Mathematical Modeling of Multicomponent Chain-Growth

966

Ind. Eng. Chem. Res. 1997, 36, 966-1015

Mathematical Modeling of Multicomponent Chain-Growth Polymerizations in Batch, Semibatch, and Continuous Reactors: A Review Marc A. Dube´ † Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5

Joa˜ o B. P. Soares and Alexander Penlidis* Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Archie E. Hamielec McMaster Institute for Polymer Production Technology, Department of Chemical Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7

A practical methodology for the computer modeling of multicomponent chain-growth polymerizations, namely, free-radical and ionic systems, has been developed. This is an extension of a paper by Hamielec, MacGregor, and Penlidis (Multicomponent free-radical polymerization in batch, semi-batch and continuous reactors. Makromol. Chem., Macromol. Symp. 1987, 10/11, 521). The approach is general, providing a common model framework which is applicable to many multicomponent systems. Model calculations include conversion of the monomers, multivariable distributions of concentrations of monomers bound in the polymer chains and molecular weights, long- and short-chain branching frequencies, chain microstructure, and crosslinked gel content when applicable. Diffusion-controlled termination, propagation, and initiation reactions are accounted for using the free-volume theory. When necessary, chain-lengthdependent diffusion-controlled termination may be employed. Various comonomer systems are used to illustrate the development of practical semibatch and continuous reactor operational policies for the manufacture of copolymers with high quality and productivity. These comprehensive polymerization models may be used by scientists and engineers to reduce the time required to develop new polymer products and advanced production processes for their manufacture as well as to optimize existing processes. 1. Introduction Mathematical models and their role in science/ engineering are points of constant debate, especially when models are employed in an industrial environment. The role of a mathematical model is often misinterpreted; as a result, we frequently blame the model, instead of blaming our own lack of understanding about a process as well as our reluctance to experiment with a process in a meaningful and systematic way. Why, then, are models useful? 1. Models enhance our process understanding since they direct further experimentation. They act as the reservoir of one’s knowledge about a process, and hence they may reveal interactions in a process that may be difficult, if not impossible, to visualize/predict solely from memory or experience, especially when many factors vary simultaneously. Since a model is a concise, compact form of process knowledge, models enhance transferability of knowledge; they may act eventually as an “inference engine”, closely resembling the train of thought of an experienced human. In a sense, mathematical modeling is the best way to find out what one does not know about a process! 2. Models are useful for process design, parameter estimation, sensitivity analysis, and process simulation. * To whom correspondence should be addressed. Phone: (519) 888-4567. Fax: (519) 746-4979. E-mail: penlidis@ cape.uwaterloo.ca. † E-mail: [email protected]. S0888-5885(96)00481-2 CCC: $14.00

The significance of these is quite obvious. A valid model allows one to test deviations from process trajectories using a simulator in lieu of running experiments. Cost effectiveness implications are also obvious. 3. Models are useful for process optimization, especially when dealing with highly nonlinear problems such as grade changes/switchovers in batch, semibatch, and continuous reactors. Extensions to recipe modifications and design are another application. 4. Models are useful for safety/venting considerations. It is very useful to be able to extrapolate to different operating conditions and anticipate “worst-case scenarios” or investigate the possible effects of process factors. In this case one may be better prepared to tackle situations that might not always be apparent from the outset. 5. Models are useful for optimal sensor selection and testing, sensor location, filtering and inference of unmeasured properties, and process control. The trends nowadays in process control are toward “model-based” control, and as the term signifies, application of advanced control techniques may not be possible without a model. 6. Finally, since a model contains process knowledge and is transferable, interactive models are extremely useful for the education and training of new (and old) personnel. In this paper, a practical methodology for the computer modeling of multicomponent chain-growth polymerizations, namely, free-radical and ionic systems, is developed. This is an extension of the paper by Hamielec © 1997 American Chemical Society

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 967

et al. (1987). The approach is general, providing a common model framework which is applicable to many multicomponent systems. Various comonomer systems are used to illustrate the development of practical reactor operational policies for the manufacture of polymers with high quality and productivity. Thorough general reviews of polymerization reactor modeling have recently been published (Penlidis et al., 1985b; Hamielec et al., 1987; Rawlings and Ray, 1988). The model developed in this paper employs the “pseudokinetic rate constant method” (Hamielec et al., 1987; Tobita and Hamielec, 1991; Xie and Hamielec, 1993a,b). Our development uses many techniques and ideas similar to those contained in Hoffman (1981), Broadhead et al. (1985), Penlidis et al. (1986), Hamielec et al. (1987), Mead and Poehlein (1988, 1989a), Rawlings and Ray (1988), Maxwell et al. (1992b), Fontenot and Schork (1992-93a,b), Xie and Hamielec (1993a,b), Casey et al. (1994), and Urretabizkaia and Asua (1994). 2. Model Development: Free-Radical Polymerizations The objective of this section is to define the equations which form a mechanistic model to simulate bulk (or suspension), solution, and emulsion free-radical homopolymerization, copolymerization, and multicomponent polymerization (three or more monomer types) in wellstirred batch, semibatch, and continuous modes. The equations presented in this section are valid for the general case of an unsteady-state CSTR. For a CSTR operating at steady state, the accumulation derivative terms can be set equal to zero to give a set of algebraic equations. For a semibatch reactor, the outflow terms should be eliminated, and for a strictly batch reactor, all inflow and outflow terms should be eliminated. However, it is usually advantageous to consider complete equations since one has the flexibility of handling all these reactor situations with a single model. The model is comprised of a set of mathematical expressions which describe the physical and chemical phenomena of polymerization. It consists of a set of differential equations that describe material and energy balances on the reaction mixture. For computational purposes, the model is split into two categories. The first category consists of bulk, suspension, and solution polymerization, while the other describes the emulsion case. In the model development, we shall examine both categories in parallel. Bulk, suspension, and solution polymerizations are characterized by the fact that all of the reaction steps proceed in a single phase. A model for a reactor carrying out such polymerizations would consist of a set of material balances describing the rates of accumulation, inflow, outflow, and disappearance by reaction of the various monomers, initiators, polymers, and other ingredients in the reactor. These polymerizations consist of initiation, propagation, termination, and transfer reactions occurring simultaneously through the full conversion range. Conventional emulsion polymerizations usually occur in three stages and are comprised of more than one phase (reactor head-space; the monomer droplets, which act as a monomer reservoir; the (continuous) aqueous phase, which can act as a locus of polymerization as well as a species transport medium; and the polymer particle phase, the main locus of polymerization). The first of the three common polymerization stages involves the nucleation (birth) of polymer particles. This can occur by either micellar or

homogeneous (coagulative) nucleation. The second stage involves the growth of the particles until the monomer droplets disappear. The third stage begins with the disappearance of the monomer droplets and continues until the end of the reaction. The emulsion polymerization model can be briefly described as follows. First, the initiation can be accomplished via a redox mechanism or via thermal decomposition of an initiator. The fate of radicals (initiator, monomeric, and oligomeric) in the water phase is propagation with dissolved monomers in the water phase, reaction with water-soluble impurities (WSI), termination in the water phase, possible recombination of initiator fragments, reaction with monomer droplets, desorption from polymer particles, reabsorption of desorbed radicals into polymer particles, capture by emulsifier micelles, and capture by polymer particles. The birth of particles can be accomplished by homogeneous (aqueous phase) nucleation, micellar nucleation, and particle coalescence. Once captured by particles (or micelles or droplets), the radicals may propagate, mutually terminate, react with monomer-soluble impurities (MSI), react with chain-transfer agent (CTA), undergo chain transfer to monomer, undergo chain transfer to polymer, and participate in internal and terminal double-bond polymerizations. The average number of radicals per particle is followed by accounting for entry/ absorption of radicals from the water phase, radicalradical termination, radical-MSI termination, and desorption of radicals into the water phase. The partitioning of monomer, monomer-soluble impurities, and CTA into the various phases is another important factor. For both the bulk/suspension/solution and emulsion models, material balances on the various components of the polymerization are used to calculate the conversion, composition, molecular weight, and, in the case of emulsion polymerizations, particle size and number. Other, not necessarily measurable, polymer properties can also be calculated or inferred. Finally, molecular weight averages dependent on termination, branching, and transfer reactions are estimated. The various symbols, subscripts and superscripts, and variables are shown in the Nomenclature section. The units of the variables are also shown therein. 2.1. Initiation. The first step in a polymerization involves the creation of highly reactive free radicals. This is accomplished in the initiation stage. Bulk, suspension, and solution polymerizations involve organicsoluble initiators such as 2,2′-azobis(isobutyronitrile) (AIBN). The initiator is decomposed into free radicals by thermal or photochemical (ultraviolet light) means. In emulsion polymerizations, there are two commonly used initiation methods. The first, redox initiation, is used for low-temperature polymerizations, while initiation by thermal decomposition is used for the higher temperature range. Andersen and Proctor (1965) suggested the following mechanism for the redox system persulfate (PS)/sodium formaldehyde sulfoxylate (SFS)/iron (Fe). k1

S2O82- + Fe2+ 98 SO4•- + Fe3+ + SO42k2

Fe3+ + RA 98 Fe2+ + X kpIj

• SO4•- + Mj 98 R1,j

(1) (2) (3)

