogeneous cases it was found that by increasing the exchange flow rate the two compartment PSDs, although identical, were yet substantially different than that calculated by the singlecompartment model. Nonhomogeneous reactor conditions were also observed by changing operational parameters of the emulsion polymerization recipe (e.g., monomer addition time and total amount of added surfactant), thus influencing the particle growth and nucleation rates. It was found that by increasing the monomer addition time monomer flooding could occur in a compartment, due to inadequate mixing, that could result in a secondary particle nucleation and the appearance of a distinct new size peak in the overall PSD in the reactor.

tot Vmax = ST(1 + 1.4κ )kBT

(A-8)

ζi = ζ0 ln((e λ4 + 1)/(e λ4 − 1))

(A-9)

(A-10)

(A-11)

and CI, Cs, and Cy are the local concentrations of initiator, anionic surfactant, and added electrolyte, respectively. The surface potential, ψi, of particle “i” will be given by ψi = ζ0 log(σi̅ +

σi̅ + 1 )

for

ψi > 0.05 J/Cb (A-12)

ψi = ζ0σκ i̅ ri /(1 + κri)

for

ψi > 0.05 J/Cb

(A-13)

Finally, the dimensionless surface charge can be expressed as σi̅ = 2π ezσi /εkBTκ

(A-1)

⎛ Vw ⎞ 2 S = 4πrm NA ⎜ ⎟ ∑ kemiCi•wCmic,w ⎝ Ve ⎠ i = 1

Cin

Concentration of species i (i = 1, 2, x, s, I) in the n phase (n = w, e, p), kmol/m3 Ci,j(t) Concentration of species “i” in compartment “j” in the nonhomogeneous model, kmol/m3 CIw,j(t) Initiator concentration in the aqueous phase, in compartment “j”, kmol/m3 Cie,j(t) Monomer concentration in the emulsion phase, in compartment “j”, kmol/m3 • Ciw Concentration of macroradicals of type i (i = 1, 2) in the aqueous phase, kmol/m3 • Cip Concentration of macroradicals of type i (i = 1, 2) in the polymer phase, kmol/m3 Cy Concentration of electrolyte, kmol/m3 CCMC Critical micelle concentration Cmic,w Micelles concentration Csw Surfactant concentration, kmol/m3 f(V) Distribution of nucleated particles, m−3 FI Total initiator volumetric feed rate, m3/s FI,i Initiator volumetric feed rate into compartment “i”, m3/s FP Total monomer pre-emulsion feed rate, m3/s FP,i Monomer pre-emulsion feed rate into compartment “i”, m3/s FPi,j Pre-emulsion feed rate of monomer “i” into compartment “j”, m3/s F12 Exchange flow from compartment “1” to “2” F21 Exchange flow from compartment “2” to “1” Gi(V) Particle growth rate function in the “i” compartment, m3/s kB Boltzman’s constant, J/K kemi Mass transfer coefficient for entry of “i” type macroradicals into the micelles, m/s kI Initiator thermal decomposition rate constant, s−1

(A-2)

⎛ ⎞ am Ae Cmic,w = ⎜CSw − CCMC − θS ⎟ NA(Vw /Ve)a′S ⎠ 4πrm 2 ⎝

(A-3)

where CSw and CCMC denote the total surfactant and the critical micelles concentration, respectively. Ae is the total particle surface area given by ne

∑ 4πrj 2Nj (A-4)

j=1

where Nj is the concentration of particles of size rj in the emulsion phase. The coagulation rate constant, β(ri,rj), between two colloidal particles will be given by the Fuchs modification of the extended Smoluchowski coagulation equation: 4 kBT (ri + rj)2 Wij−1 3 rri jμ

(A-5)

The stability ratio, Wij, depends on the total interaction potential, Vtot, according to the following equation:29,38,39 tot Wij = (ri + rj)/4κrri j exp(Vmax /kBT )

(A-6)

Vtot max,

(A-14)

Notation

where rm is the radius of the micelles. The concentration of micelles can be obtained from an overall surfactant surface coverage balance:

The maximum total interaction potential,

κ = [(8π e 2NAI )/(εkT )]1/2

I = 3C I + Cs + Cy

2

β(ri , rj) =

(A-7)

where I is the total ionic strength of the medium given by

The micellar particle nucleation rate can be expressed in terms of the micelles concentration, Cmic,w, and radicals concentration, C•iw, in the aqueous phase.