(Symbols not explained in the text are given in the

968 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

detailed Nomenclature section.) In the above mechanism, Mj refers to monomer of type j; S2O82- represents the persulfate initiator, I; RA is the reducing agent (in this case SFS), often referred to as activator; SO4•- is the initiator fragment that reacts with monomer, Mj, • to produce primary radicals R1,j , i.e., radicals of chain length 1 ending in monomer j. k1 is the rate constant for the oxidation reaction, k2 is the rate constant for the reduction reaction, and kpIj is the propagation rate constant for the addition of monomer j to an initiator fragment. Radicals are generated in the water phase by the reaction between the initiator (PS) and a complex of Fe2+ and ethylenediamine tetrasodium acetate (EDTA) (see eq [REF:eqn:ini1]). The complexation of iron and EDTA reduces the effective concentration of Fe2+ and prevents undesirable side reactions. Thus, the initiator is reduced to a negatively charged free radical, while the Fe2+ is oxidized to Fe3+ (see eq 1). The SFS then reduces Fe3+ to Fe2+ (see eq 2). The free radical, SO4•-, reacts with monomer j to form radicals of chain-length unity (see eq 3). Broadhead et al. (1985) used a similar scheme to describe redox initiation for styrene/butadiene (SBR) polymerization. Performing material balances for the initiator (I or PS) and the reducing agent (RA) gives

dNI NI k1NINFe2+ ) FI,in - vout dt VT Vw

(4)

dNRA NRA k2NRANFe3+ ) FRA,in vout dt VT Vw

(5)

where Ni is the number of moles of component i, Fi,in is the inflow of component i into the reactor in mol min-1, Vw is the volume of the water phase, VT is the total volume of the reaction mixture, and vout is the volumetric flow out of the reactor. The time dependence of all terms involved in the equations is not shown for the sake of brevity. One should note that k1 and k2 are “effective rate constants”, since other (unknown) elementary reaction steps may be occurring but have been assumed to have a negligible contribution to the overall initiation rate. This will simplify our set of equations. Further simplification occurs when we apply the reactor stationary-state hypothesis to ferrous and ferric ions to provide the ion concentrations in terms of total iron concentration as follows:

NFe2+ )

k2NFeNRA k1NI + k2NRA

(6)

where

NFe ) NFe2+ + NFe3+

(7)

dNFe NFe ) FFe,in v dt VT out

(8)

and

For the case of thermal decomposition of a persulfate initiator, the commonly accepted mechanism is (Sarkar et al., 1988) kd

S2O82- 98 2SO4•-

(9)

kpIj

• SO4•- + Mj 98 R1,j

(10)

where kd is the initiator decomposition rate constant. This mechanism results in the following material balance for the moles of initiator:

NI dNI ) FI,in - vout - kdNI dt VT

(11)

Finally, the overall rate of initiation, RI, is

RI ) k1

NFe2+ [I] + 2fkd[I] Vw

(12)

where [I] is the initiator concentration and f is the initiator efficiency factor. Equation 12 is implemented into the computer model as shown above and includes the rate of initiation by redox means and thermal means. However, when one of the methods is activated, the term representing the other option becomes negligible. The acceleration of the decomposition of potassium persulfate (KPS) by “free” sodium dodecyl sulfate emulsifier has been reported by Okubo et al. (1991). Sarkar et al. (1990) reported the acceleration of KPS decomposition due to the addition of vinyl acetate (VAc) monomer, but no emulsifier effect was detected. Considerations such as these may be responsible for some discrepancies between model predictions and experimental data. Often, the initiator efficiency is considered to be constant. However, in a high-viscosity regime the initiator efficiency may decrease significantly (GarciaRubio and Mehta, 1986; Russell et al., 1988b; Zhu et al., 1990a). Since initiator decomposition occurs in the water phase for conventional emulsion polymerizations, it is likely that initiator fragments are not subject to a high-viscosity environment and the initiator efficiency is thus held constant (Adams et al., 1990). One can then argue that, if the initiator efficiency changes, this “represents” water-soluble impurity effects. For bulk and solution polymerizations, however, the following semiempirical equation is used to describe the changing initiator efficiency when the free volume of the reaction mixture (VF) becomes less than a critical free volume, VFcrif:

f ) fo exp(-C(1/VF - 1/VFcrif))

(13)

where fo is the initial initiator efficiency and C is a parameter which modifies the rate of change of the efficiency. The critical free volume, VFcrif, is dependent on temperature and initiator type. The initiator efficiency typically becomes diffusion-controlled at very high conversions (>80 wt %). The calculation of the free volume, VF, will be discussed later. In the model, therefore, eqs 4-7 and 11-13 are used directly to describe the initiation step. 2.2. Water Phase Reactions. The discussion in this section is restricted to emulsion polymerizations. Once the radicals are generated in the water phase, they can then go on to propagate with monomer, react with various other species in the reaction mixture, and nucleate particles. Two approaches for particle nucleation are commonly employed nowadays: homogeneous (coagulative or aqueous phase) nucleation and heterogeneous (micellar) nucleation. Several representative articles dealing with these mechanisms follow: Fitch


and Tsai (1970, 1971), Hansen and Ugelstad (1978, 1982), Poehlein et al. (1986), Maxwell et al. (1991, 1992b), and Casey et al. (1994). Another interesting, slightly different nucleation case has recently been described by Lepizzera and Hamielec (1994). We now examine the fate of the radical species in the water phase: 1. The radicals may react with monomers in the water phase: kpIj

• R•I + Mj 98 R1,j kpij

• • Rn,i + Mj 98 Rn+1,j

(14) (15)

where the rate constants of propagation are used to define reactivity ratios as follows:

rij )

kpii kpij

(16)

R•n,i represents a radical of chain length n ending in monomer i. Equation 14 represents the reaction between initiator fragments and monomers to form primary radicals. Equation 15 represents propagation. Equation 15 describes the addition of monomer j to a growing radical chain of length n ending in monomer i. The reaction, proceeding with a rate constant of kpij, results in a radical chain of length n + 1 ending in monomer j. For the case of a terpolymerization, there are nine different propagation reactions and six separate reactivity ratios. In principle, kpii in eq 16 may be obtained from homopolymerization data for each monomer type, while the six reactivity ratios may be calculated from the three binary copolymerizations involved in the terpolymerization. Copolymerization is described using four propagation reactions and two reactivity ratios. The propagation reactions shown above represent terminal model kinetics. That is, the reactivity of a radical center is assumed to depend only upon the monomer unit bound in the polymer chain on which it is located. Other alternative models could also be considered and are discussed later. 2. The radicals may react with water-soluble impurities: kzj

• + WSI 98 P(WSI) Rn,i

ktw

• • Rn,i + Rm,j 98 Pm+n or Pm + Pn

where Pm+n is a dead polymer molecule of chain length m + n. Note that in eq 18 termination may occur either by combination of the radical chains or by disproportionation. ktw is often considered to be negligible (Urretabizkaia et al., 1992; Urretabizkaia and Asua, 1994); however, in systems containing highly watersoluble monomers (Sarkar et al., 1988), or depending on the emulsifier concentration (Song and Poehlein, 1988b), water-phase termination may be significant. 4. The initiator fragments may recombine. This phenomenon is taken into account by use of the efficiency factor, f, for the initiation step or by the use of an “effective” initiator decomposition rate constant. 5. The radicals may be captured by monomer droplets. Most modeling efforts ignore this phenomenon due to the fact that the surface area of the polymer particles and micelles is far greater than that of the monomer droplets. Thus, the likelihood of a radical species entering the monomer droplets is minimal. This assumption can be tested by the use of electron microscopy to determine if any abnormally large particles exist and, if so, how many. Droplet nucleation may occur should extremely high shear rates be used during mixing along with the appropriate emulsifier concentration. Also, this phenomenon usually occurs in the presence of alcohol groups with ionic emulsifiers. The rate of radical capture by monomer droplets can be defined as • Rcmd ) kcmd[RTOT ]wdrop[drops]

(19)

where [R•TOT]wdrop is the total concentration of radicals able to enter a micelle, a particle, and/or a droplet (the calculation of this quantity would be similar to that for [R•TOT]wmic and [R•TOT]wpar in eqs 30 and 31, respectively), and [drops] is the concentration of droplets. In the examples cited later in this paper, the rate constant for the capture of radicals by monomer droplets, kcmd, is set to zero. For the interested reader, the case of monomer droplet polymerization has been described by Ugelstad et al. (1973, 1974), Hansen and Ugelstad (1979), Song and Poehlein (1988a,b), and Fontenot and Schork (1992-93a,b). 6. Radicals may desorb from polymer particles at the following rate:

(17)

where P(WSI) is a dead molecule and kzj is the rate constant for reaction of water-soluble impurity, j, with a monomer i-ended radical. The use of kzj as an overall rate constant, regardless of which monomer radical the impurity is reacting with, is a simplification to our model. It is assumed that the rate of reaction of WSI’s does not depend on the radical type. Water-soluble impurities may cause an induction period by consuming large amounts of free radicals. Typical examples of such impurities are oxygen and other commonly used monomer inhibitors with a considerable water solubility at the conditions of the polymerization (e.g., hydroquinones). The effects of impurities on polymerization rate and quality are poorly understood, yet they are one of the most important sources of variation in an industrial setting (Huo et al., 1988; Penlidis et al., 1988; Chien and Penlidis, 1994a,b; Dube´ and Penlidis, 1997). 3. The radicals may terminate upon encountering another radical:

(18)

Fdes )

j kdesNpn NAVw

(20)

where Np is the number of polymer particles per liter of water, n j is the average number of radicals per particle, NA is Avogadro’s number, and Vw is the total volume of water. Several authors have reported expressions for the desorption rate constant, kdes, all of which are based on the same principle. Information may be found in Nomura et al. (1971a), Ugelstad and Hansen (1976), Nomura and Harada (1981), Rawlings and Ray (1988), Mead and Poehlein (1989b), Asua et al. (1989), and Casey et al. (1994). Desorption is a phenomenon restricted to small molecules, usually as a result of chain transfer to monomer or monomer-soluble impurity or chain-transfer agent. However, the use of relatively large CTA molecules (e.g., n-dodecylmercaptan) in many polymerizations would obviate the need to include the chain transfer to CTA in the desorption equation. The desorption rate constant is discussed in detail later.


7. Desorbed radicals may reabsorb into the polymer particles. Expressions for the reabsorption of previously desorbed radicals back into the particles have been put forth by Poehlein et al. (1986). This phenomenon, while not included in the desorption equation itself, is included in the water-phase radical balances shown later. 8. Radicals may be captured by micelles according to • Rcm ) kcm[RTOT ]wmicAm/Vw

(21)

where kcm is the rate constant of capture by micelles and Am is the total free micellar area. The radical concentration [R•TOT]wmic is the concentration of radicals in the water phase which can enter micelles; it does not include those radicals which, due to their size, electric charge, and hydrophilicity, cannot be captured by micelles. The derivation of [R•TOT]wmic is shown later; it is defined in eq 30. Equation 38 gives the rate of particle nucleation by the micellar mechanism. 9. Radicals may be captured by particles at the following rate: • ]wparAp/Vw Rcp ) kcp[RTOT

(22)

where kcp is the rate constant for capture by polymer particles and Ap is the total surface area of the polymer particles. The radical concentration [R•TOT]wpar is the concentration of radicals in the water phase which can enter particles. The same arguments explained for micellar capture of radicals in the water phase apply here as well. [R•TOT]wpar is defined later in eq 31. 2.3. Particle Nucleation. The aspect of emulsion polymerization that generates the most discussion and conflict is the debate regarding the nature of particle nucleation (Richards et al., 1989; Dunn, 1992; Hansen, 1992, 1993). In this model, both micellar and homogeneous particle nucleation mechanisms are accounted for. The assumption that particle sizes are monodisperse is also employed in the following discussion; accounting for a distribution of particle sizes is discussed in the next section. Reviews of the polymer particle formation mechanisms may be found in Ugelstad and Hansen (1976) and Hansen and Ugelstad (1982). Micellar and homogeneous nucleation can both play a significant role in particle formation (Fitch and Tsai, 1971; Fitch, 1981). The relative importance of homogeneous nucleation increases with the solubility of monomer in the water phase, while at high emulsifier levels, micellar nucleation dominates due to the high surface areas and rapid radical absorption rates from the water phase. Particleparticle coalescence has been neglected in modeling by most workers. Notable exceptions include Min and Ray (1974), Hansen and Ugelstad (1978), Morbidelli et al. (1983), and Song and Poehlein (1988a,b). It has been difficult to develop a general model for radical absorption into micelles and polymer particles and desorption from polymer particles (Hansen and Ugelstad, 1982; Nomura, 1982). We now proceed to form the expression for homogeneous particle nucleation. In this paper, a combination of two methods, along with modifications to allow for multicomponent polymerizations and inhibitors, was implemented into the model. The first method, originally proposed by Hansen and Ugelstad (1978), is more rigorous but requires knowledge of parameters that are not readily known for all polymer systems. The second

method, that of Fitch and Tsai (1970, 1971), involves the use of grouped parameters, thus simplifying the structure of the equations but not necessarily rewarding us with less uncertainty. According to the first method: Nz

N

RI )

kpIj[Mj]w + ∑kzi[WSI]i) ∑ j)1 i)1

[R•I](

(23)

where N is the total number of monomers in the system and Nz is the total number of WSI’s in the system. The above expression equates the rate of radical generation to the rate of radical disappearance. In other words, we are invoking a steady-state hypothesis. On the lefthand side of eq 23 we have the overall rate of initiation (see RI of eq 12). The right-hand side of eq 23 shows the disappearance of the initiator radicals by propagation with monomer j in the water phase (see kpIj[Mj]w) and by reaction with water-soluble impurities (see kzi[WSI]i). [R•I] is the concentration of initiator radicals in the water phase. [Mj]w is the concentration of monomer j in the water phase. We are making the assumption that initiator radicals will not terminate (this is accounted for by initiator efficiency) and the visualization that radicals of this size will not enter particles nor micelles nor monomer droplets due to electrostatic forces and the hydrophilicity of such radicals. Next, we perform balances on radicals of chain length 1, ending in monomer i: Nz

N

• ]( kpIi[Mi]w[R•I] ) [R1,i

∑ j)1

kpij[Mj]w +

kzi[WSI]i + ∑ i)1 • ]w) (24) ktw[RTOT

The above equation now includes termination with other radicals in the water phase (see ktw[R•TOT]w in eq 24). The creation of radicals of chain length 1 by desorption is not included in the above balance. The desorbed radicals are of a different nature compared to primary radicals or oligomers in that they are a result of transfer to monomer and transfer to chain-transfer agent reactions inside the polymer particles. Hence, desorbed radicals have an electric charge different from the other radicals in the water phase and can be recaptured by the particles. In fact, the desorbed radicals will not move far beyond the particle from which they desorbed. That is, it can be argued that the desorbed radicals will not necessarily enter the bulk of the water phase but will stay in the vicinity of the particle surface. However, if there is a large amount of desorption, a significant amount of desorbed radicals may stray from the particles. We now perform balances on radicals of chain length k (2 e k e (jcr/2)), ending in monomer i. jcr/2 is the critical chain length at which oligomers may be captured by micelles, particles, and/or droplets. N

(

∑ j)1

N

• kpji[Rk-1,j ])[Mi]w

)

kpij[Mj]w + ∑ j)1

• [Rk,i ]( Nz

• kzi[WSI]i + ktw[RTOT ]w) ∑ i)1

(25)

Now a balance is performed on radicals of chain length k (jcr/2 + 1 e k e (jcr - 1)), ending in monomer


i. jcr is the critical chain length at which oligomers will precipitate and form a polymer particle (homogeneous nucleation) given that there is remaining free emulsifier. We also see that all capture mechanisms (micellar, particle, droplet) have now been included (see the terms kcmAm/Vw, kcpAp/Vw, and kcmd[drops], respectively, in eq 26). The inclusion of the capture mechanisms reflects the visualization that the oligomeric radicals have now grown enough so that there are no repulsion barriers (i.e., the charge at the end of the oligomer is no longer strong enough to create a repulsion from micelles, particles, or droplets). N

(

kcpAp/Vw • [Rk,i ] + kcmAm/Vw + kcpAp/Vw k)(jcr/2)+1 N kcpAp/Vw • [Rjcr,i ] (31) kcmAm/Vw + kcpAp/Vw + kh i)1

N

• [RTOT ]wpar

)

jcr-1

∑ ∑ i)1 ∑

Finally, the concentration of radicals in the water phase that may undergo homogeneous nucleation is defined by

N

• • kpji[Rk-1,j ])[Mi]w ) [Rk,i ](∑kpij[Mj]w + ∑ j)1 j)1 • kzi[WSI]i + ktw[RTOT ]w + kcmAm/Vw + kcpAp/Vw + ∑ i)1

kcmd[drops]) (26) Equations 23-26 represent a reasonable and practical way to handle differing monomer solubilities in water. Our visualization that the capture of radicals by micelles and particles begins at about a length of jcr/2 units has been independently supported by Poehlein (1990), Maxwell et al. (1991, 1992b), and Kshirsagar and Poehlein (1994). jcr is calculated as a weighted function of the instantaneous composition of the polymer formed in the water phase (Fjw): N

jcr )

jcrjFjw ∑ j)1

(27)

Fjw is described later in eqs 132-134 as Fj. The concentration of radicals of chain length jcr ending in monomer i is N

• kpji[Rjcr-1,j ])[Mi]w ∑ j)1

kh • [Rjcr,i ] (32) kcmAm/Vw + kcpAp/Vw + kh i)1 N

• [RTOT ]whom )

Nz

( • [Rjcr,i ])

Next, the concentration of radicals in the water phase that may be captured by particles is given by

(28)

∑

Am represents the total free micellar surface area, that is, the surface area created by the emulsifier remaining after the coverage of droplets and particles. Am is given by

Am ) ([S]t - [S]CMC)VwSaNA - Ap - Ad

(33)

Ap ) (πNp)1/3(6Vp)2/3

(34)