Ve Vw

ζiζj

⎛ e ψi / ζ0 + 1 ⎞ 2k T ⎟ζ0 = B λ4 = κδs + ln⎜ ψ / ζ 0 i ze ⎝e + 1⎠

APPENDIX: PARTICLE NUCLEATION, GROWTH, AND AGGREGATION RATE FUNCTIONS The growth rate of a particle of volume V will be given by the following equation:29

Ae =

ri + rj

where ST is a constant. The inverse Debye double-layer thickness, κ, and the zeta potential of a particle of radius ri, ζi, will be given by the following equations:38

■

m MWi ⎞ 1 ⎛ ⎜⎜∑ R pie ⎟ G (V ) = φq Ne ⎝ i = 1 ρi ⎟⎠

2rri j

will be given by 12294



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αP

Propagation rate constant for monomer “i” m3/ (kmol·s) kpij Cross-propagation rate constant, m3/(kmol·s) ktmij Chain transfer rate constant of radicals of type “i” to monomer of type j, m3/(kmol·s) ktii Termination rate constant for monomer “i”, m3/ (kmol·s) ktii0 Termination rate constant for monomer “i”, m3/ (kmol·s) at zero conversion ktij Termination rate constant of a radical of type “i” with a radical of type “j”, m3/(kmol·s) ktsi Chain transfer rate constant of macroradicals of type “i” to CTA molecules X, m3/(kmol·s) Kiwp Partition coefficient of species “i” (i = 1, 2, X) in the aqueous and polymer phases Kimp Partition coefficient of species “i” (i = 1, 2, X) in the monomer and polymer phases MWi Molecular weight of monomer i ni(V,t) Number volume density function in compartment “i”, m−6 n(V,t) Number volume density in the single-compartment model, m−6 nT(V,t) Total number volume density in the two-compartment model, m−6 n̅ Average number of radicals per particle Ne Number particle concentration in the emulsion phase, no./m3 NA Avogadro’s number, kmol−1 r Particle radius, m rm Micelle radius, m n̅ Average number of radicals per particle NCLD Number Chain Length Distribution RIw,j Initiator decomposition rate in the aqueous phase in the “j” compartment Rpi Polymerization rate of monomer “i” in the polymer phase, kmol/(m3·s) Rpie Polymerization rate of monomer “i” in the emulsion, kmol/(m3·s) Rpie,j Propagation rate of monomer “i” in the emulsion phase in the “j” compartment Rpp,j Total polymerization rate in the polymer phase in compartment “j” Rpw,j Total polymerization rate in the aqueous phase in compartment “j” Rwi Polymerization rate of monomer “i” in the aqueous phase, kmol/(m3·s) S Nucleation rate in m−3 s−1 Si(t) Nucleation rate in the “i” compartment, m−3 s−1 t Time, s T Temperature, K tadd Addition time for monomer feed stream, s U,V Particle volume, m3 V1(t) Volume of compartment “1”, m3 V2(t) Volume of compartment “2”, m3 VT(t) Total emulsion volume at time t, m3 Ve Volume of the emulsion phase, m3 Vw Volume of the aqueous phase, m3 Vtot Total interaction energy between two particles, J W Stability ratio X Chain transfer agent z Valence kpii

Fraction of monomer pre-emulsion feed stream to compartment “1” αm Surface area occupied by a surfactant molecule, m2/ molecule a′s Surface area occupied by a surfactant molecule on a particle, m2/molecule βi Aggregation rate kernel in compartment “i”, m3/s δS Stem layer thickness ε Dielectric constant ζi Zeta potential of a particle of radius ri θs Particle surface coverage κ Inverse electric double-layer thickness λ(t) Ratio of the volume of compartment “1” to the overall volume ρi Density of monomer (i = 1, 2), homopolymer (i = q1, q2), initiator (i = I), and surfactant (i = s), kg/m3 σ̅i Dimensionless surface charge of a particle of radius ri σi Surface charge of a particle of radius, ri φq Polymer volume fraction in the polymer phase ψi Surface potential of a the particle of radius, ri Subscripts and Superscripts

1 2 e I m p s w x y

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

REFERENCES

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Greek Symbols

αI

“Live” polymer chains ending in a MMA unit “Live” polymer chains ending in a BuA unit Emulsion phase Initiator Number of monomers Polymer phase Surfactant Aqueous phase Chain transfer agent Electrolyte

Fraction of initiator feed stream to compartment “1” 12295



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Nonhomogeneous Mixing Population Balance Model for the

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