[S]t is the total concentration of emulsifier in the reactor, [S]CMC is the critical micelle concentration, Sa is the area occupied by an emulsifier molecule, Ad is the area of monomer droplets, and Ap is the total surface area of polymer particles. As the reaction proceeds, the total surface area of polymer particles quickly becomes very much greater than the total surface area of monomer droplets, so Ad is neglected as a simplification to our model. In eq 34, Vp is the total volume of polymer particles. Equation 34 is based on the assumption that the polymer particles are spherical. Sa (see eq 33) is affected by the polarity of the adsorbing surface and is therefore affected by persulfate initiators (Ali and Zollars, 1985). kh is the homogeneous nucleation rate constant defined by Fitch and Tsai (1971) as

(

(kcmAm/Vw + kcpAp/Vw + kcmd[drops])

A balance on the total amount of radicals in the water phase (excluding initiator radicals) gives N jcr-1

• [RTOT ]w )

∑ ∑ [Rk,i• ] i)1 k)1

kcmAm/Vw • [Rk,i ] + kcmAm/Vw + kcpAp/Vw k)(jcr/2)+1 N kcmAm/Vw • [Rjcr,i ] (30) kcmAm/Vw + kcpAp/Vw + kh i)1

N

jcr-1

∑ ∑ i)1 ∑

)

LAp 4Vw

(35)

L in the above expression represents the critical radical diffusion length and is given by Einstein’s diffusion law:

(29)

Recall that radicals of chain length jcr/2 + 1 to jcr can be captured by micelles, particles, and droplets. Also, radicals of chain length jcr can undergo homogeneous nucleation. Droplet nucleation is neglected as a simplification to our model. Thus, a balance on the concentration of radicals in the water phase able to be captured (radicals of chain length > jcr/2) yields the following equations. First, the concentration of radicals in the water phase that may be captured by micelles is given by • ]wmic ) [RTOT

kh ) kho 1 -

L)

(

2Dwjcr kpMwsat

)

1/2

(36)

Mwsat is the saturation concentration of monomer in the water phase. The rate of change of the number of particles is described by the following equation:

dNpVw Vw dNhom dNmic ) FNp,in - Np vout + Vw + Vw dt VT dt dt kFNp2Vw (37) FNp,in is the inflow of particles (i.e., in the case of a seeded emulsion polymerization), while the second term on the right-hand side of eq 37 (Np(Vw/VT)νout) represents the outflow of particles from the reactor. The rates of homogeneous and micellar particle generation are described by the third and fourth terms on the right-hand


side of eq 37, respectively. kFNp2Vw represents a crude attempt to take particle coalescence into account (Ugelstad and Hansen, 1976; Hansen and Ugelstad, 1978, 1979). Song and Poehlein (1988a,b), Fontenot and Schork (1992-93a), and Chern and Kuo (1996) discuss particle coagulation in more detail. This is a difficult subject, still under investigation, and hence, especially due to lack of specific data, we decided to neglect the term in further simulations with the model. The rate of micellar nucleation, dNmic/dt, is given by

dNmic • ) NAkcm[RTOT ]wmic/rmic dt

∫

t

R (t,τ) V(t,τ) Np(t,τ) dτ 0 p

(39)

(40)

where Rp(t,τ) is the consumption rate of monomer at time t in polymer particles born at time τ. Np(t,τ) dτ is the number of polymer particles in the reactor at time t which were born at time τ. The total polymerization rate, Rp(t,τ) in these particles may be expressed as

Rp(t,τ) )

j (t,τ) kp[M]p(t) n NAV(t,τ)

dt

N

V(t,τ)

MWjRpj(t,τ)) ∑ F (t) j)1

)(

(42)

p

N

V(t,τ) ) Vp(t,τ) +

Vjmp(t,τ) ∑ j)1

(43)

N

The homogeneous nucleation rate constant, kh, tends toward zero as the area of polymer particles, Ap, increases (see eq 35). This is because there is a higher probability for an oligomer to be captured by a preexisting particle rather than form a new particle by homogeneous nucleation. 2.3.1. Emulsion Particle Size Distribution (PSD) Calculation. If one wishes to relax the assumption of a monodisperse particle size distribution, it is necessary to account for different classes or ages of particles as shown below. In the first instance, it is assumed that statistical broadening can be neglected and, thus, polymer particles born at time τ with volume V0 will all have the same volume (V(t,τ)) at some later time t (at least for those particles which have not left the reactor in the case of a CSTR). Calculation of PSD with exact correction for statistical broadening requires the solution of a large number of partial differential equations (Behnken et al., 1963; Sundberg, 1979; Kiparissides and Ponnuswamy, 1981; Rawlings and Ray, 1988; Storti et al., 1989), and this appears to be impractical at the present time except for special cases (the number of polymer particles containing three or more radicals is zero). The consumption rate of monomer in the reactor Rp(t) is given by

Rp(t) )

dVp(t,τ)

kpijφi(t))[M]p(t) n j (t,τ) ∑ i)1

fj(t) (

(38)

As can be seen from eq 30, when the surface area of free micelles, Am, goes to zero, micellar nucleation is halted. The expression for the rate of generation of particles by homogeneous nucleation becomes

dNhom • ) NAkh[RTOT ]whomVw dt

The volumetric growth rate of polymer in a polymer particle is given by

Rpj(t,τ) )

(44)

NAV(t,τ)

where Vjmp(t,τ) is the volume of monomer j in the particle born at time τ and φi is the mole fraction of radicals in the particles ending in monomer i (see eq 48). Equation 44 is defined by eqs 126 and 86 but with the dependence on t and τ shown in the variables. The following algebraic relationship may be used to calculate the volumetric growth rate for polymer particles born at times τ > 0.

dVp(t,τ) n j (t,τ) dVp(t,0) ) dt dt n j (t,0)

(45)

In this manner, one can calculate the full particle size distribution. 2.4. Organic Phase Reactions. This section describes reactions in bulk and solution polymerization as well as those reactions occurring in the bulk phase in emulsion polymerization once the radicals have been captured by particles (or micelles or monomer droplets). 1. The radicals may propagate (as shown in eq 15): kpij

• • + Mj 98 Rn+1,j Rn,i

(46)

For multicomponent polymerizations (e.g., in this case, terpolymerization), the overall propagation pseudokinetic rate constant can be defined as (Hamielec et al., 1987; Tobita and Hamielec, 1991; Xie and Hamielec, 1993a,b) N N

kpo )

∑ ∑kpijφifj i)1 j)1

φ1 ) (kp21kp31f12 + kp21kp32f1f2 + kp23kp31f1f3)/ψ

(47) (48)

φ2 ) (kp12kp31f1f2 + kp12kp32f22 + kp13kp32f2f3)/ψ (49) φ3 ) 1 - φ1 - φ2

(50)

ψ ) (kp12kp31f12 + kp21kp32f1f2 + kp23kp31f1f3 + kp12kp31f1f2 + kp12kp32f22 + kp13kp32f2f3 + kp12kp23f2f3 + kp13kp21f1f3 + kp13kp23f32) (51)

(41)

where kp is defined in eqs 47 and 84, [M]p is the concentration of monomer in the polymer particles, and n j is the average number of radicals per particle. The rates of polymerization for the individual species Rpj(t,τ) may be found using Rp(t,τ) and reactivity ratios (see eq 126).

where φi is the mole fraction of radicals in the particles ending in monomer i and fj is the mole fraction of monomer j in the particle or bulk phase. The φi’s are calculated as above by invoking the quasi-steady-state assumption (QSSA) for the radicals. The QSSA is not valid at high conversion levels, but the conversion and M h n (number-average molecular weight) results will not be significantly affected and the M h w (weight-average


molecular weight) predictions will be affected only slightly (Achilias and Kiparissides, 1994). The pseudokinetic rate constants for multicomponent polymerization described throughout this paper are in the context of terminal kinetics. Tobita and Hamielec (1991) have derived equivalent expressions for pseudo-kinetic rate constants in the context of the penultimate copolymerization model. The pseudo-kinetic rate constant method is perhaps the only practical way of handling complex multicomponent polymerization modeling. Equations 48-51, as shown above, have been developed for the terpolymerization case. The reduction of eqs 48-51 to copolymerization and homopolymerization and their extensions to higher multicomponent systems should be clear. 2. The radicals may mutually terminate: ktc

• • Rn,i + Rm,j 98 Pm+n ktd

• • Rn,i + Rm,j 98 Pm + Pn

(52) (53)

Equation 52 represents termination by combination, whereas eq 53 represents termination by disproportionation. The two termination rate constants are defined by the overall termination rate constant, kt, and γ, the ratio of termination by disproportionation to the overall termination:

γ ) ktd/kt

(54)

kt ) ktc + ktd

(55)

Describing an overall termination pseudo-kinetic rate constant in multicomponent polymerizations has been a source of constant debate. In this study, several different methods were attempted, but each method had limitations. The overall termination pseudo-kinetic rate constant was calculated as N N

kto )

∑ ∑ktoijφiφj i)1 j)1

(56)

For the bulk/solution case, the cross-termination rate constants were defined as

ktoij ) ktoiFi + ktojFj

(57)

where Fi, the instantaneous polymer composition, is defined later in eqs 132-134. The cross-termination rate constants in the emulsion case were defined as

ktoij ) (ktoiktoj)1/2

(58)

There has been little agreement as to how the crosstermination rate constants should be defined. Therefore, both eqs 57 and 58 were implemented for future comparison. An overall γ was calculated using a weighted average of the homopolymerization γ’s based on the instantaneous polymer composition. 3. The radicals may react with monomer-soluble impurities such as hydroquinone and tert-butylcatechol (TBC), which are commonly added to the fresh monomer by the suppliers due to their radical scavenging properties: kfmsi

• + MSI 98 P(MSI) Rn,j

(59)

where P(MSI) is a dead polymer molecule. kfmsi is the rate constant for the reaction of monomer-soluble impurities with radicals ending in monomer j. The use of kfmsi as an overall rate constant, regardless of which monomer the impurity is reacting with, is a simplification to our model. It is assumed that the rate of reaction of MSI’s does not depend on the monomer type. In emulsion polymerizations, MSI’s are transferred into the particles with monomer(s) during monomer diffusion. In the bulk/solution case, the MSI’s are in the same phase as the initiator, thus causing an induction time. The effect of monomer-soluble impurities, usually ignored or at least poorly understood, can be quite pronounced on the overall reaction rate, eventually affecting particle growth (in emulsions), particle nucleation (in emulsions), and molecular weight (Huo et al., 1988; Penlidis et al., 1988; Chien and Penlidis, 1994a,b; Dube´ and Penlidis, 1997). The impurity effect on particle growth can be described as follows. If a MSI partitions into the growing polymer particles, it scavenges the free radicals. Thus, the particle growth rate is slowed down considerably. This leads to a prolonged particle nucleation stage because the micellar emulsifier, which is used to stabilize the particles, is consumed more slowly. Therefore, the average lifetime of a micelle is extended and more particles are nucleated. Since the rate of polymerization is proportional to the number of particles, a significant increase in the rate may result (Dube´ and Penlidis, 1997). 4. The radicals may react with a chain-transfer agent such as a mercaptan: kfctaj

• Rn,j + RSH 98 HPn,j + RS•

(60)

where RSH represents the chain-transfer agent which loses a labile hydrogen to the growing radical chain. kfctaj is the rate constant for transfer to a CTA molecule for the monomer j-ended radical type. The product of the above reaction, RS•, continues propagating with monomer(s), thus lowering the average molecular weight. The importance of these reactions is stressed by the findings of Broadhead et al. (1985), who showed that transfer to CTA dominated the SBR polymerization. In the case of multicomponent polymerization, an overall chain transfer to CTA pseudo-kinetic rate constant (kfcta) is defined as N

kfcta )

kfctajφj ∑ j)1

(61)

5. The radicals may undergo chain transfer to monomer: kfmij • Rn,i + Mj 98 Pn,i + M•j

(62)

In this case, the active radical center is transferred to a monomer molecule which may propagate further, thus lowering the overall molecular weight. Chain transfer to monomer is an important reaction. It greatly affects the molecular weight and is a precursor to terminal double-bond reactions and desorption. For example, the transfer reaction to VAc results in a stable radical which reinitiates slowly; thus, desorption from polymer particles is possible (Litt, 1993). As previously with kfcta, an overall pseudo-kinetic rate constant for chain transfer to monomer (kfm) is defined


as

N N

k*p* )

N N

kfm )

∑ ∑kfmijφifj i)1 j)1

(63)

6. The radicals may undergo chain transfer to polymer: kfpij

• • + Rm,j 98 Pn,i + Pm,j Rn,i

(64)

Transfer to polymer does not necessarily occur at the end of a dead polymer molecule. Thus, the species type at the end of the radical chain is not important. The presence of chain transfer to polymer reactions has been reported for BA and VAc polymerizations (Scott and Senogles, 1970, 1974; Friis et al., 1974; Hamielec, 1981; El-Aasser et al., 1981, 1983; Penlidis et al., 1985a; Dube´ et al., 1991b; Lovell et al., 1991, 1992). Lovell et al. (1991, 1992) reported that the transfer to polymer reaction for BA occurred by the abstraction of a tertiary hydrogen from a BA unit in the polymer chain. Friis et al. (1974) reported that transfer to polymer and transfer to monomer reactions dominate the VAc polymerization. The chain transfer to polymer occurs at the methyl hydrogen of VAc. The overall pseudo-kinetic rate constant for transfer to polymer (kfp) is N N

kfp )

∑ ∑kfpijφiFh j i)1 j)1

(65)

The cumulative polymer composition, F h j, is defined later in eq 135. In a batch reactor when there is significant compositional drift, the use of the above equation is not strictly valid and one would have to resort to the cumbersome method of moments for the complete set of chemical equations for polymerization. In many practical circumstances compositional drift is either small or maintained small (batch reactors and reactivity ratio pairs which give small compositional drift, semibatch reactors where compositional drift is controlled to low levels and well-mixed continuous stirred-tank reactors with micromixing where only statistical spreading of composition occurs). 7. The radicals may undergo reactions with internal and terminal double bonds. Once again, an effective rate constant will be used. The rate constant for terminal double-bond reactions is k*p and that for internal double-bond reactions is k*p*. k* p • • + Pm 98 Rm+n,i Rn,i

(66)

k* p* • • + Pm 98 Rm+n,i Rn,i

(67)

Terminal double-bond reactions yield trifunctional branch points, whereas internal double-bond reactions yield tetrafunctional branch points. Hence, an overall pseudokinetic rate constant for terminal (k*p) and internal double-bond (k* p*) reactions are defined as N N

k*p )

∑ ∑k*pijφiFh j i)1 j)1

(68)

∑ ∑k*p*ijφiFh j i)1 j)1

(69)

The importance of the terminal double-bond reaction has been documented for the BA and VAc polymerizations (Scott and Senogles, 1970, 1974; Friis et al., 1974; Hamielec, 1981; El-Aasser et al., 1981, 1983; Penlidis et al., 1985a; Dube´ et al., 1991b; Lovell et al., 1991, 1992). As in the case for transfer to polymer, the validity of the above pseudo-kinetic rate constant equations comes into question in the presence of strong compositional drift. 2.5. Diffusion-Controlled Rate Constants. When discussing the various reactions taking place in a polymerization, many of the parameters were referred to as rate constants. This is somewhat of a misnomer as these so-called rate constants (e.g., kp, kt) vary with the viscosity of the reaction medium. Thus, in a bulk system, the viscosity increases due to the increase in polymer concentration and, hence, affects the rate of propagation and termination. In the emulsion case, the viscosity in the main locus of polymerization (i.e., the particles) is high from the onset of the reaction due to the high polymer concentration. Thus, the rates of termination and propagation may be diffusion-controlled even at low conversion levels. The general chemical equations for diffusion-controlled termination were expressed earlier in eqs 52 and 53. The termination constants ktc and ktd can be redefined as ktc(n,m) and ktd(n,m), respectively. This illustrates their dependence, in general, on the chain • lengths n and m of the polymer radicals, R•n,i and Rm,j (see eqs 52 and 53) but not on the polymeric radical type (i and j denote the monomer type on which the radical center is located on the end of the polymer chain). One might expect that two backbone radical centers would mutually terminate at a significantly lower rate. The dependence of this rate on the position of the radical centers in the polymer backbone is not clear and the modeling of this effect is beyond the scope of this investigation. In other words, we limit the treatment of polymer radical/polymer radical termination to linear chain systems where termination rates are controlled by radical centers on chain ends. The termination rate constants for linear chain systems also depend on the mass concentration, molecular weight distribution of the accumulated dead polymer, and polymerization temperature. When most of the dead polymer chains (chains without a radical center) are made via chain transfer to a small molecule (such as chain-transfer agent, monomer, etc.) chain-length dependence of the termination rate constant may not be an important issue (accurate MWD’s of the dead polymer may be calculated without a knowledge of this chain-length dependence). The calculation of the total polymer radical concentration and then the rate of polymerization would require at most a single number-average termination constant (k h tN). However, when a significant number of dead polymer chains are made via polymer radical/ polymer radical termination, then to calculate the full MWD or higher molecular weight averages (M h w, M h z, etc.), one must account for chain-length dependence when it is operative. When calculating MWD and its averages, chain-length dependence can be accounted for in different ways. One can use an approximate expression to account for the dependence of kt(n,m) on chain


lengths n and m (kt(n,m) ) ktc(n,m) + ktd(n,m)). Some of the earlier studies to use this approach include Benson and North (1962), Duerksen and Hamielec (1967), Duerksen (1968), and others and more recent studies by Cardenas and O’Driscoll (1976), Ito (1980, 1981), Soh and Sundberg (1982a), Coyle et al. (1985), Russell et al. (1992), and O’Shaughnessy and Yu (1994a,b). These attempts to find a general and effective expression for kt(n,m) based on theoretical and experimental considerations are to be lauded even though to date an expression for kt(n,m) which can be considered effective and useful for commercial polymer reactor modeling is not available. In the meantime, it is recommended that a less general approach using averages of kt(n,m) be used (Boots, 1982; Olaj et al., 1987; Zhu and Hamielec, 1989) for reactor modeling. The use of kt(n,m) averages permits one to calculate the rate of polymerization, M h n, M h w, and possibly M h z of both accumulated dead polymer and dead polymer produced “instantaneously”. Individual empirical correlations for k h tN, k h tW, and k h tZ versus polymer concentration, temperature, and other significant variables as shown by Zhu and Hamielec (1989) are required. Methods based on averages have yet to be comprehensively evaluated although Vivald-Lima et al. (1994a) have recently compared the effectiveness of the CCS (Chiu et al., 1983) and MH (Marten and Hamielec, 1979) models. The number-average termination constant is given by ∞

k h tN )

∞

k h t(n,m) φ•nφ•m ∑ ∑ n)1m)1

(70)

and the termination rate of polymeric radicals, Rt, by

h tNYo2 Rt ) k

(71)

where φ•n is the mole fraction of polymeric radicals of chain length n. The use of kh tN to calculate M h w, M h z, and higher molecular weight averages will give estimates that are smaller than the true values when chain-length dependence is significant. For the correct calculation of higher molecular weight averages one should use k h tw, k h tz, etc. Examples are given in Zhu and Hamielec (1989). Anseth et al. (1994a) have made a comprehensive experimental investigation of the effect of volume relaxation on free-radical cross-linking kinetics. This phenomenon may explain observations made by Stickler (1983) of the effect of initiator level on limiting conversions for polymerization of methyl methacrylate in the absence of cross-linking monomers. At higher polymerization rates, the actual volume is greater than the equilibrium volume, permitting higher limiting conversions to be reached. Anseth et al. (1994b) have also recently shown dominance of termination by reaction diffusion in highly cross-linked systems. Recent papers by Vivaldo-Lima et al. (1994b) and Hutchinson (19923) have accounted for diffusion-controlled termination as affected by cross-linking. The validity of the stationary-state hypothesis (SSH) was tested by direct experimentation for the homopolymerization of methyl methacrylate and for the copolymerization of methyl methacrylate and cross-linker ethylene glycol dimethacrylate using ESR for the first time (Zhu et al., 1990b). The SSH was shown to be valid for homopolymerization of methyl methacrylate and for low levels of cross-linking. For high levels of crosslinking, however, the SSH is clearly not valid.

For chemically-controlled termination, kt equals kh t, and these are given by equations such as eq 56. For isothermal polymerization and small compositional drift, kt can be treated as a constant. The instantaneous number-average molecular weight is given by N

M hn)

MWi ∑ (R i)1

Rpi tc/2

(72)

+ Rtd)

where Rpi is the rate of polymerization of monomer i and Rtc and Rtd are rates of termination by combination and disproportionation, respectively. As mentioned earlier, the modeling to be discussed herein will use kh t exclusively, with chain-length dependence neglected in calculating M h w, M h z, etc. Under diffusion-controlled termination a single termination rate constant can be employed to model the rate of multicomponent polymerization and molecular weight development. In other words, the monomer type on which the radical center is located is unimportant. However, when dealing with polymer radicals with long branches, the self-diffusion coefficient of the polymer molecule will depend not only upon chain length or the number of monomer units in the chain but also upon the number of long branches, branch lengths, and their location on the polymer molecule. The position of the radical center (on the chain end or somewhere on the backbone) should also affect the termination rate. Termination under these conditions is clearly very complex, and a good deal of empiricism is required to model termination reactions for polymer radicals with long branches. The self-diffusion coefficient should also depend upon the distribution of monomer types in the polymer chains. This is another complicating factor when compositional drift is important. Diffusion-controlled rate constants are modeled in this paper using the free-volume approach (Marten and Hamielec, 1979; Soh and Sundberg, 1982a-c; Hamielec et al., 1987). In order for the termination reaction to occur, two macroradical chains must approach each other via translational diffusion. Next, the diffusion of the chain segments containing the active centers toward each other occurs. This is termed segmental diffusion. Finally, the termination reaction takes place. Thus, three diffusion-control intervals can be described. In the early stages of an isothermal, batch multicomponent polymerization (typically at conversions less than about 10% in bulk polymerization), where termination might be chemically controlled, compositional drift is usually small and hence a single constant termination constant, kt (kt ) ktc + ktd), may be used to model the rate of polymerization and molecular weight development to the conversion where termination becomes diffusion-controlled. However, when relatively high molecular weight polymers are being produced at low monomer conversions, the termination rate may be controlled by segmental diffusion. Bhattacharya and Hamielec (1986), Jones et al. (1986), and Yaraskavitch et al. (1987) have applied a model for segmental diffusion-controlled termination after Mahabadi and O’Driscoll (1977). The rate constant for termination controlled by segmental diffusion is given by

ktseg ) kto(1 + δc)

(73)

where kto is given by eq 56 shown earlier, δ is a parameter dependent on the molecular weight of the polymer radicals as well as solvent quality (monomers


plus solvent if present), and c is the mass concentration of accumulated polymer in the reaction mixture. As the polymer concentration increases in a good solvent, the coil size of polymer chains decreases, increasing the gradient of concentration of the radical center across the coil and thus the diffusion rate of the radical center. The consequence of this is that the termination rate constant increases with monomer conversion to the point where translational diffusion of the center of mass of the polymer radical coils controls the termination rate and then kt begins to fall very rapidly. The contribution of δc is rather small in many cases (Bhattacharya and Hamielec, 1986; Jones et al., 1986; Hamielec et al., 1987; Yaraskavitch et al., 1987) and could be neglected in most reactor calculations. The next interval, translational diffusion control, occurs up to high polymer concentrations (∼85-90% conversion in bulk). It is in this stage that the gel effect or autoacceleration occurs. Using the free-volume approach, the point at which the reaction becomes translational diffusion-controlled is identified by a temperature-dependent term, K3, defined by Marten and Hamielec (1979, 1982) as m K3 ) M h wcrit exp(A/VFcrit)

Vi

(0.025 + Ri(T - Tgi)) ∑ V i)1

VF )

(75)

T

where i represents the monomer, polymer, and solvent. Ri is the difference in the thermal expansion coefficients for species i above and below its glass transition temperature, Tgi. T is the polymerization temperature, Vi is the volume of each species, and VT is the total volume of the reaction mixture. The translational diffusion-controlled termination rate parameter, kT, is given by

( ) ( (

kT ) ktcrit

M h wcrit M hw

n

ktrd )

))

1 1 exp -A VF VFcrit

(76)

where ktcrit is the value of kt when eq 74 is satisfied. n is an adjustable parameter.

8πNAδD 1000

(77)

where

(74)

where A and m are adjustable parameters but hopefully independent of temperature, radical initiation rate, and monomer and polymer concentrations over wide ranges of these polymerization conditions. This has been proven so for both homo- and copolymerization by Stickler et al. (1984) and others (Bhattacharya and Hamielec, 1986; Jones et al., 1986; Yaraskavitch et al., 1987). M h wcrit is the accumulated weight-average molecular weight at the monomer conversion at which the termination rate is translational diffusion-controlled. M h wcrit = Mwcrit, the instantaneous weight-average molecular weight, and thus there is a direct connection with the size of the polymer radicals. VFcrit is the critical free volume corresponding to that conversion. K3 was found to have an Arrhenius temperature dependence for both homo- and copolymerization (Stickler et al., 1984; Bhattacharya and Hamielec, 1986; Jones et al., 1986; Yaraskavitch et al., 1987). The frequency factor had a small dependence on chain composition for the p-methylstyrene/acrylonitrile system (Yaraskavitch et al., 1987). In multicomponent polymerizations, the free volume is given by N

For the modeling of homopolymerizations and copolymerizations, the adjustable parameters have been arbitrarily set equal to 0.5 for m and 1.75 for n (Marten and Hamielec, 1979, 1982; Garcia-Rubio et al., 1985; Bhattacharya and Hamielec, 1986; Jones et al., 1986; Yaraskavitch et al., 1987). Panke (1986) has shown that setting m ) n ) 0.5 gives an equally good fit to rate and molecular weight data for the homopolymerization of methyl methacrylate. These values for m and n also give the correct response when methyl methacrylate is polymerized in a batch reactor with a high molecular weight heel. Finally, in extremely viscous environments, the reaction diffusion-control termination rate constant, ktrd, becomes significant. Two methods of describing this regime were employed in the model. The first (Stickler method) uses an expression by Stickler et al. (1984) and is given by

δ)

D)

( ) 6Vm πNA

1/3

nslo2 k [M] 6 p

(78)

(79)

and NA is Avogadro’s number, δ is the reaction radius, D is the reaction diffusion coefficient, Vm is the molar volume, ns is the number of monomer units in one polymer chain segment, and lo is the length of the monomer unit. The second method (RNG method) involves what is termed “residual termination”. Russell et al. (1988a) defined upper and lower bounds for the residual termination rate parameter as

4 ktres,min ) πkp[M]a2σ 3

(80)

8 ktres,max ) πkp[M]a3jc1/2 3

(81)

and

where a is the root-mean-square end-to-end distance per square root of the number of monomer units, σ is the Lennard-Jones diameter, and jc is the entanglement spacing of pure polymer, measured in monomer units. When attempting to model multicomponent polymerizations, options to use residual monomer mole fraction weighted averages or overall values for a, σ, and jc were made available. The lower limits of the RNG method and the Stickler method give similar results. The upper and lower bounds are handled in the model with the following expression:

ktrd ) ktres,minx + ktres,max(1 - x)

(82)

where x is the conversion. As will be discussed later, the reaction diffusion (or residual termination) concept may not be perfectly correct. Recent ideas regarding trapped radicals may have to be incorporated into future modeling efforts (Zhu et al., 1990a).


The point at which the reaction changes from translational diffusion control to reaction diffusion control can be somewhat ambiguous. Thus, the following overall kt is defined to circumvent this problem:

kt ) ktseg + kT + ktrd

(83)

Equation 83 has been previously employed to handle the various diffusion-control equations for kt by Marten and Hamielec (1979) and Soh and Sundberg (1982ac). Vivaldo-Lima et al. (1994a) showed that this approach (which they termed as a “serial” approach) resulted in as good, if not better, prediction of various polymerization data compared to the “parallel” approach such as that used by Russell (1994). The parallel approach involves summing the reciprocals of each kt expression from each diffusion-control interval. This is discussed further by Gao and Penlidis (1996). As will be discussed later, the prediction of the termination rate is still under investigation. Recent publications by Zhu et al. (1990a), Tobita and Hamielec (1991), Achilias and Kiparissides (1992), Maxwell and Russell (1993), Buback et al. (1994), Russell (1994), and Tobita (1994a,b) have shown some insight into solving this problem while at the same time demonstrating a wide variety of approaches. For free-radical polymerizations below the glass transition temperature of the polymer being synthesized, reaction mechanisms at very high monomer conversions can be very complex due to the decrease in the diffusion coefficients of small molecules: primary radicals and monomer. These decreases in mobility will have one or more of the following consequences: (1) decrease of the initiation efficiency; (2) the propagation reaction becomes diffusion-controlled; (3) radical pair formation (Zhu and Hamielec, 1989). An experimental observation for such polymerizations is the reduction in conversion rate to almost zero, even though appreciable concentrations of monomer and initiator exist in the polymerization mixture. Most polymerization models permit the propagation constant to fall with monomer conversion while fixing the initiator efficiency at a constant value. This is in effect equivalent to letting the product f1/2kp fall with monomer conversion to fit the rate of polymerization. There are practical reasons for doing this because it is very difficult to measure f and kp separately without the use of electron spin resonance spectroscopy (ESR). The use of data for polymerization rate and for molecular weights of accumulated polymer can, in principle, permit one to estimate f and kp separately as has been done before (Hui and Hamielec, 1968; Duerksen and Hamielec, 1968). However, for the bulk polymerization of methyl methacrylate with the very pronounced diffusion-controlled termination followed by a glassy effect and the associated broad MWD with both low and high molecular weight tails on the distribution, accurate molecular weight measurements are extremely difficult if not impossible to obtain. ESR is the preferred method of measuring both f and kp at high monomer conversions for bulk polymerization of monomers such as methyl methacrylate. Zhu and Hamielec (1989) did ESR/rate of polymerization measurements and found that f and kp fall significantly with conversion at about thesame monomer conversion, while Ballard et al. (1986) observed that for the same monomer (methyl methacrylate) but with emulsion polymerization, kp became diffusion-controlled at higher monomer conversions. This interesting result may be due to the swelling of surface layers of polymer particles by water in

emulsion polymerization. Sulfate groups dragged into the interior of polymer particles may be responsible for this. In the event that f falls sooner than kp, the dead polymer produced in the time interval between the fall in f and kp would have very high molecular weights. This should affect high-order molecular weight averages. This, however, has not been confirmed experimentally for MMA bulk polymerization. Molecular weights are difficult to measure as already mentioned, and there is the possibility that chain transfer to monomer might intervene to put a cap on molecular weights generated. For a discussion of phenomena such as radical pair formation and heterogeneous effects during glassy-state transition, one should refer to the original paper by Zhu and Hamielec (1989). In order to model diffusion-controlled propagation, i.e., when the polymerization temperature is less than the glass transition temperature of the polymer being synthesized, the propagation rate constants are then given by

( (

kpij ) kpijo exp -B

1 1 VF VFpcrit

))

(84)

where i is the radical type where the active center is located, j is the monomer type being added to the polymer chain, B is an adjustable parameter which should depend on monomer molecule type, and VFpcrit is the critical free volume where the propagation reaction of monomer j adding to a polymer chain ending in monomer i becomes diffusion-controlled. One might expect these critical free volumes to depend on polymer radical type as well as monomer type with propagation rates which are greater, becoming diffusion-controlled at lower monomer conversions. It is also reasonable to expect that diffusion-controlled propagation rate constants for the same monomer are equal (kpij ) kpji). This can be used to reduce the number of adjustable parameters. When dealing with multicomponent polymerizations, the diffusion-control expressions shown above were simplified somewhat by taking weighted averages of many of the parameters. The parameter δ from the segmental diffusion-controlled termination rate expression (see eq 73) was averaged by the cumulative polymer composition. A weighted average of the parameter A from eq 74 by the instantaneous polymer composition (Fi) was included as an option in the simulation package. As well, the sum of the K3 parameters of each monomer was included as an option. The option for the VFpcrit parameter in the diffusion-controlled propagation rate expression (see eq 84) was a weighted average based on the residual monomer mole fraction. Alternatively, overall values for A, K3, and VFpcrit were input. Although several parameters were mentioned above as being adjustable, only the latter three (A, K3, VFpcrit) were adjusted to fit the data. As will be discussed later, they were not adjusted for each experiment. 2.6. Free-Radical Concentrations. In order to calculate the reaction rates, one must also know the concentration of free radicals in the reactor. For the bulk, suspension, and solution case, one may use the overall concentration of free radicals in the reactor, but for the emulsion case, one requires an average number of radicals per particle. 2.6.1. Bulk/Suspension/Solution Case. The concentration of free radicals in the bulk, suspension, and solution cases is given by

978 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 Nz

∑ i)1

( Yo )

• • kcpAp([RTOT ]wpar + [RTOT ]des)VpNA2

Nz

kzi2[MSI]i2 + 4ktRI)1/2 -

∑ i)1

kzi[MSI]i

2kt

R) (85)

n j Np VpNA

(86)

where n j is the average number of radicals per polymer particle. There have been several methods proposed to estimate n j (Stockmayer, 1957; O’Toole, 1965; Ugelstad et al., 1967; Ballard et al., 1981a; Huo et al., 1988; Li and Brooks, 1993). One can, in principle, write a balance around a polymer particle assuming a steady state, where the disappearance of radicals from the particle is equal to the entry of radicals into the particle. The phenomena to be accounted for in the balance are radical entry (absorption, capture) from the aqueous phase, radical-radical termination in the particle, reaction of the radical with monomer-soluble impurities in the particle, and desorption of radicals into the water phase. The balance is performed for Nn, the number of particles with n radicals: • • ]wpar + [RTOT ]des)[Nn] + kcp([RTOT kt(n) (n - 1)[Nn]/vp + (kdes + kfmsi[MSI]p)n[Nn] ) • • kcp([RTOT ]wpar + [RTOT ]des)[Nn-1] + kt(n + 2)(n + 1)[Nn+2]/vp + (kdes + kfmsi[MSI]p)(n + 1)[Nn+1] (87)

where [R•TOT]des is the concentration of desorbed radicals in the water phase given by • ]des ) [RTOT

FdesVw kcpAp

(88)

vp in eq 87 is the average volume of a particle or Vp/Np. (Also note that Nn and Np in eqs 87, 90, and 91 represent the number of particles, not the number of particles per liter of water.) The first term (on the left-hand side) in eq 87 represents the capture of free radicals (including desorbed radicals) from the water phase (see also eq 31). The second term describes termination in the particles, while the third term represents radical desorption and reaction with monomer-soluble impurities. The solution of the above recursive equation was originally proposed as a modified Bessel function; Ugelstad et al. (1967) proposed the following partial fraction expansion as an approximation:

n j)

2R 2R

m+1+ m+2+ where

(89)

R m+

(90)

and

Equation 85 describes the initiation of radicals by initiator decomposition and the consumption of free radicals by reaction with impurities. 2.6.2. Emulsion Case. In the emulsion case the overall concentration of free radicals in the particle phase is

Yo )

Np2kt

2R m + 3 + ...

m)

(kdes + kfmsi[MSI]p)VpNA Npkt

(91)

Equation 89 requires approximately 10 levels of fractions to obtain a converged value for the average number of radicals per particle. More recently, Li and Brooks (1993) presented a more universal model (does not assume a steady state) for estimating n j . An investigation of the Li and Brooks (1993) approach is recommended in future modeling efforts. In the computer implementation, we decided to adopt the method of Ugelstad et al. (1967) due to its simple form and reduced amount of computational effort. 2.7. Radical Desorption. We will now look at the rate of free-radical desorption from the polymer particles (restricted to the emulsion polymerization case). In the most general case, desorption occurs when a small radical traverses from the particle to the aqueous phase. The desorbed radical is usually assumed to be no more than 1 monomer unit in length. The formation of these monomeric radicals in the particles is a result of chain transfer to monomer, to impurity, or to chain-transfer agent. In our case, radicals formed as a result of chain transfer to chain-transfer agent will be assumed not to desorb since typical CTAs are relatively large and insoluble molecules (e.g., n-dodecylmercaptan) (Nomura et al., 1982). It is commonly accepted that desorbed radicals will not move far beyond the particle from which they desorbed. That is, it can be argued that the desorbed radicals will not necessarily enter the bulk of the water phase but will rather stay in the vicinity of the particle surface. However, if there is a large amount of desorption, a significant amount of desorbed radicals may stray from the particles. Thus, while the following model equations are rigorous, there are certain underlying assumptions that should always be checked with specific process data, in order to explain possible discrepancies. Expressions describing desorption rate constants continue to be developed and reported in the literature. For large particles, desorption may be negligible; however, in the early stages of a reaction (i.e., when the particles are small) or when the rate of absorption of radicals by the particles is slow, desorption may be important. In the following equations, any reactions with monomer soluble impurities were assumed to yield nonreactive products. Several groups have developed expressions for the desorption rate constant. Their expressions attempted to give the desorption rate constant as a function of particle diameter. Mead and Poehlein (1989b) derived particle-size-independent expressions for kdes. They also rederived the equations put forth by other research groups while incorporating particle-size independence. The notion of a nonuniform distribution of radicals in the particles is applicable to large particles and in cases where chain-transfer reactions do not dominate (Chern and Poehlein, 1990). It is due to the hydrophilicity of end groups (initiator fragments) on the oligomers being captured. de la Cal et al. (1990) also discussed the nonuniform distribution of radicals. As a simplification


to our model, particle-size-independent expressions were not used. Recently, Fontenot and Schork (1992-93a) used the model developed by Nomura et al. (1976) and incorporated ideas by Mead and Poehlein (1989a,b) to account for nonuniform monomeric radical concentrations in particles. Fontenot and Schork (1992-93a) were dealing with large particles and droplet nucleation. The first desorption expression that we will examine is that of Ugelstad and Hansen (1976):

kdes )

[

12Dw kfm kp′ (a + D /D )d 2 w p p

]

(92)

In eq 92, kfm is the transfer to monomer rate constant, kp′ is the rate constant for reinitiation of oligomeric radicals from desorbed monomeric radicals, Dw is the diffusivity of monomer radicals in the aqueous phase, a is the partition coefficient for monomer radicals between the aqueous and particle phases, Dp is the diffusivity of monomeric radicals in the particles, and dp is the particle diameter. Harada et al. (1971) and Nomura et al. (1971a) have theoretically derived an expression for the desorption rate coefficient, kdes, using a stochastic approach and have successfully applied it to vinyl acetate emulsion polymerization. Their result is the following:

(

j+ kdes ) n

kp[M]pmddp2 12Dwδ

)( -1

kfm kfcta[CTA]p + + kp kp[M]p j) RI(1 - n

)

Npkp[M]pn j

(

)

6Dw mdDp

(

kp[M]pmddp2 12Dwδ

)

kfm[M]p

( )

12Dwδ kfm kdes ) m d 2 kp d p

)

+ 0.0017x

(97)

(98)

where

Dm )

D p Dw mdDp + Dw

(99)

and

(

ktr ) ktrm 1 +

)

ktrt[CTA]p ktrm[M]p

(100)

ktrt is the chain transfer to CTA rate constant, [CTA]p is the concentration of CTA in the particle, ktrm is the chain transfer to monomer rate constant, dm is the monomer density, φ is the monomer volume fraction in the particle, Mwt is the monomer molecular weight, and r is the particle radius. Asua et al. (1989) offered the following expression:

(95)

Nomura and Harada (1981) derived exactly the same expression as in eq 95 using a deterministic approach. If n j is much less than 0.1 during the whole range of conversions (which is not always the case), then the following expression may be used:

2

ktr 3Dm kp kdes ) 3DmMwt + r2 dmkpφ

(94)

-1

x ((1 -1 -0.19x )

Previous efforts in the literature to simulate emulsion polymerizations (Harada et al., 1971; Nomura et al., 1971b; Friis and Nyhagen, 1973; Nomura et al., 1976) made use of eq 96 and assumed an average constant value for δ throughout the conversion range. Thus, in our case, eqs 94 and 95 were used. This approach is identical to the one used by Penlidis (1986) to model vinyl acetate and vinyl chloride emulsion polymerizations. Furthermore, in order to accommodate the multicomponent aspect of the polymerization, the parameters Dw and md were calculated as weighted functions of the mole fraction of monomer in the particles. Rawlings and Ray (1988) presented an expression that included chain transfer to CTA. However, their use of an overall chain-transfer rate constant would enable one to modify other groups’ equations similarly.

-1

where Dp is the diffusivity of monomeric radicals in the particles. In eq 93, the term RI(1 - n j )/Npkp[M]pn j represents desorption of initiator radicals. In our model development, it was assumed that initiator radicals would not be captured by the particles due to their hydrophilic nature and electric charge; hence, this term may be safely neglected. As discussed above, radicals formed as a result of chain transfer to CTA will not desorb due to the size of the CTA assumed in this case. Thus, kfcta[CTA]p/kp[M]p in eq 93 can be neglected. Equation 93 can then be written as

j+ kdes ) n

Dp ) Dpo

kp[M]p (93)

In eq 93, md is the partition coefficient for monomeric radicals (same as Ugelstad and Hansen’s “a” and equivalent to 1/Kjwp in eq 110), dp is the average particle diameter, Dw is the diffusivity of monomer radicals in the aqueous phase, kfm is the transfer to monomer rate constant, and

δ) 1+

The parameter δ (see eq 94) can be described as a lumped coefficient or, alternatively, as the ratio of film mass-transfer resistance to overall mass-transfer resistance (Nomura and Harada, 1981). At the start of the polymerization, Dw and Dp are of the same order of magnitude because the particles are saturated with monomer. Thus, δ assumes a value close to unity. As the conversion and, hence, the viscosity inside of the particles increases, Dp decreases markedly and, hence, kdes decreases. An empirical model of the decrease in Dp was used in the model (Friis and Hamielec, 1977):

Ko kdes ) kfm[M]p βKo + kp[M]p

(101)

where

Ko ) (96) and

12Dw/mddp2 1 + 2Dw/mdDp

(102)


β)

kp[M]w + ktw[R•]w kp[M]w + ktw[R•]w + kaNTφw w/NA

(103)

ktw is the rate constant for termination in the water phase, [M]w is the concentration of monomer in the water phase, ka is the radical absorption (by particles) rate coefficient, NT is the total number of particles per unit volume of aqueous phase, φw w is the volume fraction of water in the continuous phase, and NA is Avogadro’s number. Finally, the particle-size-independent expression derived by Mead and Poehlein (1989b) is shown below:

kdes′ )

(

a+

kfm × 12Dw(π/6)2/3 kp′

)

Dw[sinh(φ)(1 + 3/φ2) - 3 cosh(φ)/φ]

kp′[M]pR2[cosh(φ)/φ - sinh(φ)/φ2] (104)

where

[

]

kp′[M]p Dp

interaction parameters would permit one to make reasonable estimates of these concentrations. There are several methods available to define partitioning of species between several phases: a rigorous thermodynamic approach, an experimental approach (based on xc, the conversion at which the monomer droplets disappear), an empirical approach, and the use of partition coefficients. Several groups in the literature employ the following equation (Dougherty, 1986):

[M]psat )

1 - xc a2Fm Mwt 1 - xc(1 - Fm/Fp)

(106)

where a2 is an adjustable parameter that has a value less than 1 for a water-soluble monomer and is equal to 1 otherwise. Fm and Fp are the densities of monomer and polymer, respectively. xc is the critical conversion at which the monomer droplets disappear. After the conversion has reached xc, one can replace xc in the above equation with the conversion level x. The thermodynamic approach presented by Guillot (1985) is as follows:

1/2

(105)

µi ) µo,i + RT(ln φi + (1 - m)φj + χijφj2) (107)

R, in the above equations, is the radius of a monomerswollen particle and r is the radial position within the monomer-swollen particle. Both the Nomura group and the Poehlein group present expressions for the desorption rate constant for copolymerization. The extension to the copolymer case from the homopolymerization equations shown above is quite straightforward. If one so desires, one may use a weighted average of two homopolymer equations. Also, one should check whether one of the comonomers is insoluble. This could lead to further simplifications and modifications to the copolymer equations. After careful evaluation of the methods presented above, we performed some tests with representative numbers and found very little difference between the various methods. Hence, we have employed the expression for kdes of Nomura and Harada (1981) (see eq 96). As an example, consider the monomers in the BA/ MMA/VAc system. VAc is known to exhibit significant radical desorption (Nomura et al., 1971a; Friis and Nyhagen, 1973; Penlidis, 1986), while MMA desorbs to a lesser extent (Ballard et al., 1981b) and the desorption for BA is considered negligible (Mallya and Plamthottam, 1989). Urretabizkaia et al. (1992) and Urretabizkaia and Asua (1994) reported that desorption was negligible for the seeded semibatch emulsion polymerization of BA/MMA/VAc. Considering the dominance of the acrylic monomers at lower conversions (

Mathematical Modeling of Multicomponent Chain-Growth

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